On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere

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1 DOI.7/s On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere Chunfeng Huang Haimeng Zhang Scott M. Robeson Received: 9 June 9 / Accepted: May International Association for Mathematical Geosciences Abstract Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential models, as well as power variograms with <α, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and power variograms with <α are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on the sphere is provided. Keywords Power variogram Spherical covariance Stable model Variogram models Introduction Global-scale processes and phenomena are of utmost importance in the geosciences. Data from global networks of in situ and satellite sensors are used to monitor a wide array of geophysical processes. Most methods and models in spatial statistics, however, are developed in Euclidean spaces R n and these methods have not been investigated as intensively in spherical coordinate systems Robeson 997). Among the most C. Huang ) Department of Statistics, Indiana University, Bloomington, IN 4748, USA huang48@indiana.edu H. Zhang Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 3976, USA S.M. Robeson Departments of Geography and Statistics, Indiana University, Bloomington, IN 4745, USA

2 commonly used covariance and variogram functions in the geosciences are the exponential, Gaussian, spherical, hole-effect, and power models Kitandis 997; Isaaks and Srivastava 989; Webster and Oliver ). Haylock et al. 8), for instance, compared these five models when developing a high-resolution data set of daily precipitation over Europe. Other geoscientists have used the exponential variogram model on the sphere to develop quantitative estimates of atmospheric carbon dioxide concentrations, global fire distributions, and oceanic mixed layer depths Alkhaled et al. 8; Carmona-Moreno et al. 5; de Boyer Montégut et al. 4). The Gaussian model on the sphere has been used to reconstruct paleoceanographic temperature Schäfer-Neth et al. 5) while Janis and Robeson 4) used a power model with spherical distances to analyze errors in minimum temperature data. Clearly, a wide range of geostatistical models are being used with databases that are coded in spherical coordinates. Before adapting Euclidean covariance and variogram models to the sphere, one must first evaluate their properties to ensure their validity. A random process is stationary on the sphere if its covariance function depends solely on the spherical angle. To be valid, the covariance function must be positive definite on the sphere Schoenberg 94; Yaglom 987). In practice, many applications also focus on intrinsically stationary processes and their variograms. While such processes have been well studied in Euclidean space Cressie 993; Chilès and Delfiner 999), the extension to the sphere needs more investigation. In Sect., we study the variogram of the intrinsically stationary process on the sphere, as well as the covariance function of the stationary process. Brownian motion on the sphere is presented as an example to demonstrate that an intrinsically stationary process may not be stationary. This answers some questions raised in Reguzzoni et al. 5). In Sect. 3, several commonly used covariance and variogram models are checked for their validity on the sphere. In particular, we show that the spherical and exponential covariance models and power variograms with <α are valid on the sphere. However, the Gaussian model, two Radon transforms of exponential model, Cauchy model, the hole-effect model and power variograms with <α are not valid on the sphere. A summary of our results can be found in Table. A discussion of strictly positive conditional negative) definiteness of the covariance variogram) functions is presented in Sect. 4. Covariance and Variogram Functions on the Sphere Consider a random process Xp), p S, where S is a unit sphere in R 3, and the location p = φ, λ) with latitude φ and longitude λ, φ π, λ<π. The process is stationary if the covariance function covxp ), Xp )) depends solely on the spherical angle θp,p ) [,π], where cos θp,p ) ) = sin φ sin φ + cos φ cos φ cosλ λ ). A real continuous function C ) is said to be a valid covariance function on the sphere if it is positive definite, namely, m m C θp i,p j ) ) a i a j i= j=

3 for any finite number of spatial locations {p i,i=,...,m} on S and real numbers {a i,i=,...,m}. The following theorem by Schoenberg 94) provides the characterization of a valid covariance function on the sphere. Theorem A real continuous function Cθ) is a valid covariance function on the sphere if and only if it is of the form Cθ) = b n P n cos θ), θ [,π], ) n= where b n, n= b n <, and P n ) are the Legendre polynomials. By orthogonality of the Legendre polynomials 7.. in Gradshteyn and Ryzhik 994), the coefficients can be obtained by b n = n + π Cθ)P n cos θ)sin θdθ, n=,,,... ) Therefore, we can investigate the positive definiteness of a function C ) by checking whether b n is non-negative, and n b n <. In Sect. 3, the validity of commonly used covariance functions are studied. Parallel to the intrinsically stationary process on R n Cressie 993), we have the following. Definition Suppose {Xp) : p S } satisfies EXp)) = μ, for all p S, and varxp ) Xp )) = γθp,p )), for all p,p S ; then X ) is said to be intrinsically stationary. The quantity γ ) is called the variogram, and γ ) is the semivariogram. Proposition The variogram γ ) is conditionally negative definite, namely m i= j= m a i a j γ θp i,p j ) ) for any finite number of spatial locations {p i,i=,...,m} on S and real numbers {a i,i=,...,m} satisfying m i= a i =. To investigate whether a function is conditionally negative definite on the sphere, we have the following theorem. Theorem A continuous function γ ) satisfying γ) = is conditionally negative definite if and only if γ ) is of the form γθ)= b n Pn cos θ) ), b n, n= and P n ) are the Legendre polynomials. b n < n=

4 Remark A general result can be found in Bochner 94), Schoenberg 94, note ), Karpushev 985) and Sasvari 994). The proof of this theorem can be obtained by directly applying Theorem 3 of Bochner 94). Remark Given a valid covariance function Cθ), one can always find a valid semivariogram γθ)= C) Cθ). The converse is not true in Euclidean space R n.for example, the power variograms do not have counterpart covariance functions Cressie 993). However, when the space is the sphere S, it is noted by Yaglom 96) that the space of valid variograms coincides with the space of valid covariances. That is, for each valid variogram γθ), θ [,π], one can construct a covariance function Cθ) = c γθ),forsomec π γθ)sin θdθ. Remark 3 If a function C h) is a valid covariance function in R 3, then the function Cθ) = C sinθ/)) is a valid covariance function on the unit sphere S.In general, any function of the form Cθ) = sinωsinθ/))) ω sinθ/) dφω), where Φ ) is a bounded non-decreasing function, is valid on a unit sphere S see Yadrenko 983, p. 76; or Yaglom 987, p. 389). Therefore, one can obtain a rich family of valid covariance functions on the sphere, see also Jun and Stein 7) and Banerjee 5). Remark 4 When C θ) and C θ) are both valid on the sphere, so are C θ)c θ) and a C θ) + a C θ), a,a. For example, since both exponential and power models see Table ) are valid, their product e θ π θ) is also valid. Remark 5 In practice, one may use weighted least squares to estimate the parameters Cressie 985) for valid covariance and variogram models on the sphere. It is clear that the stationary process is intrinsically stationary. However, the converse is not true even though the spaces of valid covariance functions and valid variogram functions coincide. For example, consider Brownian motion Xp) on S Gangolli 967) such that cov Xp ), Xp ) ) = θp,p ) + θp,p ) θp,p ) ), p,p S, where p is a fixed arbitrary point for example, north pole). It is clear that this process is not stationary; however, the variance of Xp ) Xp ), var Xp ) Xp ) ) = θp,p ), depends solely on θp,p ). That is, the process Xp) is intrinsically stationary, but not stationary. Note that this process has the power variogram γθ)= θ α with α =, which is shown to be valid on the sphere in Sect Validity of Covariance/Variogram Functions on the Sphere In this section, we check the validity of some commonly used covariance and variogram functions. A summary of all results is presented in Table.

5 Table Validity positive definiteness) of covariance functions on the sphere, a>, θ [,π] Model Covariance function Validity Spherical a 3θ θ3 a 3 ) θ a) Yes Stable exp{ a θ )α } Yes for α, ] No for α, ] Exponential exp{ a θ )} Yes Gaussian exp{ a θ ) } No Power a c θ/a) α Yes for α, ] No for α, ] Radon transform of order e θ/a + θ/a) No Radon transform of order 4 e θ/a + θ/a + θ /3a ) No Cauchy + θ /a ) No Hole-effect sin aθ/θ No a When α, ], power model is valid on the sphere for some c π θ/a) α sin θdθ 3. Spherical Model A spherical covariance function has the form { 3θ a Cθ) = θ 3, a 3 if θ a;, if θ>a, where a> is a scale parameter. This model is valid in R, R and R 3 Cressie 993). To check whether it is valid on the sphere, we compute the coefficients b n in ) through the following sine expansion of the Legendre polynomials Hobson 93, p. ; also in Gradshteyn and Ryzhik 994): π 4 P ncos θ)= 4 n { sinn + )θ n + ) n + n + 3) } sinn + 3)θ +. 3) More compactly, P n cos θ)= k= D nk sinn + k + )θ, for <θ<π, where D nk = 4 4 n 3 k ) n + ) n + k) >. 4) π 3 5 n + ) k n + 3) n + k + ) Then, b n = n + π Cθ) D nk sinn + k + )θ sin θdθ= n + D nk b nk, k= where, for integer k, b nk = π Cθ)sinn + k + )θ sin θdθ= gn + k) gn + k + )). Here, we define, for x>, gx)= a k= 3θ a + θ 3 a 3 ) cos xθ dθ. To

6 show that b n for all n, it is sufficient to show that b nk. Evaluating the integral for gx) and then taking its first derivative leads to g x) = 6 sin ax x 5 a 3 ax cos ax ). This indicates that gx), x >, is non-increasing, implying that b nk = [gn + k) gn + k + )]. Next we show that n b n <. We first apply the mean value theorem to obtain b nk C for k, where C n+k) 3 > is a constant independent of n and k for notational simplicity, the same C will be used throughout this section, even if each instance of C may not be the same). In addition, observing that 3 k ) n + ) n + k) k n + ) n + k + ) < and applying the Sterling s approximation to factorials for sufficiently large n, we have D nk 4 4 n π 3 5 n + ) = 4 n n!) n n!) C. π n + )! n + Hence, for n large, b n C n + ) n+ k= C n+k) 3, implying n 3/ n b n <. Therefore, the spherical covariance function is valid on the sphere. 3. Stable Model and Power Model In this subsection, we consider the stable covariance function Chilès and Delfiner 999) { ) θ α } Cθ) = exp, θ [,π], a>, α, ] 5) a and the power variogram function Cressie 993) γθ)= ) θ α, a >, α, ]. 6) a Both functions are invalid when α> but valid when α, ] in R n Yaglom 987). The exponential and Gaussian models are special cases of the stable model 5), corresponding to α = and α =, respectively. Note that the function γ ) is conditionally negative definite if and only if e λγ ) is positive definite for any λ> Schoenberg 938). Therefore, it is equivalent to check the validity of the stable covariance function and the power variogram function. In addition, by Remark in Sect., it is also equivalent to check the power variogram function and the function Cθ) = c ) θ α, a >, α, ], 7) a for some c π θ/a)α sin θdθ.wetermcθ) in 7) the power covariance function. In summary, it is equivalent to check the validity among these three covariance and variogram functions.

7 We focus our discussion on validity of these functions when α, ]. Wood 995) and Gneiting 998) show that the stable covariance function is not valid on the circle when α, ]. Therefore, the stable covariance function is not valid on the sphere, since a valid covariance function on the sphere will always be valid on the circle Schoenberg 94). When α, ], inappendix we show that the power covariance function 7) is valid on the sphere. Our findings are summarized in Table. The stable covariance function and power variogram are valid when α, ], but not valid when α, ]. In particular, the exponential covariance function exp{ θ/a} is valid, and equivalently the power variogram γθ) = θ/a is also valid. However, the Gaussian covariance function exp{ θ/a) } is not valid see also Gneiting 999). Note that power models with α> are not uncommon in practice. Janis and Robeson 4) estimated over 3 period-of-record variograms using the power model, with approximately 6% having α>. 3.3 Other Models In this subsection, we check the validity of the following covariance functions: Radon transforms of exponential model orders and 4), Cauchy model, and hole-effect model. The computations are carried out by directly checking some of the coefficients b n in ) Radon Transforms of Exponential Model The following two Radon transforms of exponential model orders and 4) can be found in Chilès and Delfiner 999, p. 85) and Gneiting 999) C θ) = e θ/a + θ/a), and C θ) = e θ/a + θ/a + θ /3a ). By direct computation, we found that b 6 < forc θ), and b < forc θ), when a =. Therefore, neither is a valid covariance function on the sphere Cauchy Model The Cauchy model has the form Chilès and Delfiner 999, p. 86) Cθ) = + θ /a ). It is also termed rational quadratic model in Cressie 993). It is valid on R n Yaglom 987, p. 365). A direct computation yields b 4 <, when a =.5. Therefore, the Cauchy model is not valid on the sphere Hole-Effect Model The hole-effect model has the form sin aθ Cθ) = c θ for some c > and scale parameter a>. A direct computation shows that b 3 < with a =. Therefore, it is not a valid covariance function on the sphere.

8 3.3.4 Matérn Class Model The Matérn class model has the form Sect. in Yaglom 987): Cθ) = c aθ) ν K ν aθ), where c >, a>, and K ν ), ν is the modified Bessel function of the second kind of order ν. It appears that the validity of the Matérn class on the sphere depends on the smoothing parameter ν. For example, the case ν = corresponds to the exponential model, a valid model on the sphere. When ν = 3, 5 and, the corresponding models are the Radon transforms of exponential family with order, order 4, and the Gaussian model, respectively, all being shown to be invalid on the sphere. Direct computation shows that the Matérn models are not valid when ν =, 3,...,. In view of the connection between the stable and the Matérn models Stein 999, p. 5), it is likely that the Matérn models are not valid when ν>/, which needs further investigation. 4 Discussion of Strictly Positive Definiteness In this section, we discuss the strictly positive conditional negative) definiteness of covariance variogram) functions in Sect. 3. A real continuous function C ) is said to be strictly positive definite on S if m i= j= m C θp i,p j ) ) a i a j > for any finite number of distinct spatial locations {p i,i=,...,m} on S and real numbers {a i, i =,...,m}. A parallel definition for strictly conditional negative definiteness can be given as well. There is a rich discussion in the literature about the strictly positive conditional negative) definiteness of a continuous function on the sphere e.g. the sufficient conditions in Xu and Cheney 99 and Schreiner 997, and the necessary conditions in Chen et al. 3). Here, we summarize our findings as the following. The sufficient condition for strictly positive definite functions on a sphere is that b n in the expansion ) should be positive with a finite number of them being zero e.g. Xu and Cheney 99; Schreiner 997). By inspecting the coefficients b n for the power model in Appendix, one can conclude that all the coefficients b n are positive when <α<, which leads to the strictly positive definiteness. In parallel, the spherical covariance function is also strictly positive definite on the sphere. The case of α = in the power model yields the coefficients 8) inappendix) to be zero for all even terms and positive for all odd terms, which does not satisfy the necessary condition in Chen et al. 3). Therefore, it is not strictly positive definite on the sphere, even though it is positive definite. Consequently, the power variogram γθ)= θ is not strictly conditionally negative definite. As an example, we consider such an intrinsic stationary process Xp), p S, at four locations: north pole p N ),

9 south pole p S ) and two points on the equator, p E and p W, such that θp E,p W ) = π. It is clear that varxp N ) + Xp S ) Xp E ) Xp W )) =. The stable model exp θ/a) α ), <α<, a>, is strictly positive definite on the sphere. The case of α =, corresponding to the exponential covariance function Cθ) = exp θ/a), a >, is also strictly positive definite, since the coefficients b n = n+ k= D nk b nk > for all n, where b nk can be shown to be b nk = a e π/a ) +n+k) a a+e π/a ) +n+k) a +n+k+) a )>, +n+k+) a )>, n even; n odd. Acknowledgements We are grateful for the comments from the associate editor and anonymous reviewers, which have helped to improve this manuscript tremendously. We would also like to express special thanks to an anonymous reviewer who suggested the investigation of strictly positive conditional negative) definiteness of covariance variogram) functions on the sphere. Chunfeng Huang s research is supported by the National Science Foundation DMS Appendix: Validity of Power Covariance Functions In this Appendix, we show that the power covariance function Cθ) is valid when α, ], where ) θ α Cθ) = c, θ [,π] a for some c π θ/a)α sin θdθ. We explore ) to show that b n for all n. First, we present the following result. Lemma Let k be an integer. If gθ) is a non-negative and non-increasing function on [,kπ], then kπ gθ)sin θdθ. The proof of Lemma is trivial, but the result will be extensively used in the calculation below. Note that this lemma was also used in Gneiting 998). First, notice that P ) =. b = π c θ/a) α) P cos θ)sin θdθ= π c θ/a) α) sin θdθ. Next, if n, by the orthogonality of P n cos θ) and sin θ, b n = n + π c θ/a) α) P n cos θ)sin θdθ = n + a α π θ α P n cos θ)sin θdθ. It is clear that the selection of a scale parameter a does not affect the sign of b n, and consequently the validity of Cθ). Without loss of generality, we assume a = throughout the rest of the Appendix.

10 When α =, by in Gradshteyn and Ryzhik 994), we have π b n = n + θp n cos θ)sin θdθ, n= m even; = n+)/ n+ n+)π { n )!! n+)/ n+ )! }, n= m + odd. 8) Therefore, b n, n=,,... To show that n b n <, we first note that n = m + : n )!! m + )! ) =! m+ m!m + )! = ) m + m+. m Applying Stirling s approximation to factorials, we have 4m + 3 b m+ C π ) C mπm + ) m mm + ) C m, for some constant C > independent of m each appearance of C above may not be the same). Hence, n b n <, implying that Cθ) = c θ is a valid covariance function for some c π θ sin θdθ= π. Consequently, the exponential covariance function exp{ θ} is valid, so is the power variogram γθ)= θ. When <α<, based on the sine expansion 3) ofp n cos θ), we can write b n = n + D nk b nk, where D nk > is given by 4), and π b nk = θ α sinn + k + )θ sin θdθ, k=,,... First, the proof of n b n < can be followed along the same lines with those for the spherical covariance function with D nk C,forlargen and k. Therefore, to k n+k prove the validity of Cθ), it is sufficient to show that b n, or in turn, that b nk for all k. Now for any n, k, by the trigonometry identity and integration by parts, π π b nk = θ α cosn + k)θ dθ θ α cosn + k + )θ dθ = n + k) α+ n+k)π n + k + ) α+ k= θ α cos θdθ n+k+)π θ α cos θdθ

11 ) = α n + k) α+ n+k)π n + k + ) α+ θ α sin θdθ ) n+k+)π + α n + k + ) α+ θ α sin θdθ. 9) n+k)π From Lemma, the first term in 9) is non-positive for any n, k. To prove that the second term is non-positive, we explore the cases of n + k being odd or even. When n + k is odd, by substitution u = θ n + k)π, the integral of second term in 9) becomes n+k+)π n+k)π θ α sin θdθ= π u + n + k)π ) α sinu) du <, implying that b nk < by Lemma. When n + k = m is an even number for some m, substituting n + k + = m + and n + k = m into 9), we have b nk = = α m) α+ α m + ) α+ m+)π αθ α sin θdθ + mπ m + ) α+ α m) α+ α m + ) α+ ) m j+)π j= jπ ) m π j= π αθ + mπ) α sin θdθ + m + ) α+ { π = α m) α+ m + ) α+ } θ + mπ)α m + ) α+ sin θdθ π = α gθ)sin θdθ, θ α sin θdθ θ + jπ) α sin θdθ ) m θ + jπ) α j= where gθ) = m) α+ m + ) α+ ) m θ + jπ) α j= m + ) α+ θ + mπ)α, <θ<π.

12 From Lemma, to prove b nk < and hence conclude the whole proof, it is now sufficient to show that gθ) is non-negative and non-increasing. To show that gθ), we recall that for <α<, θ α is a decreasing function gθ) m) α+ m + ) α+ = ) m θ + mπ) α j= θ + mπ)α m + ) α+ m m) α+ m + m + ) α+ ) θ + mπ) α. To show that gθ) is non-increasing, we take the first derivative of gθ) with respect to θ, and note that α < dgθ) dθ { = α ) m) α+ m + ) α+ } θ + mπ)α m + ) α+ { α ) m) α+ m + ) α+ ) m θ + jπ) α j= ) m θ + mπ) α j= } θ + mπ)α m + ) α+ = α ) α+ m α m + ) α θ + mπ) α. Therefore, gθ) is non-increasing, concluding the proof. References Alkhaled AA, Michalak AM, Kawa SR, Olsen SC, Wang JW 8) A global evaluation of the regional spatial variability of column integrated CO distributions. J Geophys Res 3:D33. doi:.9/7jd9693 Banerjee S 5) On geodetic distance computations in spatial modeling. Biometrics 6:67 65 Bochner S 94) Hilbert distances and positive definite functions. Ann Math 4: Carmona-Moreno C, Belward A, Malingreau JP, Hartley A, Garcia-Algere M, Antonovskiy M, Buchshtaber V, Pivovarov V 5) Characterizing interannual variations in global fire calendar using data from Earth observing satellites. Glob Change Biol : Chen D, Menegatto V, Sun X 3) A necessary and sufficient condition for strictly positive definite functions on spheres. Proc Am Math Soc 3: Chilès JP, Delfiner P 999) Geostatistics. Wiley, New York Cressie N 985) Fitting variogram models by weighted least squares. Math Geol 7: Cressie N 993) Statistics for spatial data. Wiley, New York

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