Computers & Geosciences 29 (2003) 183–193
Software for computing five existing types of deterministically modified integration kernel for gravimetric geoid determination$ W.E. Featherstone* Western Australian Centre for Geodesy, Curtin University of Technology, G.P.O. Box U1987, Perth 6845, Australia Received 29 January 2002; received in revised form 10 July 2002; accepted 11 July 2002
Abstract Five of the more frequently cited and used deterministic modifications to Stokes’s formula for gravimetric geoid determination are summarised in a self-consistent framework. FORTRAN77 software and its limitations are described to compute these modified kernels, which can easily be implemented in gravimetric geoid determination software in place of the original spherical Stokes kernel. An example from Australia is used to demonstrate the superiority of one of these deterministically modified kernels over the spherical Stokes kernel for a regional gravimetric geoid computation. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Geoid determination; Modified kernels; Truncation error reduction; Filtering; Geodesy
1. Introduction Stokes’s (1849) solution (to a spherical approximation) of the geodetic boundary-value problem requires a global integration of gravity anomalies to compute the separation ðNÞ between the geoid and geocentric reference ellipsoid at a single point. The geoid–ellipsoid separation has many applications (e.g. Van!ıc& ek and Christou, 1994), the most notable of which is the transformation of global positioning system (GPS)derived ellipsoidal heights to orthometric heights (e.g. Featherstone, 2001). However, the incomplete global coverage and/or unavailability of accurate terrestrial gravity data precludes a precise gravimetric determination of the geoid using Stokes’s formula. To circumvent this, Molodensky (1958; cited in Molodensky et al., 1962) proposed an approach to $
Code on server at http://www.iamg.org/CGEditor/index.html *Tel.: +61-8-9266-2734; fax: +61-8-9266-2703. E-mail address:
[email protected] (W.E. Featherstone).
reduce the truncation error that occurs when terrestrial gravity data are used within a spherical cap of limited spatial extent about each computation point in Stokes’s formula. This is achieved using a deterministic modification of Stokes’s integration kernel. However, Molodensky’s approach did not receive a great deal of attention at that time because of the contemporaneous availability of global geoid undulations derived from the analysis of the orbits of artificial Earth satellites. These satellite-derived global geopotential models offer a superior information source of the low-frequency component of the geoid. When used in conjunction with regional terrestrial gravity data via a truncated form of Stokes’s integral (e.g. Vincent and Marsh, 1973; Rapp and Rummel, 1975), this can also reduce the truncation error. However, modifications to Stokes’s kernel not only reduce the truncation error in the computed geoid, but also adapt it to yield some arguably preferable high-pass filtering properties (e.g. Van!ıc& ek and Featherstone, 1998). Over recent decades, several modifications to Stokes’s kernel have been presented. These include deterministic approaches (e.g. Molodensky et al., 1962;
0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 8 - 3 0 0 4 ( 0 2 ) 0 0 0 7 4 - 2
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
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de Witte, 1967; Wong and Gore, 1969; Meissl (1971); Heck and Gruninger, . 1987; Van!ıc& ek and Kleusberg, . 1987; Van!ıc& ek and Sjoberg, 1991; Featherstone et al., 1998; Evans and Featherstone, 2000) and stochastic . approaches (e.g. Wenzel, 1982; Sjoberg, 1984,1991; . . Van!ıc& ek and Sjoberg, 1991; Sjoberg and Hunegnaw, 2000). The stochastic modifications will not be considered here because reliable estimates of the error variances of the Earth’s gravity data are not currently known in all areas. However, the many different modifications of Stokes’s kernel have not been tested in the same region using the same data. Instead, the type of kernel used appears to be a largely subjective preference. Therefore, this paper presents software to compute five of the deterministically modified kernels, which can easily be implemented in existing software. This may lead to improved regional gravimetric geoid models, as will be shown to be the case for Australia.
2. Basic theory
0
where k ¼ R=4pg; R is the spherical Earth radius, g is the normal gravity evaluated on the surface of the reference ellipsoid (as is demanded by Bruns’s formula; see Heiskanen and Moritz, 1967, Section 2–13), c and a are the spherical distance and azimuth angle from the computation point, respectively, and Sðcos cÞ is the spherical Stokes kernel, which has the orthogonal series expansion Sðcos cÞ ¼
NM ¼ c
M X n¼2
2 Dgn ; n1
ð4Þ
where c ¼ 2pk ¼ R=2g and Dgn is the nth degree surface spherical harmonic of the gravity anomalies (defined below). The complementary residual gravity anomalies ðDgM Þ are evaluated by subtracting the gravity anomalies implied by the degree-M expansion of the same global geopotential model from the terrestrial gravity anomalies ðDgÞ according to DgM ¼ Dg
M X n¼2
Dgn ¼
N X
Dgn :
N X 2n þ 1 Pn ðcos cÞ n1 n¼2
for 0pcpp;
ð2Þ
where Pn ðcos cÞ is the nth degree Legendre polynomial. The closed trigonometric form of Eq. (2) is c c Sðcos cÞ ¼ csc 6 sin þ 1 cos c 2 2 c c þ sin2 5 þ 3 ln sin 2 2 for 0pcpp:
ð3Þ
ð5Þ
n¼Mþ1
In Eqs. (4) and (5) n X GM an Dgn ¼ 2 ðn 1Þ ðdC% nm cos ml r r m¼0 þ S% nm sin mlÞ P% nm ðcos yÞ;
The combined use of a global geopotential model and terrestrial gravity data, often termed the remove– compute–restore technique, is considered by some as a standard approach in regional gravimetric geoid determination. The low-frequency geoid undulations from a global geopotential model ðNM Þ are partly refined (see Van!ıc& ek and Featherstone, 1998) and extended into the higher frequencies by a global integration of complementary residual terrestrial gravity anomalies ðDgM Þ as follows: Z 2p Z p Sðcos cÞDgM sin c dc da; ð1Þ N ¼ NM þ k 0
In Eq. (1), the low-frequency geoid undulations ðNM Þ are provided by the spherical harmonic coefficients that define the global geopotential model according to
ð6Þ
where GM is the product of the Newtonian gravitational constant and the mass of the solid Earth, oceans and atmosphere, ðr; y; lÞ are the spherical polar coordinates of the computation point, a is the length of the semimajor axis of the geocentric reference ellipsoid, C% nm and S% nm are the fully normalised geopotential coefficients of degree n and order m; and P% nm ðcos yÞ are the fully normalised associated Legendre functions. In Eq. (6), dC% nm denotes that the even zonal harmonics of the normal gravity field of the geocentric reference ellipsoid must be subtracted from the even zonal harmonics of the global geopotential model. In the above and subsequent derivations, the geocentric reference ellipsoid is assumed to have been chosen such that the zero and first degree terms are inadmissible (Heiskanen and Moritz, 1967, Section 2–6); that is, Dg0 ¼ Dg1 ¼ 0: In practice, the integration domain in Eq. (1) is truncated due to the limited availability of terrestrial gravity data over the entire Earth (due to one or both of restricted field access or data confidentiality), as well as the increase in computational efficiency that arises from working with a smaller integration area. It is acknowledged that fast Fourier transform (FFT)-based geoid computations often use a rectangular data area (e.g. Schwarz et al., 1990), but the derivations of the truncation error and kernel modifications are simplified for a spherical cap (cf. Neyman et al., 1996) and are considerably easier to implement in practice, even in FFT software (Featherstone and Sideris, 1998). Therefore, a spherical cap bound by the spherical distance c0 ð0oc0 opÞ will be used throughout the sequel, beyond which the kernel is simply set to zero.
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
In this scenario, the geoid height from Eq. (1) is approximated by the following truncated integral: Z 2p Z c0 Sðcos cÞDgM sin c dc da: ð7Þ N# ¼ NM þ k 0
0
with the corresponding truncation error of (i.e. Eq. (1) minus Eq. (7)) Z 2p Z p dN ¼ N N# ¼ k Sðcos cÞDgM sin c dc da: ð8Þ 0
c0
This truncation error is expressed globally (i.e. over the whole sphere) by Z 2p Z p dN ¼ k Kðcos cÞ DgM sin c dc da ð9Þ 0
0
upon the introduction of the error kernel defined by ( 0 for 0ocpc0 ; Kðcos cÞ ¼ ð10Þ Sðcos cÞ for c0 ocpp: In the derivations that follow, it is convenient to introduce the variable y ¼ cos c; where the truncation radius of the spherical cap is denoted by y0 ¼ cos c0 : Accordingly, the spherical cap occupies the region 0pcpc0 corresponding to y0 pyp1; and the region outside the cap over the remainder of the sphere is c0 ocpp; or equivalently 1pyoy0 : This notation will be used interchangeably throughout the remainder of the paper. Returning to Eq. (10) and using the orthogonality relations for Legendre polynomials over the sphere (e.g. Heiskanen and Moritz, 1967, Section 1–13) the error kernel may be written as the following spectral series expansion: N X 2n þ 1 KðyÞ ¼ Qn ðy0 ÞPn ðyÞ; ð11Þ 2 n¼0 where the expansion (truncation) coefficients are (e.g. Molodensky et al., 1962) Z 1 Z y0 Qn ðy0 Þ ¼ KðyÞPn ðyÞ dy ¼ SðyÞPn ðyÞ dy: ð12Þ 1
1
Inserting Eqs. (5) and (11) into Eq. (9) and performing the global integrations (e.g. Heiskanen and Moritz, 1967, Section 2–5), gives the following spectral series representation of the truncation error: N X dN ¼ c Qn ðy0 Þ Dgn ; ð13Þ n¼Mþ1
which can be evaluated using a global geopotential model (Eq. (6)) if M þ 1oMmax ; the maximum degree of expansion of the global geopotential model. The derivation of Eq. (13) assumes that the global geopotential model is error free to spherical harmonic degree M; which is not the case. In general, the errors increase with increasing degree, which provides a rationale for using a lower degree reference field. Other
185
. authors (including and cited in Sjoberg and Hunegnaw, 2000) consider the case where both the truncation error and global geopotential model error are reduced through a stochastic kernel modification. However, the variances of the global geopotential model are also global and so do not necessarily represent the area under investigation. Therefore, and as the terms Dgn supplied through observations are invariant, the basis of reducing the truncation error lies either in the reduction in magnitude or improvement in decay rate, or both, of the truncation coefficients. This is best achieved by a deterministic modification of the integration kernel. In the sequel, Meissl (1971) deterministic modification to the spherical Stokes kernel (Eqs. (2) and (3)) will be presented, followed by the spheroidal Stokes kernel, which is implicit in the generalised Stokes scheme . introduced by Van!ıc& ek and Sjoberg (1991) and which satisfies its own boundary-value problem (Martinec and Van!ıc& ek, 1996). The use of the spheroidal Stokes kernel is preferable because of its high-pass filtering properties (Van!ıc& ek and Featherstone, 1998), together with the fact that satellite-derived global geopotential models are the best source of low-frequency gravity field information. Accordingly, more emphasis is placed on the presentation of modified kernels that are based upon the spheroidal Stokes kernel. These include the deterministic modifications proposed by Wong and Gore (1969), Heck and Gruninger . (1987), Van!ıc& ek and Kleusberg (1987) and Featherstone et al. (1998), which will be placed in a self-consistent framework. 3. The spherical Stokes kernel 3.1. Meissl’s modification The Qn ðy0 Þ truncation coefficients (Eq. (12)) govern the rate at which the spectral series of the truncation error associated with the spherical Stokes kernel (Eq. (13)) converges. Moreover, the rate of decay of these truncation coefficients is determined by the mathematical smoothness properties of the error kernel KðyÞ in Eq. (10). This is a well-known result from the properties of series expansions in terms of classical orthogonal polynomials, and Fourier series in general (e.g. Gottleib and Orszag, 1977). However, the error kernel in Eq. (10) is a discontinuous function at the cap radius, but can be made continuous by modifying the kernel through a simple subtraction. This approach was first noted by Meissl’s (1971) and is elaborated upon in some detail by Jekeli (1980). Meissl’s (1971) modification to the spherical Stokes kernel is given by ( Sðcos cÞ Sðcos c0 Þ for 0pcpc0 ; * Sðcos cÞ ¼ ð14Þ 0 for c0 ocpp;
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
186
where the tilde is used to distinguish this as the Meissl modification scheme; a notation that will be called upon later. Eq. (14) is used in Eq. (7) to give the following approximation of the geoid height when using the remove–compute–restore technique, noting that Meissl’s approach was originally proposed for the spherical Stokes formula: Z 2p Z c0 * N* ¼ NM þ k Sðcos cÞDgM sin c dc da: ð15Þ 0
0
4. The spheroidal Stokes kernel (Wong and Gore’s modification) The definition of the spheroidal Stokes kernel, or Wong and Gore’s (1969) modification (see the clarification below), involves removing the low-degree Legendre polynomials ð2pnpPÞ from the spherical Stokes kernel (Eqs. (2) and (3)), which yields SP ðcos cÞ ¼ Sðcos cÞ
This gives a corresponding Meissl truncation error of Z 2p Z p * Kðcos cÞDgM sin c dc da: dN* ¼ N N* ¼ k 0
0
ð16Þ The error kernel associated with Meissl’s modification is * Kðcos cÞ ¼
(
Sðcos c0 Þ for 0pcpc0 ; Sðcos cÞ
for c0 ocpp;
ð17Þ
which is now a continuous function across the cap radius, and takes the constant value of Sðcos c0 Þ within the spherical cap. The expansion series in terms of Legendre polynomials for the Meissl error kernel is * KðyÞ ¼
N X 2n þ 1 * Qn ðy0 ÞPn ðyÞ; 2 n¼0
ð18Þ
where the Meissl truncation coefficients are given by Z 1 Z y0 * * ðyÞ dy ¼ ð19Þ KðyÞP SðyÞP Q* n ðy0 Þ ¼ n n ðyÞ dy 1
1
and the spectral representation of the Meissl truncation error (Eq. (16)) becomes dN* 1 ¼ c
N X
¼
Q* n ðy0 ÞDgn :
ð20Þ
n¼Mþ1
Jekeli (1980) shows that the truncation coefficients Q* n ðy0 ÞpOðn2 Þ as n-N; thus establishing the improvement in the decay rate over the Qn ðy0 Þ coefficients (Eq. (12)), which satisfy Qn ðy0 ÞpOðn1 Þ as n-N: Although the corresponding series for the truncation error (Eq. (20)) converges at an improved rate over Eq. (13), it does not necessarily follow that the rootmean-square (rms) error is decreased, as is emphasised by Jekeli (1980, 1981). Jekeli (1981) and Smeets (1994) also note that the Meissl truncation error is not necessarily smaller than the spherical Stokes truncation error, especially for the low-degree gravity field. However, this is overcome by using a global geopotential model to provide the low-degree gravity field via the remove–compute–restore technique (Eq. (15)) such that the series expansion of the truncation error (Eq. (20)) begins at M þ 1:
P X 2n þ 1 Pn ðcos cÞ n1 n¼2
N X 2n þ 1 Pn ðcos cÞ n1 n¼Pþ1
for 0pcpp; ð21Þ
where P corresponds to the degree of spheroidal kernel (or Wong and Gore modification). When P ¼ 1; Eq. (21) degenerates to the spherical Stokes kernel (Eq. (2)). There is a subtle difference between the spheroidal Stokes kernel (e.g. Van!ıc& ek and Kleusberg, 1987; Van. c& c& ek and Sjoberg, 1991) and the Wong and Gore (1969) modification. When P ¼ M (i.e. the degree of the geopotential model used in the remove–compute–restore scheme), Eq. (21) is the spheroidal Stokes kernel that is implicit to the generalised Stokes scheme (Van!ıc& ek and . Sjoberg, 1991; Martinec and Van!ıc& ek, 1996). When PoM; then Eq. (21) is the Wong and Gore (1969) modification of the spherical Stokes kernel. The case of P > M will not be considered here since additional terms arise that are not necessary, and thus impractical, to compute. Introducing Eq. (21) into Eq. (1) and decomposing the integration domain between the spherical cap and remainder of the sphere gives Z 2p Z c0 N ¼ NM þ k SP ðcos cÞDgM sin c dc da þ k
0
Z
2p
Z
0
0 p
SP ðcos cÞDgM sin c dc da:
ð22Þ
c0
As for the spherical Stokes formula, and where PpM; the geoid height is approximated in practice by the following truncated integral: Z 2p Z c0 SP ðcos cÞDgM sin c dc da ð23Þ N# P ¼ NM þ k 0
0
with a corresponding spheroidal (if P ¼ M) or Wong and Gore (if PoM) truncation error of Z 2p Z p KP ðcos cÞDgM sin c dc da dNP ¼ N NP ¼ k 0
¼c
N X
0
fQP gn ðy0 ÞDgn ;
ð24Þ
n¼Mþ1
where fQP gn ðy0 Þ are the spheroidal (Wong and Gore) truncation coefficients, and the spheroidal (Wong and
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
Gore) error kernel is defined as ( 0 for 0pcpc0 ; KP ðcos cÞ ¼ SP ðcos cÞ for c0 ocpp;
ð25Þ
which is a discontinuous function across the spherical cap radius (cf. Eq. (10)). Therefore, the expansion coefficients fQP gn ðy0 Þ of the spheroidal (Wong and Gore) error kernel decay and hence the truncation error (Eq. (24)) converges at the rate of O ðn1 Þ: Fig. 1 shows the spheroidal Stokes kernel (Eq. (21)) for P ¼ 26 and 257 compared to the spherical Stokes kernel (Eq. (3)). The oscillation of the spheroidal Stokes kernel is due to the removal of the low-frequency Legendre polynomials, and this oscillation increases as the degree of spheroidal modification ðPÞ increases (cf. Fig. 1). Van!ıc& ek and Featherstone (1998) demonstrate that the spheroidal Stokes kernel is a perfect high-pass filter when using a global integration (i.e. c0 ¼ p in Eq. (15)), but the power of the filter diminishes when using a limited integration domain (Eq. (23)). As such, leakage of low-frequency errors from the terrestrial gravity data into the geoid solution will occur. Importantly, however, the amount of leakage is less than that would occur when using the truncated spherical Stokes kernel (for the same cap radii). Therefore, when recalling that the low-frequency geoid is best determined from the satellite-only components of a global geopotential model, the spheroidal Stokes kernel is considered superior to the spherical Stokes kernel. While increasing the degree of spheroidal modification increases the amount of high-pass filtering (thus counteracting to some extent the effect of using a truncated integration cap), the increased oscillation of
187
the kernel causes errors in the numerical solution of the discretised integral term. Specifically, a smooth integration kernel is required if its central value is to be representative of each discretised element in the numerical solution of the integral term. If the kernel varies rapidly through the discretised elements, as in the case of a high-degree spheroidal Stokes kernel, the central value for each element will be unrepresentative. Therefore, as well as for the reasons that the terrestrial gravity data are sometimes a better source of medium- and shortwavelength geoid information whereas the satellite-only global geopotential model is the better source of longwavelength geoid information, it is recommended that only low degrees of spheroidal (Wong and Gore) modification are used. 4.1. The Meissl-modified spheroidal Stokes kernel (Heck and Gruninger’s modification) . Heck and Gruninger . (1987) present a Meissl type of modification to the spheroidal Stokes kernel (Eq. (21)). To make the error kernel (defined below) a continuous function, the value of the spheroidal kernel at the truncation radius is subtracted from the spheroidal Stokes kernel inside the cap; this is S* P ðcos cÞ ( SP ðcos cÞ SP ðcos c0 Þ ¼ 0
for 0pcpc0 ; for c0 ocpp;
where the tilde is again used to distinguish this as a Meissl-type modification scheme, but now applied to the spheroidal kernel.
400 350 300
SP (ψ) (dimensionless)
250 200 150 (c) 100 (b) (a)
50 0 -50 -100 -150 0
1
ð26Þ
2
ψ (degrees)
3
4
Fig. 1. Spheroidal (Wong and Gore) kernels for (a) P ¼ 257; (b) P ¼ 26; and (c) spherical Stokes kernel.
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
188
Eq. (26) is introduced into Eq. (23) to give the following approximation of the geoid: Z 2p Z c0 N* P ¼ NM þ k S* P ðcos cÞDgM sin c dc da ð27Þ 0
minimisation to the upper bound of the truncation error. Under this scheme, the Molodensky-modified spheroidal Stokes (Van!ıc& ek and Kleusberg) kernel is defined as
0
8 < S ðcos cÞ PL 2k þ 1t ðcos c ÞP ðcos cÞ P k k 0 k¼2 2 SPL ðcos cÞ ¼ : 0
with a corresponding truncation error of Z 2p Z p dN* P ¼ N N* P ¼ k K* P ðcos cÞDgM sin c dc da 0
¼c
N X
0
fQ* P gn ðy0 ÞDgn ;
ð28Þ
n¼Mþ1
. truncation where fQ* P gn ðy0 Þ are the Heck and Gruninger coefficients, and the corresponding error kernel is ( SP ðcos c0 Þ for 0pcpc0 ; ð29Þ K* P ðcos cÞ ¼ SP ðcos cÞ for c0 ocpp; which is now a continuous function across the cap radius. Accordingly, the truncation coefficients for the expansion series of the error kernel (Eq. (28)) now satisfy the improved convergence rate of Oðn2 Þ: Coupled with the use of the remove–compute– restore scheme (Eq. (27)), this reduces the truncation error. Heck and Gruninger . (1987) propose an alternative to the simple subtraction in Eq. (26), where the degree of spheroidal modification is chosen such that the spheroidal Stokes kernel (Eq. (21)) is zero at the truncation radius (cf. Fig. 1). This satisfies the requirement of a continuous error kernel, and thus an improved convergence rate for the truncation error. However, it does place a constraint on the values that can be chosen for the parameters P and c0 : Since the latter value is normally driven by considerations of data availability, a high degree of spheroidal modification would have to be used for small areas (see Fig. 1). This is, in turn, problematic because of the high-pass filtering properties of the kernel, as well as the practical difficulties encountered when using discretised numerical integration (discussed earlier).
for 0pcpc0 ;
where the superscript L indicates a Molodensky-type modification of degree L: The cases of L > M and L > P will not be considered here because additional terms arise that are not necessary, and thus impractical, to compute. The Van!ıc& ek and Kleusberg modification coefficients tk ðcos c0 Þ ð2pkpLÞ are determined for pre-selected values of c0 and L by minimising the L2 norm of the error kernel (Eq. (36)). This generates the following system of ðL 1Þ linear equations (Van!ıc& ek and Kleus. berg, 1987; Van!ıc& ek and Sjoberg, 1991): L X 2k þ 1 tk ðy0 Þenk ðy0 Þ ¼ Qn ðy0 Þ 2 k¼2
L X 2k þ 1 enk ðy0 Þ for 2pnpL; k1 k¼2
where Qn are given by Eq. (12), and Z y0 enk ðy0 Þ ¼ Pn ðyÞPk ðyÞ dy Z1p ¼ Pn ðcos cÞPk ðcos cÞsin c dc
Van!ıc& ek and Kleusberg (1987) apply Molodensky’s (1958; cited in Molodensky et al., 1962) modification of the spherical Stokes kernel to the spheroidal Stokes kernel (Eq. (21)), which is also described in Van!ıc& ek and . Sjoberg (1991). This modification strategy applies a
ð31Þ
ð32Þ
c0
both of which can be evaluated using Paul’s (1973) algorithms. It is also emphasised that the coefficients enk ðcos c0 Þ and hence tk ðcos c0 Þ depend upon the truncation radius ðc0 Þ; which must be selected before the Van!ıc& ek and Kleusberg modification is made. In practice, and for LpM and LpP; the geoid height is approximated by Z 2p Z c0 NPL ¼ NM þ k SPL ðcos cÞDgM sin c dc da; ð33Þ 0
0
which yields a corresponding truncation error of Z 2p Z p KPL ðcos cÞDgM sin c dc da dNPL ¼ N NPL ¼ k 0
4.2. The Molodensky-modified spheroidal Stokes kernel (Van!ıc&ek and Kleusberg’s modification)
ð30Þ
for c0 ocpp;
¼c
N X
0
fQLP gn ðy0 ÞDgn ;
ð34Þ
n¼Mþ1
where the Van!ıc& ek and Kleusberg truncation coefficients are given by (cf. Eq. (31)) fQLP gn ðy0 Þ ¼
L X 2k þ 1 tk ðy0 Þenk ðy0 Þ 2 k¼2
ð35Þ
W.E. Featherstone / Computers & Geosciences 29 (2003) 183–193
and the associated error kernel is ( 0 for 0pcpc0 ; KPL ðcos cÞ ¼ SPL ðcos cÞ for c0 ocpp:
L dN* LP ¼ N N* P * ¼ k
ð36Þ
¼c
N X
Z
2p 0
189
Z
p 0
fQ* LP gn ðy0 ÞDgn ;
L K* P * ðcos cÞDgM sin c dc da
ð39Þ
n¼Mþ1
The Van!ıc& ek and Kleusberg kernel for L ¼ P ¼ M ¼ 20 and c0 ¼ 61 has been used in the computation of gravimetric geoid models for Canada (Van!ıc& ek et al., 1987,1990), South East Asia (Kadir et al., 1999), and is currently being trialled in Australia by the author.
where fQ* LP gn ðy0 Þ are the Featherstone et al. truncation coefficients, and the associated error kernel is the continuous function ( SPL ðcos c0 Þ for 0pcpc0 ; L * KP ðcos cÞ ¼ ð40Þ SPL ðcos cÞ for c0 ocpp:
4.3. The hybrid Meissl–Molodensky modified spheroidal Stokes kernel (Featherstone, Evans and Olliver’s modification)
This kernel modification strategy is considered by Featherstone et al. (1998) to be an advance upon the previous deterministic modifications (described in this paper) because it combines all the perceived advantages of each. It was used with c0 ¼ 11 and P ¼ L ¼ 20 and M ¼ 360 in the remove–compute–restore technique to compute the 1998 Australian gravimetric geoid model, AUSGeoid98 (Featherstone et al., 2001). Accelerated rates of convergence of the truncation error have been proposed by Evans and Featherstone (2000), but these will not be considered here because they have not yet been used in practice to compute regional gravimetric geoid models.
As for the spherical and spheroidal Stokes kernels, a Meissl type of modification may be applied to the Van!ıc& ek and Kleusberg (1987) kernel (Eq. (30)), which results in the deterministic kernel modification strategy originally presented by Featherstone et al. (1998). Again, this increases the rate of convergence of the Van!ıc& ek and Kleusberg truncation coefficients from O ðn1 Þ to O ðn2 Þ and hence reduces the truncation error further when used in conjunction with the remove– compute–restore scheme. The Featherstone et al. modified kernel is defined as S* LP ðcos cÞ ( SPL ðcos cÞ SPL ðcos c0 Þ ¼ 0
5. Software description for 0pcpc0 ; for c0 ocpp;
ð37Þ
where the tilde is again used to identify this as a Meissl type of modification scheme. The coefficients tk ðcos c0 Þ ð2pkpLÞ are determined for a given c0 from the system of equations given by Eq. (31) and used in Eq. (30) to compute SPL ðcos cÞ: Similarly, SPL ðcos c0 Þ is evaluated and subtracted from Eq. (30) in the region 0pcpc0 : Alternatively, and as for the Heck and Gruninger . (1987) modification, the values of L and c0 can be chosen such that the kernel in Eq. (37) is zero at the truncation radius. This makes the corresponding error kernel (defined below) a continuous function without the need to apply a subtraction (cf. Eq. (37)). In addition to restrictions outlined for the Heck and Gruninger . kernel, this approach requires some iteration for the selection of the truncation radius since the Van!ıc& ek and Kleusberg modification coefficients are a function of c0 : In practice, and for LpM and PpM; the geoid is approximated by N* LP ¼ NM þ k
Z
2p 0
Z 0
c0
S* LP ðcos cÞDgM sin c dc da ð38Þ
with a corresponding truncation error of
A FORTRAN77 program (MODKERN.f1) has been developed to compute the deterministically modified kernels described by Eqs. (3), (14), (21), (26), (30) and (37). When used in stand-alone mode, the program requests user input for the degree of spheroidal (Wong and Gore) modification ðPÞ; truncation radius ðc0 Þ beyond which the kernel is set to zero, type of kernel modification required, increment of c at which the kernel values will be computed, the maximum value of c for the computations, and the output filename. Alternatively, the latter options can be removed by the user so that the program can be adapted to be inserted as a subroutine to replace the kernel in the user’s geoid computation software. The closed-form expression of the spherical Stokes kernel is evaluated from Eq. (3) for any c0 ð0oc0 ppÞ: Because all the kernels become singular at the computation point, alternative strategies have to be used for the numerical solution of the integral terms, which are described in, for example, Heiskanen and Moritz (1967) and Nov"ak et al. (2001). Accordingly, there is no need to compute the kernels at c ¼ 0: All subsequent deterministic modifications are then applied to the closed form of the spherical Stokes kernel (Eq. (3)). The Meissl kernel (Eq. (14)) is evaluated by simply subtracting the 1
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From the above considerations, the Featherstone et al. (1998) kernel will also be subject to numerical instabilities for small values of c0 and large values of L: Accordingly, and based on the arguments presented earlier, these values should not be used. That is, if small values of c0 are to be used due to restrictions on terrestrial gravity data coverage and availability, then large values of L should not be used. As stated, this is also desirable because of the relative errors in the satellite-derived and terrestrial gravity data, as well as the problem with numerically integrating a rapidly oscillating kernel function. The Featherstone et al. (1998) kernel is evaluated simply by subtracting the value of the Van!ıc& ek and Kleusberg kernel at the truncation radius from the Van!ıc& ek and Kleusberg kernel inside the cap; outside the cap, it is zero.
value of the spherical Stokes kernel at the truncation radius from the spherical Stokes kernel inside the cap; outside the cap, it is set to zero. The spheroidal (Wong and Gore) modified Stokes kernel (Eq. (21)) is computed by subtracting the weighted sum of Legendre polynomials up to and including degree P from the closed form of the spherical Stokes kernel (Eq. (3)). The subroutine for computing the Legendre polynomials was taken from Press et al. (1983) and the results verified against the numerical values listed in Abramowitz and Stegun (1965). The same subroutines are used to compute the value of the spheroidal Stokes kernel at the truncation radius, which is subtracted from the spheroidal Stokes kernel inside the integration cap to yield the Heck and Gruninger . modified kernel (Eq. (26)). Again, the kernel is set to zero outside the cap. The computation of the Van!ıc& ek and Kleusberg modified kernel is more involved because of the need to compute the tk ðcos c0 Þ modification coefficients (Eq. (31)), which in turn depend on the spherical Molodensky coefficients [Qn ðcos c0 Þ] and the Paul (1973) coefficients [enk ðcos c0 Þ]. The tk ðcos c0 Þ modification coefficients tend to become numerically unstable for small values of c0 and large values of L: This is demonstrated using the determinant of the matrix that is inverted to yield the values of tk ðcos c0 Þ; which tends to unity as the defining equation system (Eq. (31)) becomes numerically unstable. Figs. 2 and 3 demonstrate that smaller values of c0 and larger values of L give the less stable estimates of tk ðcos c0 Þ as their determinants are close to unity. In addition to the earlier arguments in favour of using a low degree of spheroidal modification, this numerical consideration overrides the choice of the parameter L for this modification scheme.
6. Case study: comparison of the spherical Stokes and Featherstone et al. modified kernels Rather than presenting an empirical comparison of all the deterministically modified Stokes kernels described above, which has already been done to some extent by Jekeli (1980) and Smeets (1994), a case study is presented between the spherical (unmodified) Stokes kernel and the hybrid Meissl–Molodensky-modified spheroidal Stokes kernel (i.e., the Featherstone et al., 1998 kernel). Rather than examining only the truncation errors, as Jekeli (1980) and Smeets (1994) have done, this study examines the performance of two deterministically modified kernels in practical regional geoid computation, which is an arguably more informative exercise. However, due to the spatially varying error characteristics of the different gravity data sources, different
1.0 0.9 0.8
matrix determinant
0.7 0.6 0.5 0.4 L=26 0.3 L=36 L=46
0.2 0.1 0.0 0
2
4
ψ o (degrees)
6
8
10
Fig. 2. Variation of Van!ıc& ek and Kleusberg modification coefficient matrix determinant (i.e., numerical stability) as function of truncation radius ðc0 Þ for differing dimension ðLÞ:
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1.0 0.9 0.8
matrix determinant
0.7 0.6 ψo=1 0.5 0.4
ψo=2
0.3
ψo=3
0.2 0.1 0.0 0
50
100
150
L
Fig. 3. Variation of Van!ıc& ek and Kleusberg modification coefficient matrix determinant (i.e., numerical stability) as function of dimension ðLÞ for differing truncation radius ðc0 Þ:
results can be expected in different areas. As such, it is recommended that users of the MODKERN.f software trial the different kernel modifications for their own data, subject to the restrictions outlined in the preceeding text. This case study is taken from the 1998 computation of the Australian geoid model, AUSGeoid98 (Featherstone et al., 2001). The validation of the applicability of the kernel modification is assessed through absolute comparisons with global positioning system (GPS) and spirit-levelling data (cf. Featherstone, 2001). The limitations of this approach are duly acknowledged (cf. Featherstone et al., 1998, 2001), notably the error budgets associated with the GPS and spirit-levelling control data. Accordingly, universal conclusions cannot be reached from only a comparison in Australia. This presents a further argument in favour of users testing the different kernel modifications in their own region. This should be relatively straightforward using the software and approaches presented herein. AUSGeoid98 uses a L ¼ P ¼ 20 Featherstone et al. (1998) kernel modification based on the belief that the low-degree satellite-derived gravity field is the best source of long-wavelength geoid undulations. Beyond degree–20, the satellite-derived spherical harmonic coefficients become heavily contaminated by noise. These choices also avoid the numerical instabilities in the practical evaluation of the tk ðcos c0 Þ coefficients (described earlier). The remove–compute–restore technique was also used to M ¼ 360 based on the EGM96 global geopotential model (Lemoine et al., 1998), which is one of the more recent models and gives slightly
improved fits to GPS-levelling and terrestrial gravity data in Australia (Kirby et al., 1998). Accordingly, the only parameter that remains to be varied and optimised for this study is the truncation radius ðc0 Þ: Featherstone et al. (2001) show that the use of the entire Australian gravity data rectangle gives a considerably worse fit to the 906 nationwide Australian GPS-levelling data than the M ¼ 360 spherical harmonic expansion of EGM96 (i.e. no regional integration), whereas the use of a limited spherical cap in conjunction with the spherical Stokes kernel improves the fit slightly. That is, the standard deviation of the fit to the GPSlevelling data is 71:113 m for the whole gravity data grid, 70:458 m for the M ¼ 360 expansion of EGM96, and 70:402 m for the spherical Stokes kernel applied over a truncation radius of 1 arc deg. Fig. 4 shows the standard deviation of the fit of different Australian geoid models (with L ¼ M ¼ 20 for the Featherstone et al. (1998) kernel, the spherical Stokes kernel, and M ¼ 360 for the EGM96 model) as a function of increasing truncation radius. From Fig. 4, the Featherstone et al. (1998) kernel gives an improved fit to the 906 GPS-levelling data over the spherical Stokes kernel for truncation radii greater than c0 C0:51: The comparatively worse fit for smaller truncation radii is to be expected based on the numerical instability in the tk ðcos c0 Þ coefficients (demonstrated earlier). Therefore, the Featherstone et al. (1998) kernel is superior to the spherical (unmodified) Stokes kernel when used in conjunction with the M ¼ 360 remove– compute–restore technique, in the Australian context, ignoring the numerical instabilities. Of course, more
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standard deviation (metres)
0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40
Modified
0.35
Unmodified
0.30 0
1
2
3
4
5
6
spherical cap radius (degrees)
Fig. 4. Standard deviation of differences between gravimetric geoid heights computed using spherical Stokes kernel (circles) and Featherstone et al. (1998) modified kernel (triangles) and GPS-AHD heights for 906 points across Australia as function of increasing truncation radius ðc0 Þ:
exhaustive tests could be conducted (perhaps in a Monte Carlo sense) to optimise all the parameters and modifications presented. Nevertheless, the results do indicate that a deterministic kernel modification gives improved fits to the ‘control’ data, thus vindicating the need for (or, at the very least, consideration of) such an approach.
7. Summary and concluding remarks This paper has presented five existing deterministic modifications to the spherical Stokes kernel in a selfconsistent framework, while also classifying them according to the originators of the ideas. The difficulties that may be encountered when practically evaluating and applying these modified kernels have been described, along with advice for their practical use. Results of empirical evaluations of two of the kernels (the spherical Stokes kernel and the Featherstone et al., 1998 modified kernel) in practical geoid determination over Australia show that this deterministic modification is effective when compared to GPS-levelling data. However, because of the peculiarities of local gravity data error characteristics, the modifications and parameters should be varied (with logical constraints due to errors in the geopotential model and access to terrestrial gravity data) for practical gravimetric geoid computations in other parts of the world. The MODKERN.f software supplied will allow this.
Acknowledgements This research has been supported financially by the Australian Research Council though large grants A49331318 and A39938040 and discovery project DP0211827. Professor Z. Martinec of Charles Univer-
sity Prague, Czech Republic, kindly supplied the routines to compute the Van!ıc& ek and Kleusberg modification coefficients and Dr. M.K. Paul of the Canadian Geodetic Survey kindly supplied the routines to compute the spherical truncation coefficients. The Australian Geological Survey Organisation (now Geoscience Australia), the Australian Surveying and Land Information Group (now the Division of National Mapping under Geoscience Australia), NASA, NIMA, Scripps Institute for Oceanography, and the surveying agencies of the State and Territory governments of Australia kindly supplied data for the computations. Thanks are also extended to the reviewers for their time and effort taken to review this manuscript.
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