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We propose an alternative weighting scheme which is based on the noise levels of the individual solutions (Sect. 5). Relative weights are derived by VCE on the solution level (Jean et al. 2018) and then consequently applied to the NEQs. The main difference between the classical VCE on the NEQ level and the alternative VCE on the solution level is that in the latter case, while correlations between the unknown parameters are lost, the error assessment does not depend on the different error modeling and absorption strategies of the individual ACs, but on the differences to a (weighted) mean of the individual solutions.
The most obvious technique to estimate optimal weights is VCE. Unfortunately, a direct application of this technique, as it is presented in Appendix A.1, leads to suboptimal results, because the inversion of the NEQs provided by the ACs does not yield realistic covariance matrices of errors in model parameters. In the frame of the Horizon 2020 project European Gravity Service for Improved Emergency Management [EGSIEM, Jäggi et al. ( 2019)], the prototype of a scientific combination service for time-variable gravity fields was established. The goal of this service is to provide consistent, reliable and validated monthly gravity fields, which are combined on the NEQ level from standardized NEQs of all associated ACs. EGSIEM ACs contributing to the combination are the Astronomical Institute of the University of Bern (AIUB), the Helmholtz Centre Potsdam, German Research Centre for Geosciences (GFZ), the Groupe de Recherche de Géodésie Spatiale (GRGS) and the Institute of Geodesy of the Technical University of Graz (IfG, former Institute for Theoretical and Satellite Geodesy, ITSG). In a nutshell, to generate a statistically optimal combination of monthly gravity fields provided by different ACs: When installed by a Worcester Bosch Accredited Installer with a Greenstar System Filter. The Greenstar 1000 and Greenstar 2000 guarantee is available as standard to both Excelerate members and non-WAI's, when fitted with any brand of system filter. Terms and conditions apply. A variant of the DDK-filter (Kusche 2007) making use of the full, monthly covariance information is applied to filter the different versions (still in spherical harmonics representation). Therefore, the calibrated error covariances of the ITSG gravity fields were used and the characteristics of the expected hydrological or oceanographic signals were taken into account. Finally, the spherical harmonic coefficients (SHC) were transformed to global grids with \(1
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For hydrological applications, monthly means of atmosphere [GAA, Flechtner and Dobslaw ( 2013)], ocean [GAB, Flechtner and Dobslaw ( 2013)] and the global isostatic adjustment (GIA) model LM17.3 Footnote 5 are subtracted. For oceanographic applications, monthly means of the atmosphere, the terrestrial water storage modeled by the WaterGAP Global Hydrological Model [WGHM; Döll et al. ( 2003)] and GIA (evaluated at the epochs of the monthly gravity fields) are subtracted.
Based on these sources of information, the user has to decide which time-series of monthly gravity fields to use. For most users, the peculiarities of the different processing approaches of the individual GRACE ACs remain unclear. Therefore, there is urgent need for a unification of gravity models like it is done for the products of other space geodetic techniques by, e.g., the International GNSS Service [IGS, Dow et al. ( 2009)], the International Laser Ranging Service [ILRS, Pearlman et al. ( 2002)], the International VLBI Service [IVS, Nothnagel et al. ( 2017)] or the International DORIS Service [IDS, Tavernier et al. ( 2005)]. While the promise of combined offerings is great, it’s easy to get them wrong. The problem is that too many companies, expecting to catch the brass ring, don’t think through exactly how to structure, market, and sell their combined offerings. Over the past three years, the authors have analyzed more than 100 winning hybrid offerings from a variety of B2B and B2C companies. Their research shows that most companies stumble in at least one of four ways: failure to differentiate, failure to scale, failure to assess markets and prices appropriately, and failure to invest in the brand. To fully take into account correlations between the individual gravity field parameters, but also between gravity field, orbit, instrument, stochastic or other model parameters, gravity fields have to be combined on the normal equation (NEQ) level. Up to now, a combination of gravity fields on the NEQ level was not possible because the individual ACs normally do not provide NEQs. This was only changed in the frame of the EGSIEM project (see Sect. 2). Monthly Earth gravity fields based on the observations of the Gravity Recovery And Climate Experiment [GRACE, Tapley et al. ( 2004)] satellite mission are an important source of information on temporal mass variations in the system Earth (Wouters et al. 2014). Monthly gravity fields are not only provided by the official GRACE Science Data System (SDS) processing centers JPL, CSR and GFZ, but also by an increasing number of independent analysis centers (ACs) worldwide.
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Four rules help managers discover which hybrid combination is most likely to be successful for their firm.
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