ANNALS OF GEOPHYSICS, 55, 5, 2012; doi: 10.4401/ag-5459 RESEARCH ARTICLES A signal-strength model for gravity-field spectrum components in a local space Liu Hongwei1,*, Wang Zhaokui2, Zhang Yulin1,2 1 Institute of Aerospace Science and Engineering, National University of Defence Technology, Changsha, China 2 School of Aerospace, Tsinghua University, Beijing, China Article history Received October 28, 2011; accepted June 11, 2012. Subject classification: Earth gravitational potential, Local area, Spectrum structure, Spherical cap harmonic analysis, Signal strength. ABSTRACT alytical form. The Earth gravity field is usually approximated In the Earth gravity-field model, potential coefficients are projections of by a spherical harmonics series, and the set of all the gravita- the Earth gravitational potential on the corresponding spherical harmonics tional potential coefficients in an infinite series expansion of by degree and order. These coefficients reflect the spectral composition of spherical harmonics represents the Earth gravity field model. the Earth gravitational potential in the whole space, but they cannot The idea that the Earth gravity potential can be ex- reflect the spectral composition in a local area. In this study, the Earth pressed as an infinite series of spherical harmonics has been gravitational potential was projected on spherical harmonics in a local widely used in geoscience research, and it is a basic method area using a window function introduced to facilitate the mathematical in Earth sciences and gravity satellite research. The literature expressions. Then a model for the spectral structure of the Earth contains conclusions about the relationships between dis- gravitational potential in the local area was established. The model can turbing potential coefficients and matter density anomalies reflect the signal strength of any gravity-field spectrum component in the inside the Earth [Jian 1994]. It was believed that lower degree local area, which adds information for the description of the gravity field potential coefficients reflect deep Earth matter density of the Earth, and which has great significance for Earth-sciences research anomalies, while higher degree potential coefficients reflect and satellite gravity measurements. Taking the third-degree coefficients, shallow Earth matter density anomalies. By using the for example, the signal-strength distributions of potential coefficients were GRACE time-varying gravity-field model, global water stor- computed. The data show that the signal-strength distributions of third- age changes were analyzed in Xuhua et al. [2006], Antarctic degree coefficients on the sphere surface are independent of the longitude. ice-sheet mass changes were studied in Guangbin et al. Among the third-degree coefficients, the zero-order, first-order and second- [2009], and application of a gravity-field model in tectonic order coefficients are stronger near the two poles of the Earth, while the detection was analyzed in Leyang et al. [2007]. At the same third-order coefficient is stronger near the equator. time, the spherical harmonics infinite series expansion of the Earth gravitational potential has been widely used in satellite 1. Introduction gravity measurements. Satellites can measure the Earth grav- The Earth gravity field is one of basic physical fields of ity field, such as CHAMP, GRACE and GOCE, which were the Earth, and it reflects information relating to matter dis- designed for recovering gravity potential coefficients in spe- tribution inside the Earth and affects physical events on the cial bands. Among these three satellites, CHAMP has meas- Earth and in the neighboring space. Therefore, research into ured the Earth gravity field using orbit perturbation theory, the Earth gravity field is always a hot issue in geodesy [Jin- which has helped to detect lower degree potential coeffi- sheng 2001, 2002]. The gravity field outside the Earth is caused cients [Reigber et al. 2003]. GRACE measured the Earth grav- by matter inside the Earth. As well as the shape of the Earth ity field by satellite-to-satellite tracking in a low Earth orbit, affecting the Earth gravity potential, this is also affected by the which helped in the detection of the lower- and middle-de- asymmetry of the matter density distribution inside the Earth gree potential coefficients [Reigber et al. 2005, Zhao et al. [Jian 1994]. Researches into the Earth shape and density are 2011]. GOCE has used gravity gradient measurements that two basic missions in Earth gravity research [Xiaoguang et al. have helped to detect middle- and higher-degree potential 2000]. Due to anomalies in the shape of the Earth and to the coefficients [Muzi and Allasio 2003]. asymmetry of the matter density distribution inside the Earth, In the Earth gravity field model, gravitational potential the Earth gravity potential cannot be expressed as a simple an- coefficients are projections of gravitational potential on the 1017 LIU HONGWEI ET AL. corresponding spherical harmonics by degree and order in the whole space, and they reflect the spectral composition of the gravitational potential in the whole space. However, gravitational potential coefficients cannot reflect the spectral composition of gravitational potential in a local space, which is a shortcoming of conventional Earth gravity-field models. Although spherical cap harmonic analysis (SCHA) can give the spectral composition of gravitational potential on a spherical cap [De Santis et al. 1999], because of less gravity potential coefficients, the spectral structure of the spherical cap area reflected by a SCHA model is limited. In the present Figure 1. Spherical coordinate system with the Earth center as its origin. study, based on the results in Slepian and Pollak [1961], the Earth gravitational potential was projected onto spherical ttn ^ - h = 121`j n + nk! harmonic functions in a local area, and by appropriate math- Cnk a d ^ + h (4) 2 2 r GM k nk! ematical expressions, the signal-strength model for the grav- ## ^^timhh i imm i ity potential spectral component in a local space was RP,,nk cos sincosdd,k X established. This model will be helpful for Earth-science re- search and gravity satellite measurements. ttn ^^21nnk+-hh! S = 1 `j (5) nk r GM a ^nk+ h! 2. A signal-strength model for gravity-field spectrum 2 2 ## ^^timhh i imm i components in a local space RP,,nk cos sinsink dd, X We establish a spherical coordinate system with the cen- ter of the Earth as the origin, and we take the z-axis and the where X, the whole spherical surface, is the integral area. X Earth rotation axis in the same direction, as is shown in Fig- can be divided into n parts, ure 1. The spherical harmonic expansion of Earth's gravita- XXXX= tional potential is: ",12...n , (6) ^timh = X V ,, (1) where, i (i=1,2,...,n) are the local areas on the spherical sur- 3 n n face, as shown in Figure 2. X can be described by a certain GM; ++/ / `j a ^ mmih ^ hE i tt1 CkSkPnkcos nk sinnk cos , nk==20 longitude and latitude range, and its center is of interest. In Equations (4) and (5), if the integral area X is changed t i m X where the definition of the spherical coordinates ( , , ) is into i, then projections of the nonspherical Earth pertur- shown in Figure 1, and G is the gravitational constant, M is bation potential R(t,i,m) on n-degree and k-monics in the X the Earth mass, a is the semi-major axis of the Earth ellipsoid, local area i can be obtained: ",C,S are the gravitational potential coefficients, and nk nk ttn n + ^nk- h! P ^hcosi is the fully normalized associated Legendre poly- C = 121`j (7) nk nk, i r GM a d ^nk+ h! nomial. The relationship between P ^h cos i and the associ- 2 2 k nk ## ^^timhh i imm i ated Legendre polynomial, P (cos i), is: RP,,nk cos sincosdd,k nk X i P ^cos ih = (2) nk ttn ^^21nnk+-hh! - = 1 `j ^nkh! 20, k = Snk, i a ^ + h (8) ^21n + h 2 P ^cos idh,.= ' 2 2 r GM nk! d ^nk+ h! nk k ! k 1, k 0 ## ^^timhh i imm i RP,,nk cos sinsink dd. X The nonspherical Earth perturbation potential function i can be given from Equation (1): Then, n n CCSS==//,.(9) nk ==nk,, ink nk i R^tim,,h = (3) i 11i 3 n n ", GM/ / `j a ^ mmi+ h ^ h It is known that C,S nk, ii nk, reflect projections of the ttCSkPnk cosk nk sinnk cos . n = 2 k = 0 nonspherical Earth perturbation potential on n-degree and i m i k-order spherical harmonics {Pnk (cos )cos k , Pnk (cos )sin m X By the orthogonality of spherical harmonics functions, k } in the local areas i, which represent the signal strengths ", the gravitational potential coefficients C,S nk nk can be de- of n-degree and k-order spherical harmonic components in X ", rived as: i, while C,S nk nk reflect the sum of these projections in the 1018 GRAVITY FIELD SPECTRUM IN LOCAL AREA } t i m t i m As is the spherical distance from ( 0, , ) to ( 0, 0, 0), the window function satisfies 0<H≤1. The window function i m is 0 only for the point ( 0, 0), and the longer the spherical dis- i m i m tance from ( , ) to ( 0, 0) is, the smaller the H is. The atten- uation factor satisfies p>0. The greater p is, the narrower H is, and the smaller p is, the wider H is. Multiplying Equations (4) and (5) by the window func- tion in Equation (14) and substituting Equation (3) gets the sine and cosine signal strength of n-degree and k-order spec- X trum components in the local area 0 where the center is X t i m Figure 2.