Journal of Geodynamics 47 (2009) 39–46 Contents lists available at ScienceDirect Journal of Geodynamics journal homepage: http://www.elsevier.com/locate/jog Methods of determining weight scaling factors for geodetic–geophysical joint inversion Caijun Xu a,b,∗ , Kaihua Ding a , Jianqing Cai b , Erik W. Grafarend b a School of Geodesy and Geomatics, Key Laboratory of Geo-space Environment and Geodesy, Ministry of Education, Wuhan University, 129 Luoyu Road, Wuhan 430079, China b Institute of Geodesy, University of Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany article info Article history: Received 18 January 2008 Received in revised form 13 June 2008 Accepted 14 June 2008 Keywords: Geodetic–geophysical joint inversion Weight scaling factor Helmert method of variance components estimation Cross validation test method abstract Geodetic–geophysical joint inversion is a hybrid inversion of different types of geodetic data, together with geophysical or seismic, geological data. In the joint inversion, weight scaling factors of different datasets are of vital importance and should be fixed properly. This paper aims to analyze the general weight scaling factor fixing methods and to study their impacts on joint inversion. The result, validated and evaluated by the cross validation test method, showed that it is not prudent to fix the inversion parameter only by considering the objective function to be a minimum and that the parameter should be determined by the actual circumstances. At last, a more reliable inversion result was obtained by using the Helmert method of variance components estimation (VCE) for the fixing of weight scaling factor. © 2008 Published by Elsevier Ltd. 1. Introduction Geodetic–geophysical joint inversion makes use of various data to extract common information, including geodetic, seismic, geo- logical and geophysical data. The theory of geodetic–geophysical joint inversion is based on Backus–Gilbert theory (Backus and Gilbert, 1967, 1968, 1970). In 1977, Matsu’ura inverted for fault parameters by using geodetic data, and brought forward the con- cept of geodetic inversion (Matsu’ura, 1977a,b). Since then, a rapid and great development has occurred, with the continuum, lin- ear form and single dataset replaced by the discrete, non-linear form and various datasets. Examples include the joint inversion of electronic distance-measuring instrument (EDM), global posi- tioning system (GPS) and very long baseline interferometry (VLBI) (Lisowski et al., 1990), the joint inversion of leveling, GPS and grav- ity (Zhao and Sjöberg, 1993; Zhao, 1995; Li et al., 2002; Li and Xu, 2005), the joint inversion of geodetic and seismic data, and the joint inversion of geodetic, seismic and geologic data (Holt and Haines, 1993, 1995; Holt et al., 2000; Williams et al., 1993; Tinnon et al., 1995; Shen-Tu et al., 1998; Shen-Tu and Holt, 1999; Xu et al., 2000, 2003, 2005; Segall and Matthews, 1997; England and ∗ Corresponding author at: School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China. Tel.: +86 27 68778805; fax: +86 27 68778371. E-mail address: cjxu@sgg.whu.edu.cn (C. Xu). Molnar, 1997; Wu et al., 2001; Kreemer et al., 2000; Wan et al., 2004). However, a problem in the geodetic–geophysical joint inver- sion remains: how should we fix weight scaling factors of different datasets so as to achieve a reliable result? As we know, dif- ferent weight scaling factors represent different contributions to the inversion result, so only careful fixing of weight scal- ing factors can lead to a correct/sensible result. This paper firstly introduces the general methods to fix weight scaling fac- tors in the geodetic–geophysical joint inversion, then discusses and analyzes them in detail on the basis of a case study in China. 2. Methods This section introduces four methods that are usually used for fixing weight scaling factors. Without the loss of generality, the joint inversion of geodetic and seismic data is taken as a case to introduce the geodetic–geophysical joint inversion. GPS data and seismic moment tensor data have been used to invert for the crustal motion velocity field and strain rates through bicubic Bessel interpolation (Shen-Tu et al., 1998; Holt et al., 2000). The relation between the horizontal velocity field u(ˆ x) and the rota- tion vector function W(ˆ x) can be described as (Haines and Holt, 1993) u(ˆ x) = r [W(ˆ x) × ˆ x] (1) 0264-3707/$ – see front matter © 2008 Published by Elsevier Ltd. doi:10.1016/j.jog.2008.06.005