Prior to weighting, the impact of the individual NEQs on the combination has to be balanced. This is achieved by empirical factors derived from the study of pairwise combinations and has to be done independently from the VCE, since the VCE is converging robustly to the same results, independent of the a priori weights used. Based on installing a new A-rated condensing boiler in England, Scotland and Wales with a programmer, room thermostat and thermostatic radiator controls (TRVs) in a gas heated detached home from an older G-rated boiler with a programmer and room thermostat. Figures are based on fuel prices as of October2023. Savings will vary depending on the size and thermal performance of your home. Source: Energy Saving Trust.

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. 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. Most firms are trying to combine products and services into innovative offerings in an effort to boost revenue and profit streams and balance cash flows. These hybrid solutions can help companies attract new customers and increase demand among existing ones by offering them superior value. Such offerings are commonplace—think Apple (the iPod product combined with the iTunes service). We therefore compute and apply additional empirical factors, which so to say equalize the available normal matrices. Those factors are estimated such that all the individual solutions contribute equally to the combined solution, independently of their quality. (This procedure is described in Sect. 5.2.) 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.

Overview

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The standard approach is to expand the gravity field in spherical harmonics and provide the weight coefficients of this expansion (L2-products) that may be transformed to global grids (L3-products) for easier use. Depending on the field of application, the grids are complemented by monthly mean values of the short-term atmosphere and ocean mass variations [so-called GAX-products, Flechtner and Dobslaw ( 2013)] to restore the nontidal signal content. Moreover, the L3-products are usually pre-filtered to reduce noise. Four rules help managers discover which hybrid combination is most likely to be successful for their firm.

The Take Away

In tough economic times, companies need new ways to innovate and drive revenues. By combining a product with a service, or vice versa, firms can improve their bottom and top lines. However, if a hybrid offering is to be successful, managers need to understand which of four potential combinations is most appropriate. 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: 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.