Maps.py

from __future__ import absolute_import
from __future__ import division
from __future__ import print_function
from __future__ import unicode_literals

from six import integer_types
from six import string_types
from collections import namedtuple
import warnings
from . import Mesh
import numpy as np
from numpy.polynomial import polynomial
import scipy.sparse as sp
from scipy.sparse.linalg import LinearOperator
from scipy.interpolate import UnivariateSpline
from scipy.constants import mu_0
from scipy.spatial import cKDTree
from SimPEG.Utils import mkvc
import properties
from . import Utils
from .Tests import checkDerivative


class IdentityMap(object):
    """
        SimPEG Map
    """

    def __init__(self, mesh=None, nP=None, **kwargs):
        Utils.setKwargs(self, **kwargs)

        if nP is not None:
            if isinstance(nP, string_types):
                assert nP == '*', (
                    "nP must be an integer or '*', not {}".format(nP)
                )
            assert isinstance(nP, integer_types + (np.int64,)), (
                'Number of parameters must be an integer. Not `{}`.'
                .format(type(nP))
            )
            nP = int(nP)
        elif mesh is not None:
            nP = mesh.nC
        else:
            nP = '*'

        self.mesh = mesh
        self._nP = nP

    @property
    def nP(self):
        """
            :rtype: int
            :return: number of parameters that the mapping accepts
        """
        if self._nP != '*':
            return int(self._nP)
        if self.mesh is None:
            return '*'
        return int(self.mesh.nC)

    @property
    def shape(self):
        """
            The default shape is (mesh.nC, nP) if the mesh is defined.
            If this is a meshless mapping (i.e. nP is defined independently)
            the shape will be the the shape (nP,nP).

            :rtype: tuple
            :return: shape of the operator as a tuple (int,int)
        """
        if self.mesh is None:
            return (self.nP, self.nP)
        return (self.mesh.nC, self.nP)

    def _transform(self, m):
        """
            Changes the model into the physical property.

            .. note::

                This can be called by the __mul__ property against a
                :meth:numpy.ndarray.

            :param numpy.array m: model
            :rtype: numpy.array
            :return: transformed model

        """
        return m

    def inverse(self, D):
        """
            Changes the physical property into the model.

            .. note::

                The *transformInverse* may not be easy to create in general.

            :param numpy.array D: physical property
            :rtype: numpy.array
            :return: model

        """
        raise NotImplementedError('The transformInverse is not implemented.')

    def deriv(self, m, v=None):
        """
            The derivative of the transformation.

            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model

        """
        if v is not None:
            return v
        if isinstance(self.nP, integer_types):
            return sp.identity(self.nP)
        return Utils.Identity()

    def test(self, m=None, num=4, **kwargs):
        """Test the derivative of the mapping.

            :param numpy.array m: model
            :param kwargs: key word arguments of
                           :meth:`SimPEG.Tests.checkDerivative`
            :rtype: bool
            :return: passed the test?

        """
        print('Testing {0!s}'.format(str(self)))
        if m is None:
            m = abs(np.random.rand(self.nP))
        if 'plotIt' not in kwargs:
            kwargs['plotIt'] = False

        assert isinstance(self.nP, integer_types), (
            "nP must be an integer for {}"
            .format(self.__class__.__name__)
        )
        return checkDerivative(
            lambda m: [self * m, self.deriv(m)], m, num=num, **kwargs
        )

    def testVec(self, m=None, **kwargs):
        """Test the derivative of the mapping times a vector.

            :param numpy.array m: model
            :param kwargs: key word arguments of
                           :meth:`SimPEG.Tests.checkDerivative`
            :rtype: bool
            :return: passed the test?

        """
        print('Testing {0!s}'.format(self))
        if m is None:
            m = abs(np.random.rand(self.nP))
        if 'plotIt' not in kwargs:
            kwargs['plotIt'] = False
        return checkDerivative(
            lambda m: [self*m, lambda x: self.deriv(m, x)], m, num=4, **kwargs
        )

    def _assertMatchesPair(self, pair):
        assert (
            isinstance(self, pair) or
            isinstance(self, ComboMap) and isinstance(self.maps[0], pair)
        ), "Mapping object must be an instance of a {0!s} class.".format(
            pair.__name__
        )

    def __mul__(self, val):
        if isinstance(val, IdentityMap):
            if (
                not (self.shape[1] == '*' or val.shape[0] == '*') and
                not self.shape[1] == val.shape[0]
            ):
                raise ValueError(
                    'Dimension mismatch in {0!s} and {1!s}.'.format(
                        str(self), str(val)
                    )
                )
            return ComboMap([self, val])

        elif isinstance(val, np.ndarray):
            if (
                not self.shape[1] == '*' and not self.shape[1] == val.shape[0]
            ):
                raise ValueError(
                    'Dimension mismatch in {0!s} and np.ndarray{1!s}.'.format(
                        str(self), str(val.shape)
                    )
                )
            return self._transform(val)

        elif isinstance(val, Utils.Zero):
            return Utils.Zero()

        raise Exception(
            'Unrecognized data type to multiply. Try a map or a numpy.ndarray!'
            'You used a {} of type {}'.format(
                val, type(val)
            )
        )

    def __str__(self):
        return "{0!s}({1!s},{2!s})".format(
            self.__class__.__name__,
            self.shape[0],
            self.shape[1]
        )

    def __len__(self):
        return 1


class ComboMap(IdentityMap):
    """
        Combination of various maps.

        The ComboMap holds the information for multiplying and combining
        maps. It also uses the chain rule to create the derivative.
        Remember, any time that you make your own combination of mappings
        be sure to test that the derivative is correct.

    """

    def __init__(self, maps, **kwargs):
        IdentityMap.__init__(self, None, **kwargs)

        self.maps = []
        for ii, m in enumerate(maps):
            assert isinstance(m, IdentityMap), "Unrecognized data type, "
            "inherit from an IdentityMap or ComboMap!"

            if (
                ii > 0 and not (self.shape[1] == '*' or m.shape[0] == '*') and
                not self.shape[1] == m.shape[0]
               ):
                prev = self.maps[-1]

                raise ValueError(
                    'Dimension mismatch in map[{0!s}] ({1!s}, {2!s}) '
                    'and map[{3!s}] ({4!s}, {5!s}).'.format(
                        prev.__class__.__name__,
                        prev.shape[0],
                        prev.shape[1],
                        m.__class__.__name__,
                        m.shape[0],
                        m.shape[1]
                    )
                )

            if np.any([isinstance(m, SumMap), isinstance(m, IdentityMap)]):
                self.maps += [m]
            elif isinstance(m, ComboMap):
                self.maps += m.maps
            else:
                raise ValueError(
                    'Map[{0!s}] not supported',
                    m.__class__.__name__
                )

    @property
    def shape(self):
        return (self.maps[0].shape[0], self.maps[-1].shape[1])

    @property
    def nP(self):
        """Number of model properties.

           The number of cells in the
           last dimension of the mesh."""
        return self.maps[-1].nP

    def _transform(self, m):
        for map_i in reversed(self.maps):
            m = map_i * m
        return m

    def deriv(self, m, v=None):

        if v is not None:
            deriv = v
        else:
            deriv = 1

        mi = m
        for map_i in reversed(self.maps):
            deriv = map_i.deriv(mi) * deriv
            mi = map_i * mi
        return deriv

    def __str__(self):
        return 'ComboMap[{0!s}]({1!s},{2!s})'.format(
            ' * '.join([m.__str__() for m in self.maps]),
            self.shape[0],
            self.shape[1]
        )

    def __len__(self):
        return len(self.maps)


class Projection(IdentityMap):
    """
        A map to rearrange / select parameters

        :param int nP: number of model parameters
        :param numpy.array index: indices to select
    """

    def __init__(self, nP, index, **kwargs):
        assert isinstance(index, (np.ndarray, slice, list)), (
            'index must be a np.ndarray or slice, not {}'.format(type(index)))
        super(Projection, self).__init__(nP=nP, **kwargs)

        if isinstance(index, slice):
            index = list(range(*index.indices(self.nP)))
        self.index = index
        self._shape = nI, nP = len(self.index), self.nP

        assert (max(index) < nP), (
            'maximum index must be less than {}'.format(nP))

        # sparse projection matrix
        self.P = sp.csr_matrix(
            (np.ones(nI), (range(nI), self.index)), shape=(nI, nP)
        )

    def _transform(self, m):
        return m[self.index]

    @property
    def shape(self):
        """
        Shape of the matrix operation (number of indices x nP)
        """
        return self._shape

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model
        """

        if v is not None:
            return self.P * v
        return self.P


class SumMap(ComboMap):
    """
        A map to add model parameters contributing to the
        forward operation e.g. F(m) = F m1 + F m2 + ...
    """
    def __init__(self, maps, **kwargs):
        IdentityMap.__init__(self, None, **kwargs)

        self.maps = []
        for ii, m in enumerate(maps):
            assert isinstance(m, IdentityMap), "Unrecognized data type, "
            "inherit from an IdentityMap or ComboMap!"

            if (
                ii > 0 and not (self.shape == '*' or m.shape == '*') and
                not self.shape == m.shape
               ):

                raise ValueError(
                    'Dimension mismatch in map[{0!s}] ({1!s}, {2!s}) '
                    'and map[{3!s}] ({4!s}, {5!s}).'.format(
                        self.maps[0].__class__.__name__,
                        self.maps[0].shape[0],
                        self.maps[0].shape[1],
                        m.__class__.__name__,
                        m.shape[0],
                        m.shape[1]
                    )
                )

            self.maps += [m]

    @property
    def shape(self):

        return (self.maps[0].shape[0], self.maps[0].shape[1])

    @property
    def nP(self):
        """Number of model properties.

           The number of cells in the
           last dimension of the mesh."""
        return self.maps[-1].shape[1]

    def _transform(self, m):

        for ii, map_i in enumerate(self.maps):
            m0 = m.copy()
            m0 = map_i * m0

            if ii == 0:
                mout = m0
            else:
                mout += m0
        return mout

    def deriv(self, m, v=None):

        for ii, map_i in enumerate(self.maps):

            m0 = m.copy()

            if v is not None:
                deriv = v
            else:
                deriv = sp.eye(self.nP)

            deriv = map_i.deriv(m0, v=deriv)
            if ii == 0:
                sumDeriv = deriv
            else:
                sumDeriv += deriv

        return sumDeriv


class SurjectUnits(IdentityMap):
    """
        A map to group model cells into an homogeneous unit

        :param list index: list of bool for each homogeneous unit

    """
    nBlock = 1  # Variable allowing to stack same Map over multiple sets

    def __init__(self, index, **kwargs):
        assert isinstance(index, (list)), (
            'index must be a list, not {}'.format(type(index)))

        super(SurjectUnits, self).__init__(**kwargs)

        self.index = index
        nP = len(self.index[0])
        self._shape = self.nBlock*nP, self.nBlock*len(self.index),

    @property
    def P(self):

        if getattr(self, '_P', None) is None:
            nP = len(self.index[0])
            # sparse projection matrix
            row = []
            col = []
            val = []
            for ii, ind in enumerate(self.index):

                row += [ii]*ind.sum()
                col += np.where(ind)[0].tolist()
                val += [1]*ind.sum()

            P = sp.csr_matrix(
                (val, (row, col)), shape=(len(self.index), nP)
            ).T

            self._P = sp.block_diag([P for ii in range(self.nBlock)])

        return self._P

    def _transform(self, m):
        return self.P * m

    @property
    def shape(self):
        """
        Shape of the matrix operation (number of indices x nP)
        """
        return self._shape

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model
        """

        if v is not None:
            return self.P * v
        return self.P



class Wires(object):

    def __init__(self, *args):
        for arg in args:
            assert (
                isinstance(arg, tuple) and
                len(arg) == 2 and
                isinstance(arg[0], string_types) and
                # TODO: this should be extended to a slice.
                isinstance(arg[1], integer_types)
            ), (
                "Each wire needs to be a tuple: (name, length). "
                "You provided: {}".format(arg)
            )

        self._nP = int(np.sum([w[1] for w in args]))
        start = 0
        maps = []
        for arg in args:
            wire = Projection(self.nP, slice(start, start + arg[1]))
            setattr(self, arg[0], wire)
            maps += [(arg[0], wire)]
            start += arg[1]
        self.maps = maps

        self._tuple = namedtuple('Model', [w[0] for w in args])

    def __mul__(self, val):
        assert isinstance(val, np.ndarray)
        split = []
        for n, w in self.maps:
            split += [w * val]
        return self._tuple(*split)

    @property
    def nP(self):
        return self._nP


class Tile(IdentityMap):
    """
        Mapping for tiled inversion
    """

    nCell = 26  # Number of neighbors to use in averaging
    tol = 1e-8  # Tolerance to avoid zero division
    nBlock = 1

    def __init__(self, *args, **kwargs):

        assert len(args) == 2, ('Mapping requires a tuple' +
                                '(MeshGlobal, ActiveGlobal),' +
                                '(MeshLocal, ActiveLocal)')
        super(Tile, self).__init__(**kwargs)
        # check if tree in kwargs
        if 'tree' in kwargs.keys():   # kwargs is a dict
            tree = kwargs.pop('tree')

            assert isinstance(tree, cKDTree), ('Tree input must be a cKDTRee')
            self._tree = tree

        self.meshGlobal = args[0][0]
        self.actvGlobal = args[0][1]

        if not isinstance(self.actvGlobal, bool):
            temp = np.zeros(self.meshGlobal.nC, dtype='bool')
            temp[self.actvGlobal] = True
            self.actvGlobal = temp

        self.meshLocal = args[1][0]
        self.activeLocal = args[1][1]

        # if not isinstance(self.activeLocal, bool):
        #     temp = np.zeros(self.meshLocal.nC, dtype='bool')
        #     temp[self.activeLocal] = True
        #     self.activeLocal = temp

        if self.nCell > self.meshGlobal.nC:
            self.nCell = self.meshGlobal.nC

        self.index = np.ones(self.actvGlobal.sum(), dtype='bool')
        self.P

    @property
    def tree(self):
        """
            Create cKDTree structure for given global mesh
        """
        if getattr(self, '_tree', None) is None:

            # if self.meshGlobal.dim == 1:
            #     ccMat = np.c_[self.meshGlobal.gridCC[self.actvGlobal, 0]]
            # elif self.meshGlobal.dim == 2:
            #     ccMat = np.c_[self.meshGlobal.gridCC[self.actvGlobal, 0],
            #                   self.meshGlobal.gridCC[self.actvGlobal, 1]]
            # elif self.meshGlobal.dim == 3:
            #     ccMat = np.c_[self.meshGlobal.gridCC[self.actvGlobal, 0],
            #                   self.meshGlobal.gridCC[self.actvGlobal, 1],
            #                   self.meshGlobal.gridCC[self.actvGlobal, 2]]

            self._tree = cKDTree(self.meshGlobal.gridCC[self.actvGlobal, :])

        return self._tree

    @property
    def activeLocal(self):
        """This is the activeLocal of the actvGlobal used in the global problem."""
        return getattr(self, '_activeLocal', None)

    @activeLocal.setter
    def activeLocal(self, activeLocal):

        if not isinstance(activeLocal, bool):
            temp = np.zeros(self.meshLocal.nC, dtype='bool')
            temp[activeLocal] = True
            activeLocal = temp

        self._activeLocal = activeLocal


    @property
    def index(self):
        """This is the index of the actvGlobal used in the global problem."""
        return getattr(self, '_index', None)

    @index.setter
    def index(self, index):
        if getattr(self, '_index', None) is not None:
            self._S = None

        if not isinstance(index, bool):
            temp = np.zeros(self.actvGlobal.sum(), dtype='bool')
            temp[index] = True
            index = temp

        self._nP = index.sum()
        self._index = index

    @property
    def S(self):
        """
            Create sub-selection matrix in case where the global
            mesh is not touched by all sub meshes
        """
        if getattr(self, '_S', None) is None:

            nP = self.actvGlobal.sum()
            nI = self.index.sum()
            assert (nI <= nP), (
                'maximum index must be less than {}'.format(nP))

            # sparse projection matrix
            S = sp.csr_matrix(
                (np.ones(nI), (np.where(self.index)[0], range(nI))), shape=(nP, nI)
            )

            self._S = S
        return self._S

    @property
    def P(self):
        """
            Set the projection matrix with partial volumes
        """
        if getattr(self, '_P', None) is None:

            if self.meshLocal._meshType == "TREE":

                actvIndGlobal = np.where(self.actvGlobal)[0].tolist()

                indL = self.meshLocal._get_containing_cell_indexes(self.meshGlobal.gridCC)

                full = np.c_[indL, np.arange(self.meshGlobal.nC)]
                # Create new index based on unique active
                # [ua, ind] = np.unique(indL, return_index=True)

                check = np.where(self.meshLocal.vol[indL] < self.meshGlobal.vol)[0].tolist()

                # Reverse inside global to local
                indG = self.meshGlobal._get_containing_cell_indexes(self.meshLocal.gridCC)

                model = np.zeros(self.meshLocal.nC)

                rows = []
                for ind in check:

                    if ind in actvIndGlobal:
                        indAdd = np.where(ind == indG)[0]
                        rows += [np.c_[indAdd, np.ones_like(indAdd)*ind]]
                        # model[indAdd] = 0.5

                # indL = indL[actv]
                if len(rows) > 0:
                    full = np.r_[full[actvIndGlobal, :], np.vstack(rows)]
                else:
                    full = full[actvIndGlobal, :]

                # model[full[:,0]]=0.5

                actvIndLocal = np.unique(full[:, 0])

                full = np.c_[np.searchsorted(actvIndLocal, full[:, 0]), np.searchsorted(actvIndGlobal, full[:, 1])]

                activeLocal = np.zeros(self.meshLocal.nC, dtype='bool')
                activeLocal[actvIndLocal] = True

                self.activeLocal = activeLocal

            else:
                indx = self.getTreeIndex(self.tree, self.meshLocal, self.activeLocal)
                local2Global = np.c_[np.kron(np.ones(self.nCell), np.asarray(range(self.activeLocal.sum()))).astype('int'), mkvc(indx)]

                tree = cKDTree(self.meshLocal.gridCC[self.activeLocal, :])
                r, ind = tree.query(self.meshGlobal.gridCC[self.actvGlobal], k=self.nCell)
                global2Local = np.c_[np.kron(np.ones(self.nCell), np.asarray(range(self.actvGlobal.sum()))).astype('int'), mkvc(ind)]

                full = np.unique(np.vstack([local2Global, global2Local[:, [1, 0]]]), axis=0)

            # Free up memory
            self._tree = None
            tree = None

            # Get the node coordinates (bottom-SW) and (top-NE) of cells
            # in the global and local mesh
            global_bsw, global_tne = self.getNodeExtent(self.meshGlobal,
                                                        self.actvGlobal)

            local_bsw, local_tne = self.getNodeExtent(self.meshLocal,
                                                      self.activeLocal)

            nactv = full.shape[0]

            # Compute intersecting cell volumes
            if self.meshLocal.dim == 1:

                dV = np.max(
                    [(np.min(
                        [global_tne[full[:, 1]],
                         local_tne[full[:, 0]]], axis=0
                      ) -
                      np.max(
                        [global_bsw[full[:, 1]],
                         local_bsw[full[:, 0]]], axis=0)
                      ), np.zeros(nactv)
                     ], axis=0
                )

            elif self.meshLocal.dim >= 2:

                dV = np.max(
                    [(np.min(
                        [global_tne[full[:, 1], 0],
                         local_tne[full[:, 0], 0]], axis=0
                       ) -
                      np.max(
                        [global_bsw[full[:, 1], 0],
                         local_bsw[full[:, 0], 0]], axis=0)
                      ), np.zeros(nactv)], axis=0
                    )

                dV *= np.max([(np.min([global_tne[full[:, 1], 1], local_tne[full[:, 0], 1]],
                                      axis=0) -
                               np.max([global_bsw[full[:, 1], 1], local_bsw[full[:, 0], 1]],
                                      axis=0)),
                              np.zeros(nactv)], axis=0)

            if self.meshLocal.dim == 3:

                dV *= np.max([(np.min([global_tne[full[:, 1], 2], local_tne[full[:, 0], 2]],
                                      axis=0) -
                               np.max([global_bsw[full[:, 1], 2], local_bsw[full[:, 0], 2]],
                                      axis=0)),
                              np.zeros(nactv)], axis=0)

            # Select only cells with non-zero intersecting volumes
            nzV = dV > 0

            self.V = dV[nzV]

            P = sp.csr_matrix((self.V, (full[nzV, 0], full[nzV, 1])),
                              shape=(self.activeLocal.sum(), self.actvGlobal.sum()))

            sumRow = Utils.mkvc(np.sum(P, axis=1) + self.tol)

            self.scaleJ = sp.block_diag([
                Utils.sdiag(sumRow/self.meshLocal.vol[self.activeLocal])
                for ii in range(self.nBlock)])

            self._P = sp.block_diag([
                Utils.sdiag(1./sumRow) * P * self.S
                for ii in range(self.nBlock)])

            self._shape = int(self.activeLocal.sum()*self.nBlock), int(self.actvGlobal.sum()*self.nBlock)

        return self._P

    def getTreeIndex(self, tree, mesh, actvCell):
        """
            Querry the KDTree for nearest cells
        """

        # if self.meshGlobal.dim == 1:

        d, indx = tree.query(mesh.gridCC[actvCell, :],
                             k=self.nCell)

        # elif self.meshGlobal.dim == 2:
        #     d, indx = tree.query(np.c_[mesh.gridCC[actvCell, 0],
        #                                mesh.gridCC[actvCell, 1]],
        #                          k=self.nCell)
        # elif self.meshGlobal.dim == 3:
        #     d, indx = tree.query(np.c_[mesh.gridCC[actvCell, 0],
        #                                mesh.gridCC[actvCell, 1],
        #                                mesh.gridCC[actvCell, 2]],
        #                          k=self.nCell)
        return indx

    def getNodeExtent(self, mesh, actvCell):

        bsw = mesh.gridCC - mesh.h_gridded/2.
        tne = mesh.gridCC + mesh.h_gridded/2.

        # Return only active set
        return bsw[actvCell], tne[actvCell]

    def _transform(self, m):
        return self.P * m

    @property
    def shape(self):
        """
        Shape of the matrix operation (number of indices x nP)
        """
        return self.P.shape

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model
        """

        self.P
        if v is not None:
            return self.scaleJ * self.P * v
        return self.scaleJ * self.P


class SelfConsistentEffectiveMedium(IdentityMap, properties.HasProperties):
    """
        Two phase self-consistent effective medium theory mapping for
        ellipsoidal inclusions. The model is the concentration
        (volume fraction) of the phase 2 material.

        The model is :math:`\\varphi`. We solve for :math:`\sigma`
        given :math:`\sigma_0`, :math:`\sigma_1` and :math:`\\varphi` . Each of
        the following are implicit expressions of the effective conductivity.
        They are solved using a fixed point iteration.

        **Spherical Inclusions**

        If the shape of the inclusions are spheres, we use

        .. math::

            \sum_{j=1}^N (\sigma^* - \sigma_j)R^{j} = 0

        where :math:`j=[1,N]` is the each material phase, and N is the number
        of phases. Currently, the implementation is only set up for 2 phase
        materials, so we solve

        .. math::

            (1-\\varphi)(\sigma - \sigma_0)R^{(0)} + \\varphi(\sigma - \sigma_1)R^{(1)} = 0.

        Where :math:`R^{(j)}` is given by

        .. math::

            R^{(j)} = \\left[1 + \\frac{1}{3}\\frac{\sigma_j - \sigma}{\sigma} \\right]^{-1}.

        **Ellipsoids**

        .. todo::

            Aligned Ellipsoids have not yet been implemented, only randomly
            oriented ellipsoids

        If the inclusions are aligned ellipsoids, we solve

        .. math::

            \sum_{j=1}^N \\varphi_j (\Sigma^* - \sigma_j\mathbf{I}) \mathbf{R}^{j, *} = 0

        where

        .. math::

            \mathbf{R}^{(j, *)} = \left[ \mathbf{I} + \mathbf{A}_j {\Sigma^{*}}^{-1}(\sigma_j \mathbf{I} - \Sigma^*) \\right]^{-1}

        and the depolarization tensor :math:`\mathbf{A}_j` is given by

        .. math::

            \mathbf{A}^* = \\left[\\begin{array}{ccc}
                Q & 0 & 0 \\\\
                0 & Q & 0 \\\\
                0 & 0 & 1-2Q
            \end{array}\\right]

        for a spheroid aligned along the z-axis. For an oblate spheroid
        (:math:`\\alpha < 1`, pancake-like)

        .. math::

            Q = \\frac{1}{2}\\left(
                1 + \\frac{1}{\\alpha^2 - 1} \\left[
                    1 - \\frac{1}{\chi}\\tan^{-1}(\chi)
                \\right]
            \\right)

        where

        .. math::

            \chi = \sqrt{\\frac{1}{\\alpha^2} - 1}

        .. todo::

            Prolate spheroids (\alpha > 1, needle-like) have not been
            implemented yet

        For reference, see
        `Torquato (2002), Random Heterogeneous Materials <https://link.springer.com/book/10.1007/978-1-4757-6355-3>`_


    """

    sigma0 = properties.Float(
        "physical property value for phase-0 material",
        min=0., required=True
    )

    sigma1 = properties.Float(
        "physical property value for phase-1 material",
        min=0., required=True
    )

    alpha0 = properties.Float(
        "aspect ratio of the phase-0 ellipsoids", default=1.
    )

    alpha1 = properties.Float(
        "aspect ratio of the phase-1 ellipsoids", default=1.
    )

    rel_tol = properties.Float(
        "relative tolerance for convergence for the fixed-point iteration",
        default = 1e-4
    )

    maxIter = properties.Integer(
        "maximum number of iterations for the fixed point iteration "
        "calculation",
        default = 50
    )

    def __init__(self, mesh=None, nP=None, sigstart=None, **kwargs):
        self._sigstart = sigstart
        super(SelfConsistentEffectiveMedium, self).__init__(mesh, nP, **kwargs)

    @property
    def tol(self):
        """
        absolute tolerance for the convergence of the fixed point iteration calc
        """
        if getattr(self, '_tol', None) is None:
            self._tol = self.rel_tol*min(self.sigma0, self.sigma1)
        return self._tol

    @property
    def sigstart(self):
        """
        first guess for sigma
        """
        return self._sigstart

    def wennerBounds(self, phi1):
        """Define Wenner Conductivity Bounds"""
        # TODO: Add HS bounds (not needed for spherical particles, but for ellipsoidal ones)
        phi0   = 1.0-phi1
        sigWup = phi0*self.sigma0 + phi1*self.sigma1
        sigWlo = 1.0/(phi0/self.sigma0 + phi1/self.sigma1)
        W = np.array([sigWlo, sigWup])

        return W

    def getQ(self, alpha):
        if alpha < 1.:
            Chi = np.sqrt((1./alpha**2.) - 1.)
            return 1./2.*(1. + 1./(alpha**2. - 1.)*(1. - np.arctan(Chi)/Chi))
        elif alpha > 1.:
            raise NotImplementedError(
                'Aspect ratios > 1 have not been implemeted'
            )
        elif alpha == 1:
            return 1./3.

    def getR(self, sj, se, alpha):
        if alpha == 1.:
            return 3.0*se/(2.0*se+sj)
        Q = self.getQ(alpha)
        return se/3.*(2./(se + Q*(sj-se)) + 1./(sj - 2.*Q*(sj-se)))

    def getdR(self, sj, se, alpha):
        Q = self.getQ(alpha)
        return (
            sj/3. *
            ( 2.*Q/(se + Q*(sj-se))**2 + (1. - 2.*Q)/(sj - 2.*Q*(sj-se))**2 )
        )

    def _sc2phaseEMTRandSpheroidstransform(self, phi1):
        """
        Self Consistent Effective Medium Theory Model Transform,
        alpha = aspect ratio (c/a <= 1)

        """

        if self.sigstart is None:
            self._sigstart = self.wennerBounds(phi1)[0]

        if not (np.all(0 <= phi1) and np.all(phi1 <= 1)):
            warnings.warn('there are phis outside bounds of 0 and 1')
            phi1 = np.median(np.c_[phi1*0, phi1, phi1*0+1.])

        phi0 = 1.0-phi1

        sige1 = self.sigstart

        for i in range(self.maxIter):
            R0 = self.getR(self.sigma0, sige1, self.alpha0)
            R1 = self.getR(self.sigma1, sige1, self.alpha1)

            den = phi0*R0 + phi1*R1
            num = phi0*self.sigma0*R0 + phi1*self.sigma1*R1

            sige2 = num/den
            relerr = np.abs(sige2-sige1)

            if np.all(relerr <= self.tol):
                if self.sigstart is None:
                    self.sigstart = sige2  # store as a starting point for the next time around
                return sige2

            sige1 = sige2
        # TODO: make this a proper warning, and output relevant info (sigma0, sigma1, phi, sigstart, and relerr)
        warnings.warn('Maximum number of iterations reached')

        return sige2

    def _sc2phaseEMTRandSpheroidsinversetransform(self, sige):

        R0 = getR(self.sigma0, sige, self.alp0)
        R1 = getR(self.sigma1, sige, self.alp1)

        num = -(sigma0 - sige)*R0
        den = (sigma1-sige)*R1 - (sigma0-sige)*R0

        return num/den

    def _sc2phaseEMTRandSpheroidstransformDeriv(self, sige, phi1):

        phi0 = 1.0-phi1

        R0 = self.getR(self.sigma0, sige, self.alpha0)
        R1 = self.getR(self.sigma1, sige, self.alpha1)

        dR0 = self.getdR(self.sigma0, sige, self.alpha0)
        dR1 = self.getdR(self.sigma1, sige, self.alpha1)

        num = (sige-self.sigma0)*R0 - (sige-self.sigma1)*R1
        den = phi0*(R0 + (sige-self.sigma0)*dR0) + phi1*(R1 + (sige-self.sigma1)*dR1)

        return Utils.sdiag(num/den)

    def _transform(self, m):
        return self._sc2phaseEMTRandSpheroidstransform(m)

    def deriv(self, m):
        sige = self._transform(m)
        return self._sc2phaseEMTRandSpheroidstransformDeriv(sige, m)

    def inverse(self, sige):
        return self._sc2phaseEMTRandSpheroidsinversetransform(sige)


###############################################################################
#                                                                             #
#                          Mesh Independent Maps                              #
#                                                                             #
###############################################################################

class ExpMap(IdentityMap):
    """
        Electrical conductivity varies over many orders of magnitude, so it is
        a common technique when solving the inverse problem to parameterize and
        optimize in terms of log conductivity. This makes sense not only
        because it ensures all conductivities will be positive, but because
        this is fundamentally the space where conductivity
        lives (i.e. it varies logarithmically).

        Changes the model into the physical property.

        A common example of this is to invert for electrical conductivity
        in log space. In this case, your model will be log(sigma) and to
        get back to sigma, you can take the exponential:

        .. math::

            m = \log{\sigma}

            \exp{m} = \exp{\log{\sigma}} = \sigma
    """

    def __init__(self, mesh=None, nP=None, **kwargs):
        super(ExpMap, self).__init__(mesh=mesh, nP=nP, **kwargs)

    def _transform(self, m):
        return np.exp(Utils.mkvc(m))

    def inverse(self, D):
        """
            :param numpy.array D: physical property
            :rtype: numpy.array
            :return: model

            The *transformInverse* changes the physical property into the
            model.

            .. math::

                m = \log{\sigma}

        """
        return np.log(Utils.mkvc(D))

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model

            The *transform* changes the model into the physical property.
            The *transformDeriv* provides the derivative of the *transform*.

            If the model *transform* is:

            .. math::

                m = \log{\sigma}

                \exp{m} = \exp{\log{\sigma}} = \sigma

            Then the derivative is:

            .. math::

                \\frac{\partial \exp{m}}{\partial m} = \\text{sdiag}(\exp{m})
        """
        deriv = Utils.sdiag(np.exp(Utils.mkvc(m)))
        if v is not None:
            return deriv * v
        return deriv


class ReciprocalMap(IdentityMap):
    """
        Reciprocal mapping. For example, electrical resistivity and
        conductivity.

        .. math::

            \\rho = \\frac{1}{\sigma}

    """
    def __init__(self, mesh=None, nP=None, **kwargs):
        super(ReciprocalMap, self).__init__(mesh=mesh, nP=nP, **kwargs)

    def _transform(self, m):
        return 1.0 / Utils.mkvc(m)

    def inverse(self, D):
        return 1.0 / Utils.mkvc(D)

    def deriv(self, m, v=None):
        # TODO: if this is a tensor, you might have a problem.
        deriv = Utils.sdiag(- Utils.mkvc(m)**(-2))
        if v is not None:
            return deriv * v
        return deriv


class LogMap(IdentityMap):
    """
        Changes the model into the physical property.

        If \\(p\\) is the physical property and \\(m\\) is the model, then

        .. math::

            p = \\log(m)

        and

        .. math::

            m = \\exp(p)

        NOTE: If you have a model which is log conductivity
        (ie. \\(m = \\log(\\sigma)\\)),
        you should be using an ExpMap

    """

    def __init__(self, mesh=None, nP=None, **kwargs):
        super(LogMap, self).__init__(mesh=mesh, nP=nP, **kwargs)

    def _transform(self, m):
        return np.log(Utils.mkvc(m))

    def deriv(self, m, v=None):
        mod = Utils.mkvc(m)
        deriv = np.zeros(mod.shape)
        tol = 1e-16  # zero
        ind = np.greater_equal(np.abs(mod), tol)
        deriv[ind] = 1.0/mod[ind]
        if v is not None:
            return Utils.sdiag(deriv)*v
        return Utils.sdiag(deriv)

    def inverse(self, m):
        return np.exp(Utils.mkvc(m))


class ChiMap(IdentityMap):
    """Chi Map

    Convert Magnetic Susceptibility to Magnetic Permeability.

    .. math::

        \mu(m) = \mu_0 (1 + \chi(m))

    """

    def __init__(self, mesh=None, nP=None, **kwargs):
        super(ChiMap, self).__init__(mesh=mesh, nP=nP, **kwargs)

    def _transform(self, m):
        return mu_0 * (1 + m)

    def deriv(self, m, v=None):
        if v is not None:
            return mu_0 * v
        return mu_0 * sp.eye(self.nP)

    def inverse(self, m):
        return m / mu_0 - 1


class MuRelative(IdentityMap):
    """
    Invert for relative permeability

    .. math::

        \mu(m) = \mu_0 * \mathbf{m}
    """

    def __init__(self, mesh=None, nP=None, **kwargs):
        super(MuRelative, self).__init__(mesh=mesh, nP=nP, **kwargs)

    def _transform(self, m):
        return mu_0 * m

    def deriv(self, m, v=None):
        if v is not None:
            return mu_0 * v
        return mu_0 * sp.eye(self.nP)

    def inverse(self, m):
        return 1./mu_0 * m


class Weighting(IdentityMap):
    """
        Model weight parameters.
    """

    def __init__(self, mesh=None, nP=None, weights=None, **kwargs):

        if 'nC' in kwargs:
            raise AttributeError(
                '`nC` is depreciated. Use `nP` to set the number of model '
                'parameters'
            )

        super(Weighting, self).__init__(mesh=mesh, nP=nP, **kwargs)

        if weights is None:
            weights = np.ones(self.nP)

        self.weights = np.array(weights, dtype=float)

    @property
    def shape(self):
        return (self.nP, self.nP)

    @property
    def P(self):
        return Utils.sdiag(self.weights)

    def _transform(self, m):
        return self.weights*m

    def inverse(self, D):
        return self.weights**(-1.) * D

    def deriv(self, m, v=None):
        if v is not None:
            return self.weights * v
        return self.P


class ComplexMap(IdentityMap):
    """ComplexMap

        default nP is nC in the mesh times 2 [real, imag]

    """
    def __init__(self, mesh=None, nP=None, **kwargs):
        super(ComplexMap, self).__init__(mesh=mesh, nP=nP, **kwargs)
        if nP is not None:
            assert nP % 2 == 0, 'nP must be even.'
        self._nP = nP or int(self.mesh.nC * 2)

    @property
    def nP(self):
        return self._nP

    @property
    def shape(self):
        return (int(self.nP/2), self.nP)

    def _transform(self, m):
        nC = self.mesh.nC
        return m[:nC] + m[nC:]*1j

    def deriv(self, m, v=None):
        nC = self.shape[0]
        shp = (nC, nC*2)

        def fwd(v):
            return v[:nC] + v[nC:]*1j

        def adj(v):
            return np.r_[v.real, v.imag]
        if v is not None:
            return LinearOperator(shp, matvec=fwd, rmatvec=adj) * v
        return LinearOperator(shp, matvec=fwd, rmatvec=adj)

    # inverse = deriv


###############################################################################
#                                                                             #
#                 Surjection, Injection and Interpolation Maps                #
#                                                                             #
###############################################################################

class SurjectFull(IdentityMap):
    """
    SurjectFull

    Given a scalar, the SurjectFull maps the value to the
    full model space.
    """

    def __init__(self, mesh, **kwargs):
        IdentityMap.__init__(self, mesh, **kwargs)

    @property
    def nP(self):
        return 1

    def _transform(self, m):
        """
            :param m: model (scalar)
            :rtype: numpy.array
            :return: transformed model
        """
        return np.ones(self.mesh.nC) * m

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: numpy.array
            :return: derivative of transformed model
        """
        deriv = sp.csr_matrix(np.ones([self.mesh.nC, 1]))
        if v is not None:
            return deriv * v
        return deriv


class SurjectVertical1D(IdentityMap):
    """SurjectVertical1DMap

        Given a 1D vector through the last dimension
        of the mesh, this will extend to the full
        model space.
    """

    def __init__(self, mesh, **kwargs):
        IdentityMap.__init__(self, mesh, **kwargs)

    @property
    def nP(self):
        """Number of model properties.

           The number of cells in the
           last dimension of the mesh."""
        return int(self.mesh.vnC[self.mesh.dim-1])

    def _transform(self, m):
        """
            :param numpy.array m: model
            :rtype: numpy.array
            :return: transformed model
        """
        repNum = self.mesh.vnC[:self.mesh.dim-1].prod()
        return Utils.mkvc(m).repeat(repNum)

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model
        """
        repNum = self.mesh.vnC[:self.mesh.dim-1].prod()
        repVec = sp.csr_matrix(
            (np.ones(repNum), (range(repNum), np.zeros(repNum))),
            shape=(repNum, 1)
        )
        deriv = sp.kron(sp.identity(self.nP), repVec)
        if v is not None:
            return deriv * v
        return deriv


class Surject2Dto3D(IdentityMap):
    """Map2Dto3D

        Given a 2D vector, this will extend to the full
        3D model space.
    """

    normal = 'Y'  #: The normal

    def __init__(self, mesh, **kwargs):
        assert mesh.dim == 3, 'Surject2Dto3D Only works for a 3D Mesh'
        IdentityMap.__init__(self, mesh, **kwargs)
        assert self.normal in ['X', 'Y', 'Z'], (
            'For now, only "Y" normal is supported'
        )

    @property
    def nP(self):
        """Number of model properties.

           The number of cells in the
           last dimension of the mesh."""
        if self.normal == 'Z':
            return self.mesh.nCx * self.mesh.nCy
        elif self.normal == 'Y':
            return self.mesh.nCx * self.mesh.nCz
        elif self.normal == 'X':
            return self.mesh.nCy * self.mesh.nCz

    def _transform(self, m):
        """
            :param numpy.array m: model
            :rtype: numpy.array
            :return: transformed model
        """
        m = Utils.mkvc(m)
        if self.normal == 'Z':
            return Utils.mkvc(
                m.reshape(
                    self.mesh.vnC[[0, 1]], order='F'
                )[:, :, np.newaxis].repeat(
                    self.mesh.nCz,
                    axis=2
                )
            )
        elif self.normal == 'Y':
            return Utils.mkvc(
                m.reshape(
                    self.mesh.vnC[[0, 2]], order='F'
                )[:, np.newaxis, :].repeat(
                    self.mesh.nCy,
                    axis=1
                )
            )
        elif self.normal == 'X':
            return Utils.mkvc(
                m.reshape(
                    self.mesh.vnC[[1, 2]], order='F'
                )[np.newaxis, :, :].repeat(
                    self.mesh.nCx,
                    axis=0
                )
            )

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model
        """
        inds = self * np.arange(self.nP)
        nC, nP = self.mesh.nC, self.nP
        P = sp.csr_matrix((np.ones(nC),
                           (range(nC), inds)
                           ), shape=(nC, nP))
        if v is not None:
            return P * v
        return P


class Mesh2Mesh(IdentityMap):
    """
        Takes a model on one mesh are translates it to another mesh.
    """

    def __init__(self, meshes, **kwargs):
        Utils.setKwargs(self, **kwargs)

        assert type(meshes) is list, "meshes must be a list of two meshes"
        assert len(meshes) == 2, "meshes must be a list of two meshes"
        assert meshes[0].dim == meshes[1].dim, ("The two meshes must be the "
                                                "same dimension")

        self.mesh = meshes[0]
        self.mesh2 = meshes[1]

        self.P = self.mesh2.getInterpolationMat(
            self.mesh.gridCC,
            'CC',
            zerosOutside=True
        )

    @property
    def shape(self):
        """Number of parameters in the model."""
        return (self.mesh.nC, self.mesh2.nC)

    @property
    def nP(self):
        """Number of parameters in the model."""
        return self.mesh2.nC

    def _transform(self, m):
        return self.P * m

    def deriv(self, m, v=None):
        if v is not None:
            return self.P * v
        return self.P

class Mesh2MeshTopo(IdentityMap):
    """
        Takes a model on one mesh are translates it to another mesh
        with consideration of topography

    """
    tree = None
    nIterpPts = 6
    P = None #: The CSR projection matrix.

    def __init__(self, meshes, actinds, **kwargs):
        Utils.setKwargs(self, **kwargs)

        assert type(meshes) is list, "meshes must be a list of two meshes"
        assert len(meshes) == 2, "meshes must be a list of two meshes"
        assert type(actinds) is list, "actinds must be a list of two meshes"
        assert len(actinds) == 2, "actinds must be a list of two meshes"
        assert meshes[0].dim == meshes[1].dim, """The two meshes must be the same dimension"""

        self.mesh  = meshes[0]
        self.mesh2 = meshes[1]
        self.actind = actinds[0]
        self.actind2 = actinds[1]
        self._createProjection()

        # Old version using SimPEG interpolation
        # self.P = self.mesh2.getInterpolationMat(self.mesh.gridCC,'CC',zerosOutside=True)

    def genActiveindfromTopo(mesh, xyztopo):
        #TODO: This possibly needs to be improved use vtk(?)
        if mesh.dim==3:
            nCxy = mesh.nCx*mesh.nCy
            Zcc = mesh.gridCC[:,2].reshape((nCxy, mesh.nCz), order='F')
            Ftopo = NearestNDInterpolator(xyztopo[:,:2], xyztopo[:,2])
            XY = Utils.ndgrid(mesh.vectorCCx, mesh.vectorCCy)
            XY.shape
            topo = Ftopo(XY)
            actind = []
            for ixy in range(nCxy):
                actind.append(topo[ixy] <= Zcc[ixy,:])
        else:
            raise NotImplementedError("Only 3D is working")

        return Utils.mkvc(np.vstack(actind))

    #Question .. is it only generated once?
    def _createProjection(self):
        """
            KD Tree interpolation onto the active cells.
        """
        if self.tree==None:
            self.tree = cKDTree(np.c_[self.mesh.gridCC[self.actind,0], self.mesh.gridCC[self.actind,1], self.mesh.gridCC[self.actind,2]])
        d, inds = self.tree.query(np.c_[self.mesh2.gridCC[self.actind2,0],self.mesh2.gridCC[self.actind2,1],self.mesh2.gridCC[self.actind2,2]], k=self.nIterpPts)
        # Not sure consideration of the volume ...
        # vol = np.zeros((self.actind2.sum(), self.nIterpPts))
        # for i in range(self.nIterpPts):
        #     vol[:,i] = self.mesh.vol[inds[:,i]]
        w = 1. / d**2
        w = Utils.sdiag(1./np.sum(w, axis=1)) * (w)
        I = Utils.mkvc(np.arange(inds.shape[0]).reshape([-1,1]).repeat(self.nIterpPts, axis=1))
        J = Utils.mkvc(inds)
        P = sp.coo_matrix( (Utils.mkvc(w),(I, J)), shape=(inds.shape[0], (self.actind).sum()) )
        # self.P = Utils.sdiag(self.mesh2.vol[self.actind2])*P.tocsc()
        self.P = P.tocsr()

    @property
    def shape(self):
        """Number of parameters in the model."""
        # return (self.mesh.nC, self.mesh2.nC)
        return (self.actind2.sum(), self.actind.sum())

    @property
    def nP(self):
        """Number of parameters in the model."""
        # return self.mesh2.nC
        return self.actind2.sum()

    def _transform(self, m):
        return self.P*m

    def deriv(self, m):
        return self.P

# class Mesh2MeshTopo(IdentityMap):
#     """
#         Takes a model on one mesh are translates it to another mesh
#         with consideration of topography

#     """
#     tree = None
#     nIterpPts = 6
#     P = None  #: The CSR projection matrix.
#     epsilon = 1e-8  # Small value to avoid 0 division in weights

#     def __init__(self, meshes, actinds, **kwargs):
#         Utils.setKwargs(self, **kwargs)

#         assert type(meshes) is list, "meshes must be a list of two meshes"
#         assert len(meshes) == 2, "meshes must be a list of two meshes"
#         assert type(actinds) is list, "actinds must be a list of two meshes"
#         assert len(actinds) == 2, "actinds must be a list of two meshes"
#         assert meshes[0].dim == meshes[1].dim, """The two meshes must be the same dimension"""

#         self.mesh = meshes[0]
#         self.mesh2 = meshes[1]
#         self.actind = actinds[0]
#         self.actind2 = actinds[1]
#         self._createProjection()

#         # Old version using SimPEG interpolation
#         # self.P = self.mesh2.getInterpolationMat(self.mesh.gridCC,'CC',zerosOutside=True)

#     def genActiveindfromTopo(mesh, xyztopo):

#         #TODO: This possibly needs to be improved use vtk(?)
#         if mesh.dim == 3:
#             nCxy = mesh.nCx*mesh.nCy
#             Zcc = mesh.gridCC[:, 2].reshape((nCxy, mesh.nCz), order='F')
#             Ftopo = NearestNDInterpolator(xyztopo[:, :2], xyztopo[:, 2])
#             XY = Utils.ndgrid(mesh.vectorCCx, mesh.vectorCCy)
#             XY.shape
#             topo = Ftopo(XY)
#             actind = []
#             for ixy in range(nCxy):
#                 actind.append(topo[ixy] <= Zcc[ixy, :])
#         else:
#             raise NotImplementedError("Only 3D is working")

#         return Utils.mkvc(np.vstack(actind))

#     #Question .. is it only generated once?
#     def _createProjection(self):
#         """
#             KD Tree interpolation onto the active cells.
#         """
#         if self.tree is None:
#             self.tree = cKDTree(np.c_[self.mesh.gridCC[self.actind, 0],
#                                       self.mesh.gridCC[self.actind, 1],
#                                       self.mesh.gridCC[self.actind, 2]])

#         d, inds = self.tree.query(np.c_[self.mesh2.gridCC[self.actind2, 0],
#                                         self.mesh2.gridCC[self.actind2, 1],
#                                         self.mesh2.gridCC[self.actind2, 2]],
#                                   k=self.nIterpPts)

#         # Not sure consideration of the volume ...
#         # vol = np.zeros((self.actind2.sum(), self.nIterpPts))
#         # for i in range(self.nIterpPts):
#         #     vol[:,i] = self.mesh.vol[inds[:,i]]
#         w = 1. / (d+self.epsilon)**2
#         w = Utils.sdiag(1./np.sum(w, axis=1)) * (w)
#         I = Utils.mkvc(np.arange(inds.shape[0]).reshape([-1, 1]).repeat(self.nIterpPts, axis=1))
#         J = Utils.mkvc(inds)
#         P = sp.coo_matrix((Utils.mkvc(w), (I, J)),
#                           shape=(inds.shape[0], (self.actind).sum()))
#         # self.P = Utils.sdiag(self.mesh2.vol[self.actind2])*P.tocsc()
#         self.P = P.tocsr()

#     @property
#     def shape(self):
#         """Number of parameters in the model."""
#         # return (self.mesh.nC, self.mesh2.nC)
#         return (self.actind2.sum(), self.actind.sum())

#     @property
#     def nP(self):
#         """Number of parameters in the model."""
#         # return self.mesh2.nC
#         return self.actind2.sum()

#     def _transform(self, m):
#         return self.P*m

#     def deriv(self, m):
#         return self.P

class InjectActiveCells(IdentityMap):
    """
        Active model parameters.

    """

    indActive = None  #: Active Cells
    valInactive = None  #: Values of inactive Cells

    def __init__(self, mesh, indActive, valInactive, nC=None):
        self.mesh = mesh

        self.nC = nC or mesh.nC

        if indActive.dtype is not bool:
            z = np.zeros(self.nC, dtype=bool)
            z[indActive] = True
            indActive = z
        self.indActive = indActive
        self.indInactive = np.logical_not(indActive)
        if np.isscalar(valInactive):
            self.valInactive = np.ones(self.nC)*float(valInactive)
        else:
            self.valInactive = np.ones(self.nC)
            self.valInactive[self.indInactive] = valInactive.copy()

        self.valInactive[self.indActive] = 0

        inds = np.nonzero(self.indActive)[0]
        self.P = sp.csr_matrix((np.ones(inds.size), (inds, range(inds.size))),
                               shape=(self.nC, self.nP)
                               )

    @property
    def shape(self):
        return (self.nC, self.nP)

    @property
    def nP(self):
        """Number of parameters in the model."""
        return int(self.indActive.sum())

    def _transform(self, m):
        return self.P * m + self.valInactive

    def inverse(self, D):
        return self.P.T*D

    def deriv(self, m, v=None):
        if v is not None:
            return self.P * v
        return self.P

###############################################################################
#                                                                             #
#                             Parametric Maps                                 #
#                                                                             #
###############################################################################


class ParametricCircleMap(IdentityMap):
    """ParametricCircleMap

        Parameterize the model space using a circle in a wholespace.

        .. math::

            \sigma(m) = \sigma_1 + (\sigma_2 - \sigma_1)\left(
            \\arctan\left(100*\sqrt{(\\vec{x}-x_0)^2 + (\\vec{y}-y_0)}-r
            \\right) \pi^{-1} + 0.5\\right)

        Define the model as:

        .. math::

            m = [\sigma_1, \sigma_2, x_0, y_0, r]

    """

    slope = 1e-1

    def __init__(self, mesh, logSigma=True):
        assert mesh.dim == 2, (
            "Working for a 2D mesh only right now. "
            "But it isn't that hard to change.. :)"
        )
        IdentityMap.__init__(self, mesh)
        # TODO: this should be done through a composition with and ExpMap
        self.logSigma = logSigma

    @property
    def nP(self):
        return 5

    def _transform(self, m):
        a = self.slope
        sig1, sig2, x, y, r = m[0], m[1], m[2], m[3], m[4]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)
        X = self.mesh.gridCC[:, 0]
        Y = self.mesh.gridCC[:, 1]
        return sig1 + (sig2 - sig1)*(np.arctan(a*(np.sqrt((X-x)**2 +
                                     (Y-y)**2) - r))/np.pi + 0.5)

    def deriv(self, m, v=None):
        a = self.slope
        sig1, sig2, x, y, r = m[0], m[1], m[2], m[3], m[4]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)
        X = self.mesh.gridCC[:, 0]
        Y = self.mesh.gridCC[:, 1]
        if self.logSigma:
            g1 = - (
                np.arctan(a * (-r + np.sqrt((X - x)**2 + (Y - y)**2)))/np.pi +
                0.5
            ) * sig1 + sig1
            g2 = (
                np.arctan(a * (-r + np.sqrt((X - x)**2 + (Y - y)**2)))/np.pi +
                0.5
            ) * sig2
        else:
            g1 = -(
                np.arctan(a * (-r + np.sqrt((X - x)**2 + (Y - y)**2)))/np.pi +
                0.5
            ) + 1.0
            g2 = (
                np.arctan(a * (-r + np.sqrt((X - x)**2 + (Y - y)**2)))/np.pi +
                0.5
            )

        g3 = a*(-X + x)*(-sig1 + sig2) / (
            np.pi*(
                a**2*(-r + np.sqrt((X - x)**2 + (Y - y)**2))**2 + 1
            ) * np.sqrt((X - x)**2 + (Y - y)**2)
        )

        g4 = a*(-Y + y)*(-sig1 + sig2) / (
            np.pi*(
                a**2*(-r + np.sqrt((X - x)**2 + (Y - y)**2))**2 + 1
            ) * np.sqrt((X - x)**2 + (Y - y)**2)
        )

        g5 = -a*(-sig1 + sig2) / (
            np.pi*(a**2*(-r + np.sqrt((X - x)**2 + (Y - y)**2))**2 + 1)
        )

        if v is not None:
            return sp.csr_matrix(np.c_[g1, g2, g3, g4, g5]) * v
        return sp.csr_matrix(np.c_[g1, g2, g3, g4, g5])


class ParametricPolyMap(IdentityMap):

    """PolyMap

        Parameterize the model space using a polynomials in a wholespace.

        .. math::

            y = \mathbf{V} c

        Define the model as:

        .. math::

            m = [\sigma_1, \sigma_2, c]

        Can take in an actInd vector to account for topography.

    """

    def __init__(self, mesh, order, logSigma=True, normal='X', actInd=None):
        IdentityMap.__init__(self, mesh)
        self.logSigma = logSigma
        self.order = order
        self.normal = normal
        self.actInd = actInd

        if getattr(self, 'actInd', None) is None:
            self.actInd = list(range(self.mesh.nC))
            self.nC = self.mesh.nC

        else:
            self.nC = len(self.actInd)

    slope = 1e4

    @property
    def shape(self):
        return (self.nC, self.nP)

    @property
    def nP(self):
        if np.isscalar(self.order):
            nP = self.order+3
        else:
            nP = (self.order[0]+1)*(self.order[1]+1)+2
        return nP

    def _transform(self, m):
        # Set model parameters
        alpha = self.slope
        sig1, sig2 = m[0], m[1]
        c = m[2:]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)

        # 2D
        if self.mesh.dim == 2:
            X = self.mesh.gridCC[self.actInd, 0]
            Y = self.mesh.gridCC[self.actInd, 1]
            if self.normal == 'X':
                f = polynomial.polyval(Y, c) - X
            elif self.normal == 'Y':
                f = polynomial.polyval(X, c) - Y
            else:
                raise(Exception("Input for normal = X or Y or Z"))

        # 3D
        elif self.mesh.dim == 3:
            X = self.mesh.gridCC[self.actInd, 0]
            Y = self.mesh.gridCC[self.actInd, 1]
            Z = self.mesh.gridCC[self.actInd, 2]

            if self.normal == 'X':
                f = (polynomial.polyval2d(Y, Z, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - X)
            elif self.normal == 'Y':
                f = (polynomial.polyval2d(X, Z, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - Y)
            elif self.normal == 'Z':
                f = (polynomial.polyval2d(X, Y, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - Z)
            else:
                raise(Exception("Input for normal = X or Y or Z"))

        else:
            raise(Exception("Only supports 2D"))

        return sig1+(sig2-sig1)*(np.arctan(alpha*f)/np.pi+0.5)

    def deriv(self, m, v=None):
        alpha = self.slope
        sig1, sig2, c = m[0], m[1], m[2:]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)

        # 2D
        if self.mesh.dim == 2:
            X = self.mesh.gridCC[self.actInd, 0]
            Y = self.mesh.gridCC[self.actInd, 1]

            if self.normal == 'X':
                f = polynomial.polyval(Y, c) - X
                V = polynomial.polyvander(Y, len(c)-1)
            elif self.normal == 'Y':
                f = polynomial.polyval(X, c) - Y
                V = polynomial.polyvander(X, len(c)-1)
            else:
                raise(Exception("Input for normal = X or Y or Z"))

        # 3D
        elif self.mesh.dim == 3:
            X = self.mesh.gridCC[self.actInd, 0]
            Y = self.mesh.gridCC[self.actInd, 1]
            Z = self.mesh.gridCC[self.actInd, 2]

            if self.normal == 'X':
                f = (polynomial.polyval2d(Y, Z, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - X)
                V = polynomial.polyvander2d(Y, Z, self.order)
            elif self.normal == 'Y':
                f = (polynomial.polyval2d(X, Z, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - Y)
                V = polynomial.polyvander2d(X, Z, self.order)
            elif self.normal == 'Z':
                f = (polynomial.polyval2d(X, Y, c.reshape((self.order[0]+1,
                     self.order[1]+1))) - Z)
                V = polynomial.polyvander2d(X, Y, self.order)
            else:
                raise(Exception("Input for normal = X or Y or Z"))

        if self.logSigma:
            g1 = -(np.arctan(alpha*f)/np.pi + 0.5)*sig1 + sig1
            g2 = (np.arctan(alpha*f)/np.pi + 0.5)*sig2
        else:
            g1 = -(np.arctan(alpha*f)/np.pi + 0.5) + 1.0
            g2 = (np.arctan(alpha*f)/np.pi + 0.5)

        g3 = Utils.sdiag(alpha*(sig2-sig1)/(1.+(alpha*f)**2)/np.pi)*V

        if v is not None:
            return sp.csr_matrix(np.c_[g1, g2, g3]) * v
        return sp.csr_matrix(np.c_[g1, g2, g3])


class ParametricSplineMap(IdentityMap):

    """SplineMap

        Parameterize the boundary of two geological units using
        a spline interpolation

        .. math::

            g = f(x)-y

        Define the model as:

        .. math::

            m = [\sigma_1, \sigma_2, y]

    """

    slope = 1e4

    def __init__(self, mesh, pts, ptsv=None, order=3, logSigma=True,
                 normal='X'):
        IdentityMap.__init__(self, mesh)
        self.logSigma = logSigma
        self.order = order
        self.normal = normal
        self.pts = pts
        self.npts = np.size(pts)
        self.ptsv = ptsv
        self.spl = None

    @property
    def nP(self):
        if self.mesh.dim == 2:
            return np.size(self.pts)+2
        elif self.mesh.dim == 3:
            return np.size(self.pts)*2+2
        else:
            raise(Exception("Only supports 2D and 3D"))

    def _transform(self, m):
        # Set model parameters
        alpha = self.slope
        sig1, sig2 = m[0], m[1]
        c = m[2:]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)
        # 2D
        if self.mesh.dim == 2:
            X = self.mesh.gridCC[:, 0]
            Y = self.mesh.gridCC[:, 1]
            self.spl = UnivariateSpline(self.pts, c, k=self.order, s=0)
            if self.normal == 'X':
                f = self.spl(Y) - X
            elif self.normal == 'Y':
                f = self.spl(X) - Y
            else:
                raise(Exception("Input for normal = X or Y or Z"))

        # 3D:
        # Comments:
        # Make two spline functions and link them using linear interpolation.
        # This is not quite direct extension of 2D to 3D case
        # Using 2D interpolation  is possible

        elif self.mesh.dim == 3:
            X = self.mesh.gridCC[:, 0]
            Y = self.mesh.gridCC[:, 1]
            Z = self.mesh.gridCC[:, 2]

            npts = np.size(self.pts)
            if np.mod(c.size, 2):
                raise(Exception("Put even points!"))

            self.spl = {"splb": UnivariateSpline(self.pts, c[:npts],
                                                 k=self.order, s=0),
                        "splt": UnivariateSpline(self.pts, c[npts:],
                                                 k=self.order, s=0)}

            if self.normal == 'X':
                zb = self.ptsv[0]
                zt = self.ptsv[1]
                flines = ((self.spl["splt"](Y) - self.spl["splb"](Y)) *
                          (Z - zb) / (zt - zb) + self.spl["splb"](Y))
                f = flines - X
            # elif self.normal =='Y':
            # elif self.normal =='Z':
            else:
                raise(Exception("Input for normal = X or Y or Z"))
        else:
            raise(Exception("Only supports 2D and 3D"))

        return sig1+(sig2-sig1)*(np.arctan(alpha*f)/np.pi+0.5)

    def deriv(self, m, v=None):
        alpha = self.slope
        sig1, sig2,  c = m[0], m[1], m[2:]
        if self.logSigma:
            sig1, sig2 = np.exp(sig1), np.exp(sig2)
        # 2D
        if self.mesh.dim == 2:
            X = self.mesh.gridCC[:, 0]
            Y = self.mesh.gridCC[:, 1]

            if self.normal == 'X':
                f = self.spl(Y) - X
            elif self.normal == 'Y':
                f = self.spl(X) - Y
            else:
                raise(Exception("Input for normal = X or Y or Z"))
        # 3D
        elif self.mesh.dim == 3:
            X = self.mesh.gridCC[:, 0]
            Y = self.mesh.gridCC[:, 1]
            Z = self.mesh.gridCC[:, 2]

            if self.normal == 'X':
                zb = self.ptsv[0]
                zt = self.ptsv[1]
                flines = ((self.spl["splt"](Y)-self.spl["splb"](Y)) *
                          (Z - zb) / (zt - zb) + self.spl["splb"](Y))
                f = flines - X
            # elif self.normal =='Y':
            # elif self.normal =='Z':
            else:
                raise(Exception("Not Implemented for Y and Z, your turn :)"))

        if self.logSigma:
            g1 = -(np.arctan(alpha*f)/np.pi + 0.5)*sig1 + sig1
            g2 = (np.arctan(alpha*f)/np.pi + 0.5)*sig2
        else:
            g1 = -(np.arctan(alpha*f)/np.pi + 0.5) + 1.0
            g2 = (np.arctan(alpha*f)/np.pi + 0.5)

        if self.mesh.dim == 2:
            g3 = np.zeros((self.mesh.nC, self.npts))
            if self.normal == 'Y':
                # Here we use perturbation to compute sensitivity
                # TODO: bit more generalization of this ...
                # Modfications for X and Z directions ...
                for i in range(np.size(self.pts)):
                    ctemp = c[i]
                    ind = np.argmin(abs(self.mesh.vectorCCy-ctemp))
                    ca = c.copy()
                    cb = c.copy()
                    dy = self.mesh.hy[ind]*1.5
                    ca[i] = ctemp+dy
                    cb[i] = ctemp-dy
                    spla = UnivariateSpline(self.pts, ca, k=self.order, s=0)
                    splb = UnivariateSpline(self.pts, cb, k=self.order, s=0)
                    fderiv = (spla(X)-splb(X))/(2*dy)
                    g3[:, i] = Utils.sdiag(alpha*(sig2-sig1) /
                                           (1.+(alpha*f)**2) / np.pi)*fderiv

        elif self.mesh.dim == 3:
            g3 = np.zeros((self.mesh.nC, self.npts*2))
            if self.normal == 'X':
                # Here we use perturbation to compute sensitivity
                for i in range(self.npts*2):
                    ctemp = c[i]
                    ind = np.argmin(abs(self.mesh.vectorCCy-ctemp))
                    ca = c.copy()
                    cb = c.copy()
                    dy = self.mesh.hy[ind]*1.5
                    ca[i] = ctemp+dy
                    cb[i] = ctemp-dy

                    # treat bottom boundary
                    if i < self.npts:
                        splba = UnivariateSpline(self.pts, ca[:self.npts],
                                                 k=self.order, s=0)
                        splbb = UnivariateSpline(self.pts, cb[:self.npts],
                                                 k=self.order, s=0)
                        flinesa = ((self.spl["splt"](Y) - splba(Y)) * (Z-zb) /
                                   (zt-zb) + splba(Y) - X)
                        flinesb = ((self.spl["splt"](Y) - splbb(Y)) * (Z-zb) /
                                   (zt-zb) + splbb(Y) - X)

                    # treat top boundary
                    else:
                        splta = UnivariateSpline(self.pts, ca[self.npts:],
                                                 k=self.order, s=0)
                        spltb = UnivariateSpline(self.pts, ca[self.npts:],
                                                 k=self.order, s=0)
                        flinesa = ((self.spl["splt"](Y) - splta(Y)) * (Z-zb) /
                                   (zt-zb) + splta(Y) - X)
                        flinesb = ((self.spl["splt"](Y) - spltb(Y)) * (Z-zb) /
                                   (zt-zb) + spltb(Y) - X)
                    fderiv = (flinesa-flinesb)/(2*dy)
                    g3[:, i] = Utils.sdiag(alpha*(sig2-sig1) /
                                           (1.+(alpha*f)**2) / np.pi)*fderiv
        else:
            raise(Exception("Not Implemented for Y and Z, your turn :)"))

        if v is not None:
            return sp.csr_matrix(np.c_[g1, g2, g3]) * v
        return sp.csr_matrix(np.c_[g1, g2, g3])


class AmplitudeMap(IdentityMap):
    """
        Amplitude Map for the Magnetic Amplitude problem.
        Takes three components vector model and transforms it
        into an amplitude
    """

    def __init__(self, mesh):
        IdentityMap.__init__(self, mesh)

        self._nP = int(mesh.dim*mesh.nC)

        start = 0
        Proj = []
        P = sp.csr_matrix((mesh.nC, mesh.dim*mesh.nC))
        for arg in range(mesh.dim):
            Proj += [Projection(self.nP, slice(start, start + mesh.nC))]
            P += Projection(self.nP, slice(start, start + mesh.nC)).P

            start += mesh.nC

        self.Plist = Proj
        self.P = P

        print(self.P.shape)
    @property
    def nP(self):
        return self._nP

    def _transform(self, m):

        nC = self.mesh.nC

        lml = np.zeros(nC)
        for P in self.Plist:
            lml += (P * m)**2.

        return lml**0.5

    def deriv(self, m, v=None):
        """
            :param numpy.array m: model
            :rtype: scipy.sparse.csr_matrix
            :return: derivative of transformed model

            The *transform* changes the model into the physical property.
            The *transformDeriv* provides the derivative of the *transform*.

        """
        A = Utils.sdiag(self._transform(m)**-1.)

        deriv = sp.csr_matrix(self.Plist[0].shape)

        for P in self.Plist:

            deriv += (A * (Utils.sdiag(P*m)*P.P))

        if v is not None:
            return deriv * v
        return deriv

###############################################################################
#                                                                             #
#                              Depreciated Maps                               #
#                                                                             #
###############################################################################


class FullMap(SurjectFull):
    """FullMap is depreciated. Use SurjectVertical1DMap instead"""
    def __init__(self, mesh, **kwargs):
        warnings.warn(
            "`FullMap` is deprecated and will be removed in future versions."
            " Use `SurjectFull` instead",
            FutureWarning)
        SurjectFull.__init__(self, mesh, **kwargs)


class Vertical1DMap(SurjectVertical1D):
    """Vertical1DMap is depreciated. Use SurjectVertical1D instead"""
    def __init__(self, mesh, **kwargs):
        warnings.warn(
            "`Vertical1DMap` is deprecated and will be removed in future"
            " versions. Use `SurjectVertical1D` instead",
            FutureWarning)
        SurjectVertical1D.__init__(self, mesh, **kwargs)


class Map2Dto3D(Surject2Dto3D):
    """Map2Dto3D is depreciated. Use Surject2Dto3D instead"""

    def __init__(self, mesh, **kwargs):
        warnings.warn(
            "`Map2Dto3D` is deprecated and will be removed in future versions."
            " Use `Surject2Dto3D` instead",
            FutureWarning)
        Surject2Dto3D.__init__(self, mesh, **kwargs)


class ActiveCells(InjectActiveCells):
    """ActiveCells is depreciated. Use InjectActiveCells instead"""

    def __init__(self, mesh, indActive, valInactive, nC=None):
        warnings.warn(
            "`ActiveCells` is deprecated and will be removed in future "
            "versions. Use `InjectActiveCells` instead",
            FutureWarning)
        InjectActiveCells.__init__(self, mesh, indActive, valInactive, nC)


class CircleMap(ParametricCircleMap):
    """CircleMap is depreciated. Use ParametricCircleMap instead"""

    def __init__(self, mesh, logSigma=True):
        warnings.warn(
            "`CircleMap` is deprecated and will be removed in future "
            "versions. Use `ParametricCircleMap` instead",
            FutureWarning)
        ParametricCircleMap.__init__(self, mesh, logSigma)


class PolyMap(ParametricPolyMap):
    """PolyMap is depreciated. Use ParametricSplineMap instead"""

    def __init__(self, mesh, order, logSigma=True, normal='X', actInd=None):
        warnings.warn(
            "`PolyMap` is deprecated and will be removed in future "
            "versions. Use `ParametricSplineMap` instead",
            FutureWarning
        )
        ParametricPolyMap(self, mesh, order, logSigma, normal, actInd)


class SplineMap(ParametricSplineMap):
    """SplineMap is depreciated. Use ParametricSplineMap instead"""

    def __init__(self, mesh, pts, ptsv=None, order=3, logSigma=True,
                 normal='X'):
        warnings.warn(
            "`SplineMap` is deprecated and will be removed in future "
            "versions. Use `ParametricSplineMap` instead",
            FutureWarning
        )
        ParametricSplineMap.__init__(
            self, mesh, pts, ptsv, order, logSigma, normal
        )


class ParametrizedLayer(IdentityMap):
    """
        Parametrized Layer Space

        .. code:: python

            m = [
                val_background,
                val_layer,
                layer_center,
                layer_thickness
            ]

        **Required**

        :param discretize.BaseMesh.BaseMesh mesh: SimPEG Mesh, 2D or 3D

        **Optional**

        :param float slopeFact: arctan slope factor - divided by the minimum h
                                spacing to give the slope of the arctan
                                functions
        :param float slope: slope of the arctan function
        :param numpy.ndarray indActive: bool vector with

    """

    slopeFact = 1e2  # will be scaled by the mesh.
    slope = None
    indActive = None

    def __init__(self, mesh, **kwargs):

        super(ParametrizedLayer, self).__init__(mesh, **kwargs)

        if self.slope is None:
            self.slope = self.slopeFact / np.hstack(self.mesh.h).min()

        self.x = [
            self.mesh.gridCC[:, 0] if self.indActive is None else
            self.mesh.gridCC[self.indActive, 0]
        ][0]

        if self.mesh.dim > 1:
            self.y = [
                self.mesh.gridCC[:, 1] if self.indActive is None else
                self.mesh.gridCC[self.indActive, 1]
            ][0]

        if self.mesh.dim > 2:
            self.z = [
                self.mesh.gridCC[:, 2] if self.indActive is None else
                self.mesh.gridCC[self.indActive, 2]
            ][0]

    @property
    def nP(self):
        return 4

    @property
    def shape(self):
        if self.indActive is not None:
            return (sum(self.indActive), self.nP)
        return (self.mesh.nC, self.nP)

    def mDict(self, m):
        return {
            'val_background': m[0],
            'val_layer': m[1],
            'layer_center': m[2],
            'layer_thickness': m[3],
        }

    def _atanfct(self, xyz, xyzi, slope):
        return np.arctan(slope * (xyz - xyzi))/np.pi + 0.5

    def _atanfctDeriv(self, xyz, xyzi, slope):
        # d/dx(atan(x)) = 1/(1+x**2)
        x = slope * (xyz - xyzi)
        dx = - slope
        return (1./(1 + x**2))/np.pi * dx

    def _atanLayer(self, mDict):
        if self.mesh.dim == 2:
            z = self.y
        elif self.mesh.dim == 3:
            z = self.z

        layer_bottom = mDict['layer_center'] - mDict['layer_thickness'] / 2.
        layer_top = mDict['layer_center'] + mDict['layer_thickness'] / 2.

        return (
            self._atanfct(z, layer_bottom, self.slope) *
            self._atanfct(z, layer_top, -self.slope)
        )

    def _atanLayerDeriv_layer_center(self, mDict):
        if self.mesh.dim == 2:
            z = self.y
        elif self.mesh.dim == 3:
            z = self.z

        layer_bottom = mDict['layer_center'] - mDict['layer_thickness'] / 2.
        layer_top = mDict['layer_center'] + mDict['layer_thickness'] / 2.

        return (
            self._atanfctDeriv(z, layer_bottom, self.slope) *
            self._atanfct(z, layer_top, -self.slope) +
            self._atanfct(z, layer_bottom, self.slope) *
            self._atanfctDeriv(z, layer_top, -self.slope)
        )

    def _atanLayerDeriv_layer_thickness(self, mDict):
        if self.mesh.dim == 2:
            z = self.y
        elif self.mesh.dim == 3:
            z = self.z

        layer_bottom = mDict['layer_center'] - mDict['layer_thickness'] / 2.
        layer_top = mDict['layer_center'] + mDict['layer_thickness'] / 2.

        return (
            -0.5*self._atanfctDeriv(z, layer_bottom, self.slope) *
            self._atanfct(z, layer_top, -self.slope) +
            0.5*self._atanfct(z, layer_bottom, self.slope) *
            self._atanfctDeriv(z, layer_top, -self.slope)
        )

    def layer_cont(self, mDict):
        return (
            mDict['val_background'] +
            (mDict['val_layer'] - mDict['val_background']) *
            self._atanLayer(mDict)
        )

    def _transform(self, m):
        mDict = self.mDict(m)
        return self.layer_cont(mDict)

    def _deriv_val_background(self, mDict):
        return np.ones_like(self.x) - self._atanLayer(mDict)

    def _deriv_val_layer(self, mDict):
        return self._atanLayer(mDict)

    def _deriv_layer_center(self, mDict):
        return ((mDict['val_layer']-mDict['val_background']) *
                self._atanLayerDeriv_layer_center(mDict))

    def _deriv_layer_thickness(self, mDict):
        return (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_thickness(mDict)
        )

    def deriv(self, m):

        mDict = self.mDict(m)

        return sp.csr_matrix(
            np.vstack([
                self._deriv_val_background(mDict),
                self._deriv_val_layer(mDict),
                self._deriv_layer_center(mDict),
                self._deriv_layer_thickness(mDict),
            ]).T)


class ParametrizedCasingAndLayer(ParametrizedLayer):
    """
        Parametrized layered space with casing.

        .. code:: python

            m = [val_background,
                 val_layer,
                 val_casing,
                 val_insideCasing,
                 layer_center,
                 layer_thickness,
                 casing_radius,
                 casing_thickness,
                 casing_bottom,
                 casing_top
            ]

    """

    def __init__(self, mesh, **kwargs):

        assert mesh._meshType == 'CYL', (
            'Parametrized Casing in a layer map only works for a cyl mesh.')

        super(ParametrizedCasingAndLayer, self).__init__(mesh, **kwargs)

    @property
    def nP(self):
        return 10

    @property
    def shape(self):
        if self.indActive is not None:
            return (sum(self.indActive), self.nP)
        return (self.mesh.nC, self.nP)

    def mDict(self, m):
        # m = [val_background, val_layer, val_casing, val_insideCasing,
        #      layer_center, layer_thickness, casing_radius, casing_thickness,
        #      casing_bottom, casing_top]

        return {
            'val_background': m[0],
            'val_layer': m[1],
            'val_casing': m[2],
            'val_insideCasing': m[3],
            'layer_center': m[4],
            'layer_thickness': m[5],
            'casing_radius': m[6],
            'casing_thickness': m[7],
            'casing_bottom': m[8],
            'casing_top': m[9]
        }

    def casing_a(self, mDict):
        return mDict['casing_radius'] - 0.5*mDict['casing_thickness']

    def casing_b(self, mDict):
        return mDict['casing_radius'] + 0.5*mDict['casing_thickness']

    def _atanCasingLength(self, mDict):
        return (
            self._atanfct(self.z, mDict['casing_top'], -self.slope) *
            self._atanfct(self.z, mDict['casing_bottom'], self.slope)
        )

    def _atanCasingLengthDeriv_casing_top(self, mDict):
        return (
            self._atanfctDeriv(self.z, mDict['casing_top'], -self.slope) *
            self._atanfct(self.z, mDict['casing_bottom'], self.slope)
        )

    def _atanCasingLengthDeriv_casing_bottom(self, mDict):
        return (
            self._atanfct(self.z, mDict['casing_top'], -self.slope) *
            self._atanfctDeriv(self.z, mDict['casing_bottom'], self.slope)
        )

    def _atanInsideCasing(self, mDict):
        return (
            self._atanCasingLength(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), -self.slope)
        )

    def _atanInsideCasingDeriv_casing_radius(self, mDict):
        return (
            self._atanCasingLength(mDict) *
            self._atanfctDeriv(self.x, self.casing_a(mDict), -self.slope)
        )

    def _atanInsideCasingDeriv_casing_thickness(self, mDict):
        return (
            self._atanCasingLength(mDict) * -0.5 *
            self._atanfctDeriv(self.x, self.casing_a(mDict), -self.slope)
        )

    def _atanInsideCasingDeriv_casing_top(self, mDict):
        return (
            self._atanCasingLengthDeriv_casing_top(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), -self.slope)
        )

    def _atanInsideCasingDeriv_casing_bottom(self, mDict):
        return (
            self._atanCasingLengthDeriv_casing_bottom(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), -self.slope)
        )

    def _atanCasing(self, mDict):
        return (
            self._atanCasingLength(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), self.slope) *
            self._atanfct(self.x, self.casing_b(mDict), -self.slope)
        )

    def _atanCasingDeriv_casing_radius(self, mDict):
        return (
            self._atanCasingLength(mDict) *
            (
                self._atanfctDeriv(self.x, self.casing_a(mDict), self.slope) *
                self._atanfct(self.x, self.casing_b(mDict), -self.slope) +
                self._atanfct(self.x, self.casing_a(mDict), self.slope) *
                self._atanfctDeriv(self.x, self.casing_b(mDict), -self.slope)
            )
        )

    def _atanCasingDeriv_casing_thickness(self, mDict):
        return (
            self._atanCasingLength(mDict) *
            (
                -0.5 *
                self._atanfctDeriv(self.x, self.casing_a(mDict), self.slope) *
                self._atanfct(self.x, self.casing_b(mDict), -self.slope) +
                self._atanfct(self.x, self.casing_a(mDict), self.slope) *
                0.5 *
                self._atanfctDeriv(self.x, self.casing_b(mDict), -self.slope)
            )
        )

    def _atanCasingDeriv_casing_bottom(self, mDict):
        return (
            self._atanCasingLengthDeriv_casing_bottom(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), self.slope) *
            self._atanfct(self.x, self.casing_b(mDict), -self.slope)
        )

    def _atanCasingDeriv_casing_top(self, mDict):
        return (
            self._atanCasingLengthDeriv_casing_top(mDict) *
            self._atanfct(self.x, self.casing_a(mDict), self.slope) *
            self._atanfct(self.x, self.casing_b(mDict), -self.slope)
        )

    def layer_cont(self, mDict):
        # contribution from the layered background
        return (
            mDict['val_background'] +
            (mDict['val_layer'] - mDict['val_background']) *
            self._atanLayer(mDict)
        )

    def _transform(self, m):

        mDict = self.mDict(m)

        # assemble the model
        layer = self.layer_cont(mDict)
        casing = (mDict['val_casing'] - layer) * self._atanCasing(mDict)
        insideCasing = (
            (mDict['val_insideCasing'] - layer) * self._atanInsideCasing(mDict)
        )

        return layer + casing + insideCasing

    def _deriv_val_background(self, mDict):
        # contribution from the layered background
        d_layer_cont_dval_background = 1. - self._atanLayer(mDict)
        d_casing_cont_dval_background = (
            -1. * d_layer_cont_dval_background * self._atanCasing(mDict)
        )
        d_insideCasing_cont_dval_background = (
            -1. * d_layer_cont_dval_background * self._atanInsideCasing(mDict)
        )
        return (
            d_layer_cont_dval_background +
            d_casing_cont_dval_background +
            d_insideCasing_cont_dval_background
        )

    def _deriv_val_layer(self, mDict):
        d_layer_cont_dval_layer = self._atanLayer(mDict)
        d_casing_cont_dval_layer = (
            -1. * d_layer_cont_dval_layer * self._atanCasing(mDict)
        )
        d_insideCasing_cont_dval_layer = (
            -1. * d_layer_cont_dval_layer * self._atanInsideCasing(mDict)
        )
        return (
            d_layer_cont_dval_layer +
            d_casing_cont_dval_layer +
            d_insideCasing_cont_dval_layer
        )

    def _deriv_val_casing(self, mDict):
        d_layer_cont_dval_casing = 0.
        d_casing_cont_dval_casing = self._atanCasing(mDict)
        d_insideCasing_cont_dval_casing = 0.
        return (
            d_layer_cont_dval_casing +
            d_casing_cont_dval_casing +
            d_insideCasing_cont_dval_casing
        )

    def _deriv_val_insideCasing(self, mDict):
        d_layer_cont_dval_insideCasing = 0.
        d_casing_cont_dval_insideCasing = 0.
        d_insideCasing_cont_dval_insideCasing = self._atanInsideCasing(mDict)
        return (
            d_layer_cont_dval_insideCasing +
            d_casing_cont_dval_insideCasing +
            d_insideCasing_cont_dval_insideCasing
        )

    def _deriv_layer_center(self, mDict):
        d_layer_cont_dlayer_center = (
            (mDict['val_layer'] - mDict['val_background']) *
            self._atanLayerDeriv_layer_center(mDict)
        )
        d_casing_cont_dlayer_center = (
            - d_layer_cont_dlayer_center * self._atanCasing(mDict)
        )
        d_insideCasing_cont_dlayer_center = (
            - d_layer_cont_dlayer_center * self._atanInsideCasing(mDict)
        )
        return (
            d_layer_cont_dlayer_center +
            d_casing_cont_dlayer_center +
            d_insideCasing_cont_dlayer_center
        )

    def _deriv_layer_thickness(self, mDict):
        d_layer_cont_dlayer_thickness = (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_thickness(mDict)
        )
        d_casing_cont_dlayer_thickness = (
            - d_layer_cont_dlayer_thickness * self._atanCasing(mDict)
        )
        d_insideCasing_cont_dlayer_thickness = (
            - d_layer_cont_dlayer_thickness * self._atanInsideCasing(mDict)
        )
        return (
            d_layer_cont_dlayer_thickness +
            d_casing_cont_dlayer_thickness +
            d_insideCasing_cont_dlayer_thickness
        )

    def _deriv_casing_radius(self, mDict):
        layer = self.layer_cont(mDict)
        d_layer_cont_dcasing_radius = 0.
        d_casing_cont_dcasing_radius = (
            (mDict['val_casing'] - layer) *
            self._atanCasingDeriv_casing_radius(mDict)
        )
        d_insideCasing_cont_dcasing_radius = (
            (mDict['val_insideCasing'] - layer) *
            self._atanInsideCasingDeriv_casing_radius(mDict)
        )
        return (
            d_layer_cont_dcasing_radius +
            d_casing_cont_dcasing_radius +
            d_insideCasing_cont_dcasing_radius
        )

    def _deriv_casing_thickness(self, mDict):
        d_layer_cont_dcasing_thickness = 0.
        d_casing_cont_dcasing_thickness = (
            (mDict['val_casing'] - self.layer_cont(mDict)) *
            self._atanCasingDeriv_casing_thickness(mDict)
        )
        d_insideCasing_cont_dcasing_thickness = (
            (mDict['val_insideCasing'] - self.layer_cont(mDict)) *
            self._atanInsideCasingDeriv_casing_thickness(mDict)
        )
        return (
            d_layer_cont_dcasing_thickness +
            d_casing_cont_dcasing_thickness +
            d_insideCasing_cont_dcasing_thickness
        )

    def _deriv_casing_bottom(self, mDict):
        d_layer_cont_dcasing_bottom = 0.
        d_casing_cont_dcasing_bottom = (
            (mDict['val_casing'] - self.layer_cont(mDict)) *
            self._atanCasingDeriv_casing_bottom(mDict)
        )
        d_insideCasing_cont_dcasing_bottom = (
            (mDict['val_insideCasing'] - self.layer_cont(mDict)) *
            self._atanInsideCasingDeriv_casing_bottom(mDict)
        )
        return (
            d_layer_cont_dcasing_bottom +
            d_casing_cont_dcasing_bottom +
            d_insideCasing_cont_dcasing_bottom
        )

    def _deriv_casing_top(self, mDict):
        d_layer_cont_dcasing_top = 0.
        d_casing_cont_dcasing_top = (
            (mDict['val_casing'] - self.layer_cont(mDict)) *
            self._atanCasingDeriv_casing_top(mDict)
        )
        d_insideCasing_cont_dcasing_top = (
            (mDict['val_insideCasing'] - self.layer_cont(mDict)) *
            self._atanInsideCasingDeriv_casing_top(mDict)
        )
        return (
            d_layer_cont_dcasing_top +
            d_casing_cont_dcasing_top +
            d_insideCasing_cont_dcasing_top
        )

    def deriv(self, m):

        mDict = self.mDict(m)

        return sp.csr_matrix(np.vstack([
            self._deriv_val_background(mDict),
            self._deriv_val_layer(mDict),
            self._deriv_val_casing(mDict),
            self._deriv_val_insideCasing(mDict),
            self._deriv_layer_center(mDict),
            self._deriv_layer_thickness(mDict),
            self._deriv_casing_radius(mDict),
            self._deriv_casing_thickness(mDict),
            self._deriv_casing_bottom(mDict),
            self._deriv_casing_top(mDict),
        ]).T)


class ParametrizedBlockInLayer(ParametrizedLayer):
    """
        Parametrized Block in a Layered Space

        For 2D:

        .. code:: python

            m = [val_background,
                 val_layer,
                 val_block,
                 layer_center,
                 layer_thickness,
                 block_x0,
                 block_dx
            ]

        For 3D:

        .. code:: python

            m = [val_background,
                 val_layer,
                 val_block,
                 layer_center,
                 layer_thickness,
                 block_x0,
                 block_y0,
                 block_dx,
                 block_dy
            ]

        **Required**

        :param discretize.BaseMesh.BaseMesh mesh: SimPEG Mesh, 2D or 3D

        **Optional**

        :param float slopeFact: arctan slope factor - divided by the minimum h
                                spacing to give the slope of the arctan
                                functions
        :param float slope: slope of the arctan function
        :param numpy.ndarray indActive: bool vector with

    """

    def __init__(self, mesh, **kwargs):

        super(ParametrizedBlockInLayer, self).__init__(mesh, **kwargs)

    @property
    def nP(self):
        if self.mesh.dim == 2:
            return 7
        elif self.mesh.dim == 3:
            return 9

    @property
    def shape(self):
        if self.indActive is not None:
            return (sum(self.indActive), self.nP)
        return (self.mesh.nC, self.nP)

    def _mDict2d(self, m):
        return{
            'val_background': m[0],
            'val_layer': m[1],
            'val_block': m[2],
            'layer_center': m[3],
            'layer_thickness': m[4],
            'x0_block': m[5],
            'dx_block': m[6]
        }

    def _mDict3d(self, m):
        return{
            'val_background': m[0],
            'val_layer': m[1],
            'val_block': m[2],
            'layer_center': m[3],
            'layer_thickness': m[4],
            'x0_block': m[5],
            'y0_block': m[6],
            'dx_block': m[7],
            'dy_block': m[8]
        }

    def mDict(self, m):
        if self.mesh.dim == 2:
            return self._mDict2d(m)
        elif self.mesh.dim == 3:
            return self._mDict3d(m)

    def xleft(self, mDict):
        return mDict['x0_block'] - 0.5*mDict['dx_block']

    def xright(self, mDict):
        return mDict['x0_block'] + 0.5*mDict['dx_block']

    def yleft(self, mDict):
        return mDict['y0_block'] - 0.5*mDict['dy_block']

    def yright(self, mDict):
        return mDict['y0_block'] + 0.5*mDict['dy_block']

    def _atanBlock2d(self, mDict):
        return (
            self._atanLayer(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope)
        )

    def _atanBlock2dDeriv_layer_center(self, mDict):
        return (
            self._atanLayerDeriv_layer_center(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope)
        )

    def _atanBlock2dDeriv_layer_thickness(self, mDict):
        return (
            self._atanLayerDeriv_layer_thickness(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope)
        )

    def _atanBlock2dDeriv_x0(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfctDeriv(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfctDeriv(self.x, self.xright(mDict), -self.slope)
                )
            )
        )

    def _atanBlock2dDeriv_dx(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfctDeriv(self.x, self.xleft(mDict), self.slope) *
                    -0.5 *
                    self._atanfct(self.x, self.xright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    0.5 *
                    self._atanfctDeriv(self.x, self.xright(mDict), -self.slope)
                )
            )
        )

    def _atanBlock3d(self, mDict):
        return (
            self._atanLayer(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope) *
            self._atanfct(self.y, self.yleft(mDict), self.slope) *
            self._atanfct(self.y, self.yright(mDict), -self.slope)
        )

    def _atanBlock3dDeriv_layer_center(self, mDict):
        return (
            self._atanLayerDeriv_layer_center(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope) *
            self._atanfct(self.y, self.yleft(mDict), self.slope) *
            self._atanfct(self.y, self.yright(mDict), -self.slope)
        )

    def _atanBlock3dDeriv_layer_thickness(self, mDict):
        return (
            self._atanLayerDeriv_layer_thickness(mDict) *
            self._atanfct(self.x, self.xleft(mDict), self.slope) *
            self._atanfct(self.x, self.xright(mDict), -self.slope) *
            self._atanfct(self.y, self.yleft(mDict), self.slope) *
            self._atanfct(self.y, self.yright(mDict), -self.slope)
        )

    def _atanBlock3dDeriv_x0(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfctDeriv(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfctDeriv(
                        self.x, self.xright(mDict), -self.slope
                    ) *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                )
            )
        )

    def _atanBlock3dDeriv_y0(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfctDeriv(self.y, self.yleft(mDict), self.slope) *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfctDeriv(self.y, self.yright(mDict), -self.slope)
                )
            )
        )

    def _atanBlock3dDeriv_dx(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfctDeriv(self.x, self.xleft(mDict), self.slope) *
                    -0.5 *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfctDeriv(
                        self.x, self.xright(mDict), -self.slope
                    ) *
                    0.5 *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                )
            )
        )

    def _atanBlock3dDeriv_dy(self, mDict):
        return (
            self._atanLayer(mDict) *
            (
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfctDeriv(self.y, self.yleft(mDict), self.slope) *
                    -0.5 *
                    self._atanfct(self.y, self.yright(mDict), -self.slope)
                ) +
                (
                    self._atanfct(self.x, self.xleft(mDict), self.slope) *
                    self._atanfct(self.x, self.xright(mDict), -self.slope) *
                    self._atanfct(self.y, self.yleft(mDict), self.slope) *
                    self._atanfctDeriv(
                        self.y, self.yright(mDict), -self.slope
                    ) *
                    0.5
                )
            )
        )

    def _transform2d(self, m):
        mDict = self.mDict(m)
        # assemble the model
        # contribution from the layered background
        layer_cont = (
            mDict['val_background'] +
            (
                mDict['val_layer'] - mDict['val_background']
            ) * self._atanLayer(mDict)
        )

        # perturbation due to the blocks
        block_cont = (
            mDict['val_block'] - layer_cont
        ) * self._atanBlock2d(mDict)

        return layer_cont + block_cont

    def _deriv2d_val_background(self, mDict):
        d_layer_dval_background = np.ones_like(self.x) - self._atanLayer(mDict)
        d_block_dval_background = (
            (-d_layer_dval_background) * self._atanBlock2d(mDict)
        )
        return d_layer_dval_background + d_block_dval_background

    def _deriv2d_val_layer(self, mDict):
        d_layer_dval_layer = self._atanLayer(mDict)
        d_block_dval_layer = (-d_layer_dval_layer)*self._atanBlock2d(mDict)
        return d_layer_dval_layer + d_block_dval_layer

    def _deriv2d_val_block(self, mDict):
        d_layer_dval_block = 0.
        d_block_dval_block = (1.-d_layer_dval_block)*self._atanBlock2d(mDict)
        return d_layer_dval_block + d_block_dval_block

    def _deriv2d_layer_center(self, mDict):
        d_layer_dlayer_center = (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_center(mDict)
        )
        d_block_dlayer_center = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock2dDeriv_layer_center(mDict) -
            d_layer_dlayer_center*self._atanBlock2d(mDict)
        )
        return d_layer_dlayer_center + d_block_dlayer_center

    def _deriv2d_layer_thickness(self, mDict):
        d_layer_dlayer_thickness = (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_thickness(mDict)
        )
        d_block_dlayer_thickness = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock2dDeriv_layer_thickness(mDict) -
            d_layer_dlayer_thickness*self._atanBlock2d(mDict)
        )
        return d_layer_dlayer_thickness + d_block_dlayer_thickness

    def _deriv2d_x0_block(self, mDict):
        d_layer_dx0 = 0.
        d_block_dx0 = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock2dDeriv_x0(mDict)
        )
        return d_layer_dx0 + d_block_dx0

    def _deriv2d_dx_block(self, mDict):
        d_layer_ddx = 0.
        d_block_ddx = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock2dDeriv_dx(mDict)
        )
        return d_layer_ddx + d_block_ddx

    def _deriv2d(self, m):
        mDict = self.mDict(m)

        return np.vstack([
            self._deriv2d_val_background(mDict),
            self._deriv2d_val_layer(mDict),
            self._deriv2d_val_block(mDict),
            self._deriv2d_layer_center(mDict),
            self._deriv2d_layer_thickness(mDict),
            self._deriv2d_x0_block(mDict),
            self._deriv2d_dx_block(mDict)
        ]).T

    def _transform3d(self, m):
        # parse model
        mDict = self.mDict(m)

        # assemble the model
        # contribution from the layered background
        layer_cont = (
            mDict['val_background'] +
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayer(mDict)
        )
        # perturbation due to the block
        block_cont = (
            (mDict['val_block'] - layer_cont) * self._atanBlock3d(mDict)
        )

        return layer_cont + block_cont

    def _deriv3d_val_background(self, mDict):
        d_layer_dval_background = np.ones_like(self.x) - self._atanLayer(mDict)
        d_block_dval_background = (
            (-d_layer_dval_background) * self._atanBlock3d(mDict)
        )
        return d_layer_dval_background + d_block_dval_background

    def _deriv3d_val_layer(self, mDict):
        d_layer_dval_layer = self._atanLayer(mDict)
        d_block_dval_layer = (-d_layer_dval_layer) * self._atanBlock3d(mDict)
        return d_layer_dval_layer + d_block_dval_layer

    def _deriv3d_val_block(self, mDict):
        d_layer_dval_block = 0.
        d_block_dval_block = (1.-d_layer_dval_block) * self._atanBlock3d(mDict)
        return d_layer_dval_block + d_block_dval_block

    def _deriv3d_layer_center(self, mDict):
        d_layer_dlayer_center = (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_center(mDict)
        )
        d_block_dlayer_center = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_layer_center(mDict) -
            d_layer_dlayer_center*self._atanBlock3d(mDict)
        )
        return d_layer_dlayer_center + d_block_dlayer_center

    def _deriv3d_layer_thickness(self, mDict):
        d_layer_dlayer_thickness = (
            (mDict['val_layer']-mDict['val_background']) *
            self._atanLayerDeriv_layer_thickness(mDict)
        )
        d_block_dlayer_thickness = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_layer_thickness(mDict) -
            d_layer_dlayer_thickness*self._atanBlock3d(mDict)
        )
        return d_layer_dlayer_thickness + d_block_dlayer_thickness

    def _deriv3d_x0_block(self, mDict):
        d_layer_dx0 = 0.
        d_block_dx0 = (
            (mDict['val_block'] - self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_x0(mDict)
        )
        return d_layer_dx0 + d_block_dx0

    def _deriv3d_y0_block(self, mDict):
        d_layer_dy0 = 0.
        d_block_dy0 = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_y0(mDict)
        )
        return d_layer_dy0 + d_block_dy0

    def _deriv3d_dx_block(self, mDict):
        d_layer_ddx = 0.
        d_block_ddx = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_dx(mDict)
        )
        return d_layer_ddx + d_block_ddx

    def _deriv3d_dy_block(self, mDict):
        d_layer_ddy = 0.
        d_block_ddy = (
            (mDict['val_block']-self.layer_cont(mDict)) *
            self._atanBlock3dDeriv_dy(mDict)
        )
        return d_layer_ddy + d_block_ddy

    def _deriv3d(self, m):

        mDict = self.mDict(m)

        return np.vstack([
            self._deriv3d_val_background(mDict),
            self._deriv3d_val_layer(mDict),
            self._deriv3d_val_block(mDict),
            self._deriv3d_layer_center(mDict),
            self._deriv3d_layer_thickness(mDict),
            self._deriv3d_x0_block(mDict),
            self._deriv3d_y0_block(mDict),
            self._deriv3d_dx_block(mDict),
            self._deriv3d_dy_block(mDict),
        ]).T

    def _transform(self, m):

        if self.mesh.dim == 2:
            return self._transform2d(m)
        elif self.mesh.dim == 3:
            return self._transform3d(m)

    def deriv(self, m):

        if self.mesh.dim == 2:
            return sp.csr_matrix(self._deriv2d(m))
        elif self.mesh.dim == 3:
            return sp.csr_matrix(self._deriv3d(m))