Uncertainties of TWS Time Series for Arbitrary Regions - Modelled vs. Formal Covariances

<p>Knowledge of the variances and covariances of gridded terrestrial water storage anomalies (TWS) as observed with GRACE and GRACE-FO is crucial for many applications thereof. For example, data assimilation into different models, trend estimations, or combinations with other data set require reliable estimations of the variances and covariances. Today, the Level-2 Stokes coefficients are provided with formal variance-covariance matrices which can yield variance-covariance matrices of the gridded data after a labourious variance propagation through all post-processing steps, including filtering and spherical harmonic synthesis. Unfortunately, this is beyond the capabilities of many, if not most, users.</p><p><br>This is why, we developed a spatial covariance model for gridded TWS data. The covariance model results in non-homogeneous, non-stationary, and anisotropic covariances. This model also accommodates a wave-like behaviour in latitudinal-directed correlations caused by residual striping errors. The model is applied to both VDK3 filtered GFZ RL06 and ITSG-Grace2018 TWS data. </p><p><br>With thus derived covariances it is possible to estimate the uncertainties of mean TWS time series for any arbitrary region such as river basins. On the other hand, such time series uncertainties can also be derived from the afore mentioned formal covariance matrices. Here, only the formal covariance matrices of ITSG-Grace2018 are used which are also filtered with the VDK3 filter. All together, we are able to compare globally the time series uncertainties of both the modelled and formal approach. Further, the modelled uncertainties are compared to empirical standard deviations in arid regions in the Arabian, Sahara, and Gobi desert where residual hydrological signal can be neglected. Both in the temporal and spatial domain they show a very satisfying agreement proving the usefulness of the covariance model for the users. </p>

Subtleties in spherical harmonic synthesis of the gravity field

<p>Spherical harmonic synthesis (SHS) can be used to compute various gravity functions (e.g., geoid undulations, height anomalies, deflections of vertical, gravity disturbances, gravity anomalies, etc.) using the 4pi fully normalised Stokes coefficients from the many freely available Global Geopotential Models (GGMs).  This requires a normal ellipsoid and its gravity field, which are defined by four parameters comprising (i) the second-degree even zonal Stokes coefficient (J2) (aka dynamic form factor), (ii) the product of the mass of the Earth and universal gravitational constant (GM) (aka geocentric gravitational constant), (iii) the Earth’s angular rate of rotation (ω), and (iv) the length of the semi-major axis (a). GGMs are also accompanied by numerical values for GM and a, which are not necessarily identical to those of the normal ellipsoid.  In addition, the value of W<sub>0,</sub> the potential of the geoid from a GGM, needs to be defined for the SHS of many gravity functions. W<sub>0</sub> may not be identical to U<sub>0</sub>, the potential on the surface of the normal ellipsoid, which follows from the four defining parameters of the normal ellipsoid.  If W<sub>0</sub> and U<sub>0</sub> are equal and if the normal ellipsoid and GGM use the same value for GM, then some terms cancel when computing the disturbing gravity potential.  However, this is not always the case, which results in a zero-degree term (bias) when the masses and potentials are different.  There is also a latitude-dependent term when the geometries of the GGM and normal ellipsoids differ.  We demonstrate these effects for some GGMs, some values of W<sub>0</sub>, and the GRS80, WGS84 and TOPEX/Poseidon ellipsoids and comment on its omission from some public domain codes and services (isGraflab.m, harmonic_synth.f and ICGEM).  In terms of geoid heights, the effect of neglecting these parameters can reach nearly one metre, which is significant when one goal of modern physical geodesy is to compute the geoid with centimetric accuracy.  It is also important to clarify these effects for all (non-specialist) users of GGMs.</p>

High-frequency analysis of Earth gravity field models based on terrestrial gravity and GPS/levelling data: a case study in Greece

The present study aims at the validation of global gravity field models through numerical investigation in gravity field functionals based on spherical harmonic synthesis of the geopotential models and the analysis of terrestrial data. We examine gravity models produced according to the latest approaches for gravity field recovery based on the principles of the Gravity field and steadystate Ocean Circulation Explorer (GOCE) and Gravity Recovery And Climate Experiment (GRACE) satellite missions. Furthermore, we evaluate the overall spectrum of the ultra-high degree combined gravity models EGM2008 and EIGEN-6C3stat. The terrestrial data consist of gravity and collocated GPS/levelling data in the overall Hellenic region. The software presented here implements the algorithm of spherical harmonic synthesis in a degree-wise cumulative sense. This approach may quantify the bandlimited performance of the individual models by monitoring the degree-wise computed functionals against the terrestrial data. The degree-wise analysis performed yields insight in the short-wavelengths of the Earth gravity field as these are expressed by the high degree harmonics.