Spatio-Spectral Assessment of Some Isotropic Polynomial Covariance Functions on the Sphere

Abstract

In gravity field modeling, covariance functions are mainly associated with least squares collocation. Prior to the implementation of least squares collocation, the characteristics of the selected analytical covariance function need to be well understood. In this contribution, we study four polynomial covariance functions, i.e., the spherical, Askey, $C^{2}$-Wendland and $C^{4}$-Wendland models. All of them are defined on the sphere and correspond to isotropic, positive definite and compactly supported functions. We examine them in the spatial and spectral domains, and assess their characteristics, such as the correlation length, the curvature parameter, the spectral maximum and the spectral decay rate. We also provide analytical expressions and numerical estimates for these parameters.

1 Introduction

Covariance functions (CFs) are routinely used in geostatistical analysis to model the stochastic behavior of spatial random fields. The study of CFs in physical geodesy is usually conducted within the context of least squares collocation for the estimation of functionals related to the Earth’s disturbing potential. Spatial CFs are categorized in several ways, depending on their mathematical properties. With respect to the surface they are defined on, CFs are classified into planar or spherical (i.e., defined on the plane or sphere, respectively), with the former ones used mostly in local-scale applications and the latter ones in global-scale applications. Regarding their structural properties (Devaraju and Sneeuw 2018), CFs can be classified based on their invariance under translation (homogeneous/non-homogeneous) and under rotation (isotropic/non-isotropic). The examination of their support length (i.e., maximum distance at which the CF is non-zero) gives rise to yet another distinction. That is, CFs with an infinite support length, which are usually called global CFs, and CFs with a finite support length, also known as local, finite or compactly supported CFs. Lastly, depending on their validity, CFs are classified as positive definite and non-positive definite, with only the former ones to provide physically meaningful modeling options.

The design of CFs and the study of their properties is an active topic of research in the field of applied mathematics (e.g., Emery et al. 2022), with applications to all branches of geosciences. In this contribution, we focus only on polynomial CFs, which are still regarded as appealing models mostly due to the simplicity of their mathematical expressions. We examine four polynomial CFs, namely, the spherical, Askey, $C^{2}$-Wendland and $C^{4}$-Wendland CFs. The spherical CF represents the normalized volume resulted from the convolution of two identical balls. The Askey covariance model was firstly used as a radial basis function by Askey (1973), who also proved its positive definiteness. The Wendland CFs (Wendland 1995) are constructed by the repeated integration of the Askey CF using the “Montée” integral operator $\mathcal {I} \{ f (r) \} = \int _{r}^{\infty } t f (t) \, \mathrm {d} t$, which produces CFs of arbitrary smoothness k (termed $C^{k}$-Wendland CFs). Based on this design principle, the Askey CF is also considered as the $C^{0}$-Wendland CF. The selection of these models is motivated from their frequent use in previous studies that primarily investigate them from a purely mathematical perspective (Gneiting 2013; Guinness and Fuentes 2016). All models correspond to isotropic, positive definite, compactly supported functions defined on the spherical surface.

The paper is structured as follows. In Sect. 2 the spatial and spectral representation of the four polynomial CFs is presented. Some alternative expressions for the evaluation of the Askey and Wendland CFs in the spatial domain are also provided. In Sect. 3 two spatial characteristics (correlation length and curvature parameter) and two spectral characteristics (spectral maximum and decay rate) are discussed. For the correlation length, curvature parameter and spectral maximum, analytical expressions are derived, whereas the spectral decay rate is evaluated numerically. Finally, Sect. 4 summarizes the most important conclusions of this study.

2 Isotropic Polynomial Covariance Functions

2.1 Spatial Representation

Since the four CFs under study correspond to isotropic functions, they only depend on the distance between two points on the spherical surface. Their adaptation from the line (or plane) to the sphere is done by replacing the Euclidean distance with the spherical distance $\psi \in [0,\pi ]$. The standard expressions used to describe them in the spatial domain are given in Table 1. All CFs depend on the variance $c_{0}$, which represents the CF value at $\psi = 0$, and the support length $\psi _{0}$ that denotes the spatial extent of the CF, i.e., the distance for which $C (\psi ) = 0, \ \forall \psi > \psi _{0}$. The Askey and Wendland CFs also depend on the shape parameter $\tau$. The numerical range of $c_{0}$, $\psi _{0}$ and $\tau$ that results in a positive definite CF on the sphere is also provided in Table 1. The variable ${\mathbf {1}}_{I}$ denotes the indicator function, given by

$$\displaystyle \begin{aligned} {} {\mathbf{1}}_{I} (\psi) = \begin{cases} 1, & \psi \in I \\ 0, & \psi \notin I \end{cases}. \end{aligned} $$

(1)

The intervals $I_{1}$ and $I_{2}$ are defined as $I_{1} = [0,\min (\psi _{0},\pi )]$ and $I_{2} = [0,\psi _{0}]$. It is also evident by the expressions in Table 1 that $C_{\mathrm {S}}$, $C_{\mathrm {A}}$, $C_{\mathrm {W2}}$ and $C_{\mathrm {W4}}$ are polynomials of order three, $\tau$, $\tau + 1$ and $\tau + 2$, respectively. Figure 1 presents some CF examples for different values of $\psi _{0}$ and $\tau$. The spherical and Askey CFs demonstrate a sharp decrease at $\psi = 0$, whereas the Wendland CFs have a smoother, Gaussian-like behavior in the same vicinity.

Fig. 1

Examples of (a) spherical, (b) Askey, (c) $C^{2}$-Wendland and (d) $C^{4}$-Wendland CFs in the spatial domain for different $\psi _{0}$ (a) and $\tau$ values (b,c,d). A variance of $c_{0} = 1$ is selected in all cases

Table 1 Isotropic polynomial covariance functions on the sphere (Gneiting 2013)

Several alternative formulations of the Askey and Wendland CFs can be found in the literature. Hubbert (2012) derived expressions in terms of the associated Legendre function of the first kind $P_{n}^{m}$ of degree n and order m, with $n,m \in {\mathbb {N}}_{0}$ and $n \geq m$. These expressions are normalized in this work so that $C (0) = c_{0}$, resulting in the following equations:

$$\displaystyle \begin{gathered} C_{\mathrm{A}} (\psi,\tau) = g_{0,\tau} \bigg( 1 - \frac{\psi^{2}}{\psi_{0}^{2}} \bigg)^{\frac{\tau}{2}} P_{0}^{-\tau} \bigg( \frac{\psi_{0}}{\psi} \bigg) {\mathbf{1}}_{I_{1}} (\psi) {} \end{gathered} $$

(2a)

$$\displaystyle \begin{gathered} C_{\mathrm{W2}} (\psi,\tau) = g_{1,\tau} \, \psi \bigg( 1 - \frac{\psi^{2}}{\psi_{0}^{2}} \bigg)^{\frac{\tau}{2}} P_{1}^{-\tau} \bigg( \frac{\psi_{0}}{\psi} \bigg) {\mathbf{1}}_{I_{2}} (\psi) {} \end{gathered} $$

(2b)

$$\displaystyle \begin{gathered} \begin{aligned} C_{\mathrm{W4}} (\psi,\tau) = g_{2,\tau} \, \psi^{2} \bigg( 1 - \frac{\psi^{2}}{\psi_{0}^{2}} \bigg)^{\frac{\tau}{2}} P_{2}^{-\tau} \bigg( \frac{\psi_{0}}{\psi} \bigg) {\mathbf{1}}_{I_{2}} (\psi), \end{aligned} {} \end{gathered} $$

(2c)

with the parameter $g_{s,\tau }$ defined as:

$$\displaystyle \begin{aligned} {} g_{s,\tau} = \frac{2^{s} \, s ! (\tau + s) !}{(2 s) !} c_{0} \psi_{0}^{-s} \end{aligned} $$

(3)

and the associated Legendre function of negative order $P_{n}^{-m}$ given by the expression (Hubbert 2012):

$$\displaystyle \begin{aligned} P_{n}^{-m} (x) = \bigg( \frac{x - 1}{x + 1} \bigg)^{\frac{m}{2}} \sum_{j = 0}^{n} \frac{(j + n)! (x - 1)^{j}}{2^{j} j! (j + m)! (n - j)!}. \end{aligned} $$

(4)

We note that the associated Legendre function $P_{n}^{m}$ includes the Condon-Shortley phase $(-1)^{m}$, therefore the relation $P_{n}^{m} = (-1)^{m} P_{n,m}$ applies, with $P_{n,m}$ being the standard definition of the associated Legendre function used in physical geodesy. Hubbert (2012) developed additional closed-form expressions for Eqs. (2a)–(2c) using analytical relations for $P_{n}^{m}$. These results were later used by Chernih and Hubbert (2014) to derive expression in standard polynomial form. The expressions of Chernih and Hubbert (2014), again normalized here so that $C (0) = c_{0}$, read:

$$\displaystyle \begin{gathered} C_{\mathrm{A}} (\psi,\tau) = \sum_{m = 0}^{\tau} k_{0,m,\tau} \psi^{m} {\mathbf{1}}_{I_{1}} (\psi) {} \end{gathered} $$

(5a)

$$\displaystyle \begin{gathered} C_{\mathrm{W2}} (\psi,\tau) = \sum_{m = 0}^{\tau + 1} k_{1,m,\tau} \psi^{m} {\mathbf{1}}_{I_{2}} (\psi) {} \end{gathered} $$

(5b)

$$\displaystyle \begin{gathered} C_{\mathrm{W4}} (\psi,\tau) =\sum_{m = 0}^{\tau + 2} k_{2,m,\tau} \psi^{m} {\mathbf{1}}_{I_{2}} (\psi), {} \end{gathered} $$

(5c)

with

$$\displaystyle \begin{aligned} {} k_{s,m,\tau} = \frac{(-1)^{m} \Gamma \big( \frac{m + 1}{2} \big) \Gamma \big( \frac{1}{2} - s \big) (\tau + s) !}{\Gamma \big( \frac{m + 1}{2} - s \big) \Gamma \big( \frac{1}{2} \big) (\tau + s - m) ! m !} c_{0} \psi_{0}^{-m} \end{aligned} $$

(6)

and with $\Gamma (x)$ denoting the gamma function (Gradshteyn and Ryzhik 2014, p. xxxii). Additional representations of the Askey and Wendland CFs, e.g., in terms of hypergeometric functions, can be found in Hubbert (2012).

2.2 Spectral Representation

The spherical harmonic coefficients $G (n)$ of an isotropic covariance function are derived via the application of the Legendre transform $\mathcal {J}$, as follows (Jekeli 2017, p. 54):

$$\displaystyle \begin{aligned} {} \begin{aligned} G (n) &= \mathcal{J} \{ C (\psi) \} \\ &= \frac{1}{2} \int_{0}^{\pi} C (\psi) P_{n} (\cos \psi) \sin \psi \, \mathrm{d} \psi, \end{aligned} \end{aligned} $$

(7)

where $P_{n}$ denotes the Legendre polynomials of degree n. Applying the linearity property of the Legendre transform to the expressions of Sect. 2.1, the spherical harmonic coefficients of the four CFs under study are given by:

$$\displaystyle \begin{gathered} G_{\mathrm{S}} (n) = c_{0} \bigg( \varPsi_{0,I_{1}} (n) - \frac{3 \, \varPsi_{1,I_{1}} (n)}{2 \psi_{0}} + \frac{\varPsi_{3,I_{1}} (n)}{2 \psi_{0}^{3}} \bigg) {} \end{gathered} $$

(8a)

$$\displaystyle \begin{gathered} G_{\mathrm{A}} (n) = \sum_{m = 0}^{\tau} k_{0,m,\tau} \, \varPsi_{m,I_{1}} (n) {} \end{gathered} $$

(8b)

$$\displaystyle \begin{gathered} G_{\mathrm{W2}} (n) = \sum_{m = 0}^{\tau + 1} k_{1,m,\tau} \, \varPsi_{m,I_{2}} (n) {} \end{gathered} $$

(8c)

$$\displaystyle \begin{gathered} G_{\mathrm{W4}} (n) = \sum_{m = 0}^{\tau + 2} k_{2,m,\tau} \, \varPsi_{m,I_{2}} (n), {} \end{gathered} $$

(8d)

where $\varPsi _{m,I}$ is the Legendre transform of the monomial $\psi ^{m}$ in I, i.e.,

$$\displaystyle \begin{aligned} {} \varPsi_{m,I} (n) = \frac{1}{2} \int_{I} \psi^{m} P_{n} (\cos \psi) \sin \psi \, \mathrm{d} \psi. \end{aligned} $$

(9)

The spherical harmonic representation of the CFs of Fig. 1 is shown in Fig. 2. Since all CFs are positive definite, the Schoenberg criteria should apply, i.e., $G(n) \geq 0$ and $\sum _{n = 0}^{\infty } G(n) < \infty$ (Schoenberg 1942). The first Schoenberg criterion can be easily noted in Fig. 2, where it is also evident that, in all cases, the coefficients $G (n)$ decrease at a constant rate in higher degrees. The coefficients of the spherical CF exhibit a strong oscillating pattern that also appears (in a much lesser extent) in the rest of the CFs for small $\tau$ values.

Fig. 2

Examples of (a) spherical, (b) Askey, (c) $C^{2}$-Wendland and (d) $C^{4}$-Wendland CFs in the spherical harmonic domain for different $\psi _{0}$ (a) and $\tau$ values (b,c,d). A variance of $c_{0} = 1$ is selected in all cases

3 Characteristics

3.1 Spatial Characteristics

The three main spatial characteristics of a CF is the variance, the correlation length and the curvature parameter (Moritz 1976). The variance is defined as the CF value at $\psi = 0$ and equals to $c_{0}$ for all the models of Table 1. The correlation length, denoted as $\xi$, represents the spherical distance at which the CF decreases to half the variance, i.e.,

$$\displaystyle \begin{aligned} {} C(\xi) = \frac{C(0)}{2}. \end{aligned} $$

(10)

Since all the CFs of Table 1 are strictly monotonic (i.e., strictly decreasing) on $[0,\psi _{0}]$, it can be deduced by the virtue of the intermediate value theorem that there exists a unique $\xi \in [0,\psi _{0}]$ satisfying Eq. (10). The determination of an analytical expression for $\xi$ is performed by solving Eq. (10) explicitly, and therefore it depends on the mathematical complexity of $C(\psi )$. For the correlation length of the spherical CF, with the aid of MATLAB’s computer algebra system (MATLAB 2020), we find the expression:

$$\displaystyle \begin{aligned} {} \xi_{\mathrm{S}} = \bigg[ \sqrt{3} \sin \bigg( \frac{2 \pi}{9} \bigg) - \cos \bigg( \frac{2 \pi}{9} \bigg) \bigg] \psi_{0} \approx 0.3473 \psi_{0}, \end{aligned} $$

(11)

whereas, for the Askey CF the following expression can be easily derived:

$$\displaystyle \begin{aligned} {} \xi_{\mathrm{A}} = \bigg( 1 - \frac{1}{\sqrt[\tau]{2}} \bigg) \psi_{0}. \end{aligned} $$

(12)

Obtaining an analytical expression for the correlation length of the Wendland CFs is not a simple task, since it requires (a) the derivation of analytical expressions for the roots of a polynomial of arbitrary order and (b) a subsequent investigation on whether these roots are real and belong to $[0,\psi _{0}]$. Regarding the first requirement, analytical expressions for the roots of polynomials up to order four exist but become too complicated to be used in practice for orders greater than two. In addition, based on the Abel–Ruffini theorem, no algebraic expressions exist for the roots of general polynomial equations of order greater than four. We are also not aware of any method that addresses the second requirement.

Instead of a rigorous analytical expression for the correlation length of $C^{2}$- and $C^{4}$-Wendland CFs, denoted as $\xi _{\mathrm {W2}}$ and $\xi _{\mathrm {W4}}$, we seek for an approximate expression that can be easily generalized for any $\tau$. In the sequel, we outline the procedure employed for deriving such an expression for $\xi _{\mathrm {W2}}$. We firstly define the scaling parameter $s_{\mathrm {W2}} \in [0,1]$ as $s_{\mathrm {W2}} = \xi _{\mathrm {W2}} / \psi _{0}$ and rewrite Eq. (10) for the $C^{2}$-Wendland function as follows:

$$\displaystyle \begin{aligned} {} (1 + \tau s_{\mathrm{W2}}) (1 - s_{\mathrm{W2}})^{\tau} = \frac{1}{2}. \end{aligned} $$

(13)

We solve Eq. (13) numerically (e.g., using the bisection method or MATLAB’s vpasolve function; The MathWorks Inc., 2022) for several $\tau$ values and only keep the real solutions in $[0,1]$, which are unique for each $\tau$. These solutions are plotted in Fig. 3 up to $\tau = 50$. It is evident that $s_{\mathrm {W2}}$ smoothly decreases for increasing $\tau$ and can be approximated quite well by the rational model:

$$\displaystyle \begin{aligned} {} s_{\mathrm{W2}} \approx \frac{\alpha_{\mathrm{W2}}}{\beta_{\mathrm{W2}} + \tau}, \end{aligned} $$

(14)

where the parameter values $\alpha _{\mathrm {W2}} = 1.679$ and $\beta _{\mathrm {W2}} = 1.350$ are estimated using ordinary least-squares. Substituting Eq. (14) into the defining expression for $s_{\mathrm {W2}}$ and solving with respect to $\xi _{\mathrm {W2}}$, we derive the following approximation:

$$\displaystyle \begin{aligned} {} \xi_{\mathrm{W2}} \approx \frac{\alpha_{\mathrm{W2}}}{\beta_{\mathrm{W2}} + \tau} \psi_{0}. \end{aligned} $$

(15)

Performing the same procedure for the $C^{4}$-Wendland CF yields:

$$\displaystyle \begin{aligned} {} \xi_{\mathrm{W4}} \approx \frac{\alpha_{\mathrm{W4}}}{\beta_{\mathrm{W4}} + \tau} \psi_{0}, \end{aligned} $$

(16)

with $\alpha _{\mathrm {W4}} = 2.330$ and $\beta _{\mathrm {W4}} = 2.312$. The corresponding values of the scaling parameter $s_{\mathrm {W4}} = \xi _{\mathrm {W4}} / \psi _{0}$ are also provided in Fig. 3, along with $s_{\mathrm {S}} \approx 0.3473$ and $s_{\mathrm {A}} = 1 - \sqrt [-\tau ]{2}$ that are directly derived from Eqs. (11) and (12), respectively. The maximum absolute error of $s_{\mathrm {W2}}$ and $s_{\mathrm {W4}}$ using the approximations of Eqs. (15) and (16) does not exceed $6 \times 10^{-5}$ and $10^{-5}$, respectively, in the examined $\tau$ range. The overall behavior of the four s groups indicates that for the same $\psi _{0}$ and $\tau$, the following inequality applies: $s_{\mathrm {S}} > s_{\mathrm {W4}} > s_{\mathrm {W2}} > s_{\mathrm {A}}$. The same inequality is also true for $\xi$; hence, the Askey and spherical models always produce a CF with the smallest and largest correlation length, respectively, and the $C^{2}$-Wendland CF always has a smaller correlation length than the $C^{4}$-Wendland CF for a given $\{ \psi _{0}, \tau \}$ pair. An example of this behavior is shown in Figs. 1c and 1d for $\tau = 6$, where the $C^{2}$-Wendland CF (red line) is sharper than the corresponding $C^{4}$-Wendland CF (blue line). A simple investigation of Eqs. (11), (12), (15) and (16) also shows that a decreasing $\psi _{0}$ or an increasing $\tau$ yields a smaller correlation length $\xi$, which corresponds to a sharper CF. This is again corroborated by the examples in Fig. 1.

Fig. 3

Scaling parameters s, and approximations (fitted rational models) for $s_{\mathrm {W2}}$ and $s_{\mathrm {W4}}$

The curvature parameter $\chi$ of a CF is defined as:

$$\displaystyle \begin{aligned} {} \chi = \kappa (0) \frac{\xi^{2}}{C (0)}, \end{aligned} $$

(17)

where $\kappa (\psi )$ is the curvature (or reciprocal radius of curvature) of $C (\psi )$, given by:

$$\displaystyle \begin{aligned} {} \kappa (\psi) = \frac{C'' (\psi)}{\Big[ 1 + \big( C' (\psi) \big)^{2} \Big]^{\frac{3}{2}}}. \end{aligned} $$

(18)

Evaluating $\kappa (0)$ using Eq. (18) and substituting to Eq. (17) results in the following expressions:

$$\displaystyle \begin{gathered} \chi_{\mathrm{S}} = 0 {} \end{gathered} $$

(19a)

$$\displaystyle \begin{gathered} \chi_{\mathrm{A}} = \frac{\tau (\tau - 1) \psi_{0}}{ \big( c_{0}^{2} \tau^{2} + \psi_{0}^{2} \big)^{\frac{3}{2}}} \xi_{\mathrm{A}}^{2} {} \end{gathered} $$

(19b)

$$\displaystyle \begin{gathered} \chi_{\mathrm{W2}} = -\frac{\tau (\tau + 1)}{\psi_{0}^{2}} \xi_{\mathrm{W2}}^{2} {} \end{gathered} $$

(19c)

$$\displaystyle \begin{gathered} \chi_{\mathrm{W4}} = -\frac{\tau^{2} + 3 \tau + 2}{3 \psi_{0}^{2}} \xi_{\mathrm{W4}}^{2}. {} \end{gathered} $$

(19d)

The zero curvature parameter (i.e., infinite radius of curvature) of the spherical CF indicates that $C_{\mathrm {S}} (\psi )$ is linear at $\psi = 0$, whereas the positive and negative curvature parameters of the Askey and Wendland CFs, respectively, show that the former is convex and the latter concave at $\psi = 0$. Finally, only $\chi _{\mathrm {A}}$ shows a dependence on $c_{0}$.

3.2 Spectral Characteristics

The two spectral characteristics discussed in this section are the magnitude of the zeroth-degree spherical harmonic coefficient and the spectral decay rate. The magnitude of the zeroth-degree coefficient $G (0)$ denotes the spectral maximum. Substituting $n = 0$ in Eq. (7) yields the expression:

$$\displaystyle \begin{aligned} {} G (0) = \frac{1}{2} \int_{0}^{\pi} C (\psi) \sin \psi \, \mathrm{d} \psi, \end{aligned} $$

(20)

which also corresponds to the average of $C (\psi )$ over the sphere. The analytical solution of Eq. (20) for the spherical CF results in:

$$\displaystyle \begin{aligned} {} G_{\mathrm{S}} (0) = \frac{c_{0} [\psi_{0}^{3} - 3 \sin{}(\psi_{0}) + 3 \psi_{0} \cos{}(\psi_{0})]}{2 \psi_{0}^{3}}. \end{aligned} $$

(21)

The integral of Eq. (20) is rewritten for the Askey CF as:

$$\displaystyle \begin{aligned} {} G_{\mathrm{A}} (0) = \frac{c_{0}}{2 \psi_{0}^{\tau}} \int_{0}^{\psi_{0}} (\psi_{0} - \psi)^{\tau} \sin \psi \, \mathrm{d} \psi \end{aligned} $$

(22)

and has the following analytical solution (Prudnikov et al. 1986, §2.5.5, eq. 1):

$$\displaystyle \begin{aligned} {} G_{\mathrm{A}} (0) = \frac{c_{0} \psi_{0}^{2}}{2 (\tau^{2} + 3 \tau + 2)} {}_1 F_{2} \Big( 1;\tfrac{\tau + 3}{2},\tfrac{\tau + 4}{2};-\tfrac{\psi_{0}^{2}}{4} \Big), \end{aligned} $$

(23)

where $_{p} F_{q} (a_{1}, \dots , a_{p};b_{1}, \dots , b_{q};x)$ is the generalized hypergeometric series (Gradshteyn and Ryzhik 2014, §9.14, eq. 1). Proceeding in the same way and using the relation of Prudnikov et al. (1986, §2.5.7, eq. 1), we derive the expression

$$\displaystyle \begin{aligned} {} \begin{aligned} G_{\mathrm{W2}} (0) ={}&G_{\mathrm{A}} (0) - \frac{ i \tau c_{0} \psi_{0}}{4} \mathrm{B} (2,\tau + 1) \times \\ {}[{}_1 &F_{1} (2;\tau + 3;i \psi_{0}) - {}_1 F_{1} (2;\tau + 3;-i \psi_{0})] \end{aligned} \end{aligned} $$

(24)

for the $C^{2}$-Wendland CF and

$$\displaystyle \begin{aligned} {} \begin{aligned} G_{\mathrm{W4}} (0) ={}&G_{\mathrm{W2}} (0) - \frac{ i (\tau^{2} - 1) c_{0} \psi_{0}}{12} \mathrm{B} (3,\tau + 1) \times \\ {}[{}_1 &F_{1} (3;\tau + 4;i \psi_{0}) - {}_1 F_{1} (3;\tau + 4;-i \psi_{0})] \end{aligned} \end{aligned} $$

(25)

for the $C^{4}$-Wendland CF, with i being the imaginary unit and $\mathrm {B} (x,y)$ the beta function (Gradshteyn and Ryzhik 2014, §8.380, eq. 1). The magnitude of $G (0)$ is presented in Fig. 4 with respect to different $\psi _{0}$ and $\tau$ values, and for $c_{0} = 1$. The corresponding magnitude for $c_{0} \neq 1$ is given by $G (0) |{ }_{c_{0} = a} = a G (0) |{ }_{c_{0} = 1}$. It is evident from Fig. 4 that $G (0)$ has a larger magnitude for increasing $\psi _{0}$ and decreasing $\tau$. The inequality $G_{\mathrm {S}} (0) > G_{\mathrm {W4}} (0) > G_{\mathrm {W2}} (0) > G_{\mathrm {A}} (0)$ also holds true for a specific triplet of $\{ c_{0},\psi _{0},\tau \}$ values.

Fig. 4

Evaluation of $G (0)$ for the (a) spherical, (b) Askey, (c) $C^{2}$-Wendland and (d) $C^{4}$-Wendland CFs. A variance of $c_{0} = 1$ is selected in all cases. The magnitude of $G (0)$ is provided in a logarithmic scale

The spectral decay rate describes how the magnitude of $G (n)$ changes for increasing degree n. It is defined in decibel per octave using the equation:

$$\displaystyle \begin{aligned} {} u = - \frac{G^{[\mathrm{dB}]} (n_{2}) - G^{[\mathrm{dB}]} (n_{1})}{\log_{2} (n_{2}) - \log_{2} (n_{1})}, \end{aligned} $$

(26)

where the coefficients $G^{[\mathrm {dB}]} (n)$ are expressed in decibels as follows:

$$\displaystyle \begin{aligned} {} G^{[\mathrm{dB}]} (n) = 20 \log \bigg( \bigg| \frac{G (n)}{G (0)} \bigg| \bigg). \end{aligned} $$

(27)

Equation (26) represents the magnitude change every time n doubles. A positive value of u indicates magnitude decay, whereas a negative value shows magnitude gain. Based on the results of Fig. 2, the decay rate of $G (n)$ is relatively small for lower degrees and converges to a maximum value, which remains constant at higher degrees. This is also evident in Fig. 5, where the spectral decay rate of the $C^{4}$-Wendland CF, denoted as $u_{\mathrm {W4}}$, is evaluated for consecutive degrees (i.e., $n_{1} = n$ and $n_{2} = n + 1$). The strong fluctuations of $u_{\mathrm {W4}}$ for $\tau = 6$ occur due to the oscillating behavior of $G_{\mathrm {W4}}$ for small $\tau$ values. Additional numerical experiments show that the convergent value of u for a specific CF is not influenced by $c_{0}$, $\psi _{0}$ or $\tau$. For $n_{1} = 100$ and $n_{2} = 200$, the decay rate of the four CFs under study is estimated in decibel per octave as follows: $u_{\mathrm {S}} = 18$, $u_{\mathrm {A}} = 18$, $u_{\mathrm {W2}} = 30$ and $u_{\mathrm {W4}} = 42$.

Fig. 5

Spectral decay rate of $C^{4}$-Wendland CF for consecutive degrees

4 Summary and Conclusions

In this contribution, we examined the spatial and spectral properties of the spherical, Askey, $C^{2}$- and $C^{4}$-Wendland CFs, which are all isotropic, positive-definite and compactly supported functions defined on the sphere.

The spatial assessment is performed by analyzing the CFs shape, correlation length and curvature parameter. The shape of the spherical and Askey CFs exhibits a sharp decay of their covariance value at $\psi = 0$, therefore these models are better suited for modeling the stochastic behavior of geophysical signals with a sharply-decreasing empirical CF at the origin. Analytical expressions are developed that allow the rigorous calculation of the spherical and Askey CFs correlation length. Due to theoretical limitations, similar rigorous expressions cannot be derived for the Wendland models. At the present time, this issue is resolved with the development of approximate expressions. The examination of the correlation length $\xi$ for the four CFs using the same set of parameters resulted in the following inequality: $\xi _{\mathrm {S}} > \xi _{\mathrm {W4}} > \xi _{\mathrm {W2}} > \xi _{\mathrm {A}}$. Analytical expressions for the evaluation of the curvature parameter of the four CFs under study are also developed. Since the curvature parameter depends on the correlation length, these expressions are again exact for the spherical and Askey CFs, and approximate for the Wendland models. The spherical CF has a zero curvature parameter, which suggests a linear decrease of the covariance values at $\psi = 0$. The positive and negative curvature parameter of the Askey and Wendland CFs, respectively, is also reflected in their convex and concave shape at the origin.

The assessment of the four CFs in the spectral domain is performed by calculating the spherical harmonic coefficients and examining the spectral maximum and spectral decay rate. All spherical harmonic coefficients are positive, as a result of the first Schoenberg criterion for positive-definite functions on the sphere. The spectral maximum $G (0)$ can be evaluated using analytical expressions that are given in terms of the hypergeometric function for the Askey and Wendland CFs. A smaller spectral maximum, which represents the mean value of the CF over the sphere, appears to be associated with a smaller correlation length. Although this connection is not mathematically proven, it is empirically corroborated by the inequality $G_{\mathrm {S}} (0) > G_{\mathrm {W4}} (0) > G_{\mathrm {W2}} (0) > G_{\mathrm {A}} (0)$, which is based on numerical evidence. The visual inspection of the CF spectrum and the evaluation of the spectral decay rate for consecutive degrees shows that the decay rate increases in low degrees and converges to a maximum value in higher degrees. This maximum value is estimated numerically in the convergence region. Results showed that the spherical and Askey CFs have a similar spectral decay rate (18 dB/octave), whereas the $C^{2}$- and $C^{4}$-Wendland CFs have higher decay rates (30 and 42 dB/octave, respectively). Additional experiments show that the maximum spectral decay rate of each CF does not depend on any of its parameters.

Due to their spatial structure, all CFs examined in this work can be used to model positively-correlated signals. In practice, empirical CFs of geophysical signals often exhibit an oscillatory behavior for large distances that can result in negative correlations. The design of compactly supported, positive-definite CFs that account for such oscillations is therefore of great need, since they can provide better modeling options.

The analysis presented in this work contributes to the general understanding of the behavior of some frequently used polynomial CFs on the sphere. The same investigation and comparison can be performed for various other models and the results can be further utilized in the context of stochastic modeling of spatial signals for geodetic applications.