On the Mátern covariance family: a proposal for modeling temporal correlations based on turbulence theory

Abstract

Current variance models for GPS carrier phases that take correlation due to tropospheric turbulence into account are mathematically difficult to handle due to numerical integrations. In this paper, a new model for temporal correlations of GPS phase measurements based on turbulence theory is proposed that overcomes this issue. Moreover, we show that the obtained model belongs to the Mátern covariance family with a smoothness of 5/6 as well as a correlation time between 125–175 s. For this purpose, the concept of separation distance between two lines-of-sight introduced by Schön and Brunner (J Geod 1:47–57, 2008a) is extended. The approximations made are highlighted as well as the turbulence parameters that should be taken into account in our modeling. Subsequently, fully populated covariance matrices are easily computed and integrated in the weighted least-squares model. Batch solutions of coordinates are derived to show the impact of fully populated covariance matrices on the least-squares adjustments as well as to study the influence of the smoothness and correlation time. Results for a specially designed network with weak multipath are presented by means of the coordinate scatter and the a posteriori coordinate precision. It is shown that the known overestimation of the coordinate precision is significantly reduced and the coordinate scatter slightly improved in the sub-millimeter level compared to solutions obtained with diagonal, elevation-dependent covariance matrices. Even if the variations are small, turbulence-based values for the smoothness and correlation time yield best results for the coordinate scatter.

Appendix: The Mátern covariance functions

A short introduction to the Mátern covariance family, also called Whittle Mátern covariance family, von Karman model (oceanography), Markov processes (geodesy) or autoregressive models (meteorology) is presented here, giving the principal features, vocabulary as well as dependencies. More details can be exemplarily found in Stein (1999), Mátern (1960), Guttorp and Gneiting (2005), Grafarend and Awange (2012). Mátern covariance functions have been concretely used by Handcock and Wallis (1994) to model meteorological fields. Fuentes (2002) also derived a non stationary family for the determination of air quality models.

A 2D autoregressive continuous process AR(1) called $Z\left( {x,y} \right)$ can be described by the stochastic differential equation:

$$\begin{aligned} \left( {\frac{\partial }{\partial x^{2}}^{2}+\frac{\partial }{\partial y^{2}}^{2}-\alpha ^{2}} \right) Z\left( {x,y} \right) =\varepsilon \left( {x,y} \right) , \end{aligned}$$

(21)

where $\varepsilon \left( {x,y} \right)$ is white noise and $\alpha$ a constant. The corresponding spectral density is given by

$$\begin{aligned} W\left( \omega \right) \propto \frac{1}{({\omega ^2 +\alpha ^{2}})^{2}}, \end{aligned}$$

(22)

with $\omega ^{2}=\omega _1^2 +\omega _2^2$ for the 2D case (Whittle 1954). The stationary covariance between two points $\mathbf{x},\mathbf{{x}'}$ for this process is:

$$\begin{aligned} C({\mathbf{x,{x}'}})=C\left( r \right) =\left( {\alpha r} \right) K_1 \left( {\alpha r} \right) , \end{aligned}$$

(23)

where $K_1$ is the modified Bessel function of $1\mathrm{st}$ order, $r=\left\| {\mathbf{x-{x}'}} \right\|$ for the isotropic case ($\left\| . \right\|$ being the norm of the vector). Whittle (1954) presented such a covariance function as a “natural spatial covariance” for the 2D case, as the exponential-based covariance functions are for one-dimensional processes.

Mátern (1960) used Whittle’s result and derived for any dimension $d$ a family of covariance functions based on an isotropic spectral density:

$$\begin{aligned} W\left( \omega \right) =\frac{2^{\nu -1}\phi \quad \Gamma \left( {\nu +d/2} \right) \alpha ^{2\nu }}{\pi ^{d/2}({\omega ^2 +\alpha ^{2}})^{\nu +d/2}}, \end{aligned}$$

(24)

where $\omega ^{2}=\omega _1^2 +\omega _2^2 +...+\omega _d^2$ is the angular frequency, $\Gamma$ the Gamma function (Abramowitz and Segun 1972) and $\nu >0,\alpha >0,\phi >0$ are constant parameters, d the dimension. The corresponding Mátern class of covariance functions is positive definite and reads:

$$\begin{aligned} C\left( r \right) =\phi \left( {\alpha r} \right) ^{\nu }K_\nu \left( {\alpha r} \right) . \end{aligned}$$

(25)

The parameter $\nu$ can be seen as a measure of the differentiability of the field (Stein 1999) thus “its smoothness”. The constant $\alpha$ indicates how the correlations decay with increasing distance. Its inverse is usually called the correlation length in Kriging.

Smoothness parameter $\nu$

Figure 13 highlights the influence of the smoothness parameter $\nu$ and the correlation length by simulating a random field/time series corresponding to the covariance function (Cressie 1993) using the eigenvalue decomposition of the corresponding Toeplitz covariance matrix (Vennebusch et al. 2010). The same random vector was used for each simulation.

Fig. 13

a Covariance function (Màtern family) with $\alpha =1$ by varying $\nu$ and b corresponding time series. The x axis was discretized of 200 equally spaced points. c Corresponding correlation function

The smoothness parameter $\nu$ was varied from 1/6 to 3. As $\nu$ increases, the time series are becoming less noisy for high frequency, the long periodic variations are predominating. The variance is decreasing with the smoothness parameter.

Correlation time

Figure 14 shows the influence of the parameter $\alpha$. It was varied from 0.25 to 1 by keeping $\nu$ constant to 1 to simulate short and long correlation times.

Fig. 14

a Covariance function (Mátern family) with $\nu =1$ by varying $\alpha$ and b the corresponding time series. The $x$ axis was discretized of 200 equally spaced points

Using the previous parametrization of the Mátern covariance family, the variance is not varying with the correlation time. The simulations of time series (Fig. 14b) highlight that changes of the correlation time are not acting on the smoothness of the field.

Other parametrizations

In the literature, further formulation of these covariance family functions are given (Handcock and Wallis 1994) where the parameter $\rho ,\rho >0$ is nearly independent of $\nu$:

$$\begin{aligned}&C\left( r \right) =\frac{2^{1-\nu }}{\Gamma \left( \nu \right) }\left( {\frac{\sqrt{2\nu }r}{\rho }} \right) ^{\nu }K_\nu \left( {\frac{\sqrt{2\nu }r}{\rho }} \right) ,\nonumber \\&\quad \qquad \hbox { with a spectral density of the form:}\nonumber \\&W\left( \omega \right) =\frac{2^{d}\pi ^{d/2}\Gamma \left( {\nu +d/2} \right) \left( {2\nu } \right) ^{\nu }}{\Gamma \left( \nu \right) \rho ^{2\nu }}\left( {\frac{2\nu }{\rho ^{2}}+\omega ^{2}} \right) ^{-\left( {\nu +d/2} \right) }. \end{aligned}$$

This parametrization is said to be more stable when estimating the parameters $\nu ,\alpha$ with the maximum likelihood method (Stein 1999). We made use of it to develop our model for GPS phase correlations.

Shkarofsky (1968) presented a more general form of the Mátern model by introducing a shape parameter $\delta >0$. The corresponding correlation function reads:

$$\begin{aligned} C\left( r \right) =\frac{1}{\delta ^{\nu }K_\nu \left( \delta \right) }\left( {\frac{r^{2}}{L^{2}}+\delta ^{2}} \right) ^{\nu /2}K_\nu \left( {\sqrt{\frac{r^{2}}{L^{2}}+\delta ^{2}}} \right) . \end{aligned}$$

Thus with $\delta =0$, the Mátern family is obtained.

If is half-integer, the covariance can be expressed in term of a product of an exponential and a polynomial of order p (Rasmussen and Williams 2006): for $\nu =1/2$, the exponential model is obtained whereas for $\nu =3/2,C\left( r \right) =\left( {1+\frac{\sqrt{3}r}{\rho }} \right) \mathrm{e}^{-\frac{\sqrt{3}r}{\rho }}$, which corresponds to a Markov process of second order and for $\nu =5/2$, a Markov process of third order.

Advantage of the Mátern family

The advantages of the Mátern covariance functions’ family for spatial interpolation are multiple, as developed in Stein (1999). Its flexibility to model the smoothness of physical processes (thus the rate of decay of the spectral density at high frequencies) is particularly useful, as well as the possibility to include non-stationarity or anisotropy (Fuentes 2002; Spöck and Pilz 2008). The degree of smoothness can be estimated a priori or being fixed in advance and the number of parameters to manage stays reasonable. The exponential ($\nu =\frac{1}{2})$ and Gaussian case ($\nu =\infty )$ are two particular cases of this family although the last one that represents an infinitely differentiable field is concretely rarely found (Stein 1999; Handcock and Wallis 1994).