An adaptive spatial model for precipitation data from multiple satellites over large regions

Abstract

Satellite measurements have of late become an important source of information for climate features such as precipitation due to their near-global coverage. In this article, we look at a precipitation dataset during a 3-hour window over tropical South America that has information from two satellites. We develop a flexible hierarchical model to combine instantaneous rainrate measurements from those satellites while accounting for their potential heterogeneity. Conceptually, we envision an underlying precipitation surface that influences the observed rain as well as absence of it. The surface is specified using a mean function centered at a set of knot locations, to capture the local patterns in the rainrate, combined with a residual Gaussian process to account for global correlation across sites. To improve over the commonly used pre-fixed knot choices, an efficient reversible jump scheme is used to allow the number of such knots as well as the order and support of associated polynomial terms to be chosen adaptively. To facilitate computation over a large region, a reduced rank approximation for the parent Gaussian process is employed.

Appendix:  Marginalizing out ν y and \(\sigma^{2}_{y}\) for estimation of spline parameters in μ y (s)

Denote by … all parameters except $\nu,\sigma^{2}_{y}$. Let P=[ϕ ₁[x(s)],ϕ ₂[x(s)],…,ϕ _k [x(s)]], S=y(s)−Pν _y . We have,

$$\begin{aligned} &p\bigl(y(\mathbf{s})|\ldots \bigr) \\ &\quad \propto \int_{\nu_y} \int _{\sigma_y^2} p\bigl(y(\mathbf{s}) | \nu_y, \sigma_y^2,\ldots\bigr) p\bigl(\nu_y | \sigma^2_y\bigr) p\bigl(\sigma^2_y \bigr) d\sigma_y^2 d\nu_y, \\ &\quad \propto \bigl(2\pi\tau^2_y\bigr)^{-k/2} \int _{\nu_y} \int_{\sigma^2_y} \bigl( \sigma^2_y\bigr)^{-\frac{n+k}{2} - a_\sigma-1 } \\ &\qquad {}\times\exp \biggl[ - \frac{1}{2 \sigma^2_y} \bigl( S^T D^{-1}S + \nu_y^T \nu_y/\tau^2_y + 2 b_\sigma\bigr) \biggr] d\sigma^2_y d \nu_y, \\ &\quad \propto \bigl(2\pi\tau^2_y\bigr)^{-k/2} \varGamma\biggl( \frac{n}{2} + a_\sigma\biggr) \\ &\qquad {} \int _{\nu_y} \biggl( \frac{S^T D^{-1}S + \nu_y^T \nu_y /\tau^2_y}{2} + b_\sigma \biggr)^{-\frac{n+m+k}{2} - a_\sigma} d\nu_y. \end{aligned}$$

Now write $S^{T} D^{-1}S + \nu_{y}^{T} \nu_{y} = \nu_{y}^{T} A \nu_{y} - 2 \nu_{y}^{T} B + C$, where $A = P^{T}D^{-1}P + \frac{ I_{k}}{\tau^{2}_{y}}$, B=P ^T D ⁻¹ S _y , $C = S_{y}^{T} D^{-1}S_{y}$. Then we have, $S^{T} D^{-1}S + \nu_{y}^{T} \nu_{y} + 2 b_{\sigma}= (\nu_{y} - \mu_{k})^{T} \varSigma ^{-1}_{k} (\nu_{y} - \mu_{k}) + c_{0k}$, where μ _k =A ⁻¹ B,Σ _k =A ⁻¹,c _0k =C−b ^T A ⁻¹ b+2b _σ . Denote d=n+2a _σ . Then

$$\begin{aligned} &p\bigl(y(\mathbf{s})|\ldots\bigr) \\ &\quad \propto \bigl(\pi\tau^2_y \bigr)^{-k/2} c_{0k}^{-\frac {d+k}{2} }\varGamma\biggl( \frac{d+k}{2} \biggr) \int_{\nu_y} \biggl[ \frac{1}{d} (\nu_y - \mu_k)^T \\ &\qquad {}\times \biggl( \frac{ c_{0k} \varSigma_k}{d} \biggr)^{-1} (\nu_y-\mu _k) + 1 \biggr]^{-\frac{d+k}{2} } d\nu_y. \end{aligned}$$

The integrand is the pdf (up to a constant) for the k-variate t distribution with mean μ _k , dispersion $\frac{ c_{0k} \varSigma_{k}}{d}$ and degrees of freedom d. Hence, we obtain the closed form expression for $p(y(\mathbf{s}) |\ldots) \propto(\tau^{2}_{y})^{-k/2} c_{0k}^{-\frac{d}{2} } |\varSigma_{k}|^{1/2}$.