Programs for Data Handling, Presentation and Gravity Field Modelling

The main programs used in this work are as follows:

GEOGRID

This a fast program for gridding randomly distributed data into a regular rectangular grid, for interpolating data in profiles, or for interpolating individual points using enhanced, fast weighted mean interpolation or collocation/kriging. The program may detrend data prior to gridding, or just fit a trend surface to data. Local collocation or weighted means may be used. Both a geographical and a UTM grid may be used. Error estimates are produced. The program may also be used to compute estimates of individually given points (this is another randomly distributed data set). The program may be used to grid very large datasets, and has been optimized for fast prediction by internal sorting algo­

Chapter 5: Terrain Correction Analysis

rithm s; a quadrant search method is used internally in the program to speed up computations. This means that at each prediction point, only a certain fixed number of nearest points are used in each of the four quadrants around the point. To speed up the quadrant search, an internal data organization is used, where an internal data grid ensur­ ing 3 points per compartment on average are used. This grid is not related to the prediction grid that may ultimately be required. A special version GEOGRIDE may be used for gross-error detection. Quantities are predicted from nearby points, and com­ pared, and the statistics computed therefrom.

TC.

The program calculates terrain, bathymetric and/or isostatic effects on gravimetric quan­ tities using several overlapping grids with terrain data of varying spacing. The space domain prism method is used. Various modes of terrain effects may be computed:

1) direct topographic effect of all masses above sea-level, 2 ) topographic/isostatic reduction using an airy-model.

3) gravimetric terrain correction (effect of topographic irregularities with respect to a Bouguer plate) - in this case with conventional inversion of sign so that the computed quantities should be added to observations.

4) residual terrain model effects, i.e. the effects of the topographic irregularities with respect to a mean surface.

The program computes gravity disturbance, deflections of the vertical and/or geoid sep­ arations (height anomalies) in units of mgal, arcsec and m respectively, or second-order derivatives in Eotvos units (1 E = 0.1 mgal/km) The computation is based on two digital elevation models, a detailed and a coarse one, which are used in the inner and outer zones respectively. The two grids are assumed to have ’common’ boundaries, which is the case if the coarse grid has been constructed from the detailed grid by averaging. The integra­ tion of the terrain effects is performed using the formulae for the gravitational effects of a homogeneous rectangular prism. Depending on the geometry and accuracy various for­ mulae are used; exact formulae, spherical harmonic expansion (me millan) or centred point mass approximation. Parameters to determine formula choice and accuracy may be set by the user in subroutine ’prism 1’. The computation may be done out to a fixed dis­ tance from the computation point or for all masses in a given area. The detailed elevation grid is used out to a specified radius, which should be at least twice the gridspacing in the outer grid. In the local neighbourhood around the point, the terrain information may be densified using a bicubic spline interpolation. If the computation point is known to be at the surface of the topography, the terrain model is modified smoothly in an inner-zone to give the correct elevation at the computation point. The curvature of the earth is taken into

Chapter 5: Terrain Correction Analysis

account to the first order. Unknown heights in the height data files may be signalled by the value 9999. The program will not output the results for points where the computations have encountered unknown height values.

TCGRID.

This is a support program to average grids for use with program TC. This program is used to produce a mean elevation grid of a file containing a DEM (standard format or ngs modified).The mean grid is obtained by simple averaging. Areas with no elevations are given value 9999. If an individual grid element is covered partly by elevations, only these will be averaged; if no data is available the unknown flag 9999 will be written. If wanted, the mean elevation grid may be low-pass filtered using a moving-average window of a certain set number of cells of the grid

TCFOUR.:

This an FFT analysis program of DEMs. Computation of terrain effects, terrain correc­ tions etc. by Fourier methods are effected. Fourier domain expressions of the integral kernels are transformed instead of the space domain analytical kernels. This program is an FFT version of program TC.

GEOFOUR.:

The program implements the flat-earth Fourier method of modelling the gravity field. The program takes as input gridded data and produces gridded output as well. In the gravity field modelling it will evaluate Stokes’ formula to produce a geoid from gravity data; it will evaluate Vening-Meinesz’s formula to compute deflections of the vertical from grav­ ity data. It can implement inverse Stokes’ to produce the gravity field from a given geoid grid. Geoid, gravity and the gravity gradient tensor values may be used as input and a variety of other quantities may be produced.

SPFOUR.:

The program calculates geoid, upward continued gravity, isostatic effects and other quan­ tities by the multiband spherical FFT method. This allows FFTs to be carried out virtually exactly on the sphere.

STOKES.:

The program computes geoid heights at given station locations, using Stokes’ formula by numerical integration from gravity data. The gravity data must be given in a gridded

Chapter 5: Terrain Correction Analysis

format. A spline densification scheme is used to handle inner zone effects efficiently in a 3 x 3 innerzone grid around each computation point. The program also implements Vening-Meinesz’s integral.

SELECT.:

This program is used for converting between different data formats, e.g. data list to grid or the opposite. The program may also be used to average out data, or thin out data in a pixel mode. The program may also be used to add random noise to data in a file. The same program has an option to select an areal window from a given data grid.

GEOIP.:

This is a general purpose bilinear or spline interpolation program. It interpolates from a grid to point values. It may add/subtract values interpolated from grids by linear or spline interpolation. The program assumes the prediction points to be reasonably close, reading in the smallest necessary subgrid to perform the required interpolations. It may also be used to add or subtract two grids of different spacings, e.g. for restoring reference effects after an FFT prediction. The program has special options where it may interpolate two grids at the same time (designed for deflections of the vertical) or may be used for "sand­ wich interpolation" between two different grids representing data in different elevations. (3-D interpolation). Geoip also produces statistics of the data reductions. This general purpose interpolation program is able to implement the UTM grids fully.

GEOID.:

This program performs simple linear interpolation and transformation for binary geoid grids. The program may also be used for some coordinate transformations between UTM, geographic and Cartesian coordinate systems. The transformation between binary and ASCII grids is carried out by GBIN. One important aspect is that it can be used to fit the geoid onto local GPS control to eliminate residual geoid errors and vertical datum dis­ crepancies as included in the ellipsoidal to orthometric height conversion.

The rest of the package contains the following software described briefly:

EC.: This program (File Comparison) is used to produce from two files a file with the sum or difference of the values in the files, along with statistics of the various effects.

Chapter 5: Terrain Correction Analysis

GCOMB.: (grid combination) is a general purpose grid manipulator. It may add/subtract, expand or overwrite grids in a number of fashions. The grids need not cover the same areas.

CRSADJ.: is used for cross-over adjustment of satellite altimeter data, with output of a set o f adjusted altim eter data. The adjustm ent is perform ed using bias or bias/tilt as parameters.

ALTSTACK.: stacks collinear tracks of altimeter observations and computes variability of the data.

POINTMASS. :The program generates a grid or data list of point mass gravitational effects. This program is most useful for checking various procedures such as conversion of geoid heights to gravity.

AZTRACK.iThe program will calculate the azimuth of a track segment, by calculating the azimuth between neighbouring points. This program is required when using line av­ erages in collocation, representing e.g. airborne gravity measurements.

CONVOLVE.: A convolution program for track data, designed e.g. for filtering predic­ tions corresponding to airborne gravimetry filters.

GEOCOL.: The program has a variety of functions of which the most important are: Computation and evaluation of a gravity field model using least squares collocation, es­ timation of the same kind of point or mean values and their error estimates. Also parameters such as datum shift or altimetry bias and tilt values may be estimated. The program may also be used purely for evaluating spherical harmonic series, calculating error degree-variances etc. It may also be used for the transformation between different datums.

EMPCOV.:The program empcov is used to produce values of an empirical covariance function,. CO VEIT. The program has two functions:(l) evaluation of an analytic expres­ sion for the covariance function of the anomalous gravity field and (2) the fitting of an

analytic expression to given empirical values.

CGVFFT.:The program calculates 1-D and 2-D covariance functions and power spectra using gridded data using Fourier analysis of gridded data files.

Chapter 5: Terrain Correction Analysis