GRAVITY DATA ANALYSIS
6.2 Gravity Data Summary and Conclusions
A preliminary analysis was carried out on the gravity data to evaluate its accuracies and distribution as well as data density. Two geopotential models OSU91A and OSU89B were used to obtain residual gravity anomalies used in the analysis. Both the free -air and Bouguer anomalies were analysed. The areas under investigation were the Zimbabwe area with some 6150 point anomalies and the southern region with 18858 points. This was the current state of data coverage as at 1994. More observations have since been made to bring the current total data points to 13948 points locally and 26220 points regionally.
Mean free-air anomalies that are used in geoid determination are derived from the point gravity anomalies by some averaging technique. There are three important requirements for these averaging techniques which are that :
(i) the gravity anomalies to be used in the prediction process should be centred, because the optimal methods are based on the assumption that the gravity anomalies considered as statistical quantities with a mean value of zero, are random and follow the normal distribution characteristics;
(ii) the gravity anomalies should be height independent; and
(iii) the gravity field in the region under consideration should be homogeneous ( i.e. should be independent of position) and isotropic (independent of direction).
Regression analysis was carried out to substantiate that the gravity anomalies are in fact height dependent. Results showed that the free air anomalies are positively correlated with height with a correlation coefficient, r, of 0.777. (maximum value of r=l for a perfect linear relationship) The local deviations from regression are caused by topographic and subsurface mass or density anomalies. The best estimate of the regression coefficient, b, was computed to be 0.088mgal per metre, with a standard error of ±0.0009. The best estimate of the regression constant was found to be -79.80mgal with a standard error of
±0.91mgal. The average height in the local area with gravity data was found to 982m, with a maximum height at a measured gravity station being 1888m and a minimum height of 161.2m. The mean value of the observed free-air gravity anomalies was computed to be
6.564mgal ±39.695mgal. The measured maximum and minimum free-air anomalies were found to be 132.565mgal and -144.319mgal respectively. The regression analysis factors become important in the process of centring the data prior to prediction. It was found that the local area is quite varied in terrain with some low flat areas and some rug ged terrain in the east of the country. This becomes important later in the process of terrain reduction for geoid determination. The expected mean value of a stochastic quantity is zero. The mean values of the observations are quite small, close to zero proving that the
Chapter 6: Gravity Data And Gridding Effects Analysis
observations can be considered as stochastic quantities in the prediction process. The measured maximum and minimum gravity values are also typical of the terrain. Region ally the statistical results show a mean of 7.971 mgal ±34.203mgal, which is also close to zero.
Analysis made on the reduced gravity anomalies showed in both cases that the maximum order and degree of 360 for the geopotential models 0SU91A & OSU89B should be used in the future. Comparison of the local data and the regional data showed that the standard deviations for the larger data set were smaller, enhancing the fact that good gravity cov erage improves the statistical properties of the data for geoid computations. Further analysis showed that the 0SU91A produced a better reference gravity field for both the local area and regionally. This means the 0SU91A geopotential model is able to repre sent the gravity field in the area better than the OSU89B model. It was thus selected for future geoid determination work. Analysis in the individual 1° blocks showed the unequal distribution of the gravity data and the need for more observations in the devoid areas in the central west and northern east region of the Zimbabwe area. In particular two blocks were found to possibly contain some erroneous data which warranted further investiga tions in that area. The tables 6.7 and 6 . 8 show the statistics for the above analysis carried
out at a later date with the current data set ZW95 with 13948 gravity points locally and 26220 for the regional data set. The results show a great improvement in the values brought about by the increase in the data coverage. Comparing results of table 6.2 with
Table 6.7 Regression Analysis results fo r the Zimbabwe Area gravity anomalies ZW95
Free-air anomalies
correlation coefficient r 0.687
best estimate (regression coeff.) b (mgalm'*) 0.070 std error of regression coeff. (mgalm'*) ±0.001 best estimate (regression constant) a (mgal) -60.80
std error of constant (mgal) ±0.63
mean height (m) 935.10±348.51 max height (m) 1888.00 min height (m) 28.80 minimum (mgal) -144.43 maximum (mgal) 127.39 range (mgal) 271.82 mean (mgal) 4.73 std dev (mgal) ± 35.55 rms (mgal) ±35.86
Chapter 6: Gravity Data And Gridding Effects Analysis
Table 6.7, there is an improvement in the mean value of the gravity anomalies from 6.56 down to 4.73. There is also an improvement of the standard deviation with the new larger data set from ±39.70 to ±35.55. There is also a slight increase in the data range between the maximum and minimum values for the gravity anomalies. There is however some loss of correlation between the gravity anomalies and topography.
Table 6.8 Statistical results o f residual ZW95 Gravity Anomalies.
Nmax=360 ORIGINAL ZW95-OSU91A ZW95-OSU89B
(units: mgals) local regional local regional local regional
mean value 4.73 6.91 4.27 1.91 5.29 2.36 Minimum value -144.43 -145.30 -85.22 -118.78 -82.72 -118.56 Maximum value 127.39 180.49 135.10 135.20 138.52 138.62 std deviation ±35.55 ±35.06 ±19.70 ±19.61 ±19.90 ±19.80 rms ±35.86 ±35.74 ±20.16 ±19.71 ±20.59 ±19.94 range 271.82 325.79 220.32 253.98 221.24 257.18
A look at Table 6 . 8 re-emphasizes the fact that the OSU91A model represents the gravity
field in the area much better than the OSU89B gravitational model, both locally as well as regionally. This is confirmed by values of 4.27 and 1.91 for 0SU91A compared to 5.29 and 2.36 for the OSU89B model for the mean of the residual values. Values of the stan dard deviations also emphasize the same effect as well as the great smoothing effect which results from the use of the global geopotential models. It was recommended to carry on the process of data collection in the devoid area to improve on data coverage for future work. The next section looks at the problem of gridding the data in order to form a homogeneous data set, prior to geoid determination.
6.3 G ridding Effects Analysis
In the process of geoid determination it is necessary to use gridded data which is more homogeneous and therefore leads to stable computation of the geoid, less prone to oscil lations and pertubations due to the data gaps present in the randomly distributed data. However this gridding generally implies that some information is lost when going from the original point data to the gridded data. On the other hand data gaps have to be filled by some suitable prediction method in advance of the geoid computation. This section looks
Chapter 6: Gravity Data And Gridding Effects Analysis
at the effects of gridding gravity data on both gravity values and geoid h eight.
6.3.1 Introduction
The computation of the geoid from gravity data requires in principle a global, continuous data coverage. However, data gaps will in practice always occur due to lack of measure ments or restriction. Apart from lacking data, other sources of error include our limited ability to take into account enough data in actual computations, the use of approximations such as the spherical or planar approximation, and the effects of topography-induced aliasing in interpolating and averaging point gravity data in rugged topography. In other words it attenuates the aliasing problems associated with insufficient sampling of gravity data caused by the high frequency gravity field generated by the topographic masses. These errors may be reduced drastically by smoothing the data in a consistent manner so that harmonicity is preserved, followed by a "desmoothing" of the prediction result. This smoothing or "remove-restore" technique has become a standard procedure in many applications. The smoothing basically consists of eliminating the influence, or effect of a high degree geopotential model which is of long wavelength, and of removing effects of the residual topography relative to a mean elevation surface, taking into account the short wavelength or high frequency gravity field variation. This residual terrain model or so-called RTM reduction may in practice be computed by, for example, prism methods, selecting a mean height surface by filtering given DEM heights to, for example, 30’ or 1 degree resolution. In this case mountains above this surface are computationally removed and valleys below "filled" with material of some assumed constant. The resulting terrain reduced field still in principle refers to the surface of the original topography height level and the advantage of the RTM reduction is that the effect on the geoid is rather small and the systematic errors in the OEMs used have less influence on long wavelength features of the geoid. [Forsberg, R. & Tscheming,C.C., 1992], [Forsberg,R., & Tscheming,CC.,
1981].
In order to study the gridding effects on geoid height prediction , the following was done: (1) comparing results of geoid calculations with precise geoid heights obtained in a se
lected area, and
(2 ) comparing the results using different methods with each other.
Specifically the following studies were done:
(1) comparing the original gravity values with values predicted from grids with varying
Chapter 6: Gravity Data And Gridding Effects Analysis
(2) comparing geoid heights obtained using grid spacings with geoid heights obtained
using original data. ■ '
It is obvious, that the denser the gravity grid, the more accurate the recovery of the original values (the recovery is naturally limited if the data is assumed to have a high noise, implying filtering of the data.) On the other hand it is of no value to create a much finer grid than what the original data quality justifies.
6.3.2 Gridding Effects Test Area And Data Used
The selected test area is a 3°x3° block in the south-east corner of Zimbabwe. The general topography is from high to low (1490m - 160m). The area had a total of 2591 gravity data points. From the random point height data values, a 1km resolution DEM was created. The figure 6.11 shows the topography of the area.
- 1 4 CZ] ABOVE 1650 Q 1500 - 1650 1350- 1500 ■ 1200 - 1350 1050 - 1200 H i 900 - 1050 75 0 - 900 600 - 750 ■ i 450 - 600 300 - 450 150- 300 BELOW 150 $ 4 0) “19
S
J I 29 I 30 I 31 I 32 I 33 I 34 Longitude (deg. E)Chapter 6: Gravity Data And Gridding Effects Analysis
Figure 6.12 shows the location of the observed free-air gravity anomalies. The available gravity database for Zimbabwe currently (1997) has a total of 13948 observation points located in the area 14S-24S and 24E-34E. This data includes classified national agency data (DGS) and unreleased commercial data (Mobil). The gravity data within the test subarea above constituted a total of 2665 points bounded by 19°S-22°S and 29°.5E- 32°.5E. These data are on the IGSN71 datum and referred to GRS80 and all have the attributes of station name,latitude, longitude, height, observed gravity value, computed free-air anomalies and Bouguer anomalies.
Gravity D o t e l o c a t i o n m o p 2 5 9 1 p o i n t s - 1 4 — 16 ^ - 1 8 cr> (U -o (U TD 3 - 2 0 - 2 2 - 2 4 I I I " 1 I 1 1 1 1 I l l " 1 1 1 -
;
- . 4 - >■y
■V
1 1 1 i l l . , 1 1 1 . 1 1 1 1 1 1 2 4 2 6 28 30 Longitude (deg)E 32 3 4Figure 6.12 Location o f the g ra vity d a ta points.
The geopotential models available for use in this project include OSU91A, [Rapp, R.H., Wang,Y.M.& Pavlis, N., 1991], OSU89B,[Rapp, R.H. & Pavlis, NK., 1990], GEM-T3, [Lerch,F., et., al., 1994], JGM-1 & 2,[Nerem, R.S., et. al.,1994] & GPM2[Wenzel, H.,1985.] From previous tests done in gravity data analysis for Zimbabwe, it was recom-
Chapter 6: Gravity Data And Gridding Effects Analysis
mended to use the geopotential model 0SU91A on the basis of good representation of the gravity field in Zimbabwe. This entails use of global models with residuals that have an empirical mean value as near to zero as possible with a small signal and small noise variances. In terms of covariance analysis this means the solution must have a smooth covariance function that is not long tailed. [Benciolini, et. al., 1990]. Geopotential mod els are used as standard reference fields for local and regional geoid determination, where removing the contribution of a high resolution geopotential model from gravity field in a local or regional area makes possible precisely the determination of the medium and short wavelength signals of the geoid height.[Rapp,RH., 1986], [Ayhan,ME., 1993], [Arabe- los,D. & Tsiavos, I.N., 1980], [Tscherning,C.C. & Forsberg, R., 1986].
The preliminary DEM used was determined from the point height data of the gravity data. The DEM was produced by the program GEOGRID using the method of weighted means prediction with 20 closest points utilized in any single point height prediction.[Forsberg,R. & Tscherning,C.C.,1990] The resultant DEM was a 1km digital terrain model over the study subarea.
6.3.3 Gridding Effects Practical Procedure
The computations are based on slightly modified forms of the GRAVSOFT software package using mainly the programs GEOGRID, GEOIP, STOKES, SELECT and TC. [Tscheming,C.C. & Forsberg, R. & Knudsen,?., 1992] and the RAPP program. The anal ysis was carried out using the author’s ANALYSIS program.
The tests were carried out as follows:
Starting with the original 2591 free-air anomalies, the data was gridded using GEOGRID to varying spacing grids from 30"x30" down to 30’x30’. Using program GEOIP, data prediction was carried out from the various grids back to original 2591 points. The data predicted from the grids was compared to the original data in analysis. Secondly, as part of gravity data analysis, using program SELECT the original 2591 data points were di vided into two files, one containing some 2036 points and the other the remaining 555 points. The values of the first file were used as sample points for the prediction i.e. using the 2036 points the data was gridded and from the grid, data was predicted at the 555 points location. The values of the second file were then used for the aposteriori estimation of the accuracy of the prediction of point gravity anomalies.
Chapter 6: Gravity Data And Gridding Effects Analysis
The next set of tests were on geoid prediction. Starting with the 2591 points, using pro gram RAPP the long wavelength gravity field contributions of gravity anomalies and geoid height were computed based on the 0SU91A model recommended in this area. Based on a 1km dem and a 30’x30’ reference surface the RTM short wavelength contri butions to the gravity field were computed using program TC. Using the standard procedure of the "remove-restore" technique geoid prediction was carried out at the start ing sample points. The same procedure was repeated using 2591 points predicted from a gridded data set to compute a geoid at the same sample points. An analysis was then carried out to determine the accuracy of the geoid height prediction, from which the effects of data gridding were assessed. The whole procedure is shown in the following schematic diagram below.
1) GRAVITY