MRR decomposition

In Fig. 4.7, the MRR decomposition is described by successively low-pass filtering a high-resolution signal ZJ level by level ”top-down“ to a low-resolution signalZj^′. This filter procedure is applied to the signal of differential gravity anomalies ∆g from case study (e), displayed in Fig. 6.15 (i). The approach was set up at level J = 11, and the background model GOCO05s subtracted up to degreel₇ =127. The latter defines the minimum resolution level j^′ = 7, cf. Eq. (5.10), since up to its maximum degreel^′_j, most of the signal is removed from the observations. Consequently, the fundamental equation (4.28) of MRR reads for the following study case (f)

∆Z₁₁(x)=∆Z_j′=7(x)+ X11

j=8

G_j(x). (6.4)

The detail signalsGj are computed according to the modeling equations (5.35) for j =8, . . . ,11. Herein, the wavelet functions ˜Ψj are formulated for gravity anomalies∆g and expanded in a series up to the maximum degree lj of the particular resolution level j according to Eq. (5.33). Since in the approach of study case (e), Blackman scaling functions are used in the synthesis, cf. Tab. 6.11, the referring Blackman wavelet functions now are applied in study case (f). They are defined by the band-pass filtering Legendre coefficients ψ^Bla_l,j according to Eq. (4.37), as displayed in Fig. 4.10 (e). In the top-down approach, the detail signals are successively subtracted from∆Z_J and yield the low-pass filtered signals

∆Z10(x)=∆Z11(x)−G11(x),

∆Z9(x)=∆Z10(x)−G10(x),

∆Z8(x)=∆Z9(x)−G9(x),

∆Z7(x)=∆Z8(x)−G8(x). Low- and band-pass filtered signals

The low- and band-pass filtered signals∆Zj andGj, as well as the initial signal∆Z11at highest resolution level, are displayed in Fig. 6.21 in terms of gravity anomalies w.r.t. GOCO05s up tol7=127. The referring statistics (range, mean value, SD) are listed in Tab. 6.16; outliers have been removed. As expected, the successive low-pass filtering of the initial signal from level J = 11 down to level j = 8 is clearly visible:

While the signal∆Z₁₁ shows a lot of geographical structures, the latter become smoother and smoother for decreasing resolution levels. The information of the low-resolution signal ∆Zj^′=7(x) is almost completely reduced by the background model, and thus, as good as no gravitational variations are visible.

The statistics (range and SD of the signals∆Z_j) establish the considerations. The range of∆Z₁₁counts about

±30 mGal, the SD around 10 mGal, whereas∆Z8ranges between±10 mGal and yields a SD of approximately 4 mGal. The remaining differences of ∆Z7 w.r.t. GOCO05s at level j^′ = 7 yield 0.29 mGal to 1.64 mGal.

For the levels j = 8, . . . ,11, the mean values of ∆Z_j are very similar; they slightly increase from around 2.18 mGal (j = 8) to 2.61 mGal (J = 11). For ∆Z7, the mean value is with 1.11 mGal only approximately half as large. As expected, the mean values ofG_j are close to zero²⁷for the levels j =9, . . . ,11, while forG8

the mean value counts 1.07 mGal, i. e. this detail signal contains information of an offset w.r.t. GOCO05s.

Since long wavelengths cannot be resolved by regional models, the appearance of offsets w.r.t. global models is explicable.

In analogy to∆Z11, the detail signal G11shows very fine structures in Fig. 6.21; the latter become rougher forG10,G9, andG8. Those signals contain the spectral information of the frequency bands, described by the Legendre coefficients from Fig. 4.10 (e). For the band-pass filtering Blackman wavelet, the frequency bands overlap due to the smoothing behavior. Consequently, the detail signals are correlated and not independent from each other. The corresponding ranges and SD values in Tab. 6.16 decrease as well together with the resolution level.

The erroneous effects, which are visible in the south-western edge and along the northern borderline in Fig. 6.15 (i) have been discussed in study case (e). They result from data gaps. The missing information is filled up by prior information, i. e. with ”zero-signal“ w.r.t. GOCO05s up tol7=127. The higher-resolution detail signalsG11and G10show corresponding erroneous structures in Fig. 6.21;G9 is still affected by the

27The detail signals are globally per definition zero.

data gap in the south-western corner. In G8 (respectively Z8), the data gaps are not longer apparent; the

”zero-signal“ from the prior information seems to be sufficient to fill them up.

Table 6.16: Study case (f): Range (min... max), mean value and SD of low- (∆Z_j ) and band- (G_j) pass filtered signals, as well as of the background model GOCO05s up tol₇=127.

∆Z_j range [mGal] mean [mGal] SD [mGal] G_j range [mGal] mean [mGal] SD [mGal]

∆Z₁₁ −33.81 to 40.42 2.61 11.169 G₁₁ −26.91 to 31.83 0.28 6.850

∆Z₁₀ −20.47 to 24.17 2.36 8.841 G₁₀ −13.67 to 13.27 0.05 4.568

∆Z₉ −13.83 to 15.97 2.31 6.697 G₉ −9.87 to 13.16 0.13 4.291

∆Z₈ −7.38 to 7.80 2.18 3.551 G₈ −8.02 to 6.51 1.07 3.389

∆Z₇ 0.29 to 1.64 1.11 0.332

GOCO05s −28.20 to 16.85 −6.16 11.161

Standard deviations

In Fig. 6.22 the referring standard deviations s∆Z_j of the low-, andsG_jof the band-pass filtered signals are visualized. Note, s∆Z8, s∆Z7andsG7refer to different color bars. The statistics (maximum, mean, and SD values) are listed in Tab. 6.17. As discussed in the context of study case (e), the values ofs∆Z11enormously increase in the corresponding south-western corner and along the northern borderline due to the edge effects.

The standard deviations reach up to around 30 mGal. Low-pass filtering the initial signal∆Z11 provokes a smoothing of the signal, as well as of the standard deviations. Consequently, the structures in Fig. 6.22 become also smoother, especially for the levels j = 8 and j = 9. The referring mean standard deviations then are smaller than 0.5 mGal, cf. Tab. 6.17.

The standard deviationss∆Z7refer to the low-resolution signal∆Z7stemming from the remaining observations which have not been completely removed by the background model. Consequently, those standard deviations could give a rough measure of the long-wavelength errors, discussed in the context of Fig. 6.20. The maximum and mean values are in the order of 0.08 mGal and 0.05 mGal. However, those estimates are obtained via error propagation from the initial standard deviations at level J = 11. A more realistic estimate of the long-wavelength errors might be obtained by setting up the modeling approach (with specifications of Tab. 6.11) at levelJ =7, see next section. In general, the standard deviations might be too optimistic, especially since correlations and realistic accuracies of the observations are not considered in the stochastic model, as discussed in the previous study cases.

Table 6.17: Study case (f): Maximum , mean value and SD of the standard deviationss∆Z_j of the low-, andsG_j of the band-pass filtered signals w.r.t. GOCO05s up tol₇=127.

Discussion of study case (f)

Besides the previously discussed weaknesses of the regional modeling approach, cf. study cases (a) – (e), the MRR decomposition in this study case (f) reinforces data gaps to be one of the major problems. As already

Figure 6.21: Study case (f): MRR decomposition of the signal∆Z₁₁of differential gravity anomalies from Fig. 6.15 (i) (w.r.t.

GOCO05s up tol₇=127) down to∆Z₇by successively subtracting band-pass filtered detail signalsG_j.

Figure 6.22: Study case (f): Standard deviationss∆Z_jof low-, andsG_jof band-pass filtered signals w.r.t. GOCO05s up tol₇=127.

assumed, the low-resolution prior information seems not appropriate to overcome data gaps in high-resolution regional gravity field models and provokes large erroneous effects. The VCE regulates the regularization, and thus, the relative weighting of the prior information w.r.t. the whole observation area∂ΩO. This regularization might be too weak for single data gaps, if most of the area is well-observed, as in study case (e), respectively (f). An alternative handling will be discussed in the sequel. Further, since the approach seems to be sensitive to data gaps, this property is useful, vice versa, in order to detect data gaps of existing gravitational models.

Observation depending weighting of prior information Instead of introducing the additional observation µ_d with unit matrix P_d = I in Eq. (5.19), the stochastic model could be adapted to the heterogeneous spatial distribution of the observations. Hereby, the main diagonal elements of Pd can vary depending on the number of observations, counted in a specific circle around each grid pointP_q ∈∂ΩC. The principle is schematically displayed in Fig. 6.23. In the left red-colored circle with radiusρaround a grid point (red cross), only two observations are available, here e. g. terrestrial measurements (green triangles). Consequently, the unknown coefficient to be estimated at this grid point, is only weakly supported by information. Additional prior information would be necessary. It could be considered by setting the corresponding value at the main diagonal of Pd smaller than one, depending on the counted number of data points. In contrast, in the right red-colored circle, there are several terrestrial (green) and satellite (gray-dashed tracks) observations and the corresponding coefficient located at the right red cross is sufficiently supported by observed signal;

less prior information is necessary and the corresponding value at the main diagonal of P_d could remain one. Consequently, the estimation model might be purposefully stabilized at data gaps. However, the spectral content of the prior information refers to the background model, and thus, appropriate high-resolution information still is missing in the resulting regional model.

Figure 6.23: Observation depending weighting of prior information depending on the number of observations (gray, green) in a specific influence zone (spherical radiusρ) of each grid point (red cross).

Detecting data gaps in existing models Vice versa to setting up a MRR decomposition by subtracting the detail signals, the same results can be obtained by successively low-pass filtering the initial signal. As mentioned in the context of Eq. (4.35) band-pass filters are generated by subtracting the Legendre coefficients of two low-pass filtering SBFs. Consequently, the subtraction of two consecutive low-pass filtered models, e. g. ∆Z_jand∆Z_j−1, yields the detail signalG_j, according to the general Eq. (4.28) of MRR (de)composition.

This aspect is of beneficial relevance, e. g. for spectrally decomposing an arbitrary given model, and thus, visualizing the signal w.r.t. different spectral domains (Schmidt et al., 2007): Data gaps can be detected at different resolution levels, if they were filled up with synthetic information. Further applications of a MRR decomposition arise, for instance, in order to analyze coarser structures as mass variations in the Earth’s interior, the density of the Earth’s crust and lithosphere (Bouman et al., 2013;Ebbing et al., 2013;Bouman et al., 2016).