SPDE simulation on sphere¶

by N. Desassis and D. Renard

The aim of this tutorial is to show how to use gstlearn to simulate the solution of

$$(\kappa^2-\Delta_{\mathcal{S}_R})^{\alpha/2}Z = \sigma\mathcal{W}$$

on the sphere $\mathcal{S}_R$ of radius $R$.

• $\Delta_{{\mathcal{S}_R}}$ is the Laplace-Beltrami operator, i.e, it acts on each point of the sphere as the usual Laplacian on the tangent plane at this point.

• $\kappa$ is the inverse of the scale parameter

• $\alpha \geq 2$ is an integer describing the smoothness of the solution.

• $\mathcal{W}$ is a Gaussian white-noise suitably normalized such as $\sigma^2$ is the variance of the solution.

In this notebook, we will define the covariance of Matérn on the sphere, as the covariance of the solution of this SPDE (other extensions of the Matérn function are possible). By analogy with the Euclidian case, its smoothness parameter will be defined by $\nu = \alpha -1$. To compute the covariance function with respect on the geodetic distance, one have to use a decomposition on the Legendre polynomial (see below).

We also treat the more general case $$P^{1/2}(-\Delta_{\mathcal{S}_R})Z = \sigma\mathcal{W}$$

where $P$ is a polynom positive on $\mathbb{R}^+$

Parametrization¶

Meshing¶

Sampling Db creation

Covariance model¶

The covariance model is Markov or Matérn.

Precision matrix¶

Simulation¶

Display the realization¶

Compute covariance (of discretized solution)¶

We use the fact that $\Sigma = Q^{-1}$ and solve $Qx = e_j$ for an arbitrary index $j$.

Get the distances

Compute the covariances

Sort for the plot

Display the result

Variogram of the realization¶

The empirical variogram is computed by using the great-circle distance.

Theoretical covariance function¶

The covariance between two points with great-circle distance $d$ on the sphere of radius $R$ is given by $$C(d) = \frac{\sigma^2}{\sum_{i=0}^\infty f(i)}\sum_{i=0}^\infty f(i) P_i\left(\cos \frac{d}{R}\right)$$

where the $P_i$'s are the Legendre polynomials computed with the following reccurence formula

$$P_0(x) = 1.$$$$P_1(x) = x$$$$P_{n+1}(x)=\frac{(2n+1)xP_n(x) - n P_{n-1}(x)}{n+1}$$

For $n\geq 0$, $$f(n) = \frac{2 (n + 1)}{ (R^2\kappa^2 + n ( n + 1))^\alpha}$$

For numerical computations, the sums are truncated at N.

For more details on the covariances on sphere, see Lantuejoul, Freulon and Renard (2019)

Evaluation¶

There is a slight difference between the theoretical variogram and the one obtained from the SPDE discretization due to a numerical error on the variance introduced by the discretization. The comparison of the covariance shows that this numerical error is rather small :

Kriging