Fast Fourier Transforms (FFT)

From a practical point of view, the two categories as described above require either very time consuming numerical integration or large matrix inversion. The need for large amounts of computation time for these methods is particularly significant when dealing with large scale computations in which large volumes of data are involved. This drawback could be overcome by using the FFT method. The FFT method of gravity field approximation is a frequency domain method in planar approximation.

In the FFT method, Stoke's integral is planarised and integrated over a Cartesian rectangular zone of gravity anomalies defined on a grid. However, many approximations are involved that decrease the amount of true information from the gravity field. A full account of FFT and their application to gravity field approximation can be found in Schwarz, et.al., (1990). This approach has been successfully implemented by Zhao, (1989), Mainville et.al., (1990), Tscheming and Forsberg, (1992) and Li and Sideris, (1994). Another similar approach has been carried out by Steward and Hipkin, (1989) who computed the geoid for the British Isles by integrating separately the Bouguer anomaly and the attraction of a terrain model and applying the FFT algorithm.

In general, the confuted geoid using the FFT technique is as:

^ i (3.26)

where F and F'^ denote the two-dimensional discrete Fourier transforms and its inverse, respectively and k is the radial frequency.

The FFT method requires gridded gravity data. Thus in this approach, a grid of gravity data is derived from observed gravity over the area of interest and gravity anomalies are transformed to the geoid separation at the same grid point by using the spectrum of the gravity field. The value of N at the points of computation

are then obtained by interpolation from the grid. Details of the FFT technique used in this study is also discussed in Sections 6.3 and 6.7.

3 .6 The Terrain Effect Evaluation

As previously mentioned in Section 3.3 (Table: 3.2), the short (or very short) wavelength variations of the gravity field come mainly from the topographic masses, i.e. rugged terrain. The influence of these masses will not be great on the geoid height, but will be considerable in the case of gravity anomalies and deflection of vertical which are much more terrain dependent. A terrain correction causes the gravity field to be much smoother and more homogeneous, as particular detail is removed prior to computations. Thus, in mountainous areas, the terrain effects completely dominate the local variation of the gravity field, and some kind of terrain reduction is indispensable when attempting gravity field modelling in such areas. The application of a good terrain reduction in mountainous areas decreases the gravity field variation to levels comparable with lowland areas, (Forsberg, 1984).

When a topographic model or height data is available, usually in the form of a Digital Elevation Model (DEM), the potential due to the terrain may be developed as a harmonic function to be subtracted firom the original function. Again, the signal retains its harmonicity. The various terrain effects in use are illustrated in Figure: 3.7. To use terrain reduced data in a remove-restore technique for gravity field modelling, the utilisation of known or assumed density models is very essential covering a given geographical area.

(A) - Topography

Mean Height Surface (C) - Residual Terrain Model

D~32 km

(B) - Isostatic

The most well known terrain reduction is the topographic reduction or 'a

complete Bouguer reduction \ consisting of the effects of a Bouguer plate minus the terrain correction. The topographic effect is well suited for geophysical work and the prediction of mean free-air anomalies, but is not applicable for reduction of geoid heights. Conventional isostatic reduction (see Section 2.7.3) formalises the prevailing tendency of the earth's topography to be compensated at depth. This type of terrain effect evaluation provide the smoothest residual fields and is easily applicable to aU the various types of gravity field data available. A drawback of the isostatic reduction is that it primarily should be global, that is, it requires height or depth data to be integrated over large regions.

An alternative to the use of the isostatic reduction is only to take the short periodic variations of the topography into account. This is done by only considering the deviations of the topography from some mean elevation surface using a Residual Terrain Model (RTM) method, see Figure: 3.7 (C). The RTM method has the advantage over the topographic and isostatic reduction methods that a fixed area is unnecessary in the calculations. Another advantage of the RTM method is that, because there is no need for any sort of isostatic compensation masses, the calculation is also quicker as fewer prisms are involved in the 'building ' of the terrain masses. Moreover, as the RTM have oscillating in positive and negative densities corresponding to the 'removal ' of mountains and 'filling ' of valleys, the effect of these will, in general, cancel out at certain distance from the calculation point. Thus, the total mass removed by using a reference surface will on average equal zero, but the major requisite is that the mass removed be a harmonic function. In this way, the recovery of only short wavelength information from the topography has the advantage that elevation data is only needed for a limited area. The application of the terrain reduction used in this study is presented in Section 6.5.

In general, the terrain effect can be quantified for the disturbing potential by:

*2 y2*2

TTc = G p j |j ia z 3 y 9 z

(3.28) *1 yi zi

where r = -^(x - )^ + (y - )^ + (z - )"

and p is the density for the element.

This computation is most naturally done using the simplest form of finite element representation of the density distribution by assuming that the density anomaly

rectangular prism, see Figure: 3.8. For terrain reductions using digital models, these prisms, for example, naturally correspond to the subdivision defined by the height data grid. For computational efficiency, the size of these sectors is increased at a distance whereby the majority of the effect is due to very near topography. The integral described in the above equation gives a global estimation. Thus, in practice, the integral need only extend to a given distance from computation point, as the effect of distant topography is negligible. For contributions at a distance r, a condensed formula is used through which equation (3.28) is modified to:

Ttc = GkJ J-3 y d x *1 yi ^

.(3.29)

where k is the surface mass density p(z2 - z%).

Pnsm

Computation

Figure: 3.8 - Terrain effect of distant prism

Terrain contributions to the gravity anomaly and geoid height are given by:

A g H = ^ - - T ^

dz r .(3.30)

and

.(3:31)

where Y is the normal gravity.