Terrain Correction Analysis Table 5.16 Terrain correction analysis

Data sources: the original 3x3arc sec dems with 1442401 pts as ref & the original 30x30arc sec dem with 130321 pts as ref Analysis of the actual gravity terrain correction on test area points test area lat

(deg) long (deg) ht(m) variation test pts sum of tc (mgal) mean tc (mgal) std dev (mgal) zimbabwe-c 20.3S-20.8S 30.3E-31.s e 459-973 113 11.435 0 . 1 0 1 0.260 montrose-e 38N-39N 107W-106W 2438-4267 99 412.081 4.162 2.495 santafe-e 35N-36N 106W-105W 1695-3291 99 120.058 1.213 0.971 pueblo-w 38N-39N 106W-105W 1645-3901 99 205.731 2.078 1.132 greeley-w 40N-41N 106W-105W 1707-4346 99 457.690 4.623 2.701

Based on the above analysis and conclusions further tests were then carried out on Terrain

1 to find out the effect on computed geoid height of using gravity anomalies with obser­

vation "noise" and also using signals with the typical terrain correction. The tests were carried out on the same 113 points used before by firstly computing the geoid height tak­ ing into account residual gravity anomalies from observed gravity anomalies and the OSU91A spherical harmonic model-derived gravity anomalies. Secondly observation noise and later some typical terrain correction was added to the 'observations' and the geoid heights were recomputed at the same 113 points. A DEM for the whole test area was used to supply the height information for the surrounding area. The DEM used was at 30"x30" data spacing. The geopotential model information was also supplied at the same density. Program RAPP was used to compute the geopotential model field in the DEM area. Program STOKES was used to compute the geoid heights using residual anomalies. For each point an inner zone of 0.00025 degrees around the computation point was used for finite spline interpolation to densify the data and overcome the singu­ larity problem at the computation point in Stokes’ equation. The outer zone used was 2.5 degrees around each computation point. A numerical analysis was done in order to find these two limits empirically by keeping one value constant and varying the other. Both the gravity anomaly observation "noise", based on statistics of gravity observations done in Zimbabwe and the typical terrain effect were created by a random number generator from program RANDOM. [Press,W., et.al., 1992]. The random numbers were generated using normal (Gaussian) distribution theory with specified mean and standard

Chapter 5: Terrain Correction Analysis

deviation. The gravity observation noise was typically from ±0.36 mgals in the first in­ stance, ±2.0mgals in the next and finally ±6.7mgals(based on mean value of observa­ tions). All the sets of observation "noise" had a mean of zero. These statistics are shown in Table 5.17.

The terrain effects were generated on the assumption that the terrain correction is always positive (a hill above the computation station and a valley below both yield the same sign of the contribution of the terrain correction) [Forsberg, R., 1994], [Forsberg, R. & Tscheming, C.C., 1981], [Heiskanen, W.A. & Moritz, H., 1993]. Values with mean of 0.25mgal and 3.25mgals were used in this case.(based on typical terrain effects in the two test areas.) Synthetic random noise and terrain corrections were used on real gravity data in an attempt to produce near-realistic spectral signatures of the data for the analysis.

Table 5.17 Synthetic Gravity Anomaly Observation Noise Signals,

(units: mgals)

total points mean std. dev. min max

130321 -0 . 0 0 1 ±0.361 -1.616 +1.562

130321 -0.003 ±2.005 -9.743 +8.610

130321 -0 . 0 1 2 ±6.722 -30.079 +29.076

Data for the above analysis are shown in the following tables. File GRAV.DAT with typical results shown in the Appendix contains the observed gravity anomalies in milli- gals, file SATGRAV.DAT also in the Appendix contains the OSU91A geopotential model data. File RESID.DAT contains the residual anomalies. Appendix 3 shows the typical "noise" in the data from the file NOISE.DAT. The file NEWGRAV.DAT con­ tains the new residual anomalies. The geoid height results are shown in Appendices 4 and 5 as files STOKES.DAT and ST0KES2.DAT. The typical terrain effect signal is shown in Appendix 6.

An analysis was then carried out comparing the original geoid computed from the avail­ able gravity data and reference model without any terrain corrections, and the new geoids, incorporating the terrain correction signals. The analysis statistics are shown be­ low in Table 5.18. The results show a mean of the geoid height residuals of about 5cm and a standard deviation of a millimetre in the case of using terrain correction effects with a

Chapter 5: Terrain Correction Analysis

mean of 0.25mgals, while using larger terrain effects with mean value of 3.25mgals gave residuals with a mean of 63cm with a 1cm standard deviation. In both cases a Imgal signal from the terrain correction gives rise to a 20cm error in computed geoid height. This shows that the terrain effect on the geoid height due to the terrain correction on the gravity anomalies in this mixed topography is quite significant for computations towards a 10cm geoid. The terrain correction computations are even more critical in more rugged terrain as in the New Mexico/Colorado areas where some terrain effect signals rise up to 50mgals or more. And in order to capture these signals, DEMs at better than 30"x30" are required for very rugged terrain. However further tests are required for extremely rugged terrain which has gravity data as well as DEM data at high densities such as 3"x3" in Zimbabwe, if and when such data becomes available.

Table 5.18 The Ejfects o f Terrain Corrections on Geoid Heights.

terrain effect computations using GRAVSOFT software

terrain effect on geoid height computations using mixed terrain h;0-1500m total no. computed sum of resid. mean(m) std dev(m) mean terrain signal(mgal)

113 113 5.228 0.049 ±0.001 0.25

113 113 71.678 0.634 ±0.010 3.25

As for the gravity observation data noise, the typical noise of ±0.36mgals in gravity ob­ servations gave a geoid error of about a millimetre. The ±2mgal "noise" gave a mean geoid height error of 3 mm and finally, the ±6.7mgal gravity "noise" gave rise to geoidal errors of about 1cm. The effect of the gravity observation noise on computed geoid heights using different noise signals is shown in Table 5.19.

Table 5.19 Ejfects o f Gravity Observation Noise on Computed Geoid Heights.

gravitational noise effect computations using GRAVSOFT software

gravitational noise effect on geoid height computations using mixed terrain h:0-1 0 0 0m

total no. computed sum of resid. mean(m) std dev(m) noise signal(mgal)

113 113 0.005 0 . 0 0 0 0 . 0 0 1 ±0.36mgal

113 113 -0.079 -0 . 0 0 1 0 . 0 0 2 ±2 . 0 mgal

Chapter 5: Terrain Correction Analysis

5.5 Terrain Corrections and Gravity Data Noise Effects Summary and Conclusions.

A summary of the rate of signal loss is shown in table 5.20.

Table 5.20 Terrain Correction Signal Loss, (mgal per arc min)

TERRAIN TYPE HEIGHT VARIATION RATE OF SIGNAL LOSS (mgal per arc sec)

terrain 1 m ixed 0 - 1500m 0.03

terrain 2 sm ooth flat 0 - 500m 0.007

terrain 3 m edium rough 500 - 1500m 0.02

terrain 4 rugged 1500 - 4300m 0.02

terrain 5 extrem ely rugged 0 - 4500m 0.09

m ontrose-e rugged 2400 - 4300m 1.20

santafe-e rugged 1500 - 3300m 0.60

pueblo-w rugged 1500 - 4000m 0.60

greeley-w rugged 1700 - 4400m 1.80

Looking at Zimbabwe in particular the effect of the topography on surface gravity was studied for various grid spacing using results obtained with the 30"x30" height grid as control, and 3"x3" control in parts of the USA. In order to obtain an estimate of aliasing effects or data capture efficiency occurring from using grid spacing larger than 30"x30", the computations were repeated on grids from l"x l" to l°x l° and the results were com­ pared to those of the 30"x30" grid on different terrain types. Now the typical terrain "white noise" obtained in medium topography(below 1000m) was found to be 0.36mgal using 113 points. Using this as typical tc signal, it was found that to get accuracies better than ±0.36mgal, the terrain has to be sampled at 6 ’ for this mixed terrain.(see fig. 5.13

p i 29), 18’ for flat terrain (fig 5.14 p i 29), 10’ for medium terrain (fig.5.15 p i 32), 8 ’ for

rough terrain and less than 30" for extremely rugged terrain (fig 5.16) Using the typical gravity anomaly observation noise for Zimbabwe gravity data taken at 2mgal level [van Gysen,H. & Merry,C.C., 1987], the terrain would have to be sampled at less than 45’ for mixed terrain and at less than 30" for rugged terrain. The USA tests have shown that a 15"xl5" grid would be required to get the 0.36mgal signal. Tests done by Schwarz, Si- deris and Forsberg (1990), using a O.lkmxO.lkm grid in the Canadian Rocky Mountains (1400-3400m), have shown that for RMS accuracies better than Imgal for gravity anom­ alies, the terrain has to be sampled at spacing of 0.5km or less (-15") in areas of similarly rough topography. Gravitational noise effect on geoid height tests done have shown that typical noise of 0.36mgal, 2mgal and 6.7mgal give errors of less than 1cm to the geoid height regardless of terrain type, (see Table 5.19). This seems to show the insensitivity of

Chapter 5: Terrain Correction Analysis

the geoid to the short wavelength noise features. [Forsberg, R. & Tscheming, C.C., 1992]. However, the terrain effect tests on geoid height in the mixed terrain have shown a mean value of Imgal signal giving rise to a 20cm geoid height effect, (table 5.17) Similar tests carried out by Tsiavos et al (1992) using a Ikm xlkm grid have shown lOmgal terrain correction giving rise to 20cm effect in geoid height. Based on these results a DEM at 9’x9’ in Zimbabwe (corresponding to O.Smgal) would be sufficient for a 10cm geoid ac­ curacy (relative to a 30"x30" DEM) for the mixed terrain. However, a better than 30" grid for the Eastern Highlands area (> 1500m) would be required to get the same level of accuracy. It is therefore recommended to use at least a 9’x9’ DEM for the mixed terrain in Zimbabwe (0-1500m) and a 30"x30" DEM or denser in the Eastern Highlands(>1500m), based on the analysis done in the test area in Zimbabwe and the 3" grid in parts of the USA

There is therefore a need for a DEM to compute the terrain corrections in Zimbabwe, without which the geoid produced would still lack the short wavelength contribution of the terrain in the computation of a 10cm geoid. Apart from extending and densifying data coverage, it is also recommended to compute a homogeneous gravity anomaly grid and to look at the related effects of noise from data gridding, using mean instead of point gravity anomalies, and the effect of low data coverage in some parts of the country before pro­ ceeding with the geoid computations. These recommendations led to the Gridding Analysis in chapter 6 and the subsequent Geophysical Gravity Smoothing Analysis in