GEOID DETERMINATION METHODS AND TECHNIQUES
3.2 Geoid Determination Techniques
3.2.1 Data Spectrum
In physical geodesy the measurements or signals are always given in some spatial form
as randomly distributed data or in some regular grid. These signals can be viewed as part
of a whole spectrum of different wavelengths or in the frequency domain, as different
frequencies. With this in mind, the gravity field can be considered as consisting of signals
of different wavelengths.[Schwarz,K.P., 1984], [Szabo,B., 1986].
3.2.1.1 Data Spectrum Wavelength Signals
The data spectrum wavelength signals are discussed next. At one end of the spectrum there are the long wavelength signals, contributing to the gravity field spectrum at any thing from 50km to several thousand km. The broad medium wavelength part of the spectrum is between 100km down to 5km. The other end of the spectrum producing sig nals from 1km down to few mm is the short to very-short w avelength. The long wavelength signals are represented by the global geopotential models like the 0SU91A. Gravimetry contributes to the medium wavelength signals. The short wavelength part of the gravity field spectrum is considered to come from the terrain corrections. [Szabo,B., 1986]. There is, however, no clear cut division between the various parts of the spectrum, as they tend to merge into one another. It is only the optimal combination of the various parts of the spectra that will lead to a more complete solution of the geoid in any local area. It is this concept that gave rise to the method of gravimetric combination through the numerical integration technique.[Zhao,S., 1989], [Sideris,M.G., 1994].
Chapter 3: Geoid Determination Methods and Techniques
As part of this research a new part of the gravity field spectrum was investigated. This was termed the medium-to-short wavelength part of the spectrum. In the chapters that follow later in this work, signals coming from the topographic density anomalies, as part of total terrain corrections were examined in great detail. [Nyamangunda,P. & Iliffe,J.C., 1996]. These signals are considered to be of wavelength from about 20km down to 1km. It must be emphasized that this signal still exists at the medium to long wavelength 5km to
1 0 0km but is not very dominant at these long wavelengths.
The data spectrum structure is shown in figure 3.4 [Nyamangunda,?. & Iliffe,J.C., 1996]
topography
^wVWyx=PeAP^^P^d^VX7^7^
1/2 degdensity
anomaly
gravity
anomaly
global
model
^ o i d 1 0 0 spectral signal contri bution Geo potential Model , Terrain Corrections Density Anomalies Gravity Anomaiies 18000 1km 3600 5km 360 Nmax 50km n 2.5min 30sec Wavelength SpectrumChapter 3: Geoid Determination Methods and Techniques
3.2.1.2 Causes of the Data Spectrum Wavelength Signal Variation.
The Earth’s gravity field is as a direct result of its mass and rotation as discussed in the Theory of the Potential in Chapter 2. The signals that can be detected or measured vary both in time and space. The main cause for the variation in the measurements is as a result of the mass and density anomaly distributions. At any given point on the Earth’s surface, the contribution of the masses and density contrasts at the various levels towards the gravity signal measured varies considerably in the power spectrum. The deeper Earth masses have been considered to give the long wavelength signal of the gravity field. Geo physicists have long been aware of lateral variations in the Earth’s crustal properties but the recognition that such variations also exist at greater depth occurred only in more recent times.
The most direct, yet ambiguous, evidence came from observations of gravity where the long wavelength anomalies point clearly to the existence of lateral variations in the den sity of the mantle. The deep Earth masses in this case refer to anything from the Moho level through the upper and lower mantle down to the Earth’s outer and inner core as shown in figure 2.3, which shows the Earth’s internal structure. This type of signal varies from medium to long wavelength in the spatial domain. This medium to long wavelength signal has naturally a very low frequency in the frequency domain as the variation in time and space is long. According to Lambeck (1988) dominant deformations of the Earth as sociated with mantle convection and plate tectonics give rise to low frequency, long wavelength gravity field variations. Thus it can only be detected by global measurements of the Earth’s gravity field. This is why most global model solutions are a result of satellite orbits perturbation analysis.[Pavlis, N.K.& Rapp,R.H., 1990]. The medium wavelength contribution in the current global models comes from the inclusion of observed surface gravimetry and satellite altimetry observations. So, in any small sized local area, there is very little variation in the long wavelength power spectrum. Large changes are only no ticeable within large areal coverage. [Rapp, R.H., 1989].
The medium wavelength power spectra is a result of the observed gravity anomalies. These are due to the mass anomalies and density contrasts in the Earth’s crust. Crustal differences are most pronounced between the continental and oceanic regions. Continen tal crust, on average, has a thickness of 30-40km, whereas, the oceanic crustal thickness is only 7-10km. The chemical compositions of the two crusts are also quite different. Al so, in Lambeck (1988), there occurs substantial regional variations in continental structure, reflecting different geologic histories of different segments of crust. This all
Chapter 3: Geoid Determination Methods and Techniques
causes variation in the observed gravity anomaly signals. The variation in the power of the gravity anomaly signal is of medium to high frequency and is affected by the visible topography as well. Topography that has been removed by physical processes changes the actual contribution of the signal at any one point on the Earth’s surface. According to Lambeck (1988) dominant deformations that cause medium to high frequency changes with time to the gravity field are the earthquake displacement fields, which are the in stantaneous expression of plate tectonics and mantle convection. In flat terrain the variation of the gravity anomaly signal is quite small in a small limited area. However, in areas of rugged terrain the variation in this signal is quite significant. [Kearsley,A.H.W., et al.,1985]. The free-air anomaly in this case is known to be highly correlated with topography. [Arabelos,D.,1989].
Gravity anomaly observations provide information on the non-hydrostatic stress in the Earth and on the response of the Earth to this stress. Hager (1984) gives an illustrative account of the gravity anomaly and geoid effect produced by a surface deformation.
Gravity data measurements have been made on a world-wide basis both by local national and private mining and geophysical companies and also by international organisations. [Balmino,G., 1993]. The acquisition of the land gravity data has been rather slow due to operational and political reasons. [Podmore,F., 1981]. As a result gravity data coverage on land as shown in most of the gravity data location maps e.g. figure 4.1, suffers primarily from huge data gaps as found in a lot of countries. In the current case of study there are huge data gaps varying from 20’ to 40’ in Zimbabwe. Gaps of tens of minutes of arc are to be found in neighbouring Mozambique. These gaps are also to be found in the other neighbouring countries in Southern Africa. [Merry,C.L.,1981].
Most of the gravity data is also to be found in valleys and low lying areas rather than in mountainous regions and high terrain. The huge data gaps create a particular problem as to getting a homogeneous gravity data set over the earth or local area of interest for gravity field geoid computations. This is mainly due to the fact that the data gaps are at longer wavelengths compared to the gravity signal as shown in figure 3.4. This is a challenge to the geodesists and geophysicists to try and predict the gravity anomalies in these data gaps. Part of this research addresses this problem by looking at the geophysical properties of the topography to determine realistic values for the gravity field in the data gaps.
The visible topography gives rise to the short wavelength part of the gravity field power spectra. [Forsberg,R., 1984]. In mountainous areas for example, the topographic effects
Chapter 3: Geoid Determination Methods and Techniques
completely dominate the local variation of the gravity field. These are the terrain correc
tions and part of what has been called the Topo-Density anomaly corrections. The
topo-density anomalies are a direct result of the variation of the geophysical characteris tics of the geology in a given area. In particular, this is a gravity anomaly arising out of the topographic density contrasts between the computation point and the surrounding geology in the area. In terms of the actual wavelength and signal variation, the topo-density anom aly signal is therefore medium to short wavelength in the power spectrum. The terrain correction, on the other hand, is a short to very short wavelength signal characterised by high frequency changes. [Forsberg,R., 1984]. Computations of terrain corrections require a Digital Elevation Model, DEM. The topo-density anomaly computations require both a DEM and a Digital Density Model, DDM.
It must be noted that theoretically, each data source contains the total wavelength spec trum with some parts of it being dominant as described above. In other words, taking the terrain corrections, say, it is the short wavelength spectrum which is dominant though the terrain corrections still contain some low frequency and medium wavelength information which is not dominant in the solution. According to Schwarz (1984) and Szabo (1986) in practice, the measuring process acts as a bandwidth pass filter limiting the range of the spectrum. As a result of this, a single data source cannot resolve the complete spectrum and it becomes necessary to combine different types of measurements to obtain a homo geneous power spectral resolution of the gravity field.