3.3.5. Influence on IceMass Balance
In this section we are concerned with the question of whether the improved accuracy of the estimated Spherical Harmonic Coefficients, SHC, by future satellite missions (e.g. GOCE), can ensure a better aerial estimation of depths to the bedrock below the ice, in particular below the Greenland Inland Ice. Gravity information obtained from the satellite missions is viewed as additional, but indirect, source of information to e.g. depths obtained from the airborne radar echosoundings (which were conducted in Greenland in late 1970's and early 1980's). The relation to the icemass balance is indirect; a better knowledge of the bedrock topography below the ice results in a better glaciological model for the icemass balance. In order to obtain a better understanding, it was decided to make an error propagation study. The investigated area covers the whole of Greenland (55^{o}N  85^{o}N and 0^{o}W  90^{o}W).
The error in the gravity field masks the information about parts of the gravity signal below the noise level. If the noise is spatially uncorrelated and centred (i.e. the mean value for each point in space is zero), it is still possible (e.g. by Wiener filtering or even by a simple moving window averaging technique), to recover gravity features of certain spatial extension. This statement is valid even if the gravity signal associated with such features is below the noise level for each point in space. Thus, the "visibility" of certain features of the signal masked by the noise depends not only on the signal to noise ratio for a particular point in space, but also on the relation between the spatial extension of the signal feature and the correlation properties of the noise.
The computer program HARMEXG.F (see chapter 2.1) is used to generate 20 realizations of noise caused by the errors in SHC, and for the area of investigation (see Sec. 1). In details, SHC of EGM96 have been perturbed 20 times according to the expected SHC error statistics of the GOCE mission. This yields 20 realizations of the field contaminated by the noise in different height levels H. (In practise, the direct evaluation of the noisy gravity field in different height levels was chosen as an alternative to the harmonic downward continuation of the noisy gravity data from the flight level, H = 300 km). Subsequently, the reference gravity field (EGM96) was computed at each height level and subtracted from the 20 realizations of the error contaminated gravity field. For the area of investigation, and for each height level H, this yields 20 realizations of the noise according to the expected SHC error statistics of the GOCE mission. Fig. 3.5.5.1 shows the reference gravity signal (EGM96) for H=3000m. contaminated by the SHC noise according to the apriori error statistics of the GOCE mission.
The noise realizations described above are given on a regular grid in geographical coordinates (spacing: dlat x dlon= 0.25^{o} x 0.50^{o}). The gravity error covariance properties are expressed as variance C_{0}(e_{g}) and the correlation length s_{1}(e_{g}). Another problem is to transform these expressions to the formal error covariance properties for the undulations of the topography. The technique is directly related to the notion of the linear approximation for topographic effects, see e.g. [Forsberg, 1984, Sec. 7.3]. It is assumed, that the transition from the ice to the bedrock in Greenland is associated with a fixed mass density contrast drho=(2670 kg/m^{3}  920 kg/m^{3}] = 1750 kg/m^{3}. In the spherical case, the linear approximation yields:
dg_{H} = (4 pi G drho) dh_{H} (1)
where G is the gravitational constant, dg_{H} denotes a harmonic downward continued gravity signal from the satellite height to the height level H (above or below the zerolevel), dh_{H} denotes the undulation of the rockice interface just below the level H.
Eq.(1) can be viewed as a linear functional relating dg_{H} and dh_{H}. Thus, the law of covariance propagation applies (Moritz, 1980). (It is important to emphasize, that we do not postulate that the sources of the gravity field are located just below the height level H. Eq. (1) will only be used for the purpose of covariance propagation, i.e. in order to determine the "gravi equivalent" covariance properties in different depths for the undulation error.) Using the functional expressed by eq. (1) the variance and the correlation lengths of the gravity error can be directly translated to the equivalent error of the interface undulation in some depth H:
C_{0}(e_{g} ) = (4 pi G drho)^{2} C_{0}(e_{h} ) (2a)
and the correlation length
s_{1}(e_{g}) = s_{1}(e_{h} ) = s_{1 } (2b)
Investigations of the noise covariance properties were carried out in the whole area of investigation regarded as one. Furthermore, the latitude dependence of the covariance properties were also investigated in this study.
It is not easy to relate the formal undulation error parameters C_{0}(e_{h}) and s_{1}(e_{h}), obtained from eqs. (2a)(2b), to the equivalent masking property of the correlated noise. Instead, a purely empirical method was used. A small subarea of the investigated area has been chosen. A rectangular initial prism of height h_{0} = [C_{0}(e_{h})]^{2 }and a horizontal width s_{0}=dx_{0}=dy_{0}=s_{1} (both in EW and NSdirections) has been chosen. The centre of mass of the prism coincides (in horizontal) with the location of the gravity grid point in the centre of the chosen subarea. In the vertical, the bottom side of the prism is located 100 m below the height level H. The mass density contrast to the surroundings is, drho = 1750 kg/m^{3}, thus representing the undulation of the rock topography below the ice in the height level H. Subsequently, the thickness (i.e. the height) of the prism is changed in steps dh= 1 m, so that h = h_{0} + Ndh, where N is a whole number (positive/negative), until the gravity response from the prism is clearly seen above the gravity noise. The obtained value of h is denoted h_{lim.}
Starting with the prism with parameters (h_{lim}, s_{0}, s_{0}) the horizontal extension of the prism are changed gradually and simultanously in both directions. The steps are ds = 0.1 km, until the gravitational attraction from the prism is no more visible above the noise level. Thus, for each depth level H the specific parameter s_{lim} is obtained. The results of the investigation show, that the question of whether certain features are visible depends mainly on h. For h<h_{lim} the prism response is not visible above the noise level, irrespectible of the horizontal extension. For h > h_{lim}, the features are clearly visible, even for features with relatively small horizontal extensions as compared to the correlation length, see Table 3.5.5.2.
The precise meaning of the parameters h_{lim }and s_{lim} is, thus, that for h>h_{lim} and s>s_{lim} the undulations of the icerock interface with spatial resolution (h,s,s) will be visible in the gravity signal. Fig.3.5.5.2 shows the gravity response in height H = 0 m of the rectangular prism (h_{0},dx_{0}, dy_{0}) = (11.4 m, 75 km, 75 km) with the bottom side located in a height level H=100 m. Fig. 3.5.5.3 shows the gravity response of the prism shown on Fig. 3.5.5.2 with the background noise. Clearly, the prism gravity response is hidden in the noise. Fig. 3.5.5.4 shows the gravity response of a prism (h_{lim},dx_{0}, dy_{0}) and the background noise and Fig.3.5.5.5 shows the gravity response of a prism (h_{lim},s_{lim}, s_{lim}) and the background noise. As explained above, Fig. 3.5.5.4 and Fig. 3.5.5.5 demonstrate the concept of the visibility of the undulations of the icerock interface in the presence of the additive correlated noise.
The computations were repeated for the error statistics of EGM96 to illustrate the improvement. The results shown in Table 3.5.5.1 indicate an improvement by a factor 4.4 in the formal standard deviation of the covariance function. However, Table 3.3.5.3 shows that the improvement in "visibility" yields a factor 6.4 for h_{lim} and a factor 710 for s_{lim}.
Table 3.3.5.1. The whole area (55N  85N and 0W  90W). Covariance properties of noise for different heights.
H (metres) 
Gravity noise, standard deviation (mgal) 
Correlation length, s1 (km)

Rockice interface undulation noise, standard deviation (metres) 


GOCE 
EGM96 
GOCE 
EGM96 
GOCE 
EGM96 
300000 3000 200 0 200 500 
0.0085 1.55 1.68 1.67 1.68 1.69 
0.1832 6.91 7.43 7.43 7.47 7.52 
135 85 75 75 75 75 
250 60 60 60 60 60 
11.42 11.40 11.46 11.55 
50.64 50.64 50.88 51.23 
The undulations of the icerock interface in Greenland are located in height levels between some 500m and 200m.The height level H = 300000 m is that of the GOCE satellite, and H = 3000 m is the height just above the highest point of the Inland Ice. Computation of the formal equivalent height undulation error statistics is not relevant for these two heights.
Table 3.3.5.1. shows the formal covariance parameters (here the standard deviations) of the error empirical covariance function.
Table 3.5.5.2. indicate that there is a signicicant and clear change of covariance properties with taltitude. There is a clear increase of error variance with latitude and a small decrease of noise variance
Table 3.3.5.2. Change of covariance properties with latitude (Height=200 m)
Latitude 
Gravity noise, standard deviation (mgal) 
correlation length s1 (km) 
Rockice interface undulation noise, standard deviation (metres) 

GOCE 
EGM96 
GOCE 
EGN96 
GOCE 
EGM96 

55EN  58EN 61EN  64EN 67EN  70EN 73EN  76EN 79EN  82EN 
1.16 1.30 1.42 1.69 2.07 
7.13 7.27 7.38 7.57 7.79 
85 70 75 80 86 
70 65 50 60 60 
7.92 8.87 9.71 11.51 14.14 
48.68 49.60 50.46 51.56 53.21 
Finally, Table 3.3.5.3. shows the visibility parameters h_{lim} and s_{lim} different heights. It should be mentioned, that a decision about which parameter to choose is somehow subjective. The difference in the effect of changing parameters of the prism is not sharp and depends on the chosen location (a grid point). For this reason, the results displayed in Table 3.3.5.3 should be used with great caution. For comparison, the excersise was repeated for EGM96 error statistics. These results were obtained using a rougher resolution for the parameters (+/0.5km for s_{lim} and +/5m for h_{lim}).
Table. 3.3.5.3. The parameters slim and hlim related to the concept of the visibility of the undulations of the icerock interface.
H (metres) 
s_{lim} (km) 
h_{lim} (metres) 

GOCE 
EGM96 
GOCE 
EGM96 

500 200 100 0 100 200 
0.2 0.2 0.3 0.4 0.6 0.8 
1.5 1.5 2.0 3.0 5.0 8.0 
43 41 40 37 34 32 
270 260 250 240 220 205 
Conclusions
The computations using the error statistics provided by GOCE and EGM96 illustrate the improvement obtainable by GOCE . An improvement by a factor 4.4 in the formal standard deviation of the covariance function of the height differences can be expected. Similarly and improvement in the visibility yields a factor 6.4 for h_{lim} and a factor 710 for s_{lim}. For GOCE compared with EGM96.
Note. The corresponding figures and investigation for other model parametrisations can be found via the project home page. http://www.gfy.ku.dk/~cct/gocestudy.html