The relation between degree-2160 spectral models of Earth’s gravitational and topographic potential: a guide on global correlation measures and their dependency on approximation effects

Abstract

Comparisons between high-degree models of the Earth’s topographic and gravitational potential may give insight into the quality and resolution of the source data sets, provide feedback on the modelling techniques and help to better understand the gravity field composition. Degree correlations (cross-correlation coefficients) or reduction rates (quantifying the amount of topographic signal contained in the gravitational potential) are indicators used in a number of contemporary studies. However, depending on the modelling techniques and underlying levels of approximation, the correlation at high degrees may vary significantly, as do the conclusions drawn. The present paper addresses this problem by attempting to provide a guide on global correlation measures with particular emphasis on approximation effects and variants of topographic potential modelling. We investigate and discuss the impact of different effects (e.g., truncation of series expansions of the topographic potential, mass compression, ellipsoidal versus spherical approximation, ellipsoidal harmonic coefficient versus spherical harmonic coefficient (SHC) representation) on correlation measures. Our study demonstrates that the correlation coefficients are realistic only when the model’s harmonic coefficients of a given degree are largely independent of the coefficients of other degrees, permitting degree-wise evaluations. This is the case, e.g., when both models are represented in terms of SHCs and spherical approximation (i.e. spherical arrangement of field-generating masses). Alternatively, a representation in ellipsoidal harmonics can be combined with ellipsoidal approximation. The usual ellipsoidal approximation level (i.e. ellipsoidal mass arrangement) is shown to bias correlation coefficients when SHCs are used. Importantly, gravity models from the International Centre for Global Earth Models (ICGEM) are inherently based on this approximation level. A transformation is presented that enables a transformation of ICGEM geopotential models from ellipsoidal to spherical approximation. The transformation is applied to generate a spherical transform of EGM2008 (sphEGM2008) that can meaningfully be correlated degree-wise with the topographic potential. We exploit this new technique and compare a number of models of topographic potential constituents (e.g., potential implied by land topography, ocean water masses) based on the Earth2014 global relief model and a mass-layer forward modelling technique with sphEGM2008. Different to previous findings, our results show very significant short-scale correlation between Earth’s gravitational potential and the potential generated by Earth’s land topography (correlation +0.92, and 60% of EGM2008 signals are delivered through the forward modelling). Our tests reveal that the potential generated by Earth’s oceans water masses is largely unrelated to the geopotential at short scales, suggesting that altimetry-derived gravity and/or bathymetric data sets are significantly underpowered at 5 arc-min scales. We further decompose the topographic potential into the Bouguer shell and terrain correction and show that they are responsible for about 20 and 25% of EGM2008 short-scale signals, respectively. As a general conclusion, the paper shows the importance of using compatible models in topographic/gravitational potential comparisons and recommends the use of SHCs together with spherical approximation or EHCs with ellipsoidal approximation in order to avoid biases in the correlation measures.

Appendix

1.1 Appendix 1: Properties of the sphEGM2008 model

Fig. 9

Top gravity disturbances from the sphEGM2008 model, synthesised in band 2 to 2240 at a constant height of 9000 m height above the reference sphere. Bottom Spherical effect $=$ discrepancies between the models EGM2008 (synthesised in band 2 to 2190 at 9000 m height above reference ellipsoid and sphEGM2008 (synthesised in band 2 to 2240 at 9000 m height above the reference sphere). Units in mGal, grids equally spaced in terms of geocentric latitudes. The figure illustrates that the differences (reflecting the effect of the different mass arrangement in ellipsoidal and spherical approximation) are rather small compared to the sphEGM2008 gravity signal (top)

The transformation described in Sect. 3.2 was applied to obtain the spherical transform of EGM2008, named sphEGM2008 in this paper. The sphEGM2008 model represents Earth’s geopotential as if the field-generating masses were arranged relative to a sphere, and not relative to an ellipsoid (which is the case for EGM2008). As the main benefit of such a “spherical” high-degree spectral model of the geopotential, it can be readily and meaningfully applied in band-limited operations, such as degree-wise syntheses or correlation coefficient computations. This is fundamentally different from EGM2008 (or any other high-degree model of the geopotential represented as SHCs and relying on ellipsoidal approximation) where its SHCs can only be used within the full bandwidth (i.e. band of degrees from 2 to 2190).

Fig. 10

Gravity disturbances synthesised in band 2 to 2000 at 4000 m height above reference. Left sphEGM2008 model, reference surface $=$ sphere, Right EGM2008, reference surface $=$ GRS80 ellipsoid. Units in mGal; grids equally spaced in terms of geocentric latitude; area shown is Northern Polar region ($80 {^{\circ }} {-} 90 {^{\circ }}$ geocentric latitude). The figure illustrates that band-limited operations such as truncation below the maximum model degree are permitted for sphEGM2008, while high-latitude striations prohibit the band-limited use of EGM2008 (as SHC representation)

Because the sphEGM2008 approach is not yet well known in the gravity field community, some exemplary results shall illustrate the differences between the SHC model representations of sphEGM2008 (this work) and EGM2008 (Pavlis et al. 2012). In all gravity syntheses presented next,

• the sphEGM2008 model coefficients are evaluated at some height above the reference sphere (sphere with radius of 6,378,137.0 m), and

• the EGM2008 model coefficients are evaluated at the same height above the reference ellipsoid (ellipsoid with semi-major axis of 6,378,137.0 m, and semi-minor axis of 6,356,752.3141 m, taken from the GRS80 parameters),

such that choice of reference surfaces and evaluation points is mutually consistent.

Figure 9a shows global gravity disturbances from the sphEGM2008 model, evaluated at 9000 m height above the reference sphere in the full bandwidth of the model (degrees 2 to 2240). These are in very close agreement with gravity disturbances of the EGM2008 model (evaluated at 9000 m height above the reference ellipsoid in the full bandwidth of degrees 2 to 2190, as is shown in Fig. 9b. The differences between the full-banded evaluations of sphEGM2008 and EGM2008 (Fig. 9b) can be interpreted as a spherical effect, reflecting the nonidentical mass arrangement in spherical approximation (sphEGM2008) and ellipsoidal approximation (EGM2008).

Fig. 11

Gravity disturbances synthesised in band 1001 to 2000 at 4000 m height above reference. Left sphEGM2008 model, reference surface $=$ sphere, right EGM2008, reference surface $=$ GRS80 ellipsoid. Units in mGal. The figure illustrates that band-limited operations such as evaluations of high degree bands are permitted for sphEGM2008, while high-latitude striations prohibit the band-limited use of EGM2008 (as SHC representation)

The differences show a North-South structure and correlate spatially with North-South-aligned gravity structures (compare Fig. 9a, b). They are small (min/max/rms $=-$0.64/+0.77/0.07 mGal) at 9000 m above the reference spheres, and would somewhat increase if the syntheses were done at the respective reference surfaces (min/max/rms $=$ −1.28/+1.82/0.09 mGal). A reduction of the differences (e.g., through modelling and correction of the spherical effect in the spectral domain) was not attempted in this work.

We note that the sphEGM2008 model features additional signals at harmonic degrees larger than 2160, which are a consequence of the windowing effect in the transformation (Eq. 13). The sphEGM2008 signal strength associated in band of degrees 2161–2190 does not exceed 0.09 mGal anywhere on the globe (at 9000 m height), which is almost one order of magnitude smaller than the spherical effect. Beyond degree 2190, the signal strength is always smaller than $5 \times 10^{-5} \, \hbox {mGal}$. All in all, the additional coefficients beyond degree 2160 can be considered to be of minor relevance in practical applications of sphEGM2008.

Figures 10 and 11 illustrate the benefits of a spherically approximated geopotential model (sphEGM2008) over an ellipsoidally approximated geopotential model (EGM2008). For the North Pole region, Fig. 10 shows gravity disturbances from both models, truncated at harmonic degree 2000. The sphEGM2008 model is seen to be free of truncation effects, while these are very clearly manifested as striations for EGM2008. We emphasise that the striations of course disappear when EGM2008 is evaluated in its full bandwidth.

Fig. 12

Gravity disturbances implied by harmonic degree 500 over Northern Europe ($10 {^{\circ }}$ to $25 {^{\circ }}$ longitude, $60 {^{\circ }}$ to $70 {^{\circ }}$ geocentric latitude), obtained from six models or combinations. Top row EGM2008 and dV_ELL_Earth2014 (both based on ellipsoidal approximation), middle row sphEGM2008 and dV_SPH_Earth2014 (both based on spherical approximation), bottom row differences EGM2008-sphEGM2008 and dV_ELL_Earth2014-dV_SPH_Earth2014). The figure shows that the striations (bottom row) produce a bias that is the reason for the higher correlation between the ellipsoidal pairs (top row) compared to the more realistic values for the spherical pairs (middle row)

Figure 11 compares band-limited gravity disturbances from sphEGM2008 and EGM2008 in harmonic degrees of 1001-2000 (spatial scales of 5.4 to 10.8 arc-min). The striations visible in Fig. 11 render any band-limited application of EGM2008 (or any other high-degree model) near the poles and in short-scale bands impossible. In contrast, sphEGM2008 is not subjected to striations (also see Sect. 3.2), so can be used in a band-limited fashion (narrow bands or degree-wise), e.g., in syntheses or correlation coefficient computations as in this paper. As a drawback of band-limited applications of sphEGM2008 model, a part of the gravity signals associated with the spherical effect (Fig. 9 bottom) are neglected. This drawback can be overcome by using the EHCs of EGM2008 in band-limited ellipsoidal harmonic syntheses, which however, was not performed in this study.

1.2 Appendix 2: The bias in degree correlation coefficients and reduction rates

To obtain insight into the reason behind the biased CC and RR values (cf. Sect. 5.2), we have investigated gravity effects in the space domain. The four models EGM2008, sphEGM2008, dV_ELL_Earth2014 and dV_SPH_Earth2014 were used to synthesise gravity disturbances implied by single spherical harmonic degrees. The syntheses were done at the surface of the reference sphere (sphEGM2008 and dV_SPH_Earth2014) and reference ellipsoid (EGM2008 and dV_ELL_Earth2014) in terms of 5 arc-min global grids equally spaced in geocentric latitude. As harmonic degree of evaluation, we have chosen degree 500 where large differences in CCs and RRs were observed (Sect. 5.2). CCs and RRs were computed as described in Hirt (2014). Figure 12 shows the computed gravity disturbances over Northern Europe. Note that dV_ELL_Earth2014 and EGM2008 were evaluated at the reference ellipsoid, while dV_SPH_Earth2014 and sphEGM2008 were evaluated at the reference sphere.

Among the gravitational and topographic potential models that rely on EA, a regional correlation of +0.89 and reduction rate of 54.1% is observed (Fig. 12a, b).

• This is larger than the corresponding values obtained among the gravitational and topographic potential models that are based on SA (CC of +0.85 and RR of 47.1%), cf. Fig. 12c, d.

• The differences between EGM2008 and sphEGM2008 (Fig. 12e) represent in approximation the windowing effect contained in the gravitational potential model EGM2008 (caused by Eq. 6). Accordingly, the differences dV_ELL_Earth2014 minus dV_SPH_Earth2014 (Fig. 12f) reveal the windowing effect contained in the topographic potential model dV_ELL_Earth2014 (caused by Eq. 9).

• Figure 12e, f shows that the windowing effect produces “striations” that are strongly correlated (in our example +0.90 and RR of 55.4%). Note that the striations tend to increase with degree and equatorial distance.

• The gravity disturbances shown in Fig. 12 (top row) are the sum of those shown in the middle row (gravity signals in spherical approximation) and bottom row (striations).

• Thus, it is the high correlation between the striations (bottom row) that drives up the correlation between the ellipsoidal model pairs (top row), compared to the spherical model pairs (middle row).

It becomes obvious that the striations produce apparent correlation, which is nonexistent in the actual gravity signals implied by harmonic degree 500 (Fig. 12, middle row). Strictly speaking, the windowing effect (Eqs. 6 and 9) introduces functional (aka geometric) correlations among the coefficients of the models based on ellipsoidal approximation, which lead to higher degree correlations than in spherical approximation.

From a global comparison instead of a regional comparison (as in Fig. 12), very similar results can be obtained. Globally, between the ellipsoidal gravitational and topographic potential models, a CC of +0.90 and RR of 57.0% is obtained. In SA, the correlation measures are lower (CC of +0.86 and RR of 48.9%). When the windowing effect is isolated in approximation (as in Fig. 12, bottom row, but here globally), a CC of +0.91 and RR of 58.1% is obtained among the two fields. These values, which are in good agreement with those obtained directly from the harmonic coefficients in Sect. 5, demonstrate that correlation measures computed in ellipsoidal approximation may be biased.

The described experiment can be repeated for all other harmonic degrees. For high degrees, e.g., degree 2000, the effect reverses, in that, a lower correlation between the gravitational and topographic potential striations biases the correlations towards values too low in case of ellipsoidal approximation: For degree 2000, SA yields a CC of +0.92 and RR of 55.2% (vs. CC of +0.86 and 47.6% in EA). For the windowing effect (as in Fig. 12 bottom row, but globally and degree 2000), a CC of +0.86 and RR of 47.6% is obtained, which is largely responsible for the values observed in EA.

Finally, it is emphasised that the striations shown in Figs. 10,  11,  12 are not to be interpreted as model errors. They are vitally important constituents of the SHCs needed to correctly represent the EGM2008 gravity signals over the complete full bandwidth (degrees 2 to 2190) when a potential model is based on ellipsoidal approximation. As shown above, the windowing effect only ever matters if the SHCs of an ellipsoidally approximated model is used in a band-limited manner at high degrees, though it should not.