Improved Parker's method for topographic models using Chebyshev ...

Geophysical Journal International Geophys. J. Int. (2017) 209, 1296–1325 Advance Access publication 2017 March 3 GJI Gravity, geodesy and tides

doi: 10.1093/gji/ggx093

Improved Parker’s method for topographic models using Chebyshev series and low rank approximation Leyuan Wu and Qiang Lin Center for Optics and Optoelectronics Research, Zhejiang University of Technology, Hangzhou 310000, China. E-mail: [email protected]

Accepted 2017 March 2. Received 2017 February 28; in original form 2016 August 3

SUMMARY We present a new method to improve the convergence of the well-known Parker’s formula for the modelling of gravity and magnetic fields caused by sources with complex topography. In the original Parker’s formula, two approximations are made, which may cause considerable numerical errors and instabilities: (1) the approximation of the forward and inverse continuous Fourier transforms using their discrete counterparts, the forward and inverse Fast Fourier Transform (FFT) algorithms; (2) the approximation of the exponential function with its Taylor series expansion. In a previous paper of ours, we have made an effort addressing the first problem by applying the Gauss-FFT method instead of the standard FFT algorithm. The new Gauss-FFT based method shows improved numerical efficiency and agrees well with space-domain analytical or hybrid analytical-numerical algorithms. However, even under the simplifying assumption of a calculation surface being a level plane above all topographic sources, the method may still fail or become inaccurate under certain circumstances. When the peaks of the topography approach the observation surface too closely, the number of terms of the Taylor series expansion needed to reach a suitable precision becomes large and slows the calculation. We show in this paper that this problem is caused by the second approximation mentioned above, and it is due to the convergence property of the Taylor series expansion that the algorithm becomes inaccurate for certain topographic models with large amplitudes. Based on this observation, we present a modified Parker’s method using low rank approximation of the exponential function in virtue of the Chebfun software system. In this way, the optimal rate of convergence is achieved. Some pre-computation is needed but will not cause significant computational overheads. Synthetic and real model tests show that the method now works well for almost any practical topographic model, provided that the assumption, that the entire topographic mass lies below the observation surface, is met. Key words: Numerical solutions; Fourier analysis; Numerical approximations and analysis; Gravity anomalies and Earth structure; Geopotential theory; Magnetic anomalies: modelling.

1 I N T RO D U C T I O N Parker (1973) introduced an efficient method for the calculation of gravity and magnetic anomalies, of 2-D and 3-D complexly layered models, based on the Fast Fourier Transform (FFT) algorithm, which is now named the Parker’s formula. In the ensuing decades, the method has played a significant role in the interpretation of gravity and magnetic fields caused by layer models with uneven boundaries. It has become a classical method in gravity and magnetic exploration (Nabighian et al. 2005a,b). Originally introduced as a forward algorithm, Parker’s formula can be directly applied to the evaluation of the gravity effect of a terrain relief, or a sediment layer, and the evaluation of the magnetic effect caused by a layer of magnetized material. Furthermore, Parker’s forward method also inspired several useful inversion algorithms mainly related to gravity and magnetic interface models. A rearrangement of Parker’s formula provides useful iterative procedures for the inversion of either the shape of the layer with known physical property (Oldenburg 1974; Pilkington & Crossley 1986), or the density and magnetization variation within a layer of known shape (Parker & Huestis 1974; Macdonald et al. 1983; Tontini et al. 2008). The former is known as the Parker–Oldenburg inversion method, which is maybe the most widely used algorithm in the mapping of the crust-mantle (Moho) interface based on gravity field observations (Block et al. 2009; Hsieh et al. 2010; Steffen et al. 2011; Bagherbandi 2012; Li et al. 2012; Jiang et al. 2012; Prasanna et al. 2013; van der Meijde et al. 2013; Benoit et al. 2014; Shin et al. 2015; Grigoriadis et al. 2016). The latter is also a widely applied inversion technique, which is frequently used in the determination of

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The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

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an equivalent density or magnetization distribution (Honsho et al. 2012; Nogi 2013; Sato et al. 2013; Bouligand et al. 2014; Tivey et al. 2014; Caratori Tontini et al. 2014; Fujii et al. 2015; Paulatto et al. 2015; Quartau et al. 2015). Besides the Parker–Oldenburg formula, a combination of Parker’s forward algorithm and Bott’s iterative procedure (Bott 1960) also provides a very useful, and mathematically less sophisticated, algorithm for the inversion of interface models such as the basement topography of deep sedimentary basins (Granser 1987), or the thickness of the earth crust (depth of the Moho interface) (Xia & Sprowl 1995; Zeng et al. 1997). This approach may converge more slowly than the Parker–Oldenburg method. However, according to Granser (1987), the high frequency stability is improved and maybe more details of the topography can be extracted. Some improvements of Parker’s formula and the corresponding inversion algorithms have been made by several authors. The density model treated by Parker (1973) is horizontally variable, but vertically constant. To offer a better approximation for the density-depth relationship within the topographic layer, which is particularly useful in the modelling of a sedimentary basin, Parker’s formula has been extended by many authors to several vertically variable density models including the exponential model (Feng et al. 1986; Granser 1987; Chappell & Kusznir 2008), the parabolic model (Shi et al. 2015; Zhang et al. 2015) and the general polynomial model (Chai & Jia 1990; Guspi 1992). In a previous paper of ours (Wu 2016), we have made two improvements to Parker’s forward method: (1) numerical precision of the forward and inverse Fourier transforms embedded in Parker’s formula and its extended forms are significantly improved by the Gauss-FFT method; (2) versatile geometric representations, including the mass line model, the mass prism model, the polyhedron model, and smoother topographic models interpolated from discrete data sets using high-order splines or predefined by analytical functions, now treated in a consistent framework. The Gauss-FFT method now works well for most practical models. However, as has been mentioned in Wu (2016), we had left unsettled the convergence problem of Parker’s formula for rough terrain models. In the original paper of Parker (1973), it has been rigorously proved that the Parker’s formula is uniformly and absolutely convergent as long as the observation plane lies entirely above the topographic material, the rate of convergence being closely related to the H , with H being the largest absolute value of the surface function and z being the vertical distance between the obmodel parameter z H H , the faster the series converges. However, when z < 1 and servation level and the z origin level. The smaller the absolute value of z H ≈ 1, which happens when the peaks of the topography approach the observation surface too closely, the algorithm fails or becomes z inaccurate. Some attempts have been made to solve this convergence problem. Parker (1995) presented an improved Fourier terrain correction method by dividing the gravitational attraction of the topographic model into two parts: a local contribution from the material within a cylinder around each observation point, and an exterior contribution from the matter outside the cylinder. Numerical quadrature methods are used to evaluate the local contribution and a series of convolutions, which are eventually evaluated using the FFT algorithm based on the convolution theorem in spectral analysis, are used to calculate the exterior component. This approach is further extended in Parker (1996) to deal with an uneven observation surface locating immediately above the topographic layer, which is more suited for terrain corrections of land or sea-floor gravity surveys. In this paper, we focus on the Parker formula convergence problem for topographic models whose peaks approach very closely to the observation surface. We still assume that the calculation surface is a level plane located above all sources. Although the algorithm offered in Parker (1995) provided a straightforward solution for such topographic models, it is more like a Forsberg-type algorithm rather than a Parker-type algorithm (Wu 2016), as the expansion into a power series is carried out in the space domain (Forsberg 1985) rather than in the Fourier domain (Parker 1973). Furthermore, the solution seems a bit cumbersome, with the elegance and simplicity of the original Parker’s formula abandoned. Here we present an alternative Parker-type solution for topographic models with high amplitudes. The central idea behind Parker’s formula is the separation of variables k and z˜ in expressions like e|k|˜z , with k the angular frequency and z˜ the surface function. The Taylor series expansion seems a convenient choice, but it may not be the optimal choice. In fact, once the topographic model has been rigorously defined, we are dealing with an approximation problem of a bivariate function f (|k|, z˜ ) = e|k|˜z on a rectangular domain (|k|, z˜ ) ∈ [min(|k|), max(|k|)] × [min(˜z ), max(˜z )]. Although the Taylor series expansion is uniformly and absolutely convergent, it works well only if H is considerably smaller than 1, which is most likely to occur when the z origin is chosen midway between the maximum and the value of z H approaches 1, the slow convergence of the Taylor series expansion gives rise minimum values of the surface function. When the value of z to unacceptable errors for certain parts of the k − z˜ plane even if the optimal z origin is chosen, which then leads to errors in the final forward results. Based on these observations, we present a modified Parker’s method using low rank approximation of the exponential function e|k|˜z for separation of variables |k| and z˜ . The low rank approximation is obtained by using the function svd overloaded by the Chebfun software system (Driscoll et al. 2014) (see Appendices A and B). The Chebfun overloaded svd algorithm can be understood as the continuous analogous of the discrete singular value decomposition (SVD) matrix factorization, which provides the optimal low rank approximation of a matrix subjected to the Frobenius norm. It is worth mentioning that Parker (1995, 1996) also applied Chebyshev series to obtain an economized version of the power series in order to reduce the number of convolutions in the outer calculation, but here we use the Chebyshev series in a very different way. The new method shows improved numerical stability and computational efficiency for synthetic and real topographic H ≈ 1. models with z

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2 PA R K E R ’ S F O R M U L A R E V I S I T E D Before introducing the new method, we first briefly recall the original Parker’s formula and the concept of an optimal z origin level (Parker 1973). Parker’s formula and its numerical aspects have been studied by several authors (Oldenburg 1974; Blakely 1996; Nagendra et al. 1996; Gomez-Ortiz & Agarwal 2005; Shin et al. 2006). Computer programmes in the FORTRAN language implementing Parker’s forward method are presented by Blakely (1996, p. 399), Nagendra et al. (1996) and Shin et al. (2006). A MATLAB version is also available in Gomez-Ortiz & Agarwal (2005). The mathematical notations used by these authors differ from each other. For consistency, we adopt mathematical notations from both Parker (1973) and a previous paper of ours (Wu 2016). In the following, two Cartesian systems of coordinates are established: r = (x˜ , y˜ , z˜ ) for source body coordinates and r0 = (x, y, z) for field coordinates, with positive x pointing north, positive y pointing east, and positive z pointing downward. The projection of r onto the x˜ − y˜ plane is denoted by r = (x˜ , y˜ ). √ The projection of r0 onto the x − y plane is denoted by r0 = (x, y). Wavenumbers (u, v) correspond to x and y spatial coordinates, with q = u 2 + v 2 . Angular wavenumbers and wave vector are also used to simplify expressions, with k = 2π u for a 2-D model and k = (k x , k y ) = (2π u, 2π v) for a 3-D model.

2.1 Parker’s formula with Taylor series expansion Here we provide an alternative derivation of Parker’s formula. For simplicity, we discuss the 2-D case first. We start  from the basic integral unit of 2-D sources, that is, the infinitely long wire model. The Fourier transform of the gravity potential φ(x, z)z=z along the x axis on 0 a constant level z = z0 , which is simply denoted as φ(x, z0 ) or φ(x) hereafter, caused by an infinitely long wire buried at (x˜ , z˜ ) is (Wu & Chen 2016): Wire : F[φ(x)] = Gλ

e2π z0 |u| −2π z˜ |u| −2πiu x˜ e e , |u|

(1)

where G is Newton’s gravitational constant, λ is the line density of the wire, and F here is the 1-D forward Fourier transform operator. Applying the angular wavenumber notation k = 2π u, eq. (1) can be simplified as: Wire : F[φ(x)] = 2π Gλ

e|k|z0 −|k|˜z −ik x˜ e e . |k|

(2)

Now if the 2-D topographic model extends horizontally with −X/2 ≤ x˜ ≤ X/2, and is vertically confined between two uneven top and bottom boundary functions: z˜ 1 (x˜ ) ≤ z˜ (x˜ ) ≤ z˜ 2 (x˜ ), with ρ(x˜ , z˜ ) as its density contrast function, applying the relation λ = ρd x˜ d˜z , and integrating eq. (2) between the limits of the topographic model, we have:   e|k|z0 X/2 z˜ 2 (x˜ ) ρ(x˜ , z˜ )e−|k|˜z e−ik x˜ d˜z d x˜ . (3) F[φ(x)] = 2π G |k| −X/2 z˜ 1 (x˜ ) For simplicity we first consider a density contrast function that varies only horizontally, which is also assumed in the original work of Parker (1973), then we have ρ(x˜ , z˜ ) = ρ(x˜ ), and eq. (3) becomes:  e−|k|˜z1 − e−|k|˜z2 −ik x˜ e|k|z0 X/2 e ρ(x˜ ) d x˜ . (4) F[φ(x)] = 2π G |k| −X/2 |k| The integral on the right-hand side (RHS) of eq. (4) has a form of a finite Fourier transform, but the integrand contains both the variables x˜ and k, hence cannot be evaluated directly through the FFT algorithm. A key step employed by Parker here separates the variables k and x˜ by approximating the exponential function e−|k|˜z using its truncated Taylor series expansion at z˜ = 0, we then have: e−|k|˜z1 − e−|k|˜z2 ≈

NT   (−|k|) j  j j z˜ 1 − z˜ 2 , j! j=1

(5)

where NT is the order of the Taylor series expansion, the j = 0 term is discarded because z˜ 10 − z˜ 20 = 0 always holds. Substituting eq. (5) into eq. (4), and changing the order of summation and integration we have: F[φ(x)] = 2π G

NT  (−|k|) j−1   j e|k|z0  j F ρ z˜ 2 − z˜ 1 . |k| j=1 j!

(6)

Once the Fourier transform of the gravity potential is obtained, Fourier-domain expressions of its first and second derivatives, which correspond to vector and tensor gravity components, respectively, can be easily obtained through the well-known relation:

l+n ∂ φ(x, z)  = (ik)l |k|n F[φ(x)], l + n ≤ 2, (7) F  ∂ x l ∂z n z=z0 which is also valid for higher-order partial and mixed derivatives (l + n ≥ 3). Therefore, the corresponding expression for the conventional gravity anomaly, that is, the vertical component of the anomalous gravity vector φ z (x), can be easily obtained via eq. (7) by setting l = 0,

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n = 1: F[φz (x)] = 2π Ge|k|z0

NT  (−|k|) j−1 j=1

j!

   j j F ρ z˜ 2 − z˜ 1 ,

(8)

which is equivalent to the eq. (5) in Parker (1973), except that we choose the positive z axis pointing downward, while Parker (1973) chose the positive z axis pointing upwards. The equivalent total-intensity magnetic expression is: T  (−|k|) j   j μ0 |k|z0  j m  f e F χ z˜ 2 − z˜ 1 , 2 j! j=1

N

F[T (x)] = −

(9)

where μ0 is the magnetic constant, χ is the distribution of magnetization within the topographic layer,  m = mˆ z + i mˆ x sgn k,  f = fˆ z + i fˆ x sgn k, with (mˆ x , mˆ z ) and ( fˆ x , fˆ z ) the unit vectors of the magnetization and the ambient field directions for a 2-D model, respectively (Blakely 1996, p. 283). The 3-D  case Parker’s formula for the gravity and magnetic fields can be obtained analogously and are not discussed in detail here. Let φ(x, y, z)z=z denote the gravity potential on the x − y plane at a constant height z = z0 , which is simplified as φ(r0 , z 0 ) or φ(r0 ) hereafter. 0 Applying the notation introduced before, we have the Fourier transform of the gravity potential φ(r0 ) as: F[φ(r0 )] = 2π G

NT   j−1  j

 (−|k|) e|k|z0  j F ρ(r ) z˜ 2 (r ) − z˜ 1 (r ) .  j! |k| j=1

(10)

Its first and second derivatives, which correspond to the 3-D gravity vector and tensor components, respectively, can be obtained through a relation analogous to eq. (7):

l+m+n φ(x, y, z)  ∂  n F[φ(r0 )], l + m + n ≤ 2, = (ik x )l (ik y )m |k| (11) F  z=z 0 ∂ x l ∂ y m ∂z n which is also valid for higher-order partial and mixed derivatives (l + m + n ≥ 3). Similarly, the corresponding expression for the vertical component of the anomalous gravity vector φz (r0 ), can be easily obtained via eq. (11) by setting l = 0, m = 0, n = 1: 

F[φz (r0 )] = 2π Ge|k|z0

NT   j−1  j

 (−|k|) j F ρ(r ) z˜ 2 (r ) − z˜ 1 (r ) . j! j=1

(12)

The 3-D case total-intensity magnetic field expression is: T   j  j

 (−|k|) μ0  j m  f e|k|z0 F χ (r ) z˜ 2 (r ) − z˜ 1 (r ) , 2 j! j=1

N

F[T (r0 )] = −

(13)

fˆ k + fˆ y k y mˆ k +mˆ y k y where m = mˆ z + i x x|k| ,  f = fˆ z + i x x|k| , with (mˆ x , mˆ y , mˆ z ) and ( fˆ x , fˆ y , fˆ z ) the unit vectors of the magnetization and the   ambient field directions for a 3-D model, respectively (Blakely 1996, p. 278). We note that in practical situations, only topographic models with finite horizontal extent need to be treated. The 1-D forward Fourier transform operator F on the RHS of eqs (6), (8) and (9) then reduces to the 1-D finite Fourier integral operator over the domain [−X/2, X/2]. Similarly, the 2-D forward Fourier transform operator F on the RHS of eqs (10), (12) and (13) also reduces to the 2-D finite Fourier integral operator over the domain [−X/2, X/2] × [−Y/2, Y/2]. Furthermore, we mention that eqs (9) and (13) are valid only when the magnitude of the anomalous field is much smaller than the ambient field. A more accurate definition of the total-intensity magnetic field is written as:

T = |B0 + δ B| − |B0 |,

(14)

where B0 is the ambient magnetic field vector and δ B is the anomalous magnetic field vector. However, this does not affect the validity of the algorithm presented above. Taking the 3-D anomalous magnetic field vector as δ B, let δ B = (δ Bx , δ B y , δ Bz ). Then these three anomalous components δBx , δBy , δBz can be obtained simply by setting the value of the unit vector ( fˆ x , fˆ y , fˆ z ) to (1, 0, 0), (0, 1, 0) and (0, 0, 1) in eq. (13), respectively. 2.2 The optimal z origin level Numerical algorithms directly based on eqs (8), (9), (12) and (13) may converge rather slowly even for topographic models with moderate  j relief. Parker (1973) proved that the series converges at least as rapidly as ∞ j=0 (H/z 0 ) , hence optimum convergence rate can be achieved by positioning the z origin level midway between the greatest and the smallest values of the topography so as to make the absolute value of H/z0 the smallest. The mathematical idea behind such a choice is in fact very simple. Considering the convergence property of the Taylor series expansion of a continuous function, in order to obtain a globally optimal approximation of the function e−|k|˜z1 − e−|k|˜z2 , its Taylor series should be

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expanded at z˜ = δ, where δ = mean(˜z 1 , z˜ 2 ) = e−|k|˜z1 − e−|k|˜z2 ≈ e−|k|δ

max(˜z 2 )+min(˜z 1 ) 2

instead of z˜ = 0. Thus we have:

NT 

(−|k|) j (˜z 1 − δ) j − (˜z 2 − δ) j , j! j=1

(15)

with δ now representing the mean depth of the double-layer topographic model. Substituting eq. (15) into eq. (4) and applying the relation F[φz (x)] = |k|F[φ(x)] we have: F[φz (x)] = 2π Ge−|k|(δ−z0 )

NT 

 (−|k|) j−1  F ρ (˜z 2 − δ) j − (˜z 1 − δ) j . j! j=1

(16)

Parker (1973) argued that the convergence rate of the algorithm is very sensitive to the choice of the z origin level, which is somehow misleading because the physical problem should not be affected by the choice of the coordinate system. It is in fact the different forms of the Taylor series expansion that really control the convergence of the algorithm. An optimal rate of convergence is achieved by eq. (16), irrespective of the choice of the z origin level. Parker (1973) suggested that an optimal z origin level be chosen at the mean depth of the topographic model. Under such circumstances, we have δ = 0, then the Taylor series expansion in eq. (15) reduces to the one in eq. (5), and the final expressions, that is, eqs (16) and (8), also become identical. Based on the analysis above, the algorithm can be further improved by applying the optimal Taylor series expansion to the exponential functions e−|k|˜z1 and e−|k|˜z2 separately (Xia & Sprowl 1995; Wu 2016), thus we have: 1

−|k|˜z 1

e

−|k|δ1

NT  (−|k|) j (˜z 1 − δ1 ) j , j! j=0

−|k|δ2

NT  (−|k|) j (˜z 2 − δ2 ) j , j! j=0

≈e

(17)

2

−|k|˜z 2

e

≈e

(18)

where δ 1 and δ 2 represent the mean depths of the upper and bottom topographic surfaces, respectively. Substituting eqs (17) and (18) into eq. (4), and applying the relation F[φz (x)] = |k|F[φ(x)] again, Parker’s formula can be rewritten as:  N T2 N T1  



(−|k|) j−1 (−|k|) j−1 |k|z 0 −|k|δ2 j −|k|δ1 j F ρ(˜z 2 − δ2 ) F ρ(˜z 1 − δ1 ) F[φz (x)] = 2π Ge −e e . (19) j! j! j=0 j=0 Note that now the j = 0 term is kept because normally we have δ 1 = δ 2 . In fact, as illustrated in Fig. 1, eq. (19) can be better understood as breaking the topographic model into three parts: (i) a layer confined between two horizontal levels δ 1 and δ 2 , which corresponds to the j = 0 term: Part I: F[φz (x)] = 2π Ge|k|z0

e−|k|δ1 − e−|k|δ2 F[ρ]; |k|

(20)

(ii) an uneven layer with mean depth δ 1 : 1

Part II: F[φz (x)] = −2π Ge−|k|(δ1 −z0 )

NT 

(−|k|) j−1 F ρ(˜z 1 − δ1 ) j ; j! j=1

(21)

(iii) an uneven layer with mean depth δ 2 : 2

−|k|(δ2 −z 0 )

Part III: F[φz (x)] = 2π Ge

NT 

(−|k|) j−1 F ρ(˜z 2 − δ2 ) j . j! j=1

(22)

We have in fact chosen two different z origins here. We place the z origin at the level δ 1 when evaluating Part II, and displace the z origin to the level δ 2 when evaluating Part III. It should also be observed that we now use N T1 and N T2 separately in eq. (19) to indicate the different convergence properties of the H1 H2 and z , with Taylor series expansions in eqs (17) and (18). The values of NT1 and N T2 are determined separately by the model parameters z 1 2 H1 and H2 being the largest absolute values of the surfaces z˜ 1 and z˜ 2 , respectively, and z1 = δ 1 − z0 , z2 = δ 2 − z0 being the distances from the mean depths of two topographic surfaces to the observation level, respectively. If the upper and lower surfaces of the topographic H1 H2 > z . model have similar roughness, that is, H1 ≈ H2 , then N T1 > N T2 is needed because z1 < z2 and z 1 2 A comparison of the rate of convergence of eqs (19) and (16) may be model dependent. Obviously, δ 1 < δ < δ 2 holds. Then the model H1 +(δ−δ1 ) H2 +(δ2 −δ) H1 H2 H H = z > z and z = z > z . Therefore, we definitely need N T > N T1 and N T > N T2 in parameter in eq. (16) satisfy z 1 +(δ−δ1 ) 1 2 −(δ2 −δ) 2 eq. (16) in order to obtain an accuracy which is comparable with the one provided by eq. (19). The computational efforts of the two algorithms, however, rely on NT and N T1 + N T2 + 2, respectively. Our intuition is that eq. (16) may require less computational efforts when the two surface

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Figure 1. Eq. (19) understood as breaking a double-layer topographic model into three parts.

functions are close to each other, but that increasing separation leads to slower converge with increased numerical burden. In general, eq. (19) is preferable in terms of numerical stability and efficiency; this will be illustrated with an example in Section 4.5. Analogous to eq. (19), the corresponding 2-D case magnetic expression is written as:  N T1 NT2 



(−|k|) j (−|k|) j μ0 |k|z0 −|k|δ2  j −|k|δ1 j m  f e F χ (˜z 2 − δ2 ) F χ (˜z 1 − δ1 ) −e F[T (x)] = − e . (23) 2 j! j! j=0 j=0 The 3-D case gravity and magnetic expressions are:  N T2 N T1  

  j−1   j−1 

j

j (−|k|) (−|k|)  0  2  1 |k|z −|k|δ −|k|δ F ρ(r ) z˜ 2 (r ) − δ2 F ρ(r ) z˜ 1 (r ) − δ1 −e e , F[φz (r0 )] = 2π Ge j! j! j=0 j=0

F[T (r0 )] = −

 N T2 N T1  

  j   j 

j

j (−|k|) (−|k|) μ0    m  f e|k|z0 e−|k|δ2 F χ (r ) z˜ 2 (r ) − δ2 F χ (r ) z˜ 1 (r ) − δ1 − e−|k|δ1 . 2 j! j! j=0 j=0

(24)

(25)

H considerably Numerical algorithms based on eqs (19), (23), (24) and (25) work stably and accurately for models with the parameter z H ≈ 1, which occurs when the peaks smaller than 1. However, as has been mentioned before, they fail or become inaccurate for models with z of the topographic functions approach the observation level very closely. To overcome this defect, we introduce in the following section a modified Parker’s method based on the low rank approximation of bivariate functions.

3 M O D I F I E D PA R K E R ’ S M E T H O D U S I N G L O W R A N K A P P R O X I M AT I O N We have briefly reviewed the derivation of Parker’s formula and the concept of an optimal z origin level, which we will refer to as the Parker–Taylor algorithm in the rest of the paper. The key observation is that we can use some approximant to the exponential functions e−|k|˜z1 and e−|k|˜z2 , in which the variables |k| and z˜ 1 or z˜ 2 become separated. The Taylor series expansion is a convenient choice, but it becomes inadequate for some models. The low rank approximation in the L2 -norm can provide an optimal approximation of a bivariate function (see Appendix B), which is clearly a better choice. In the following, we present a modified Parker’s method based on the low rank approximations of the bivariate functions e−|k|˜z1 and e−|k|˜z2 . The open-source Chebfun software tool, for numerical computation with Chebyshev series,

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is applied for the numerical realization of the low rank approximation (Driscoll et al. 2014). See Appendices A and B for an elementary introduction of the Chebfun tool and its extension to 2-D functions (Chebfun2). Starting from eq. (4), let h 1 (x˜ ) = z˜ 1 (x˜ ) − z 0 ,

(26)

h 2 (x˜ ) = z˜ 2 (x˜ ) − z 0

(27)

denote the vertical distances between the upper and bottom surface functions and the observation level, respectively. Then eq. (4) can be written as:    2π G X/2 F[φ(x)] = ρ(x˜ ) e−|k|h 1 − e−|k|h 2 e−ik x˜ d x˜ . (28) |k|2 −X/2 Since we are dealing with a topographic model with its observation level above all sources, we have:

max , 0 < h 1 ∈ h min 1 , h1

(29)

max 0 < h 2 ∈ h min , 2 , h2

(30)

where = min(˜z 1 ) − = max(˜z 1 ) − z 0 , and h2 . In the meantime, we have: z 0 , h max 1

h min 1

h min 2

= min(˜z 2 ) −

z 0 , h max 2

= max(˜z 2 ) − z 0 are the upper and lower bounds of h1 and

|k| ∈ [0, K ],

(31)

π where K = x is the Nyquist frequency that can be determined once the grid interval of the field points has been settled. As a result, the upper and lower bounds of all three parameters h1 , h2 and |k| can be obtained once the topographic model has been rigorously defined, regardless of the choice of the z origin of the coordinate system. Applying the low rank approximation to the functions e−|k|h 1 and e−|k|h 2 we have: 1

−|k|h 1

e

≈

NC 

σ ja ϕ aj (h 1 )ψ aj (|k|),

max , (|k|, h 1 ) ∈ [0, K ] × h min 1 , h1

(32)

σ jb ϕ bj (h 2 )ψ bj (|k|),

max , (|k|, h 2 ) ∈ [0, K ] × h min 2 , h2

(33)

j=1 2

−|k|h 2

e

≈

NC  j=1

where {σ ja } j=1,2,···,N 1 and {σ jb } j=1,2,···,N 2 are two sequences of non-increasing singular values corresponding to the low rank approximations C C in eqs (32) and (33), respectively. The sets {ψ aj (|k|)} j=1,2,···,N 1 and {ψ bj (|k|)} j=1,2,···,N 2 are orthonormal functions in L2 ([0, K]); the sets C C max max min 2 {ϕ aj (h 1 )} j=1,2,···,N 1 and {ϕ bj (h 2 )} j=1,2,···,N 2 are orthonormal functions in L 2 ([h min 1 , h 1 ]) and L ([h 2 , h 2 ]) respectively. C C The idea of a low rank function approximation using the Chebfun2 overloaded function svd is illustrated in Fig. 2. The svd of a Chebfun2 object (also called a cmatrix, which is in fact a bivariate function defined on a rectangle domain [a, b] × [c, d]) returns a quasimatrix U composed of column vectors of orthogonal chebfun objects, a diagonal matrix  with decreasing singular values, and a quasimatrix V composed of row vectors of orthogonal chebfun objects (Townsend & Trefethen 2015). The process can be carried out very quickly with little computational efforts. The functions ϕ aj (h 1 ), ψ aj (|k|) with j = 1, 2, . . . , NC1 , and ϕ bj (h 2 ), ψ bj (|k|) with j = 1, 2, . . . , NC2 are all represented by chebfun objects, they can be evaluated instantly at arbitrary values of |k|, h1 or h2 within the approximation domain through the barycentric interpolation formula (see Appendix A). Substituting eqs (32) and (33) into eq. (28), and applying the relation F[φz (x)] = |k|F[φ(x)], we obtain a new modified Parker’s formula:  N1  NC2 C      2π G  σ ja ψ aj (|k|)F ρ(x˜ )ϕ aj [h 1 (x˜ )] − σ jb ψ bj (|k|)F ρ(x˜ )ϕ bj [h 2 (x˜ )] . F[φz (x)] = (34) |k| j=1 j=1 We call this approach the Parker-LRA algorithm. The equivalent 2-D magnetic expression is  N1  NC2 C      μ0  a a a b b b m  f σ j ψ j (|k|)F χ (x˜ )ϕ j [h 1 (x˜ )] − σ j ψ j (|k|)F χ (x˜ )ϕ j [h 2 (x˜ )] . F[T (x)] = − 2 j=1 j=1

(35)

To extend the algorithm to 3-D gravity and magnetic cases, two things should be noted. First, the low rank approximation is now applied    as: to the functions e−|k|h 1 and e−|k|h 2 , with the lower and upper bounds of the variable |k|   ∈ [0, K x2 + K y2 ], (36) |k|

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Figure 2. The low rank function approximation of a bivariate function, illustrated as the continuous analogous of the SVD matrix factorization. π π where K x = x , K y = y are Nyquist frequencies along the two orthogonal directions; Second, we have mentioned that the functions ϕ aj , ψ aj , b b ϕ j , ψ j are all represented by Chenfun objects, they can be evaluated with matrices as input and output as well as vectors, hence the 3-D algorithm is a straightforward extension of the 2-D algorithm. The final expression for the 3-D gravity and magnetic cases are

2π G F[φz (r0 )] =  |k|



1

NC 

 σ ja ψ aj (|k|)F



ρ(r )ϕ aj [h 1 (r )]

j=1

μ0 m  f F[T (r0 )] = − 2



2

−

NC 

 σ jb ψ bj (|k|)F

   b ρ(r )ϕ j [h 2 (r )] ,

(37)

j=1



1

NC 

 σ ja ψ aj (|k|)F

j=1

 NC2      a b b  b χ (r )ϕ j [h 1 (r )] − σ j ψ j (|k|)F χ (r )ϕ j [h 2 (r )] .

(38)

j=1

It should be observed that although we have applied the low rank approximation separately to the functions e−|k|h 1 and e−|k|h 2 in eqs (32) and (33), treating them as two different functions in the derivation above, they are in fact, the same bivariate function f(x, y) = e−xy defined max max min on two different rectangular domains: 1 = [0, K ] × [h min 1 , h 1 ] and 2 = [0, K ] × [h 2 , h 2 ], respectively. Then apparently there is an −xy alternative solution by performing the low rank approximation to the function f(x, y) = e on a domain, say , covering both domains 1 and 2 . Since min max min max min max ⊂ h1 , h2 , (39) h1 , h1 ∪ h2 , h2 let

max ,  = [0, K ] × h min 1 , h2

(40)

we have: 1 ∪ 2 ⊂ .

(41)

Applying the low rank approximation to the function f(|k|, h) = e−|k|h on the domain : e−|k|h ≈

NC  j=1

σ j ϕ j (h)ψ j (|k|),

(|k|, h) ∈ ,

(42)

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L. Wu and Q. Lin Table 1. Model parameters of synthetic and real models. min(˜z 1 −z 0 ) x

Const

H2 H H1 z ( z 1 ; z 2 ) ∞, 43 , 25 , 47 2 2.02 ≈ 0.99 4+D 2 2 10+D ( 5 ; 5+D ), D = 2,4,16,32 4+D 2 2 4.04+D ( 2.02 ; 2.02+D ), D = 1,2,3,4

Const

0.9083, 0.9519

2,1

Geometry

ρ

TB

2-D mass prism

Const

MB

2-D mass prism

Const

DTB

2-D mass prism

Const

DMB

2-D mass prism

Helens

3-D mass prism

Model

3

z1 , z2 Fig. 3

2

Fig. 7

3

Fig. 14(a)

2

Fig. 14(b) Fig. 17

then instead of eq. (34) we have:  NC   2π G  σ j ψ j (|k|)F ρ(x˜ ) ϕ j [h 1 (x˜ )] − ϕ j [h 2 (x˜ )] , F[φz (x)] = |k| j=1

(43)

where {σ j } j=1,2,···,NC is a sequence of non-increasing singular values, differing from both sequences {σ ja } and {σ jb } in eq. (34); {ψ j (|k|)} j=1,2,···,NC max form a set of orthonormal functions in L2 ([0, K]), and{ϕ j (h)} j=1,2,···,NC form a set of orthonormal functions in L 2 ([h min 1 , h 2 ]). The analogous 2-D magnetic and 3-D gravity and magnetic expressions corresponding to eq. (43) are:  NC   μ0  m  f σ j ψ j (|k|)F χ (x˜ ) ϕ j [h 1 (x˜ )] − ϕ j [h 2 (x˜ )] , (44) F[T (x)] = − 2 j=1  NC   2π G   F[φz (r0 )] = σ j ψ j (|k|)F ρ(r ) ϕ j [h 1 (r )] − ϕ j [h 2 (r )] , (45)  |k| j=1

F[T (r0 )] = −

 NC    μ0  m  f σ j ψ j (|k|)F χ (r ) ϕ j [h 1 (r )] − ϕ j [h 2 (r )] . 2 j=1

(46)

The relationship between eqs (43) and (34) is similar to the relationship between eqs (16) and (19). To achieve comparable numerical accuracy, we need NC > NC1 and NC > NC2 because the low rank approximation is now performed on a rectangular domain  ⊃ 1 and  ⊃ 2 . However, the computational efforts of the two algorithms rely on NC and NC1 + NC2 , respectively. Consequently, a comparison of the numerical efficiency between eqs (43) and (34) may also be model dependent; this will be tested with an example in Section 4.5. 4 NUMERICAL TEST In this section, we test the numerical efficiency of the presented method using synthetic and real topographic models. Model parameters are summarized in Table 1. Space-domain analytical or hybrid analytical-numerical methods are used as precise references. The Gauss-FFT method is applied to all Fourier-domain methods to ensure the numerical accuracies of both the forward and inverse Fourier transforms. We H ≈ 1. focus mainly on topographic models with their peaks approaching the observation level very closely, that is, with the model parameter z Convergence behaviour of the Parker-LRA algorithm is tested and compared to the original Parker’s formula with Taylor series expansion. We use objective criteria similar to those used in Wu (2016) for comparison of numerical precision and computational time costs. For comparison of numerical precision, we provide statistical properties of the misfits between either the Parker–Taylor algorithm or the Parker-LRA algorithm and space-domain solutions. These statistical properties include the maximum and minimum misfits, the root mean square (rms) error, and the relative root mean square (rrms) error, which is computed as: ˆ = rrms(φ)

ˆ rms(φ − φ) , rms(φ)

(47)

where φ can be any gravity or magnetic component calculated by a space-domain solution, and φˆ refers to the same component calculated either by the Parker–Taylor algorithm or the Parker-LRA algorithm. For comparison of computational time costs, the accelerations of the Parker–Taylor algorithm and the Parker-LRA algorithm, which are defined as: tc tc τtay = , τlra = , (48) ttay tlra are used, where tc , ttay , tlra represent the time costs of the space-domain classical method, the Parker–Taylor algorithm and the ParkerLRA algorithm, respectively. Algorithm parameters and accelerations of the Parker–Taylor algorithm and the Parker-LRA algorithm for all numerical examples are summarized in Table 2. 4.1 Building appropriate synthetic models As has been discussed before, both the Parker–Taylor algorithm and the Parker-LRA algorithm make two approximations: (1) the numerical evaluations of both the forward and inverse Fourier transforms; (2) the approximation of the bivariate exponential function using the summation

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Table 2. Algorithm parameters, accelerations and errors of the Parker-LRA algorithm and the Parker–Taylor algorithm for synthetic and real models. Model TB MB DTB DMB Helens

Parker-LRA NC (NC1 ; NC2 ) 10 26 – Figs 16(a) and (b) 15

τ lra 0.67 8.5 – Fig. 16(c) 1438

Parker–Taylor N T (N T1 ; N T2 ) 13 100 Figs 15(a) and (b) – 30

M

Error

4 4 4 4 4

Figs 4–6 Figs 8, 10 and 11 Figs 15(a) and (b) Figs 16(a) and (b) Fig. 18, Table 3

τ tay 4 3.9 Fig. 15(c) – 1272

Figure 3. The bump model (TB) with different z origin levels.

of products of two univariate functions. These two approximations lead to three major error sources: (1) truncation error due to the truncation of the Fourier transform at some cut-off frequency (the Nyquist frequency); (2) quadrature error due to the numerical evaluation of the continuous Fourier integral; (3) approximation error due to the Taylor series expansion or the low rank approximation of the bivariate function. In this paper we aim to prove that the Parker-LRA algorithm has better numerical performance than the original Parker–Taylor algorithm H ≈ 1. To verify this, other in reducing the approximation error so that it can work stably and efficiently even for topographic models with z two error sources need to be suppressed. Since the Gauss-FFT method is used, quadrature error can be fully reduced by a sufficient large number of Gauss nodes M. To control the truncation error, some results from a previous paper of ours (Wu & Chen 2016) may be useful here. z 1 )−z 0 , which is the ratio of the It has been rigorously proved in Wu & Chen (2016) that an appropriate relevant model parameter min(˜x minimum source depth to the sampling interval, can be used to estimate quantitatively the upper bound of the relative truncation error. For z 1 )−z 0 ≥ 2, the relative truncation error is about 10−3 for vector gravity components and a 2-D forward problem, when the parameter min(˜x min(˜ z 1 )−z 0 ≥ 3, the upper bounds of the relative truncation errors reduce to about 10−4 and about 10−2 for tensor gravity components; for x 10−3 for vector and tensor gravity components, respectively. For a 3-D forward problem, if a square grid is used, that is, x = y, then z 1 )−z 0 ≥ 2, the relative truncation error is less than 10−3 for vector gravity components and less than 10−2 for tensor when the parameter min(˜x z 1 )−z 0 ≥ 3, the corresponding upper bounds reduce to 10−5 and 10−4 for vector and tensor gravity components, gravity components; when min(˜x respectively. See fig. 2 in Wu & Chen (2016) for more details. Moreover, it should be noted that these are just theoretical upper bounds, in a practical modelling case, the actual value of the relative truncation error may be much smaller than its theoretical upper bound. Since our major concern is to deal with topographic models whose peaks approach the observation level closely, it seems that truncation z 1 )−z 0 rather than the error may become the major error source of both Fourier-domain methods. However, it is the model parameter min(˜x H parameter z that determines the magnitude of the relative truncation error. It should also be understood that when near-source fields are to be calculated, it is natural that the model itself should be depicted in detail, that is, represented using more densely sampled height values, otherwise the ‘precise’ calculation of the corresponding field at some observation point close to the source is just meaningless.

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Figure 4. Comparison of convergence behaviours of the Parker–Taylor algorithm with different z origin levels and the Parker-LRA algorithm for the modelling of vector and tensor gravity components (a) φ x , (b) φ z , (c) φ xx or φ zz and (d) φ xz caused by the model TB in Fig. 3. z 1 )−z 0 H Based on these observations, we use topographic models with z ≈ 1 and min(˜x ≥ 1 to prove the superiority of the Parker-LRA algorithm over the Parker–Taylor algorithm in numerical accuracy and stability. Since now that both the quadrature error and the truncation error of Fourier-domain methods are suppressed, the approximation error becomes the dominant factor of the misfits between either the Parker– Taylor algorithm or the Parker-LRA algorithm and the referenced space-domain solution. We will also examine directly the approximation errors of different approximants, including the Taylor series expansion with different z origins and the low rank approximation obtained from the Chebfun2 overloaded svd function.

4.2 The bump model H ≈ 1, we first consider a simple 2-D synthetic model: the bump model, namely the model TB in Before treating topographic models with z Tables 1 and 2, which is a single-layer topographic model that has been used by several authors in studying Parker’s forward formula and the corresponding Oldenburg’s inversion formula (Oldenburg 1974; Nagendra et al. 1996; Shin et al. 2006). As shown in Fig. 3, the bump is 20 km wide at the base; it has a height of 4 km and an assumed density contrast of 1.0 g cm−3 ; the vertical distance from the minimum

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Figure 5. (a) Forward results of the vector gravity fields φ x and φ z caused by the model TB in Fig. 3 using the PS-Space method, (b) misfits between the PS-Fourier method and the PS-Space method, (c) misfits between the Parker–Taylor algorithm with NT = 13 and the PS-Space method, (d) misfits between the Parker-LRA algorithm with NC = 13 and the PS-Space method.

depth of the bump to the observation level is 3 km. The model is represented by a series of juxtaposed 2-D prisms with fixed bottom depth and varied top depth (a mass prism model); each prism has a width of 1 km. The gravity effects of this bump, including the vector and tensor components, are to be calculated at 128 equidistant points (also 1 km apart) by using both space-domain and Fourier-domain approaches. To show how the choice of the z origin affects the convergence rate of the Parker–Taylor algorithm, the z origin of the coordinate system is not provided explicitly. Four candidates are provided, namely the observation level A, the minimum depth (level B), the mean depth (level H (see Fig. 3). The C) and the maximum depth (level D) of the bump model, leading to very different values of the model parameter z implementation of the Parker-LRA algorithm has nothing to do with the choice of the z origin, here we simply test its numerical performance z 1 )−z 0 = 3 for this model, which corresponds to upper bounds for topographic models with moderate amplitudes. In the meantime, we have min(˜x of the relative truncation errors 10−4 and 10−3 for vector and tensor gravity components, respectively. The actual errors, as has been noted before, may be much smaller. To explore more thoroughly the origin of the misfits between space-domain and Fourier-domain solutions, which are composed of three different error sources mentioned above, four methods are used to calculate both the vector and tensor gravity components caused by this model: (1) a prism summation method with each prism evaluated using space-domain analytical solutions, called the PS-Space method; (2) a prism summation method with each prism evaluated using Fourier-domain Gauss-FFT solutions, called the PS-Fourier method; (3) the Parker–Taylor algorithm with different z origins evaluated by the Gauss-FFT method; (4) the Parker-LRA algorithm evaluated by the Gauss-FFT method. The PS-Space method produces the precise referenced results. If the quadrature error is fully suppressed by the Gauss-FFT method with big enough M value, the misfit between the PS-Fourier method and the PS-Space method is actually the truncation error; and

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Figure 6. (a) Forward results of the tensor gravity fields φ xx , φ xz and φ zz caused by the model TB in Fig. 3 using the PS-Space method, (b) misfits between the PS-Fourier method and the PS-Space method, (c) misfits between the Parker–Taylor algorithm with NT = 13 and the PS-Space method, (d) misfits between the Parker-LRA algorithm with NC = 8 and the PS-Space method.

the misfit between either the Parker–Taylor algorithm or the Parker-LRA algorithm and the PS-Space method is composed of the truncation error and the approximation error, with the latter introduced either by the Taylor series approximation or the low rank approximation. Figs 4(a) and (b) compare the convergence behaviours of the Parker–Taylor algorithm with four different z origin levels and the ParkerLRA algorithm for the modelling of vector gravity components φ x and φ z caused by the bump model in Fig. 3. The results confirm the H , the faster the Parker–Taylor algorithm converges. The optimal conclusions in Parker (1973) that the smaller the value of the parameter z H = 25 and is the smallest among four choice of the z origin is the mean depth of the topography (level C), which provides a value of z candidates. Convergence rate of the Parker-LRA algorithm is comparable to the Parker–Taylor algorithm with optimal z origin for this model. The Parker-LRA algorithm converges a bit slower than the Parker–Taylor algorithm at first, but it reaches the minimum error level earlier. Similar results are obtained for the tensor gravity components φ xx , φ xz and φ zz , as is shown in Figs 4(c) and (d). One obvious difference is that for tensor components the Parker-LRA algorithm converges faster than the Parker–Taylor algorithm with optimal z origin level from the very beginning (when NC , NT ≥ 3), and it also reaches the minimum error level considerably faster (NC = 8) than the Parker–Taylor algorithm (NT = 13). It should be mentioned that since φ zz = −φ xx always holds, we have rrms(φ xx ) = rrms(φ zz ), the results of these two tensor components are identical. We also mention, with reference to Fig. 4, that when the observation level (level A) is chosen as the z origin, H = ∞ on account of z = 0. The Parker–Taylor algorithm is then seen to diverge at first, but ultimately converge and reach we have z minimum error level when NT = 70. The Gauss-FFT method with M = 4 proves to be capable to fully suppress the quadrature error. We applied it to evaluate all three Fourier-domain methods. Fig. 5(a) shows the forward results of the vector gravity fields φ x and φ z using the PS-Space method, which are

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Figure 7. A modified bump model (MB).

used as the precise reference. Figs 5(b)–(d) show the misfits between the PS-Space method and the PS-Fourier method, the Parker–Taylor algorithm with NT = 13, the Parker-LRA algorithm with NC = 10, respectively. Identical error distributions are generated by the three different Fourier-domain approaches. For this model, therefore, both the Parker–Taylor and the Parker-LRA algorithms introduce negligible approximation errors, leaving the truncation error as the dominant error source. The truncation errors oscillate across the profile and exhibit maxima near x = 0 km, where the true field functions change most rapidly (Wu & Chen 2016). Forward results of the tensor gravity components φ xx , φ xz and φ zz show similar properties, as is shown in Fig. 6. Identical error H is distributions are generated by three different Fourier-domain approaches, which again demonstrates that when the model parameter z considerably smaller than 1, both the Parker–Taylor and the Parker-LRA algorithms work stably and accurately. The relative truncation error level for the tensor components is about 10−6 , which is larger than that of the vector components (about 10−8 ) on account of containing more high-frequency harmonic components. The computational work and time required to calculate this bump model are almost negligible for both space-domain and Fourier-domain methods. For 2-D models, the computational effort is not the major concern, especially when the terrain is represented by sparsely sampled height values. Still, the time costs of different approaches are provided for reference. All algorithms were designed to evaluate all five vector and tensor gravity components in a single programme, and were implemented on the same computer. The PS-Space method costs 0.04 s; the Parker–Taylor algorithm with NT = 13 costs 0.01 s; the Parker-LRA algorithm with NC = 10 costs 0.06 s. The corresponding accelerations are τ tay = 4 and τ lra = 0.67. The Parker-LRA algorithm costs the most due to the pre-computation of the low rank approximation. Nevertheless, the pre-computation effort will become negligible when 3-D models are to be considered.

4.3 A modified bump model We now consider a modified bump model, namely the model MB in Tables 1 and 2. As shown in Fig. 7, the bump is 2 km wide at the base; it has a height of 4 km and an assumed density contrast of 1.0 gm cm−3 ; the vertical distance from the minimum depth of the bump to the observation level is now reduced to only 0.02 km. The model is now depicted by much more densely sampled height values, and is represented by a series of juxtaposed 2-D prisms with fixed bottom depth and varied top depth (also a mass prism model); each prism has a width of 0.01 km and the gravity effects of this bump, including the vector and tensor components, are to be calculated at 1000 equidistant points (also 0.01 km apart) within the horizontal region [−5, 5] by using both space-domain and Fourier-domain approaches. Such a model may be encountered in a practical terrain correction problem for airborne surveys, where the flight level is only slightly above the terrain and a densely sampled digital terrain model is used to provide high-precision terrain gravity or magnetic effects. It may also be suitable for gravity or magnetic terrain corrections in marine surveys, where the peaks of the sea mountains approach the sea level very closely. Again, four different approaches mentioned above, including the PS-Space method and three Fourier-domain methods, are implemented for comparison. The Gauss-FFT method with M = 4 once more proves to be able to fully suppress the quadrature error. The optimal z origin H 2 = 2.02 ≈ 0.99 (see Fig. 7). Theoretical level for the Parker–Taylor algorithm is chosen at the mean depth of the bump, reflecting a value of z upper bounds of the relative truncation errors are 10−3 for vector gravity components and 10−2 for tensor gravity components for this model z 1 )−z 0 = 2 (Wu & Chen 2016). with the parameter min(˜x Fig. 8 compares the convergence behaviours of the Parker–Taylor algorithm with optimal z origin level and the Parker-LRA algorithm for the modelling of vector and tensor gravity components caused by the modified bump model in Fig. 7. The results confirms the conclusion H ≈ 1, the Parker–Taylor algorithm converges very slowly. This inefficiency becomes even more pronounced for in Parker (1973) that when z tensor gravity components. On the other hand, it can be observed that the Parker-LRA algorithm converges much faster than the Parker–Taylor algorithm. For a low rank approximation of rank NC = 26, the corresponding rrms error levels are below 10−8 for vector components and

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Figure 8. Comparison of convergence behaviours of the Parker–Taylor algorithm with optimum z origin level and the Parker-LRA algorithm for the modelling of vector and tensor gravity components (a) φ x , (b) φ z , (c) φ xx or φ zz and (d) φ xz caused by the model MB in Fig. 7.

below 10−5 for tensor components. For the Parker–Taylor algorithm, the corresponding rrms errors are still above 10−4 for vector components and are above 10−2 for tensor components even for a degree NT = 100 Taylor series expansion. To provide more details of the convergence behaviours of the Parker–Taylor algorithm and the Parker-LRA algorithm, suppose we write the Parker–Taylor algorithm as: T =

NT 

Ti ,

(49)

i

where T is the final sum of the series and Ti represent a single contribution. In a similar way, the Parker-LRA algorithm is written as: L=

NC 

Li ,

(50)

i

where L is the final sum of the series and Li represent a single contribution. Fig. 9 shows some single contributions Ti and Li , i = 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, based on the modelling of the φ zz component caused by the model MB. It can be observed that the magnitude of Li reduces much faster than that of Ti , with L15 at the magnitude of 10−3 Eotvos and L30 at the magnitude of 10−8 Eotvos, much smaller than the corresponding truncation error level for this model (about 0.005 Eotvos for the φ zz component). On the other hand, the magnitude of Ti reduces rather slowly, with T30 still several magnitudes above the truncation error level. Fig. 10 shows the forward results of the vector gravity fields φ x and φ z calculated by the PS-Space method and the misfits generated by the PS-Fourier method, the Parker–Taylor algorithm with NT = 100 and the Parker-LRA algorithm with NC = 26, respectively. The errors

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Figure 9. Some single contributions Ti and Li , i = 1, 2, 3, 4, 5, 10, 15, 20, 25, 30, selected from the summation series of the Parker–Taylor algorithm N N T = i T Ti and the Parker-LRA algorithm L = i C L i , respectively. Based on the modelling of the φ zz component caused by the model MB in Fig. 7, units in Eotvos.

produced by the Parker-LRA algorithm are identical to those produced by the PS-Fourier method, which proves that the approximation errors have been fully reduced by the low rank approximation. On the other hand, the Parker–Taylor algorithm exhibits errors of magnitudes much larger than the truncation error level, which are clearly originated from the approximation error of the Taylor series expansion. Forward results of the tensor gravity components φ xx , φ xz and φ zz show similar behaviour, as is shown in Fig. 11. Identical error distributions are generated by the Parker-LRA algorithm with NC = 20 and the PS-Fourier method because the approximation errors produced by the low rank approximation are smaller than the truncation error level. On the contrary, the Parker–Taylor algorithm converges rather slowly and generates errors of several tens of Eotvos near the bump peak even for a value of NT = 100, which may become unacceptable in a precise terrain correction computation. The amount of computational work calculating this modified bump model increases considerably, since the modified model is composed of 1000 prisms and there are 1000 field points to be evaluated. Once again, all algorithms are designed to evaluate all five vector and tensor gravity components in a single programme. The PS-Space method costs 2.46 s; the Parker–Taylor algorithm with NT = 100 costs 0.63 s; the

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Figure 10. (a) Forward results of the vector gravity fields φ x and φ z caused by the model MB in Fig. 7 using the PS-Space method; (b) misfits between the PS-Fourier method and the PS-Space method; (c) misfits between the Parker–Taylor algorithm with NT = 100 and the PS-Space method; (d) misfits between the Parker-LRA algorithm with NC = 26 and the PS-Space method.

Parker-LRA algorithm with NC = 26 costs 0.29 s. The corresponding accelerations are τ tay = 3.9 and τ lra = 8.5. The Parker-LRA algorithm shows better performance than the Parker–Taylor algorithm both in numerical accuracy and computational efficiency.

4.4 Comparison of approximants We have shown that two different approaches to approximate the exponential function, that is, the Taylor series expansion and the low rank approximation, lead to two algorithms with very different numerical behaviours, the Parker–Taylor algorithm and the Parker-LRA algorithm. For topographic models with moderate amplitudes, such as the model TB in Fig. 3, both algorithms work stably and accurately. However, H ≈ 1 are to be evaluated, such as the model MB in Fig. 7, the Parker-LRA algorithm is when topographic models with the parameter z numerically superior to the Parker–Taylor algorithm both in accuracy and efficiency. We now look directly into the two approximants and their approximation errors. For the Parker-LRA algorithm, the approximant is: e−|k|h ≈

NC  j=1

σ j ϕ j (h)ψ j (|k|).

(51)

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Figure 11. (a) Forward results of the tensor gravity fields φ xx , φ xz and φ zz caused by the model MB in Fig. 7 using the PS-Space method; (b) misfits between the PS-Fourier method and the PS-Space method; (c) misfits between the Parker–Taylor algorithm with NT = 100 and the PS-Space method; (d) misfits between the Parker-LRA algorithm with NC = 20 and the PS-Space method.

For the Parker–Taylor algorithm, the approximant is: e−|k|h ≈ e−|k|δh

NT  (−|k|) j j=0

j!

(h − δh ) j =

NT  1 (h − δh ) j (−|k|) j e−|k|δh , j! j=0

(52)

where h = z˜ − z 0 and δ h = mean(h) is the mean value of the function h. Note that the Taylor series can also be understood as some kind of low rank approximation by letting σ j = j!1 , ϕ j (h) = (h − δ h )j and ψ j (|k|) = (−|k|) j e−|k|δh . The difference is that {ψ j (|k|)} j=1,2,···,NC and{ϕ j (h)} j=1,2,···,NC form two sets of orthonormal functions, while the sets {(−|k|) j e−|k|δh } j=1,2,···,NT and {(h − δh ) j } j=1,2,···,NT certainly do not. The approximation domain for the model TB is tb = (|k|, h) ∈ [0, π ] × [3, 7], the approximation domain for the model MB is mb = (|k|, h) ∈ [0, 100π ] × [0.02, 4.02]. Fig. 12 compares the convergence behaviours of the approximants in eqs (51) and (52) approximating the bivariate function e−|k|h on these two rectangular domains. The low rank approximation returns a rank 13 diagonal matrix  tb for the domain tb containing 13 significant singular values; and a rank 33 diagonal matrix  mb for the domain mb containing 33 significant singular values. For a value of NC > rank, we simply mean that NC = rank. It can be observed in Fig. 12 that the low rank approximation converges faster than the Taylor series expansion in both cases, especially for the domain mb corresponding to the model MB. The Taylor series expansion also show some advantages on the approximation domain tb by reaching machine precision when NT = 30. However, it fails on the domain mb , with the rrms error about 9.3 per cent when NT = 100. The approximation errors of the low rank approximation for these two domains are 10−13 and 10−11 , respectively, which are insignificant comparing to other error sources such as the truncation error.

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Figure 12. Comparison of convergence behaviours of two different approximants in eqs (51) and (52) approximating the bivariate function e−|k|h on two rectangular domains: (a) tb = (|k|, h) ∈ [0, π ] × [3, 7] related to the model TB in Fig. 3; (b) mb = (|k|, h) ∈ [0, 100π ] × [0.02, 4.02] related to the model MB in Fig. 7.

Fig. 13 shows in detail the 2-D distributions of the approximation errors introduced by the Taylor series expansion and the low rank approximation approximating the bivariate function e−h|k| on these two domains tb and mb . For the domain b , which corresponds to the H = 25 , both approximants converge perfectly. For the domain mb , which corresponds to the model MB with a model TB with a value of z H 2 value of z = 2.02 ≈ 1, the low rank approximation still converges uniformly and globally, while the Taylor series expansion fails to do so by introducing errors of considerable magnitudes in certain parts of the |k| − h plane.

4.5 Double-layer models So far we have tested the Parker-LRA algorithm and the Parker–Taylor algorithm using two single-layer models: the model TB and the model MB. Now we turn to double-layer models. We have in fact provided two sets of expressions for double-layer models: eqs (16) and (19) for the Parker–Taylor algorithm; eqs (43) and (34) for the Parker-LRA algorithm. The computational efforts of eqs (16) and (19) rely on NT and N T1 + N T2 + 2, respectively, from now on we call them the Parker–Taylor-I algorithm and the Parker–Taylor-II algorithm, respectively. Analogously, the computational efforts of eqs (43) and (34) rely on NC and NC1 + NC2 , respectively, we call them the Parker-LRA-I algorithm and the Parker-LRA-II algorithm, respectively. In order to gain some insight into the convergence properties of algorithms based on different expressions, two synthetic double-layer models are constructed, namely the models DTB and DMB in Tables 1 and 2. As shown in Fig. 14, these two models are obtained by adding a bottom surface z˜ 2 = z˜ 1 + D, for different values of D, to the models TB and MB in Figs 3 and 7, respectively. It should be noted that although such double-layer models with constant thickness can be evaluated using simplified expressions (Parker 1973), here we treat them as general double-layer models with uncorrelated top and bottom surfaces. Fig. 15 compares convergence behaviours of two Parker–Taylor algorithms with increasing values of NT or N T1+2 = N T1 + N T2 + 2, respectively, based on the forward results of the φ z and φ zz components caused by the double-layer model DTB in Fig. 14(a) with four different values of D = 2, 4, 16, 32. The Parker–Taylor-I algorithm is implemented with NT = 1: 1: 50. For the Parker–Taylor-II algorithm, as mentioned above, we can regard the topographic model as consisting of three parts (see Fig. 1). Since the value of D does not affect the top surface z˜ 1 , we choose a fixed value of N T1 = 15 that can fully reduce the approximation error (see Fig. 4). We then let N T2 = 1 : 1 : 33, and N T1+2 = 18 : 1 : 50 in algorithm Parker–Taylor-II, and plot the resulting error levels. The results confirm our intuition: when the two surface functions are close to each other (D = 2, 4), the Parker–Taylor-I algorithm converges faster; however, as the two surface functions move far away (D = 16, 32), the Parker–Taylor-II algorithm converges more rapidly, while the Parker–Taylor-I algorithm slows down significantly. Accelerations of both algorithms are shown in Fig. 15(c), which confirms that the computational efforts of the Parker–Taylor-I algorithm and the Parker–Taylor-II algorithm rely on NT and N T1 + N T2 + 2, respectively. Fig. 16 compares convergence behaviours of two Parker-LRA algorithms with increasing values of NC or NC1+2 = NC1 + NC2 , respectively, based on the forward results of the φ z and φ zz components caused by the double-layer model DMB in Fig. 14(b) with four different values of D = 1, 2, 3, 4. The Parker-LRA-I algorithm is implemented with a series of NC values picked in the range [1, 80]. The Parker-LRA-II

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Figure 13. Approximation errors of (a) the Taylor series expansion with NT = 30 and (b) the low rank approximation with NC = 13 for the approximation of the bivariate function e−h|k| on the domain tb = (|k|, h) ∈ [0, π ] × [3, 7] related to the model TB in Fig. 3; approximation errors of (c) the Taylor series expansion with NT = 100 and (d) the low rank approximation with NC = 33 for the approximation of the bivariate function e−h|k| on the domain mb = (|k|, h) ∈ [0, 100π ] × [0.02, 4.02] related to the model MB in Fig. 7.

algorithm is implemented with a series of (NC1 , NC2 ) pairs that match the condition NC = NC1 + NC2 , with NC1 = NC2 if NC is an even number, and NC1 = NC2 + 1 if NC is an odd number. The situation of the Parker-LRA algorithms is quite different from that of the Parker–Taylor algorithms. For all four values of D = 1, 2, 3, 4, the Parker-LRA-I algorithm converges faster than the Parker–LRA-II algorithm. Note that the D values chosen here are much smaller than those used in testing the two Parker–Taylor algorithms. We made such a choice because the horizontal extent of this model is only 10 km, much smaller than that of the model DTB in Fig. 14(a) (128 km). Furthermore, if the bottom layer goes deeper, it would be unnecessary to depict it with such a densely sampled grid (x = 0.01 km). Comparison of the accelerations of both algorithms (see Fig. 16c) generally agrees with the intuition that the computational efforts of the Parker-LRA-I algorithm and the Parker–LRA-II algorithm rely on NC and NC1 + NC2 , respectively. The Parker–LRA-II algorithm runs a bit slower because it requires two low rank approximations. Although we have mentioned that the time cost of the low rank approximation is very small (about 0.1 s for this model), it does become notable for a 2-D model, especially when NC is small.

4.6 3-D model test We now test the Parker-LRA algorithm for 3-D topographic models. The spherical patch of 3 × 3 SRTM digital terrain data covering a part of the Mount St. Helens area, which has been used in Wu (2016) for testing the Gauss-FFT method, is adopted again here to test the

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Figure 14. (a) A double-layer model (DTB) obtained by adding a bottom surface z˜ 2 = z˜ 1 + D to the model TB in Fig. 3; (b) a double-layer model (DMB) obtained by adding a bottom surface z˜ 2 = z˜ 1 + D to the model MB in Fig. 7

Parker-LRA algorithm. Several modifications are made to the original model. First, the topography is now represented using SRTM1 data with 1 × 1 spatial resolution instead of the original SRTM3 data with 3 × 3 spatial resolution; second, a mass prism model is used instead of a polyhedron model; finally, the computation plane is now chosen at levels only slightly above the topography to make the value of the H closer to 1. parameter z As shown in Fig. 17, the elevation range is [1301, 2525] m, the topographic model is bounded above by the surface function −2.525 ≤ z˜ 1 ≤ −1.301 km, and below by the horizontal plane z˜ 2 = −1.301 km. The patch is approximated as a plane with grid intervals given, in kilometres, by x˜ = 1  = 0.0309,  y˜ = 1  cos ϕm = 0.0214, where ϕ m is the mean latitude of the patch and  is the arcsec-to-kilometre conversion factor for a sphere of radius, R = 6371 km. For a more thorough comparison between the Parker-LRA algorithm and the Parker– Taylor algorithm, the computation plane is chosen at two elevations z0 = 2586.8 m and z0 = 2555.9 m, which correspond to model parameters min(˜z 1 )−z 0 z 1 )−z 0 H H = 2, z = 0.9083 and min(˜x = 1, z = 0.9519, respectively. A constant density ρ = 2.67 g cm3 is assumed for the topographic x mass; the model is named Helens in Tables 1–3. The grid size is now 180 × 180, and the space-domain algorithm becomes rather slow, therefore, only two orthogonal profiles along the longitude W122◦ 11 29.5 and the latitude N46◦ 11 29.5 (A and B in Fig. 17) are evaluated using space-domain analytical solutions and used as the precise references. For the Parker-LRA algorithm and the Parker–Taylor algorithm, the whole plane is evaluated, with the vector and tensor gravity fields along these two profiles extracted for comparison. Fig. 18 compares the convergence behaviours of the Parker-LRA algorithm and the Parker–Taylor algorithm for the modelling of vector and tensor gravity components on the profile A at the elevation z0 = 2586.8 m caused by the Helens model in Fig. 17. The Parker-LRA algorithm clearly has better numerical performance. The low rank approximation for this model returns a rank 25 diagonal matrix . Thus for

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Figure 15. Comparison of the convergence behaviours of two Parker–Taylor algorithms with increasing values of NT or N T1+2 = N T1 + N T2 + 2, respectively, based on the forward results of the double-layer model DTB in Fig. 14(a) with four different values of D = 2, 4, 16, 32. (a) Error level change of the φ z component; (b) error level change of the φ zz component; (c) accelerations of both algorithms.

a value of NC > 25, we simply mean that NC = 25. When NC = 10, the rrms errors reduce to less than 10−4 for all nine gravity components, while the Parker–Taylor algorithm with NT = 20 still leaves the rrms errors of six tensor gravity components above 10−3 . The Parker-LRA algorithm converges to the minimum error level when NC = 15, while the Parker–Taylor algorithm leaves the rrms errors of 6 tensor gravity components above 10−4 when NT = 30. The accelerations of both algorithms are shown in Fig. 18(c), with τ lra = 1438 when NC = 15 and τ tay = 1272 when NT = 30. Statistics of the errors of all vector and tensor gravity components along two profiles A and B, and at the two elevations mentioned above, using the Parker-LRA algorithm (NC = 15) and the Parker–Taylor algorithm (NT = 30) are summarized in Table 3. All numerical results indicate that the Parker-LRA algorithm can provide improved numerical accuracy while retaining fast computation speed.

5 DISCUSSION 5.1 A hybrid Parker–Taylor-LRA algorithm It can be observed in Figs 12 and 13 is that although the Taylor series expansion converges very slowly for a layer with the model parameter H H ≈ 1, it converges quickly and stably for a deep layer with the model parameter z considerably smaller than 1, obtaining even better z numerical accuracy than the low rank approximation. Moreover, the Parker–Taylor algorithm is mathematically less sophisticated and more easily programmed. Therefore, a hybrid Parker–Taylor-LRA algorithm may be a better solution for multi-layer models, where the Parker-LRA algorithm is used to evaluate the top layer which approaches the observation surface closely, and the Parker–Taylor algorithm is used to evaluate other deep-buried layers. The hybrid Parker–Taylor-LRA algorithm, for a 2-D gravity forward problem due to a double-layer model,

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Figure 16. Comparison of the convergence behaviours of two Parker-LRA algorithms with increasing values of NC or NC1+2 = NC1 + NC2 , respectively, based on the forward results of the double-layer model DMB in Fig. 14(b) with four different values of D = 1, 2, 3, 4. (a) Error level change of the φ z component; (b) error level change of the φ zz component; (c) accelerations of both algorithms.

can be written as: 



 NC NT  

1  (−|k|) j−1 −|k|(δ2 −z 0 ) j F ρ(˜z 2 − δ2 ) F[φz (x)] = 2π G σ j ψ j (|k|)F ρ(x˜ )ϕ j [h 1 (x˜ )] + e . |k| j=1 j! j=0

(53)

Extension of this hybrid algorithm to multi-layer models, and to the 2-D magnetic and 3-D gravity and magnetic cases, can be understood analogously.

5.2 Better Parker-LRA algorithm for vector components By comparing Figs 4(a) and (b) with Figs 4(c) and (d), and similarly, by comparing Figs 8(a) and (b) with Figs 8(c) and (d), one can observe that the Parker-LRA algorithm converges faster for tensor gravity components than for vector gravity components. Such a phenomenon might be caused by the target bivariate function that we chose to approximate. Take the vector component φ z (x) and the tensor component φ zz (x) for example, the Parker-LRA algorithm for evaluating the φ z (x) component is based on eq. (43), and by applying the well-known relation in

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Figure 17. 1 × 1 SRTM1 digital elevation data covering a 3 × 3 patch in the Mount St. Helens area. The elevation range is [1301, 2525] m. Two computation planes are at elevations z0 = 2586.8 m and z0 = 2555.9 m. A and B are two orthogonal profiles along the longitude W122◦ 11 29.5 and the latitude N46◦ 11 29.5 , respectively.

eq. (7), the corresponding φ zz (x) expression is:  NC    σ j ψ j (|k|)F ρ(x˜ ) ϕ j [h 1 (x˜ )] − ϕ j [h 2 (x˜ )] . F[φzz (x)] = 2π G

(54)

j=1

Comparing eq. (54) with eq. (43), we suspect that the slower convergence of the Parker-LRA algorithm for vector gravity components may be 1 , which is not covered by the low rank approximation. In fact, if we apply the low rank approximation to the function caused by the factor |k| e−|k|h |k|

instead of the function e−|k|h , then instead of eqs (42) and (43) we have:

C  e−|k|h ≈ σˇ j ϕˇ j (h)ψˇ j (|k|), |k| j=1

N

(|k|, h) ∈ ,

(55)

and F[φz (x)] = 2π G

NC 

   ˇ σˇ j ψ j (|k|)F ρ(x˜ ) ϕˇ j [h 1 (x˜ )] − ϕˇ j [h 2 (x˜ )] ,

(56)

j=1

where {σˇ j } j=1,2,···,NC , {ψˇ j (|k|)} j=1,2,···,NC and {ϕˇ j (h)} j=1,2,···,NC are new sets of singular values and orthonormal functions corresponding to −|k|h the low rank approximation of the function e |k| . We mention that the low rank approximation in eq. (55) has a singularity when k = 0. This would not be a problem for the Gauss-FFT method, because this singular point is not picked by its shifted spectra (Wu 2016). Thus, we can simply change the approximation domain  in eq. (40) to

max ˇ = [|k|min , K ] × h min , (57)  1 , h2 where |k|min is the minimum value of |k| picked by the Gauss-FFT method, which can be determined by the number of Gauss nodes M and the grid interval. Fig. 19 compares the convergence behaviours of three algorithms: the Parker–Taylor algorithm with optimum z origin level; the ParkerLRA algorithm with the low rank approximation applied to the function e−|k|h ; and the Parker-LRA algorithm with the low rank approximation −|k|h applied to the function e |k| based on the modelling of vector and tensor gravity components caused by the model MB in Fig. 7. Note that the

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L. Wu and Q. Lin Table 3. Statistics of the errors forward modelling the 3-D model Helens at two elevations z0 = 2586.8 m and z0 = 2555.9 m along two profiles A and B using the Parker-LRA algorithm and the Parker–Taylor algorithm. AU indicates the amplitude unit. Model

Gravity

Parker–Taylor algorithm

Parker-LRA algorithm

AU

Min

Max

rms

rrms

Min

Max

rms

rrms

Helens z0 = 2586.8 m min(˜z 1 )−z 0 =2 x H = 0.9083 z Profile A

φx φy φz φ xx φ yy φ zz φ xy φ xz φ yz

−1.5e-03 −4.9e-04 −1.2e-03 −3.6e-01 −2.2e-01 −8.6e-01 −1.8e-01 −5.8e-01 −2.3e-01

1.3e-03 6.2e-04 1.6e-03 4.0e-01 5.6e-01 5.8e-01 2.5e-01 3.4e-01 3.0e-01

8.3e-04 7.3e-05 5.1e-04 5.6e-02 6.1e-02 9.8e-02 3.0e-02 6.7e-02 3.4e-02

3.4e-05 3.9e-05 1.1e-05 3.2e-04 3.5e-04 3.1e-04 1.1e-03 3.1e-04 7.1e-04

−1.5e-03 −3.0e-06 4.9e-04 4.6e-03 8.9e-05 −5.6e-03 −1.5e-04 −3.4e-04 −2.6e-04

1.3e-03 −2.2e-06 4.9e-04 5.4e-03 5.0e-04 −5.0e-03 1.8e-04 3.2e-04 2.5e-04

8.1e-04 2.6e-06 4.9e-04 5.0e-03 3.2e-04 5.3e-03 4.3e-05 5.0e-05 3.8e-05

3.4e-05 1.4e-06 1.0e-05 2.9e-05 1.9e-06 1.7e-05 1.6e-06 2.3e-07 8.1e-07

mGal mGal mGal Eotvos Eotvos Eotvos Eotvos Eotvos Eotvos

Helens z0 = 2586.8 m min(˜z 1 )−z 0 =2 x H z = 0.9083 Profile B

φx φy φz φ xx φ yy φ zz φ xy φ xz φ yz

−3.6e-03 −4.0e-03 −7.7e-03 −8.0e-01 −1.4e+00 −4.2e+00 −1.6e+00 −1.8e+00 −2.1e+00

6.1e-03 4.6e-03 4.9e-03 2.5e+00 1.8e+00 2.1e+00 1.7e+00 3.3e+00 2.4e+00

8.8e-04 7.5e-04 1.2e-03 3.3e-01 2.6e-01 5.6e-01 2.7e-01 4.5e-01 3.6e-01

1.5e-04 3.0e-05 2.0e-05 1.1e-03 1.2e-03 1.1e-03 3.3e-03 2.8e-03 1.3e-03

−1.4e-04 −5.3e-05 4.7e-04 −1.7e-03 −1.3e-03 −2.7e-02 −2.9e-03 −1.4e-02 −5.7e-03

−1.3e-04 5.0e-05 4.9e-04 2.4e-02 3.0e-03 2.8e-03 1.7e-03 3.5e-03 6.6e-03

1.4e-04 3.1e-05 4.8e-04 5.5e-03 4.3e-04 5.9e-03 3.6e-04 1.9e-03 8.1e-04

2.3e-05 1.2e-06 8.2e-06 1.9e-05 2.0e-06 1.2e-05 4.4e-06 1.2e-05 3.0e-06

mGal mGal mGal Eotvos Eotvos Eotvos Eotvos Eotvos Eotvos

Helens z0 = 2555.9 m min(˜z 1 )−z 0 =1 x H = 0.9519 z Profile A

φx φy φz φ xx φ yy φ zz φ xy φ xz φ yz

−6.1e-03 −2.3e-03 −8.3e-03 −2.1e+00 −1.2e+00 −5.0e+00 −8.6e-01 −3.3e+00 −1.3e+00

3.3e-03 2.9e-03 6.4e-03 2.2e+00 3.1e+00 3.3e+00 1.3e+00 2.0e+00 1.5e+00

1.0e-03 3.2e-04 1.1e-03 2.7e-01 3.3e-01 5.3e-01 1.4e-01 3.5e-01 1.6e-01

4.2e-05 1.6e-04 2.1e-05 1.5e-03 1.9e-03 1.6e-03 4.3e-03 1.5e-03 3.1e-03

−1.5e-03 −7.9e-06 4.5e-04 −1.2e-02 −7.5e-03 −2.9e-02 −7.3e-03 −1.3e-02 −5.3e-03

1.3e-03 2.8e-06 4.8e-04 2.2e-02 7.7e-03 1.7e-02 5.2e-03 2.0e-02 4.9e-03

8.1e-04 2.7e-06 4.7e-04 5.4e-03 1.1e-03 5.9e-03 8.5e-04 2.3e-03 7.6e-04

3.3e-05 1.4e-06 9.5e-06 2.9e-05 6.4e-06 1.8e-05 2.7e-05 1.0e-05 1.4e-05

mGal mGal mGal Eotvos Eotvos Eotvos Eotvos Eotvos Eotvos

Helens z0 = 2555.9 m min(˜z 1 )−z 0 =1 x H z = 0.9519 Profile B

φx φy φz φ xx φ yy φ zz φ xy φ xz φ yz

−1.8e-02 −2.2e-02 −4.4e-02 −4.7e+00 −7.1e+00 −2.6e+01 −8.7e+00 −9.5e+00 −1.3e+01

3.4e-02 2.5e-02 2.1e-02 1.6e+01 1.1e+01 1.1e+01 9.8e+00 1.9e+01 1.4e+01

4.4e-03 3.6e-03 5.6e-03 1.9e+00 1.4e+00 3.2e+00 1.5e+00 2.4e+00 2.0e+00

7.2e-04 1.4e-04 9.3e-05 6.2e-03 6.1e-03 5.9e-03 1.5e-02 1.3e-02 6.9e-03

−8.4e-04 −2.6e-04 −5.5e-04 −3.5e-01 −1.8e-01 −1.4e+00 −2.1e-01 −1.0e+00 −3.3e-01

5.0e-05 2.7e-04 8.0e-04 1.2e+00 2.6e-01 4.2e-01 1.6e-01 2.7e-01 3.6e-01

1.7e-04 4.6e-05 4.8e-04 1.4e-01 2.9e-02 1.6e-01 2.6e-02 1.1e-01 4.4e-02

2.8e-05 1.8e-06 7.8e-06 4.3e-04 1.3e-04 2.9e-04 2.7e-04 6.0e-04 1.5e-04

mGal mGal mGal Eotvos Eotvos Eotvos Eotvos Eotvos Eotvos

blue and green lines are identical to those in Fig. 8, except that now only the parts NC , NT ∈ [1, 50] are shown instead of NC , NT ∈ [1, 100]. The results confirms our intuition: optimal convergence rate is obtained for vector gravity components when the low rank approximation is −|k|h applied to the function e |k| , and for tensor gravity components when the low rank approximation is applied to the function e−|k|h . Therefore, this procedure works more efficiently if we simply need the vector or the tensor gravity components; it will cause more computational work if we need both the vector and the tensor components. Besides, some 3-D models are also implemented to test these two different approaches. In our experience, although the convergence behaviours of the 3-D algorithms are similar to the 2-D ones, the 3-D case Parker-LRA algorithm based on the low rank approximation of −|k|h the function e |k| seems to be more time consuming. The reason for this is still unknown, it might be related to the property of the function e−|k|h |k|

. Therefore, in general we recommend the Parker-LRA algorithm based on the low rank approximation of the function e−|k|h .

6 C O N C LU S I O N S We have tested the numerical performance of a new algorithm, the Parker-LRA algorithm, against the original Parker–Taylor algorithm for the H considerably smaller modelling of gravity effects caused by 2-D and 3-D topographic models. For topographic models with the parameter z than 1, both algorithms work accurately and efficiently; however, when topographic models with their peaks approaching the observation H ≈ 1, the Parker-LRA algorithm presented in this paper converges much plane too closely are to be evaluated, that is, when the parameter z faster than the Parker–Taylor algorithm and shows superior performance both in numerical accuracy and computational efficiency. For double-layer models, both the Parker–Taylor algorithm and the Parker-LRA algorithm have two different approaches, depending on whether the two exponential functions are approximated simultaneously or separately using either the Taylor series expansion or the low

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Figure 18. Comparison of the convergence behaviours of (a) the Parker-LRA algorithm and (b) the Parker–Taylor algorithm for the modelling of vector and tensor gravity components on the profile A at the elevation z0 = 2586.8 m caused by the model Helens in Fig. 17; (c) accelerations of both algorithms.

rank approximation. Our tests show that in general, the Parker–Taylor-II algorithm works more stable than the Parker–Taylor-I algorithm, especially when the two surface functions are far away from each other; and the Parker-LRA-I algorithm with one single low rank approximation converges faster than the Parker-LRA-II algorithm, the former also has simpler expressions and is more easily programmed than the latter. There are two areas not addressed in this paper, but will be studied in our future works: (1) only topographic models with vertically constant density are treated in this paper, extensions of the new method to vertically variable models such as the exponential and the general polynomial models, are also possible; (2) all synthetic and real models are represented by the mass prism model in this paper, extensions of the new method to more geometric representations, including the polyhedron model and smoother topographic models, are straightforward and might be useful in studying the relationship between the modelling error of a near-source field and the different geometric representations of the source. A more important extension of the Parker-LRA algorithm presented may be the modification of Forsberg-Type algorithms, as the low rank approximation can be applied not only to the Fourier-domain kernel function e|k|˜z , but also to the space-domain kernel function | r −1r | . 0 Therefore, although the method presented here works only when the simplification that the calculation surface is a level plane above all sources is met, it probably can be generalized to the calculation on an arbitrary uneven surface; this will also be studied in future works. Moreover, the method can also be applied to other problems like that of upward continuation to an uneven surface, or the iterative reduction of the potential fields on an uneven surface to a plane. Finally, it is worth mentioning that the new expressions introduced in this paper may inspire some new inversion algorithms, which might be useful in the inversion of either the shape of the layer with known physical property, or the density and magnetization variation within a layer of known shape.

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Figure 19. Comparison of convergence behaviours of three algorithms: the Parker–Taylor algorithm with optimum z origin level; the Parker-LRA algorithm −|k|h with the low rank approximation applied to the function e−|k|h ; the Parker-LRA algorithm with the low rank approximation applied to the function e |k| ; based on the modelling of vector and tensor gravity components (a) φ x , (b) φ z , (c) φ xx or φ zz (d) φ xz caused by the model MB in Fig. 7.

AC K N OW L E D G E M E N T S The authors are very grateful to reviewer Horst Holstein for his helpful comments and valuable suggestions to improve the manuscript significantly. This study was funded by the National Natural Science Foundation of China under Grant No. 41504089 and the National Basic Research Program of China (973 Program) under Grant No. 2013CB329501. REFERENCES Bagherbandi, M., 2012. A comparison of three gravity inversion methods for crustal thickness modelling in Tibet plateau, J. Asian Earth Sci., 43(1), 89–97. Benoit, M.H., Ebinger, C. & Crampton, M., 2014. Orogenic bending around a rigid Proterozoic magmatic rift beneath the Central Appalachian Mountains, Earth planet. Sci. Lett., 402, 197–208. Berrut, J.P. & Trefethen, L.N., 2004. Barycentric Lagrange interpolation, SIAM Rev., 46(3), 501–517. Blakely, R.J., 1996. Potential Theory in Gravity and Magnetic Applications, Cambridge Univ. Press.

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L. Wu and Q. Lin

APPENDIX A: CHEBFUN Here we provide a brief introduction to Chebfun. Much of the relevant material can be found collected in the Chebfun-based book entitled Approximation Theory and Approximation Practice (Trefethen 2013). Chebfun is an open-source software system written in MATLAB for numerical computation with functions. A chebfun is a function of one variable defined on an interval [a, b]. The syntax for chebfuns is almost exactly the same as the usual MATLAB syntax for vectors, with the familiar MATLAB commands for vectors overloaded in natural ways. Thus, for example, whereas sum(f) returns the sum of the entries when f is a vector, it returns a definite integral when f is a chebfun (Driscoll et al. 2014). The aim of Chebfun is to ‘feel symbolic but run at the speed of numerics’. More precisely, the vision of Chebfun is to achieve for functions what floating-point arithmetic achieves for numbers: rapid computation in which each successive operation is carried out exactly apart from a rounding error that is very small in relative terms (Trefethen 2007). The implementation of Chebfun is based on the mathematical fact that smooth functions can be represented very efficiently by polynomial interpolation in Chebyshev points (Chebyshev points of the second kind): x j = cos( jπ/n),

0 ≤ j ≤ n,

(A1)

or equivalently, by expansions in Chebyshev polynomials: f (x) =

∞ 

ak Tk (x),

(A2)

k=0

where ak is the kth Chebyshev expansion coefficient and Tk is the kth Chebyshev polynomial. There is a one-to-one linear mapping between values at Chebyshev points and Chebyshev expansion coefficients, and this mapping can be applied in O(nlog n) operations with the aid of the fast Fourier transform (FFT) or the fast cosine transform (Trefethen 2013). For a simple function, interpolation in 20 or 30 Chebyshev points often suffice, but the process is stable and effective even for functions complicated enough to require 1000 or 1 000 000 points. Chebfun makes use of adaptive procedures that aim to find the right number of points automatically so as to represent each function to roughly machine precision, that is, about 15 digits of relative accuracy (Driscoll et al. 2014). A chebfun object is represented by its value at Chebyshev points or by the equivalent Chebyshev expansion coefficients. The curve between the data points is the polynomial interpolant, which can be evaluated by the barycentric interpolation formula (Salzer 1972; Berrut & Trefethen 2004; Trefethen 2013). This method of evaluating polynomial interpolants is stable and efficient even if the degree is in the millions (Higham 2004). The barycentric formula for polynomial interpolation in an arbitrary set of n + 1 points {xj } is written out as (Trefethen 2013, p. 36): Theorem A.1 (Barycentric interpolation formula). The polynomial interpolant through data {fj } at n + 1 points {xj } is given by  n n   λj λj f j , p(x) = x − xj x − xj j=0 j=0

(A3)

with the special case p(x) = fj if x = xj for some j, where the weights {λj } are defined by λj = 

1 k= j (x j

− xk )

.

(A4)

For Chebyshev points, the weights {λj } are wonderfully simple: they are equal to (−1)j times the constant 2n − 1 /n, or half this value for j = 0 and n, thus the formula above reduces to (Trefethen 2013, p. 36): Theorem A.2 (Barycentric interpolation in Chebyshev points). The polynomial interpolant through data {fj } at the Chebyshev points (eq. (A1)) is  n n   (−1) j (−1) j f j , (A5) p(x) = x − xj x − xj j=0 j=0 with the special case p(x) = fj if x = xj . The primes on the summation sighs signify that the terms j = 0 and j = n are multiplied by 1/2. The formula A5 is used to evaluate a chebfun object f at an arbitrary x within the approximation domain [a, b] simply by the command f(x). The function f can also be evaluated with a vector or matrix as input parameter, the output being the function evaluation of every element of the input data structure. It is extraordinarily effective, even if n is in the thousands or millions, and even if f must be evaluated at thousands or millions of points (Trefethen 2013, p. 37).

A P P E N D I X B : C H E B F U N 2 A N D L O W R A N K F U N C T I O N A P P R O X I M AT I O N Chebfun2 is the part of Chebfun that deals with functions of two variables defined on a rectangle [a, b] × [c, d]. Chebfun2 exploits the observation that many functions of two variables can be well approximated by low rank approximants. A rank 1 function, also known as separable, is of the form u(y)v(x), and a rank k function is one that can be written as the sum of k rank 1 functions. Smooth functions tend

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to be well approximated by functions of low rank. Chebfun2 determines low rank function approximations automatically by means of an algorithm that can be viewed as an iterative application of Gaussian elimination with complete pivoting (Townsend & Trefethen 2013).

B1 Low rank matrix approximation using SVD Before introducing the low rank function approximation using Chebfun2 objects, a brief review of the low rank matrix approximation using SVD would be helpful. Most of the contents provided here can be found in the book Manning et al. (2008). Given an M × N matrix C and a positive integer k, we wish to find an M × N matrix C k of rank at most k, so as to minimize the Frobenius norm of the matrix difference X = C − C k , defined to be   M N   2 Xi j . (B1) |X| F =  i=1 j=1

Thus, the Frobenius norm of X measures the discrepancy between C k and C; our goal is to find a matrix C k that minimizes this discrepancy, while constraining C k to have rank at most k. If r is the rank of C, clearly C r = C and the Frobenius norm of the discrepancy is zero in this case. When k is far smaller than r, we refer to C k as a low-rank approximation. The singular value decomposition (SVD) can be used to solve the low-rank matrix approximation problem using a three-step procedure: 1. Given C, construct its SVD in the form as: C = UV T ,

(B2)

where U, V are orthogonal matrices, and  is a diagonal matrix with entries σ 1 , σ 2 , . . . , σ r stored in a descending order σ 1 ≥ σ 2 ≥ ··· ≥ σ r > 0. 2. Derive from  the matrix  k formed by replacing by zeros the r − k smallest singular values on the diagonal of . 3. Compute and output C k = U k V T as the rank-k approximation to C. The rank of Ck is at most k: this follows from the fact that  k has at most k non-zero values. Since the effect of small eigenvalues on matrix products is small, it seems plausible that replacing these small eigenvalues by zero will not substantially alter the product, leaving it ‘close’ to C. In fact, it can be proved that this procedure yields the matrix of rank k with the lowest possible Frobenius error. ⎛ ⎞ 12 r  σ j2 ⎠ . (B3) min |C − Z| F = |C − C k | F = ⎝ Z |rank( z )=k j=k+1

B2 Low rank function approximation using Chebfun2 Low rank function approximation can be viewed as the continuous analogous of the low rank matrix approximation. Given a continuous bivariate function f (x, y) : [−1, 1]2 → R, the optimal rank k approximation in the L2 -norm is written as: f (x, y) =

∞ 

σ j ϕ j (y)ψ j (x),

(B4)

j=1

where σ 1 , σ 2 , . . . , is a non-increasing real sequence of singular values, and the sets {ψ 1 (x), ψ 2 (x), . . . , } and {ϕ 1 (y), ϕ 2 (y), . . . , } are orthonormal functions in L2 ([−1, 1]). Each term in eq. (B4) is an ‘outer product’ of two univariate functions, called a rank 1 function. The optimal rank k approximation to f in the L2 -norm can be found by truncating eq. (B4) after k terms, ⎛ ⎞ 12 k ∞   f (x, y) ≈ f k (x, y) = σ j ϕ j (y)ψ j (x), || f − f k || L 2 ([−1,1]2 ) = ⎝ σ j2 ⎠ . (B5) j=1

j=k+1

The convergence rates for the singular values have a similar flavour to univariate approximation theory: the smoother the function f, the faster the sequence σ 1 , σ 2 , . . . , decays. As illustrated in Fig. 2, the Chebfun2 overloaded function svd is used to implement the low rank function approximation. Numerically, a rank k approximation to f can be computed by sampling it on an n × n Chebyshev tensor grid, taking the matrix of sampled values, and computing its matrix singular value decomposition. The first k singular values and left and right singular vectors form the optimal rank k approximation to the sampled matrix in the discrete 2-norm. The singular vectors are then represented by chebfun objects, which can be evaluated very quickly at arbitrary values within the approximation domain through the barycentric interpolation formula (See Appendix A). More materials about Chebfun2 can be found in (Townsend & Trefethen 2013, 2015).