Local relationships between the disturbing density, the disturbing potential and the disturbing gravity of the Earth’s gravity field Z.L. Fei, M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 e-mail: zlfei@ucalgary.ca; Tel.: +1 403 220 4113; Fax: +1 403 284 1980 Received: 6 April 1998 / Accepted: 16 June 1999 Abstract. A function having some properties of a wavelet and being harmonic around a given point in R 3 is defined, and three models showing the local relationships between the disturbing density, the dis- turbing potential and the disturbing gravity are estab- lished by using the function as the kernel function of the integrals in the models. The local relationship has two meanings. One is that we can evaluate with a high accuracy the integrals in the models by using mainly high-accuracy and high-resolution data in a local area. The other is that we can obtain a stable solution with high resolution when we invert the integrals in the models because of the rapid decrease of the kernel function of the integrals. As a result, with these models we evaluate one quantity with high resolution, in a band limited by the maximum degree of a set of geopotential coecients or by the resolution (spacing) of the local data, from another quantity (or quantities) in a local area, and the resulting solution is stable. Key words. Earth’s gravity field  Local relationships  Density  Potential  Gravity 1 Introduction With the advance of gravimetric techniques, the amount of global gravity data obtained on the Earth’s surface and at aircraft and satellite altitudes is increasing, and the accuracy and resolution of the data are constantly improving. Therefore it becomes very important to utilize these data for determining the gravity potential outside the Earth and for researching the distribution of the Earth’s density. Because of the complexity of the Earth, the problem of researching its density and gravity potential is con- verted to the problem of researching the small disturbing density and the disturbing potential, by using a so-called standard Earth which is an approximation of the real Earth. In the past, many models showing the relation- ships between the disturbing density, the disturbing potential and gravity data have been established (Heis- kanen and Moritz 1967; Guan and Ning 1981). These traditional models essentially involve an integral equa- tion as follows: Y P Z r K P; QX QdQ 1 where X and Y respectively represent the input and output data; r is the Earth’s surface, or the geoid, or the surface of the Bjerhammar sphere, etc.; and the kernel K P; Q satisfies the relationship lim lP;Q!1 K P; Q= o or P l 1 P; Q 6 0 2 where lP; Q is the distance between P and Q, and r P is the radius vector of P. For example, in Stokes’ formula, Y and X represent the disturbing potential T and the gravity anomaly Dg respectively, r is the geoid, and K P; Q is Stokes’ function. Since o or P l 1 P; Q vanishes slowly as lP; Q!1, it follows from Eq. (2) that K P; Q also vanishes slowly as lP; Q!1. Therefore, when computing Y from X in Eq. (1), the integral must be evaluated in a larger area, thus making the collection of data and the computation of the integral dicult; and when computing X from Y in Eq. (1), besides the need for data in a larger area, the stability of the solution X declines as its resolution increases, thus restricting the resolution of the solution X (Fei 1994; Keller 1995). Correspondence to: Z.L. Fei Journal of Geodesy (1999) 73: 534–542