Final Drafts/Module 0, a quick overview.ipynb

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     "source": [
      "What is inversion?"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For those with little or no background or experience, the concept of inversion can be intimidating. Many available resources assume a certain amount of background already, and for the uninitiated, this can make the topic difficult to grasp. Likewise, if material is too general and simplistic, it is of little use to someone who has some understanding but is looking for deeper insight. This suite of tutorials will begin in earnest from the first module, but if you have no background, then this is the place to start. I will assume that you have had some basic linear algebra and matrix theory, and little else. I will also assume, at least at first, that you may have forgotten a few of these things along the way. So let's begin from zero, with the absolute simplest terms:<br>\n",
      "\n",
      "In algebra, when we want to solve an equation for an unknown variable $x$, say in the equation:<br>\n",
      "\n",
      "\\begin{equation}\n",
      "ax=b \n",
      "\\end{equation} <br>\n",
      "where $a$ and $b$ are real numbers, we can do so, quite simply, by multiplying both sides of the equation by the *inverse* of $a$, that is $a^{-1}$. Our result $x = b/a$, is, in a very simple way, the solution to an inverse problem. And even though this example is trivial, inversion is simply an expansion of this process to larger systems of equations with more unknowns. Expanding ever so slightly, we do an analogous process by obtaining the inverse of a matrix. For example, say we have a system of equations that we represent in a square matrix $A$, a vector of unknowns $x$, and a vector of known values $b$. What we constuct is the matrix equation <br> \n",
      "\n",
      "\\begin{equation}\n",
      "Ax=b \n",
      "\\end{equation} <br>\n",
      "\n",
      "and provided an inverse for $A$ exists (that is, if $det(A) \\neq 0$, or if you prefer $A^{-1}A = I$), the solution can be found by multiplying both sides by the inverse of $A$\n",
      "\n",
      "\\begin{equation}\n",
      "A^{-1}Ax= Ix = x = A^{-1}b \\\\\n",
      "x = A^{-1}b\n",
      "\\end{equation} <br>\n",
      "\n",
      "If you are the type of person that likes to see things through simple examples, consider the following. Say we have a set of two equations and two unknowns:\n",
      "\n",
      "\\begin{equation}\n",
      "2x +0y = 4 \\\\\n",
      "0x +3y = 6 \\\\\n",
      "\\end{equation} <br>\n",
      "\n",
      "Now, because this is a trivial example, we can see from inspection that $x=2$ and $y=2$. But for the purpose of seeing how things are put together, let's solve this using matrices anyway. We can assemble the coefficients for each equation into the matrix equation:\n",
      "\n",
      "\\begin{equation}\n",
      "\\begin{bmatrix}\n",
      "    2 &0\\\\\n",
      "    0 &3\\\\    \n",
      "\\end{bmatrix}\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "x\\\\\n",
      "y\n",
      "\\end{array}\n",
      "\\right]\n",
      "=\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "4\\\\\n",
      "6\n",
      "\\end{array}\n",
      "\\right]\n",
      "\\end{equation}<br>\n",
      "\n",
      "This is an equation of the form $Ax=b$, and we can solve this by multiplying both sides of the equation by $A^{-1}$:\n",
      "\n",
      "\\begin{equation}\n",
      "\\begin{bmatrix}\n",
      "    \\frac{1}{2} &0\\\\\n",
      "    0 &\\frac{1}{3}\\\\    \n",
      "\\end{bmatrix}\n",
      "\\begin{bmatrix}\n",
      "    2 &0\\\\\n",
      "    0 &3\\\\    \n",
      "\\end{bmatrix}\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "x\\\\\n",
      "y\n",
      "\\end{array}\n",
      "\\right]\n",
      "=\n",
      "\\begin{bmatrix}\n",
      "    \\frac{1}{2} &0\\\\\n",
      "    0 &\\frac{1}{3}\\\\    \n",
      "\\end{bmatrix}\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "4\\\\\n",
      "6\n",
      "\\end{array}\n",
      "\\right]\n",
      "\\end{equation}<br>\n",
      "\n",
      "Multiplying this out on both sides gives us (unsurprisingly):<br>\n",
      "\n",
      "\\begin{equation}\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "x\\\\\n",
      "y\n",
      "\\end{array}\n",
      "\\right]\n",
      "=\n",
      "\\left[\n",
      "\\begin{array}{c}\n",
      "2\\\\\n",
      "2\n",
      "\\end{array}\n",
      "\\right] \n",
      "\\end{equation} <br>\n",
      "\n",
      "Again, this is a trivial example. Now we will build up the notation a little. Thus far I have used the standard matrix-vector equation ($Ax=b$) that you have likely seen in a basic linear algebra textbook. In this set of tutorials I will use slightly different notation, and our matrix equation will take the form $Gm=d$. Of course, the choice of symbols for matrices and vectors is arbitrary, but in the context that we are going to proceed, $G$, $m$, and $d$ will take on new meaning. We represent the right hand side of the equation by $d$, which is the *data* that we measure in the field, $m$ which represents our *model parameters* (the unknowns that we seek), and $G$, which is our *forward operator* that operates on the model to produce the data. Generally, the governing equations that go into the physical system are represented by our matrix $G$. And even more generally, our matrix $G$ is a special case of a more general idea of a forward operator, $F[m]$. For a geophysical survey the a broader representation of our problem, for the ith data point would be\n",
      "\n",
      "\\begin{equation}\n",
      "F_i[m] = d_i + n_i\n",
      "\\end{equation}\n",
      "\n",
      "where $F_i[m]$ is the forward operator, $d_i $ is the measured data and $n_i$ is additive noise. This distinction is important as it adds a degree of richness to our scenario. Here are a few points to consider:\n",
      "\n",
      "<ul>\n",
      "<li>First  $F[m]$ does not need to be linear. You may recall that a transformation is considered *linear* if the following property holds: <br>\n",
      "\n",
      "\\begin{equation}\n",
      "T(\\alpha u+ \\beta v) = \\alpha T(u) + \\beta T(v)\n",
      "\\end{equation} <br>\n",
      "In the most general case our forward operator $F[m]$ can be linear or nonlinear.\n",
      "\n",
      "<li> In the case where  $F[m]$ is a matrix operation (our $G$, above), it might be a square matrix that is not invertible.\n",
      "<li> $F[m]$  might represent an  overdetermined system (i.e. $G$ is not square matrix, but is a \"tall\" matrix).\n",
      "<li> $F[m]$  might represent an underdetermined system (i.e. $G$ is not square matrix, but is a \"wide\" matrix).\n",
      "<li> In geophysical applications, the number of values is large, and the doing these operations directly on large data sets is impractical.\n",
      "<li> In geophysics we deal with physical systems, and physical systems will always have noise in the data, which is accounted for in the $n_i$ term above.\n",
      "<li> We will need to estimate the uncertainties in our data, again, because we are dealing with physical systems.\n",
      "<li> And it can never be emphasized enough, so I will introduce it now: geophysics problems produce nonunique solutions. Obtaining a solution will involve optimization theory and will require implementing what we call a \"regularization.\" The solution that we get will depend heavily, very heavily,  on our choice of regularization. This choice is critical as it makes the difference between arriving at a solution that makes logical (and geological) sense, and complete gibberish. \n",
      "</ul>\n",
      "\n",
      "\n",
      "The end goal of this set of tutorials is to understand and learn how to manage some of the complexities that go into the inversion process. I will, whenever possible, try to build intuition by first using simple systems and trivial solutions, and then develop these into more complex (and more useful) versions of the same.\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ]
    },
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     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Forward modeling vs. inversion\n"
     ]
    },
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Forward modeling and inversion are opposite processes. When the model of the Earth is known and we want to calculate the data, we call this a forward model. Forward modeling answers the question: given that I know the physics and have a model of the earth, what will the measured data look like? In the matrix-vector terms we used above, $Gm=d$, the forward model has two knowns, $G$ and $m$, and one unknown, $d$. Calculating the data is done by applying our forward operator $G$ to our model parameters, $m$. Inversion, as I have stated above already, takes measured data $d$ and a known forward operator $G$ and solves for the model parameters $m$. Put slightly differently, given that we have (1) field survey observations, (2) an estimate about the errors in these observations, (3) a physical representation of the earth, and (4) the ability to produce a forward model, the goal of the inversion is to produce a reasonable model that generated data. <br>\n",
      "\n",
      "\n",
      "\n",
      "\n"
     ]
    },
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     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Types of Inversion Problems"
     ]
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     "cell_type": "markdown",
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     "source": [
      "Various methodologies for performing geophysical inversion have been developed. There are two broad classes of inversion: \"Parametric\" methods and \"Generalized\" inversion methods. \n",
      "\n",
      "\n",
      "**Parametric methods**\n",
      "\n",
      "These inversion methods involve finding a model of the earth which is described using only a few parameters. The solutions require that there be fewer parameters than there are data values so that the problem is formally \"over-determined.\" A few examples of parametric models are:\n",
      " - **Buried object:** parameters could be depth to a sphere (or cylinder), a radius of a sphere or radius and length of a cylinder, and the physical property contrast between the object and host rocks. \n",
      " - **Layered earth:** parameters are layer thicknesses and physical property values.\n",
      " - **A buried sheet:** parameters might be depth to the top of sheet, it's dip, strike, thickness, and the physical property contrast between the sheet and host rocks.\n",
      "\n",
      "**Generalized inversion methods**\n",
      "\n",
      "This second class of inversion methods allows the earth's model to be more realistically complex, which means that more parameters than data points are permitted. Such problems are mathematically referred to as \"under-determined\". Most solutions to this more general form of the geophysical inversion problem involve three steps, which can be explained as follows: \n",
      "\n",
      "1. Represent the earth with many cells so that complex distributions of physical properties can be simulated. In practice, the earth is divided into thousands or millions of cells of fixed geometry. Each cell has a constant, but unknown, value. The parameters we seek are the physical property values for these cells.\n",
      "\n",
      "2. Design a model objective function. This is a mathematical quantity which measures the \"size\" of any solution. It is a single number. A priori information about the earth can be incorporated into the objective function. Usually the model objective has different components. One will make it \"close\" to a supplied reference model, others may control \"smoothness\" in various spatial directions. Mathematical optimization theory is used to find a solution that minimizes the objective function. The resultant solution will have minimum structure. This will be a good choice since it will tend to show large scale important features rather than a great deal of extraneous structure that can result from noisy observations.\n",
      "\n",
      "3. The final solution must also acceptably reproduce the field observations. Our final optimization problem is to find that model which minimizes our model objective function subject to the constraint that the measured data are adequately reproduced.\n",
      "\n",
      "\n",
      "In practice a number of inversions, with different reasonable objective functions, should be carried out so the interpreter has some insight about the range of earth models that can acceptably reproduce the field data. Error statistics about the data will determine how closely the reproduced data matches the real measured data. The fact that these error statistics are often poorly known is a second good reason for performing several inversions before settling upon a preferred model. \n",
      "\n",
      " <!--- This is directly from the new GPG -->"
     ]
    },
    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "Inversion as a workflow"
     ]
    },
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     "cell_type": "markdown",
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     "source": [
      "Even though in the back of our minds we are really just solving a matrix equation, the complications that arise in practical situations will require inversion to take the form of a multi-step process, which is often represented as a workflow.\n",
      "\n",
      "<img src=\"figures/workflow.jpg\" width=\"40%\" height=\"40%\"> <br>\n",
      "\n",
      "Each subsequent module will begin to discuss one of the boxes in the workflow, and we will build up the entire process, step by step, until completion. The first module of this tutorial, then, will begin with setting up the problem\u2014taking a continous distribution of a physical property, discretizing it, and putting it on a mesh. "
     ]
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