Triangulated spherical splines

for geopotential reconstruction∗

M. J. Lai, C. K. Shum, V. Baramidze, P. Wenston, and S. C. Han

Abstract

We present an alternate mathematical technique We describe how than contem-porary spherical harmonics to approximate the geopotential on the Earth surfaceusing triangulated spherical spline functions which are smooth piecewise sphericalharmonic polynomials over spherical triangulations. The new method is capableof multi-spatial resolution modeling and could thus enhance spatial resolutions forregional gravity field inversion using data from space gravimetry missions such asCHAMP, GRACE or GOCE. First, we propose to use the minimal energy spher-ical spline interpolation to find a good approximation of the geopotential at anorbital surfacethe orbital altitude of the satellite. Then we explain how to usethe solution expression of Laplace’s equation on the Earth’s exterior to computea spherical spline to approximate the geopotential on the Earth surface.

We propose a domain decomposition technique which enable tocan compute an

approximation of the minimal energy spherical spline interpolation on the orbital

surface and a multiple star technique to compute the spherical spline approxima-

tion by the collocation method. We prove that the spherical spline constructed

by means of the domain decomposition technique converges to the minimal en-

ergy spline interpolation. We also prove that the spline geopotential on the Earth

surface approximates the geopotential on that surface. We have implemented the

two computational algorithms and applied them to compute the geopotential on

the Earth surface CHAMP in a simulated study generating CHAMP geopoten-

tial observations at the satellite altitude using EGM96 (n=90). Comparison with

the traditional spherical harmonic expansion (EGM96 model) is given in several

study regions. Our numerical evidence demonstrates the algorithms are effective

and efficient produce a viable alternative of regional gravity field solution poten-

tially exploiting data from space gravimetry missions. The major advantage of our

method is that it allows us to compute the geopotential over the regions of interest

as well as enhancing the spatial resolution commensurable with the characteristics

of satellite coverage, which could not be done using a global spherical harmonic

representation.

Keywords: CHAMP · Geopotential · · Spherical Splines · Minimal EnergyInterpolation · Domain Decomposition

*) M. J. Lai is the corresponding aurthor. His email address is mjlai@math.uga.edu. Hisphone and fax numbers are (706)542-2065 and (706)542-5907. The results in this paperare based on the research supported by the National Science Foundation under the grantNo. 0327577. The second author is associated with the School of Earth Sciences, OhioState University, Columbus, OH 43210, The third author is associated with Department ofMathematics, Western Illinois University, Macomb, IL61455, the fouth author is associatedwith Department of Mathematics, The University of Georgia, Athens, GA 30602, and thefifth author is associated with Ohio State University, now at NASA Goddard Space FlightCenter, Greenbelt, MD, 20771.

2

Advances in the measurement of the gravity have with modern free-fallmethods reached accuracies of 10−9g (∼ 1 ν Gal or 10 nm/s2), allowing themeasurement of effect of mass changes within the Earth interior or the geo-physical fluids to the commensurate accuracy, and surface height change mea-surements to 3 mm relative to the Earth center of mass [Forsberg, Sideris,Shum, 2005]. During this Decade of the Geopotential, satellite missions al-ready in operation (CHAMP [Reigber et al., 2004] and GRACE [Tapley et al.,2004] gravimetry), or will be launched (GOCE gradiometry [Rummel et al.,1999]) will provide an opportunity towards global synoptic mapping of theseclimate-sensitive mass fluxes, providing an important constraint for climatechange studies. The geopotential function V is defined on R3 such that thegradient ∇V is the gravitational field. Traditionally, V is reconstructed by us-ing spherical harmonic functions (cf. e.g., [Tapley et al., 2002] to model datafrom the U.S. German Gravity Recovery and Climate Experiment, GRACE,mission). Recently, several new methods were proposed. Spherical waveletmethods were studied in the literature, [Freeden and Schreiner, 1998], [Free-den, Michel, and Nutz, 2002] and results were surveyed in [Freeden, Maier,and Zimmerman, 2003]. [Freeden and Schreiner, 2005], [Fengler, Freeden,Kohlhaas, Michel, Peters, 2007]. See spherical wavelets discussed in [Freeden,Gervens, and Schreiner, 1998]. Other spherical wavelet techniques includePoisson multipole wavelets [Chambodut et al., 2005, Panet et al., 2006] andBlackman spherical wavelets [Schmidt et al. 2005, 2006]

Other methods include a pre-processing of the normal equations from theleast squares method of spherical harmonic polynoimals by using singularvalue decomposition (cf. [Schmidt, 1999]), representation of gravity fieldbased on numerical solution of boundary value problems by boundary elementmethod (cf. [Klees, 1999]), and modeling gravity field from short kinematicarcs [Mayer-Gurr, Eicker, and Ilk, 2006]). including mass concentrations(mascon) [e.g., Rowlands et al. 2005] and the energy conservation methodto estimate the gravity field using satellite in-situ geopotential (CHAMP) ordisturbance potential (GRACE) [e.g., Han et al., 2006] based on a regionalapproachinversion with stochastic least squares collocation and 2D-FFT [Hanet al., 2003]. Several spline functions including triangulated spherical splineswere suggested for modeling geopotential in [Jekelli, 2005]. However, nocomputational results were presented there.

In this paper, we propose to use triangulated spherical splines to computean approximation of the geopotential. The triangulated spherical splinesover the unit sphere S

2 were introduced and studied by Alfeld, Neamtu

1

and Schumaker in a series of three papers [Alfeld, Neamtu, Schumaker’96a],[Alfeld, Neamtu, Schumaker’96b], and [Alfeld, Neamtu, Schumaker’96c].These spline functions are smooth piecewise spherical harmonic polynomi-als over triangulation of the unit spherical surface S

2. Basic properties oftriangulated spherical splines are summarized in [Lai and Schumaker, 2007].They can have locally supported basis functions which are completely differ-ent from the spherical splines defined in [Freeden, Gervens, and Schreiner,1998]. A straightforward computational method to use these triangulatedspherical splines for scattered data fitting and interpolation is given in ourearlier paper [Baramidze, Lai, and Shum’06]. We explain how to use tri-angulated spherical splines directly without constructing locally supportedbasis functions like finite elements for constructing fitting and/or interpolat-ing spherical spline functions from any given data locations and values. Inthis direct method, we explain how to use an iterative method to solve someconstraint minimization problems with smoothness conditions and interpo-lation conditions as constraints. In addition, the approximation property ofminimal energy spherical spline interpolation can be found in [Baramidze andLai’05]. The approximation property implies that the triangulated sphericalspline interpolation by using the minimal energy method gives an excellentapproximation of sufficiently smooth functions over the surface of the unitsphere.

However, when using the minimal energy method to find spherical splineinterpolation of the geopotential, the matrix associated with the method isvery large for the given large amount of the geopotential measurements from asatellite. In this study, we choose to use the simulated satellite data of in situgeopotential values measurements (in m2/s2) which was computed for thegravity mission satellite, The CHAllenging Minisatellite Payload (CHAMP)(cf. [Reigber et al. 2004]). CHAMP is a German geodetic satellite, launchedon July 15, 2000, with a circular orbit at an altitude of 450 km and orbitalinclination of 87. In Figures 1 and 2, we show a set of CHAMP potentialdata locations for two day period (two methods for these data locations areshown). From Figure 2, it is clear that the locations of the geopotentialare scattered around and hence any tensor product of two univariate func-tions may not work well with such scattered data locations. On its orbit for30 days, using a ”true” geopotential model, EGM96 (which is complete tospherical harmonic degree 360, nmax=90 is used), total amount of simulateddata is 86,400.

2

Figure 1. Geopotential Data Locations

0 30 60 90 120 150 180 210 240 270 300 330

−60

−30

0

30

60

Latitude

Long

titud

e

Figure 2. Planar View of Geopotential Data LocationsWe propose a new computational method called a domain decomposition

technique to compute an approximation of the global minimal energy splineinterpolation. This technique is a generalization of the same technique inthe planar setting (cf. [Lai and Schumaker, 2003]). It enables us to do thecomputation in parallel and hence, effectively reduce the computational time.

Since the purpose of the research to reconstructing the geopotential is tofind a good approximation of the geopotential values on the Earth surface,we use the above approximation of the geopotential on the satellite orbitto approximate the geopotential on the Earth surface. To this end, we re-call the classic theory of geopotential (cf., e.g. [Heiskanen and Moritz’67]).The geopotential function V defined on the R3 space outside of the Earthsatisfies the Laplace’s equation with Dirichlet boundary condition on theEarth surface. If the boundary values V (u) or V (Re, θ, λ) are known for allu = (x, y, z) over the surface of the imaginary sphere with mean Earth radius

3

Re = 6371.138km with x = Re sin θ cosλ, y = Re sin θ sinλ, and z = Re cos θ,the solution of Laplace’s equation outside the sphere can be explicitly givenin terms of spherical harmonics or in terms of Poisson integral. That is,the solution V to the exterior problem (|u| =

√x2 + y2 + z2 > Re) can be

represented in terms of an infinite sum:

V (u) =

∞∑

n=0

(Re

|u|

)n

Yn(θ, λ), (0.1)

where

Yn(θ, λ) =2n+ 1

4π

∫ 2π

λ′=0

∫ π

θ′=0

V (Re, θ′, λ′)Pn(cosψ) sin θ′dθ′dλ′ (0.2)

are spherical harmonics, Pn are Legendre polynomials, and

cosψ = cos θ cos θ′ + sin θ sin θ′ cos(λ− λ′).

We also know Poisson integral representation of the solution V for |u| >Re, i.e.,

V (u) = Re|u|2 − R2

e

4π

∫ 2π

λ′=0

∫ π

θ′=0

V (Re, θ′, λ′)

`3sin θ′dθ′dλ′, (0.3)

where ` =√

|u|2 − 2|u|Re cosψ +R2e is the distance from u to v with |v| = Re

and angles θ′, λ′.It is known that the geopotential V is infinitely differentiable. By

the approximation property of the minimal energy spline interpolation (cf.[Baramidze and Lai’05]), the spherical interpolatory spline SV of the geopo-tential measurement data at the in-situ orbital surface at Ro := Re + 450kmaltitude is a very good approximation of the geopotential V (see Section 2).Intuitively, we may replace V by SV in (0.3). That is,

SV (u) ≈ Re|u|2 − R2

e

4π

∫ 2π

λ′=0

∫ π

θ′=0

V (Re, θ′, λ′)

`3sin θ′dθ′dλ′, (0.4)

Next we approximate V on the surface of the Earth in the above formulaby using triangulated spherical splines. Let S0

d(4e) be the space of all contin-uous spherical splines of degree d over 4e which is induced by the underlying

4

triangulation 4 of SV . We find sV ∈ S0d(4e) solving the following collocation

method:

SV (u) = Re|u|2 − R2

e

4π

∫ 2π

λ′=0

∫ π

θ′=0

sV (Re, θ′, λ′)

`3sin θ′dθ′dλ′, for u ∈ Dd

4,

(0.5)where Dd

4 is a set of domain points on the orbital surface (to be precise later).The reason we use S0

d(4e) is that the geopotential on the surface of the Earthmay not be a very smooth function. It should be approximated by sphericalsplines in S0

d(4e) better than by spline functions in Srd(4e) with r ≥ 1.

We need to show that the linear system (0.5) above is invertible for certaintriangulations. Then we continue to prove that sV yields an approximationof the geopotential V on the Earth’s surface (see Section 4). To compute sV ,we shall use a so-called multiple star technique so that the computation canbe done in parallel. In Section 4 we shall explain this technique and showthat the numerical solution from the multiple star technique converges.

The above discussions outline a theoretical basis for approximating geopo-tential at any point on the surface ReS

2 by using triangulated sphericalsplines. Our triangulated spherical spline approach is certainly different fromthe traditional and classic approach by spherical harmonic polynomials. Forexample, in [Han, Jekeli, and Shum’02], a least squares method is used todetermine the coefficients in the spherical harmonic expansion up to degreen = 70 to fit the CHAMP measurements. The total number of coefficients is70 × 71/2 = 4, 970. The rigorous estimation of these coefficients potentiallyrequires many hours of CPU time of a supercomputer back to ten years ago.When evaluating the geopotential at any point, all these 4,970 terms have tobe evaluated since each harmonic basis function Yn is globally supported overthe sphere. As the degree n of spherical harmonics increases, spherical har-monic polynomials Yn oscillate more and more frequently and the evaluationof Yn with large degree n is very sensitive to the accuracy of the locations.

The advantages of triangulated spherical splines over the method of spher-ical harmonic polynomials are as follows:

(1) our spherical spline solution is an interpolation of the given geopotentialdata measurements instead of a least squares data fitting. Due tocomputer capacity, we are not able to interpolate the data values withinthe given accuracy. In our computation, the standard deviation of these86,400 values is 0.018m2/s2 while the least squares fit has the standarddeviation about 0.5m2/s2.

5

(2) our solution is solved in parallel in the sense that the solution is dividedinto many small blocks and each small block is solved independentlywhile in the least squares method, the observation matrix is dense andof large size and hence, is very expensive to solve;

(3) our solution can be efficiently evaluated at any point since only a fewterms (e.g., ≤ 21 terms for spherical splines of degree 5) which maybenonzero at any point.

(4) Our algorithms allow us to compute an approximation of the geopoten-tial over any region ω on the Earth surface directly from the measure-ments of a satellite on the orbital level. That is, to compute sV overω ⊂ ReS

2, we need SV (u) for u ∈ Ω on RoS2 (see Section 4). Note

that Ω is corresponding to an enlarged region on ReS2 covering ω. To

compute SV over Ω on the orbital surface, we use the measurementsfrom a satellite over a larger region starq(Ω) (see Section 3).

Let us summarize our approach as follows: First we compute a sphericalspline interpolation SV of geopotential values at these 86, 400 data locationsover the orbital surface by using the minimal energy method. It is one of keysteps. Although computing a minimal energy interpolant for such a large dataset requires a significant memory storage and high speed computer resources,we shall use a domain decomposition technique to overcome this difficulty.We shall explain the technique and computation in §3. Next we solve (0.5)to find a spherical spline approximation sV on the surface of the Earth. Wefirst show that sV is a good approximation of V and then discuss how tocompute sV by using a multiple star technique. All these will be given in §4.Finally in §5 we conclude that our triangular spherical splines are effectiveand efficient for computing the geopotential on the Earth surface.

1 Preliminaries

1.1 Spherical Spline Spaces

Given a set P of points on the sphere of radius 1, we can form a triangu-lation 4 using the points in P as the vertices of 4 by using the Delaunaytriangulation method. Alternatively, we can use eight similar spherical tri-angles to partition the unit sphere S

2 and denote the collection of triangles

6

by 40. Then we uniformly refine 40 by connecting the midpoints of threeedges of each triangle in 40 to get a new refined triangulation 41. Then werepeat the uniform refinement to get 42,43, .... See Figure 3 for a uniformtriangulation over a surface of the Earth. In this paper, we will assume thatthe triangulationtriangle is regular in the sense that: 1) any two triangles do not intersecteach other or share either a common vertex or a common edge; 2) every edgeof 4 is shared by exactly two triangles.

Figure 3. A uniform triangulation of the sphereLet

Srd(∆) = s ∈ Cr(S2), s|τ ∈ Hd, τ ∈ ∆

be the space of homogeneous spherical splines of degree d and smoothnessr over ∆. Here Hd denotes the space of spherical homogeneous polynomialsof degree d (cf. [Alfeld, Neamtu and Schumaker’96]). This spline space canbe easily used for interpolation and approximation on sphere if any splinefunction in Sr

d(∆) is expressed in terms of spherical Bernstein Bezier polyno-mials and the computational methods in [Baramidze, Lai, and Shum’06] areadopted.

To be more precise, we write each spline function s ∈ Srd(∆) by

s =∑

T∈∆

∑

i+j+k=d

cTijkBTijk, (1.1)

where BTijk is called spherical Bernstein basis function which is only supported

7

on triangle T and cTijk are coefficients associated with BTijk. More precisely,

let T = 〈v1, v2, v3〉 be a spherical triangle on the unit sphere with nonzeroarea. Let b1(v), b2(v), b3(v) be the trihedral barycentric coordinates of apoint v ∈ S2 satisfying

v = b1(v)v1 + b2(v)v2 + b3(v)v3.

We note that the linear independence of the vectors v1, v2 and v3 ∈ R3 implythat b1(v), b2(v), and b3(v) are uniquely determined. Clearly, b1(v), b2(v),and b3(v) are linear functions of v. It was shown in [Alfeld, Neamtu, andSchumaker’96] that the set

BTijk(v) =

d!

i!j!k!b1(v)

ib2(v)jb3(v)

k, i+ j + k = d (1.2)

of spherical Bernstein-Bezier (SBB) basis polynomials of degree d forms abasis for Hd restricted to the unit spherical surface S2. More about sphericalsplines can be found in [Lai and Schumaker, 2007].

1.2 Minimal Energy Spline Interpolations

Next we briefly explain one of the computational methods presented in[Baramidze, Lai and Shum’06]. Suppose we are given values f(v), v ∈ Pof an unknown function f on a set P. Let

Uf := s ∈ Srd(4) : s(v) = f(v), v ∈ P

be the set of all splines in S ⊆ Srd(4) that interpolate f at the points of P.

Then a commonly used way (cf. [Fasshauer and Schumaker’98]) to create anapproximation of f is to choose a spline Sf ∈ Uf such that

Eδ(Sf) = mins∈UfEδ(s), (1.3)

where Eδ is an energy functional:

Eδ(f) =

∫

S2

(|∂2

∂x2fδ|

2 + |∂2

∂y2fδ|

2 + |∂2

∂z2f)δ|2

+|∂2

∂x∂yfδ|

2 + |∂2

∂x∂zfδ|

2 +|∂2

∂y∂zfδ|

2

)dθdφ, (1.4)

8

where, since f is defined only on S2, we first extend f into a function fδ

defined on R3 to take all partial derivatives and then restrict them on theunit spherical surface S

2 for integration. Here, we use Eδ for δ = 0 or δ = 1to denote the even and odd homogenuous extensions of f . In the rest of thepaper, we should fix δ = 1.

We refer to Sf in Eq. (1.3) the (global) minimal energy interpolatingspline. To compute Sf , we use the coefficient vector c consisting of cTijk, i +

j + k = d, T ∈ 4 (see (1.1)) to represent each spline function s ∈ S−1d (4),

where S−1d (4) denotes the space of LL piecewise spherical polynomials of

degree d over triangulation 4 without any smoothness. To ensure the Cr

continuity across each edge of 4, we impose the smoothness conditions overevery edge of 4. Let M denote the smoothness matrix such that

Mc = 0

if and only if s ∈ Srd(4). Note that we can assemble interpolation conditions

into a matrix K, according to the order in which the coefficient vector c isorganized. Then Kc = F is the linear system of equations such that thecoefficient vector c of a spline s interpolates f at the data sites V.

The problem of minimizing (1.3) over Srd(4) can be formulated as follows

(cf. [Baramidze, Lai and Shum’06]):

minimize Eδ(s), subject to Mc = 0 and Kc = F.

To simplify the data management we linearize the triple indices of BB-coefficients cijk as well as the indices of the basis functions Bd

ijk. By usingthe Lagrange multipliers method, we solve the following linear system

E K ′ M ′

K 0 0M 0 0

cηγ

=

0F

0

. (1.5)

Here γ and η are vectors of Lagrange multipliers, K ′ and M ′ denotes thetransposes of K and M , respectively, and the energy matrix E is definedas follows. E = diag(ET , T ∈ ∆) is a diagonally block matrix. Each blockET = (emn)1≤m,n≤d(d+1)/2 is associated with a triangle T ∈ 4 and containsthe following entries

emn :=

∫

T

(♦BTm(v)) · ♦(BT

n (v))dσ(v), (1.6)

9

where Bm and Bn denote SBB polynomial basis functions BTijk of degree d

corresponding to the order of the linearized triple indices (i, j, k), i+j+k = d.Here, ♦ denotes the second order derivative vector, i.e.,

♦f =

(∂2

∂x2f,

∂2

∂y2f,

∂2

∂z2f,

∂2

∂x∂yf,

∂2

∂x∂zf,

∂2

∂y∂zf

)(1.7)

and · denotes the dot product of two vectors in Eq. (1.6).Note that E is a singular matrix. The special linear system is now solved

by using the iterative method: Writing the above singular linear system Eq.(1.5) in the following form

[A L′

L 0

] [cλ

]=

[FG

],

where A = E and L = [K;M ] are appropriate matrices. The system can besuccessfully solved by using the following iterative method (cf. [Awanou andLai’05]) [

A L′

L −εI

] [c(`+1)

λ(`+1)

]=

[F

G− ελ(`)

],

for l = 0, 1, 2, · · · , where ε > 0 is a fixed number, e.g. ε = 10−4, λ(`) isiterative solution of a Lagrange multiplier with λ0 = 0 and I is the identitymatrix. The above matrix iterative steps can in fact be rewritten as follows:

(A+1

εL′L)c(l+1) = AFc(l) +

1

εL′G

with c(0) = 0. It is known that under the assumption that A is symmetric andpositive definite with respect to L, the vectors c(`) converge to the solutionc in the following sense: there exists a constant C such that

‖c(k+1) − c‖ ≤ Cε‖c(k) − c‖

for all k (cf. [Awanou and Lai’05]). Since A may be of large size, we shall in-troduce a new technique to make the computational method more affordablein the next section.

The approximation properties of minimal energy interpolating sphericalsplines are studied in (cf. [Baramidze and Lai’05]). Let us state here brieflythat for the homogeneous spherical splines of degree d under certain assump-tions on triangulation 4 we have

‖Sf − f‖∞,S2 ≤ C|4|2|f |2,∞,S2

10

for f ∈ C2(S2) and d odd, and

‖Sf − f‖∞,S2 ≤ C ′|4|2|f |2,∞,S2 + C ′′|4|3|f |3,∞,S2

for f ∈ C3(S2) and d even. Here |4| denotes the size of triangulation, i.e.,the largest diameter of the spherical cap containing triangle T for T ∈ 4.Here, |f |2,∞,S2 denotes the maximum norm of all second order derivatives off over S

2. Since geopotential G is the solution of Laplace’s equation, it isinfinitely many differentiable. It follows that SG approximates G very wellby the above result.

2 Approximation of Geopotential over the

Orbital Surface

2.1 Explanation of the domain decomposition tech-

nique

Since the given simulated set of in situ geopotential measurements collectedby CHAMP during 30 days amounts to 86, 400 locations and values, com-puting a minimal energy interpolant for such a large set requires a significantamount of computer memory storage and high speed computer resources. Toovercome this difficulty, we use a domain decomposition technique which willbe used to approximate the minimal energy spline interpolant.

The domain decomposition method can be explained as follows. Dividethe spherical domain S

2 into several smaller non-overlapping subdomainsΩi, i = 1, ..., n along the edges of existing triangulation 4 of Ω. For example,we may choose each triangle of 4 is a subdomain. Fix q ≥ 1. Let starq(Ωi) bea q-star of subdomain Ωi which is defined recursively by letting star0(Ωi) = Ωi

andstarq(Ωi) := ∪T ∈ 4, T ∩ starq−1(Ωi) 6= ∅ (2.1)

for positive integers q = 1, 2, · · · , .Instead of solving the minimal energy interpolation problem over the en-

tire spherical surface, we solve the minimal energy spherical spline interpola-tion problem over each q-star domain starq(Ωi) by using the spherical splinespace Si,q := Sr

d(starq(Ωi)) for i = 1, 2, · · · , n. Let sf,i,q be the minimalenergy solution over starq(Ωi). That is, let

Uf,i,q := s ∈ Si,q, s(v) = f(v), ∀v ∈ starq(Ωi) ∩ P.

11

Then sf,i,q ∈ Uf,i,q is the spline satisfying

Ei,q(sf,i,q) = mins∈Uf,i,qEi,q(s), , (2.2)

where

Ei,q(s) :=∑

T∈starq(Ωi)

∫

T

♦(s) · ♦(s)dσ (2.3)

with ♦ being defined in (1.7). It can be shown that sf,i,q|Ωiapproximates

the global minimal energy spline (1.3) Sf |Ωivery well. That is, we have

Theorem 2.1 Suppose we are given data values f(v) over scattered datalocations v ∈ P for a sufficiently smooth function f over the unit sphere. LetSf be the minimal energy interpolating spline satisfying (1.3). Let sf,i,k bethe minimal energy interpolating spline over starq(Ωi) satisfying (2.2). Thenthere exists a constant σ ∈ (0, 1) such that for q ≥ 1

||Sf − sf,i,q||∞,Ωi≤ C0σ

q(tan|4|

2)2(C1|f |2,∞,S2 + C2||f ||∞,S2), (2.4)

if f ∈ C2(S2) and d is odd. Here C0, C1 and C2 are constants depending on dand β = |4|/ρ4, where ρ4 denotes the smallest radius of the inscribed capsof all triangles in 4. If f ∈ C3(S2) and d is even

||Sf − sf,i,q||∞,Ωi≤ C0σ

q(tan|4|

2)2(C3|f |2,∞,S2 + C4|f |3,∞,S2 + C3||f ||∞,S2),

(2.5)for positive constants C4 and C5 depending on d and β.

One significant advantage of the domain decomposition technique is thatsf,i,q can be computed over subdomain starq(Ωi) independent of sf,j,q forj 6= i. Thus, the computation can be done in parallel. Usually, we chooseeach triangle in 4 as a subdomain. We use sf,i,q to approximate Sf overΩi. The collection of sf,i,q|Ωi

is a very good approximation of Sf over Ω. Ifthe computation for each subdomain requires a reasonable time, so is theapproximation of the global solution.

The proof of Theorem 3.1 is quite technique in mathematics. We omitthe detail here. For the interested reader, see [Baramidze, 2005] and [Laiand Schumaker, 2003]. In the following subsection we present some numer-ical experiments to demonstrate the convergence of local minimial energyinterpolatory splines to the global one.

12

2.2 Computational Results on the Orbital Surface

We have implemented our domain decomposition techinque for the recon-struction of geopotential over the orbital surface in both MATLAB and FOR-TRAN. To make sure that our computational algorithms work correctly, wefirst choose several spherical harmonic functions to test and verify the accu-racy of the computational algorithm. Then we apply our algorithm to thegiven data set from CHAMP. The following numerical evidence demonstratethe effectiveness and efficiency of our algorithm.

First of all we illustrate the convergence of the minimal energy interpo-lating spline to some given test functions:

f1(x, y, z) = r−9 sin8(θ) cos(8φ)f2(x, y, z) = r−11 sin10(θ) sin(10φ)f3(x, y, z) = r−16 sin15(θ) sin(15φ)f4(x, y, z) = 789/r + f3(x, y, z)

where r =√x2 + y2 + z2. All of them are harmonic. Let 4 be a trian-

gulation of the unit sphere which consists of 8 congruent triangles with 6vertices. We then uniformly refine it several times as described in Section2 to get new triangulations 41,42,43, · · · , . That is, 4n is the uniformrefinement of 4n−1. Thus, 41 consists of 18 vertices and 32 triangles, 42

contains 66 vertices and 128 triangles, 43 has 258 vertices and 512 triangles,44 consists of 1026 vertices and 2048 triangles and 45 contains 4098 verticesand 8172 triangles.

Recall that S15(4n) is the C1 quintic spherical spline space over triangu-

lation 4n. We choose

r = 1.05 ≈Re + 450

Re,

where Re = 6, 371.388km is the radius of the Earth and 450 represents thehight of the orbital surface of CHAMP above the surface of the Earth.

The minimal energy spline functions in S15(4n) with n = 4 and n = 5

interpolates 16,200 points equally spaced grid points over [−π, π]× [0, π]. Tocompute these spline interpolants, we use the domain decomposition tech-nique. The following numerical results are based on the domain decompo-sition technique with q = 3 and Ωi being triangles in 4n as described inSection 3.1.

13

Then we estimate the accuracy of the method by evaluating the splineinterpolants and the exact functions over 28,796 points almost evenly dis-tributed over the sphere and then computing the maximum absolute valueof the differences and computing the standard deviation

std =

√∑28796i=1 (s(pi) − f(pi))2

28795

where s and f stand for spline interpolant and function to be interpolatedand pi stands for points over the surface at the orbital level. The standarddeviations and maximum errors are listed in Table 1.

Table 1: Maximum errors of C1 quintic interpolatory splines for variousfunctions

f\i ∆4 ∆4 ∆5 ∆5

std max. errors std max. errorsf1 5.091e− 04 1.242e− 02 6.566e− 06 1.269e− 04f2 6.959e− 04 1.715e− 02 9.556e− 06 1.827e− 04f3 1.2313 × e− 03 3.011e− 02 2.756e− 05 3.549e− 04f4 1.213 × e− 03 3.010e− 02 2.753e− 05 3.549e− 04

From Table 1, we can see that the spherical interpolatory splines approx-imate these standard functions very well on the spherical surface with radiusr = 1.05. This example also shows that our domain decomposition techniqueworks very well. The computing time is 30 minutes for finding spline inter-polants in S1

5(∆4) and 2 hours for S15(45) using a SGI computer (Tezro) with

four processes with 2G memory each.Let us make a remark. Although these functions may be approximated

by using spherical harmonics better than spherical splines, the main point ofthe table is to show how well spherical splines can approximate. Intuitively,the geopotential does not behave nicely as these test functions and it ishard to approximate by one spherical harmonic polynomial. Instead, bybreaking the spherical surface S

2 into many triangles, triangulated sphericalsplines, piecewise spherical harmonics may have a hope to approximate thegeopotential better.

Next we compute interpolatory splines SG over the given set of datameasurements of the geopotential on the orbital surface. We first compute

14

an minimal energy interpolatory spline using the data locations and valuesover the two day period. The spline space S1

5(44) is used, where triangulation44 consists of 1026 points and 2048 triangles. Although the interpolatoryspline fits the first two day’s measurements (5760 locations and values) tothe accuracy 10−6, the standard deviation of the spline over the 30 daymeasurement values is 0.60 m2/s2.

Next we compute the minimal energy interpolatory spline in S15(45)

which interpolates 23032 data locations and values over an eight-day pe-riod. The standard deviation of the spline at all 86,400 data locations andvalues of 30 days is 0.018m2/s2. This shows that the minimal energy splinefits the geopotential over the orbital surface very well. In Fig. 4, we show thegeopotential measurements (after a normalization such that the normalizedgeopotential values are all bigger than the mean radius of the Earth) and theinterpolatory spline surface around the Earth. The normalized geopotentialand the spline surface are plotted in 3D view.

15

Fig. 4. Normalized geopotential values over the Earth andC1 quintic spherical spline interpolatory surface

3 Approximation of Geopotential on the Earth’s

Surface

3.1 The Inverse Problem

Let SV be the spherical spline approximation of the geopotential V on theorbit. Recall from the previous section that SV approximates V very well. Wenow discuss how we can compute spline approximation sV of the geopotentialV on the Earth surface.

Let 4e be a triangulation on the unit sphere induced by the triangulation4 on the orbital spherical surface used in the previous section. Let sV ∈

S0d(4e) be a spline function sV =

∑

T∈4e

∑

i+j+k=d

cTijkBTijk solving the following

collocation problem

SV (u) = Re|u|2 −R2

e

4π

∫ 2π

λ′=0

∫ π

θ′=0

sV (θ′, λ′)

`3sin θ′dθ′dλ′, for u ∈ Dd

4, (3.1)

where Dd4 is the collection of domain points of degree d on 4, i.e.,

Dd4 := ξlmn =

lv1 +mv2 + nv3

‖lv1 +mv2 + nv3‖2, T = 〈v1, v2, v3〉 ∈ 4, l +m+ n = d.

16

More precisely, Eq. (3.1) can be written as follows: Find coefficients cTijk suchthat

∑

T∈4

∑

i+j+k=d

cTijkRe|u|2 −R2

e

4π

∫

T

BTijk(θ

′, λ′)

`3sin θ′dθ′dλ′,= SG(u), u ∈ Dd

∆.

(3.2)Note that we use continous spherical spline space S0

d(4e) since the geopo-tential is not very smooth on the surface of the Earth.

We need to show that the collocation problem (3.1) above has a uniquesolution as well as sV is a good approximation of V on the Earth surface. Tothis end, we begin with the following

Lemma 3.1 Let f be a function in L2(S2). Define

F (|u|, θ, φ) =|u|2 − 1

4π

∫ 2π

θ′=0

∫ π

φ′=0

f(θ′, φ′)

`3sin θ′dθ′dφ′, ∀θ, φ. (3.3)

Suppose that for all |u| = R > 1, F (u) = 0. Then f = 0.

Proof: It is clear that F is a harmonic function which decays to zero at∞. We can express F in an expansion of spherical harmonic functions as in(0.1) and (0.2). Now F (u) ≡ 0 implies that the coefficients in the expansionhave to be zero. That is, by using (0.2),

2n+ 1

4π

∫ 2π

θ′=0

∫ π

φ′=0

f(θ′, φ′)Pn(cosψ) sin θ′dθ′dφ′ = 0, ∀n ≥ 0.

Thus, f ≡ 0. This completes the proof. ♠

This is just say that if a solution of the exterior Poisson equation is zeroover whole layer |u| = R0, it is a zero harmonic function.

Theorem 3.2 There exists a triangulation 4e such that the minimization(3.1) has a unique solution.

Proof: If the minimization (3.1) has more than one solution, then theobservation matrix associated with (3.1) is singular. Thus there exists aspline s0 ∈ Sr

d(∆e) such that

∫ 2π

θ′=0

∫ π

φ′=0

s0(θ′, φ′)

`3sin θ′dθ′dφ′ = 0, (3.4)

17

for all θ, φ which are associated with the domain points Dd4 of degree d. That

is, the points u ∈ R3 with length Ro and angle coordinates (θ, φ) are domainpoints in Dd

41. Without loss of generality, we may assume that ‖s0‖2 = 1. Let

us refine 4 uniformly to get 41. Write 4e,1 to be the triangulation inducedby 41. If the linear system in (3.1) replacing 4 by 41 is not invertible, thereexists a spline s1 ∈ S0

d(4e,1) such that ‖s1‖2 = 1 and

∫ 2π

θ′=0

∫ π

φ′=0

s1(θ′, φ′)

`3sin θ′dθ′dφ′, (3.5)

for those angle coordinates (θ, φ) such that vectors u ∈ R3 with length Ro

and angle coordinates (θ, φ) are domain points in Dd41

.In general, we would have a bounded sequence s0, s1, · · · , in L2(ReS

2).It follows that there exists a subsequence sn′ which converges weakly to afunction s∗ ∈ L2(ReS

2). Then

0 =

∫ 2π

θ′=0

∫ π

φ′=0

s∗(θ′, φ′)

`3sin θ′dθ′dφ′, ∀(θ, φ). (3.6)

By Lemma 4.1, we would have s∗ ≡ 0 which contradicts to ‖s∗‖2 = 1. Thiscompletes the proof. ♠

Using the above Theorem 4.2. we can compute a spline approximationsV of V over certain triangulations. Next we need to show sV is a goodapproximation of V on the Earth’s surface. Recall Ro = Re + 450km withRe being the mean radius of the Earth. Let

V (Ro, θ, φ) = ReR2

o −R2e

4π

∫ 2π

θ′=0

∫ π

φ′=0

sV (θ′, φ′)

`3sin θ′dθ′dφ′, (3.7)

for all (θ, φ). In particular, V (Ro, θ, φ) agrees SV (θ, φ) for those angle co-ordinates (θ, φ) that their associated vectors u ∈ Dd

4 by Eq. 3.1. That is,

SV is also an interpolation of V . Thus SV is a good approximation of V byLemma 3.3 to be discussed later and thus,

‖V − V ‖∞,RoS2 ≤ ‖V − SV ‖∞,RoS2 + ‖SV − V ‖∞,RoS2

is very small, where the maximum norm ‖ · ‖∞,Rois taken over the surface of

the sphere with radius Ro.

18

In addition, we shall prove that

‖V − sV ‖∞,ReS2 ≤ C‖V − V ‖∞,RoS2 . (3.8)

by using the open mapping theorem (cf. [Rudin, 1967]). Indeed, define asmooth function

L(f)(θ, φ) := ReR2

o − R2e

4π

∫ 2π

θ′=0

∫ π

φ′=0

f(Re, θ′, φ′)

`3sin θ′dθ′dφ′ (3.9)

for all (θ, φ). Let H = L(f)(θ, φ), f ∈ L2(ReS2), where L2(ReS

2) is thespace of all square integrable functions on the surface of the sphere ReS

2. Itis clear that H be a linear vector space. If we equip H with the maximumnorm, H is a Banach space.

Then L(f) is a bounded linear map from L2(ReS2) to H which is 1 to 1

by Lemma 4.1. Since L is also an onto map from L2(ReS2) to H . By the

open mapping theorem (cf. [Rudin’63]), L has a bounded inverse. Thus, wehave Eq. (3.8). We remark that this is different from the integral operator.Indeed, our SV at the orbital surface and sV at the surface of the Earth haveno radial part. They are just defined on the subdomain with |u| = Ro and|u| = Re respectively. That is, from SV we can not downward continuationto get sV at r = Re or Re/r = 1. We have to solve (4.1) in order to get theapproximation on the surface of the Earth. Certainly, the constant for theboundedness in the discussion above may be dependent on 450km.

By Theorem 3.4, we have

‖V − SV ‖∞,RoS2 ≤ C|4|2,

where |4| denotes the size of triangulation 4. Thus we only need to estimate

‖SV − V ‖∞,RoS2 . To this end, we first note that V (u) = SV (u) for u ∈ Dd4.

The following Lemma (see [Baramidze and Lai’05] for a proof) ensures the

good approximation property of SV to V .

Lemma 3.3 Let T be a spherical triangle such that |T | ≤ 1 and supposef ∈W 2,p(T ) vanishes at the vertices of T , that is f(vi) = 0, i = 1, 2, 3. Thenfor all v ∈ T ,

|f(v)| ≤ C tan2(|T |

2)|f |2,∞,T (3.10)

for some positive constant C independent of f and T .

19

It follows that

|SV (u) − V (u)| ≤ C tan2(|∆|

2)(‖SV ‖2,∞,RoS2 + ‖V ‖2,∞,RoS2).

Recall that ‖SV ‖2,∞,RoS2 ≤ C‖V ‖2,∞,RoS2 and ‖V ‖2,∞,RoS2 ≤ C‖SV ‖2,∞,RoS2 .Therefore we conclude the following

Theorem 3.4 There exists a spherical triangulation 4 of the surface of thesphere ReS

2 such that the solution sV of the linear system (3.1) approximatesthe geopotential V on the surface of Earth in the following sense

‖sV − V ‖∞,ReS2 ≤ C|4|2 (3.11)

for a constant C dependent on the geopotential V on the orbital surface.

3.2 A Computational Method for the Solution of the

Inverse Problem

Finally we discuss the numerical solution of the linear system (3.1). Clearly,when the number of data locations increases, so is the size of linear system.It is expensive to solve such a large linear and dense system. Let us describethe multiple star technique as follows. For each triangle T ∈ 4e, let star`(T )be the `-star of triangle T . We solve cTijk, i + j + k = d by considering thesublinear system which involves all those coefficients ctijk, i + j + k = d and

t ∈ star`(T ) for a fixed ` > 1 using the domain points u ∈ star`(T ). That is,we solve

∑

t∈star`(T )

∑

i+j+k=d

ctijkRe|u|2 − R2

e

4π

∫

t

Btijk(v)

|u−Rev|3dσ(v) = SG(u), (3.12)

for u ∈ Dd4 ∩ star`(T ). We solve (3.12) for each T ∈ 4e. Clearly this can be

done in parallel. Let us now show that the solution from the multiple startechnique converges to the original solution as ` increases. To explain theideas, we express the system in the standard format:

Ax = b

with A = (aij)1≤i,j≤n, x = (x1, · · · , xn)T and b = (b1, · · · , bn)T . Note thatentries aij have the following property:

aij = O(1

|i− j|3 + 1)

20

since coefficients in (3.2) is∫

T

BTijk

|u−v|3dσ(v) for some triangle T and (i, j, k)

with i+j+k = d. For domain points u on 4 of degree d, the distance |u−v|is increasing when u locates far away from v ∈ T . Our numerical solution(3.12) can be expressed simply by

∑

|i−i0|≤N`

aij xi = bj , |j − i0| ≤ N`

for i0 = 1, · · · , n, where N` is an integer dependent on `. If ` increases, sodoes N`. We need to show that xi converges to xi as ` increases. To this end,we assume that ‖x‖∞ is bounded and the submatrices

[aij ]|i−i0|≤N`,|j−j0|≤N`

have uniform bounded inverses for all i0. Letting ei = xi − xi,∑

|i−i0|≤N`

aijei = −∑

|i−i0|>N`

aijxi, |j − i0| ≤ N`.

Then the terms in the right-hand side can be bounded by

|∑

|i−i0|>N`

aijxi| ≤ C∞∑

j=N`+1

1

1 + |j|3≤ C

1

1 +N2`

.

and hence,

|ei| ≤MC1

1 +N2`

for all i. The above discussions lead to the following

Theorem 3.5 Let cTijk be the solution in (3.12) using the multiple star tech-

nique. Then cTijk converge to cTijk as the number ` of the star`(T ) increases.

3.3 Computational Results on the Earth Surface

In this subsection we use spherical splines to solve the inverse problem asdescribed in Section 4. We first wrote a FORTRAN program to solve Eq.(3.2) directly. We tested our program for the following spherical harmonicfunctions

f1(x, y, z) = sin8(θ) cos(8φ)

21

f2(x, y, z) = sin15(θ) sin(15φ)f3(x, y, z) = 789 + sin15(θ) sin(15φ)

in spherical coordinates. Clearly, F1(x, y, z) = r−9f1(x, y, z), F2(x, y, z) =r−16f2(x, y, z), and F3(x, y, z) = 789/r + r−16f2(x, y, z) are natural homo-geneous extension of f1, f2, and f3, where r2 = x2 + y2 + z2. We use thetriangulations 4n over the unit sphere as explained in the previous Sectionand spherical spline spaces S1

d(4n) and n = 0, 1, 2 and d = 3, 4, 5. Supposethat the function values of Fi at r = 1.05 with domain points of 4n aregiven. We compute the spline approximation si on the surface of the sphereby

Fi(u) =1

4π

∫

S

si(v)

|u− v|3dσ(v)

where u = 1.05(cos θ sinφ, sin θ sinφ, cosφ) for (θ, φ) as we explained in Sec-tion 4. We then evaluate si at 5760 points almost evenly distributed over thesphere and compare them with the function values of fi at these points. Themaximum errors are given in Tables 2, 3, 4 below for d = 3, 4, 5. From thesetables we can see that the numerical values from our program approximatethese standard spherical harmonic polynomials pretty well.

Table 2: Maximum errors of C1 cubic splines over various triangulationsf\∆i ∆0 ∆1 ∆2

f1 0.35138 0.06905 0.003720f2 1.36733 0.22782 0.049460f3 2.13489 0.81975 0.165639

Table 3: Maximum errors of C1 quartic splines over various triangulationsf\∆i ∆0 ∆1 ∆2

f1 3.3684e− 01 4.63305e− 02 3.72039e− 03f2 1.423358 1.2708e− 01 1.4788e− 02f3 1.49262 0.41301 0.11598

22

We are not able to compute the approximation over refined triangulations4n with n = 4 and 5 since the linear system is too big large for our computerwhen we solve (4.2) directly. Thus we have to implement the multiple starmethod described in Section 4.2. That is, we implemented (4.10) in FOR-TRAN and we can solve (4.10) for each triangle T . We need to explain ourimplementation a little bit more. To make each submatrix associated with atriangle is invertible for any triangulation, we actually used a least squarestechnique. That is, we solves

ATAx = AT b,

with rectangular matrix A. In fact we choose more domain points in eachtriangle than the domain points of degree d. Our discussion of the multiplestar method in Section 4 can be applied to this new linear system. That is,Theorem 4.5 holds for this situation.

In the following we report the numerical experiments based on the mul-tiple star technique for computing the geopotential one triangle at a time.We first present the convergence for the three test functions f1, f2, and f3.We consider 43 with 258 vertices and 512 triangles and choose 5 trianglesT65, T158, T209, T300, T400. We use C0 cubic spline functions and ring number` = 4, 5, 6. By feeding Fi(x, y, z) with r = 1.05 into the FORTRAN programwe compute spline approximation si of fi at r = 1. In Table 5 we list themaximum errors and maximal relative errors which are computed based on66 almost equally spaced points over triangle T158.

Similarly, we list the maximal absolute errors and maximal relative errorsover triangle T209 in Table 6. The maximal absolute and relative errors arecomputed based on 66 almost equally spaced points over triangle T209.

The maximal absolute and relative errors over other T65, T300, T400 havethe similar behaviors. We omit them to save space here.

Next we compute the geopotential on the Earth surface using the simu-lated in-situ geopotential measurements observed bygenerated for the grav-

Table 4: Maximum errors of C1 quintic splines over various triangulationsf\∆i ∆0 ∆1 ∆2

f1 1.5857e− 01 1.2766e− 02 9.19161e− 04f2 4.5208e− 01 2.9861e− 02 2.3973e− 03f3 1.99722 0.18698 0.10227

23

ity mission satellite, CHAMP (cf. [Reigber et al. 19982004]). In orderto check the accuracy of our numerical solution, we compare it with thesolution obtained from the traditional spherical harmonic series with de-gree 12090. We used the CHAMP data (from EGM96 model with 1m2/s2

random noises) at a fixed satellite orbit 450km above the mean equatorialradius of the Earth. Using the traditional spherical harmonic series with ra-dius Re/r = 1, we compute the geopotential at the Earth surface at (θi, φj)with θi = −89 + 2(i − 1), i = 1, · · · , 90 and φj = −180 + 2(j − 1), j =1, 2, · · · , 180 which we refer as the ”exact” solution.

We first use our FORTRAN program to compute a spline approximationbased on the given measurements from the CHAMP (in the model EGM’96with 1m2/s2 noises) and compute a spline solution at the surface of theEarth (the surface of the mean radius of the Earth) to compare with the”exact” solution. We compute the spline solution restricted to 8 trianglesT65, T156, T158, T159, T160, T209, T300, T400. We have to use the multiple starmethod in order to solve the large linear system. Consider the numericalresult from ` = 6 as our spline solution of the geopotential at the surface ofthe Earth.

There are 157 (θi, φj)’s falled in these 8 triangles and the relative errorsof spline approximation and the ”exact” solution are plotted in Fig. 5.

0 20 40 60 80 100 120 140 1600

0.02

0.04

0.06

0.08

0.1

0.12

Fig. 5. Values of Relative Errors

Let us take a closer look at triangle T158. There are 19 (θj, φj)’s falled

24

in T158. The standard deviation s of the spline approximation against the”exact” solution is s = 6.288 and maximum of the relative errors is 3.84%.The coefficients of determination R2 is 98.1%. That is, 98.1% of the totalvariation is explained by the spline model. 63.16% of the errors is withinone standard deviation of zero and 100% of the errors is within two standarddeviation of zero. These indicate the errors are normally distributed. Thecoefficient of variation is 5.46% which shows that the standard deviation isnot too large comparing to the average of the ”exact” solutions. Thus thespline method is reasonably accurate for prediction of the geopotential valuesat other locations within the triangle. Similar for the other triangles.

It should be noted that the truth solution is directly computed fromspherical harmonic coefficients (EGM96) at the Earths surface. A more faircomparision would have been generating the truth solution using a regionaldownward continuation from orbital altitude (e.g., using Poisson integrals),to compare with the spline regional solutions. The comparisons done here isfor convenience and proof of concept of the proposed alternate gravity fieldinversion numerical methodology.

4 Conclusion

In this paper we proposed to use triangular spherical splines to approximatethe geopotential on the Earth surface to assess its feasibility as an alternatemethod for regional gravity field inversion using data from satellite gravime-try measurements. A domain decomposition technique and a multiple startechnique are proposed to realize the computational schemes for approxi-mating the geopotential. In particular, our computational algorithms areparallalizable and hence enables us to model regional gravity field solutionsover the triangular regions of interest. Thus our algorithms are efficient. Thecomputational results show that triangular spherical splines for the geopoten-tial over the orbital surface at the height of a satellite is reasonable accuracy.The computational results for the geopotential at the Earth surface are ef-fective in approximation the ”exact” geopotential over some triangles. Thesecomputational algorithms can be adapted to approximate the gravity fieldusing GRACE and GOCE measurements (e.g., disturbance potential andgravity gradient measurements at orbital altitude, respectively). However,over other triangles, the approximation are relatively worse, indicating ourcomparison studies may not be fair to the spline technique and that further

25

improvement in both the theory and numerical computation is warranted.

Acknowledgement: The authors want to thank Ms. Jin Xie for her helpusing computers at the Ohio State University and Dr. Lingyun Ma for herhelp performing many numerical experiments at the University of Georgia.This study is funded by NSFs Collaboration in Mathematical Geosciences(CMG) Program.

The authors would also like to thanks two annoymous referees for theircomments and suggestions which make the paper more readable.

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Table 5: Errors of C0 cubic splines over T158

f\` ` = 4 ` = 5 ` = 6f1(maximum errors) 0.0270 0.00587 0.01018f2(maximum errors) 0.0429 0.0388 0.0367f3(maximum errors) 65.19 20.31 16.04f3(relative errors) 8.26% 2.57% 2.03%

Table 6: Errors of C0 cubic splines over T209

f\` ` = 4 ` = 5 ` = 6f1(maximum errors) 0.0892 0.0403 0.0114f2(maximum errors) 0.0594 0.0633 0.0669f3(maximum errors) 247.9 56.66 37.68f3(relative errors) 31.4% 7.18% 4.77%

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