Contains software for filtering (destriping) GRACE Stokes coefficients.
DDK Filtering of GRACE Stokes coefficients
updated 15 October 2012 (Added DDK4 and DDK5 to package) updated 29 January 2015 (Added DDK6 to DDK8 to package)
DDK5d9: filtered with inverse signal degree power law 5e9deg^4 (DDK8) weakest smoothing
DDK1d10: filtered with inverse signal degree power law 1e10deg^4 (DDK7) |
DDK5d10: filtered with inverse signal degree power law 5e10deg^4 (DDK6) |
DDK1d11: filtered with inverse signal degree power law 1e11deg^4 (DDK5) |
DDK5d11: filtered with inverse signal degree power law 5e11deg^4 (DDK4) |
DDK1d12: filtered with inverse signal degree power law 1e12deg^4 (DDK3) |
DDK1d13: filtered with inverse signal degree power law 1e13deg^4 (DDK2) |
DDK1d14: filtered with inverse signal degree power law 1e14deg^4 (DDK1) strongest smoothing
The filter matrix and its coefficients
The block diagonal filter coefficients can be found in the directory data/DDK
If you don't want to be bothered with the storage scheme and just want to use octave/matlab to filter your coefficients just go here.
The filter coefficients are based on a block diagonal approximation of the normal system for the month august in 2003. Together with a signal variance model which behaves as a degree dependent power law one can construct a filter matrix W which is also block diagonal:
| W_Cl0 0 0 .. .. 0 0 | | | | 0 WCl1 0 .. .. 0 0 | | | | 0 0 WSl1 .. .. 0 0 | | | W = | : : : \ .. 0 0 | | | | : : : : \ 0 0 | | | | 0 0 0 0 0 WCl120 0 | | | | 0 0 0 0 0 0 WSl120| l=2-120 l=2-120 l=2-120 l=120-120 l=120-120 ord=0 ord=1 ord=1 ord=120 ord=120 COS COS SIN COS SIN block 1 2 3 240 241
The sides of W are arranged such that orders and sine or cosine coefficients are grouped in square block diagonal matrices.
W_Clm: means an order block with Cosine coefficients of order m which varies over the degree l. Its side has size lmax-max(lmin,m)+1.
For our case lmax=120 and lmin=2 meaning that blocks WSl120 and WCl120 have only one entry.
A filtered set of a spehrical harmonic set SH can be obtained by matrix multiplication:
SH_filt= W * SH
Where the ordering scheme of the input vector SH follows that of the matrix.
File format of DDK files
Only the block diagonal part of the filter matrix is stored. This causes a dramatic reduction of storage space. When reading in the data one should NOT expand the matrix to its full size as this will need more than 3Gb of storage. One should therefore apply the filter blockwise.
Binary Data files may be read in matlab/octave with the script read_BINslow.m
On the matlab prompt type:
dat = version: 'BINV2.1 ' type: 'BDFULLV0' descr: 'ANISOTROPIC FILTER matrix with power law regularization (alpha*l^pow) nval1: 14637 nval2: 0 pval1: 1180123 pval2: 0 nblocks: 241 ints_d: [6x24 char] ints: [6x1 double] dbls_d: [2x24 char] dbls: [2x1 double] side1_d: [14637x24 char] blockind: [241x1 double] pack1: [1180123x1 double]
The variable dat is a structure array with the following fields: version: version of binary file type ( not relevant) type: Type of the matrix, 'BDFULLV0' stands for a block diagonal matrix with no associated vectors
descr: Short description of the data content
nval1: size of the FULL matrix
nval2: unused yet
pval1: size of the stored matrix (packed matrix is stored in a vector with length pval1)
pval2: unused yet
nblocks: amount of diagonal blocks
ints_d: description of the integer meta data
ints: integer meta data dbls_d: description of the double meta data
dbls: double meta data
side1_d: description of the elements of the side of the FULL matrix. In case of spherical harmonics the description tag is TSN DEGORD or TCN DEGORD where DEG is the three digit degree and ORD is the 3 digit order of the coefficient. One can retrieve the degree and order from the character array by the matlab command
ord=str2num(dat.side1_d(:,8:10)). TCN denotes a cosine coefficient and TSN a sine coefficient.
blockind: array with indices of the last row(or column) of each block. For example: block n is the submatrix:
blockind(0) is not present but should be considered zero. The blocks are stored in the order as shown in the matrix above. Thus block 1 corresponds to WCl0 and block 241 corresponds to WSl120
pack1: Block diagonal matrix in packed form. Blocks are stored sequentially in column major order. So block n starts at:
nstart = 1+ sum(sz(i)^2), for i=1 .. n-1. Where the
sz(i) is the size of the ith block side:
sz(i)=blockind(i)-blockind(i-1) and ends at:
nend= nstart+ sz(n))^2 In matlab one can reshape the section of pack1 into a square matrix by the command:
As a check for a correct read one can display the first 5 values of the pack1 matrix by issuing the following commands on the matlab prompt:
>> format long
ans = 0.999995413476564 0.000000054102768 0.000000668145433 -0.000000013374350 0.000000077291901
The repository contains some m-scripts to filter coefficients with the anisotropic filter. The script testfilter.m executes some tests and compares the results with independent filter results. The required m-functions are located here.
In essence, the first step is to load the filter coefficients, e.g.:
with which then coefficients can be filtered in a second step:
Note that the coefficients are stored in (lower triangular) matrices which have rows corresponding to degrees and columns to the order. Taking into account the matlab style 1 indexing, a cosine coefficient,
CNM, with degree and order N,M is thus referenced as
References and further reading:
Decorrelated GRACE time-variable gravity solutions by GFZ, and their validation using a hydrological model Kusche, J.; Schmidt, R.; Petrovic, S.; Rietbroek, R. Journal of Geodesy, DOI: 10.1007/s00190-009-0308-3
For information on the GAC/GAA/GSM/GAD/GAB products and their use refer to the GRACE documentation: http://isdc.gfz-potsdam.de/index.php?name=UpDownload&req=viewdownload&cid=4