Change of Support in Spatial Variance-Based Sensitivity Analysis

Abstract

Variance-based global sensitivity analysis (GSA) is used to study how the variance of the output of a model can be apportioned to different sources of uncertainty in its inputs. GSA is an essential component of model building as it helps to identify model inputs that account for most of the model output variance. However, this approach is seldom applied to spatial models because it cannot describe how uncertainty propagation interacts with another key issue in spatial modeling: the issue of model upscaling, that is, a change of spatial support of model output. In many environmental models, the end user is interested in the spatial average or the sum of the model output over a given spatial unit (for example, the average porosity of a geological block). Under a change of spatial support, the relative contribution of uncertain model inputs to the variance of aggregated model output may change. We propose a simple formalism to discuss this issue within a GSA framework by defining point and block sensitivity indices. We show that the relative contribution of an uncertain spatially distributed model input increases with its correlation length and decreases with the size of the spatial unit considered for model output aggregation. The results are briefly illustrated by a simple example.

Appendices

Appendix A: Proof of the Relation Between Site Sensitivity Indices and Block Sensitivity Indices

As mentioned in Sect. 2, we assume that the first two moments of Y(x) exist. The ratio of block sensitivity indices gives (Eq. (5))

$$ \frac{S_{Z}(v)}{S_{\mathbf{U}}(v)} = \frac{\operatorname{Var} ( \mathbb{E} [Y_v\mid Z(\mathbf{x}) ] )}{\operatorname{Var} ( \mathbb{E} [Y_v\mid \mathbf{U} ] )}. $$

(14)

The conditional expectation of block average Y _v given Z(x) gives

Thus, we have $\operatorname{Var} ( \mathbb{E} [Y_{v}\mid Z ] ) = \operatorname{Var} ( 1/\vert v\vert\int_{v}\mathbb{E}_{Z}Y(\mathbf {x})\,d\mathbf {x} ) = \sigma^{2}_{v}$ (definition of $\sigma^{2}_{v}$). Moreover, the conditional expectation of block average Y _v given input U gives

$\mathbb{E} [ Y(\mathbf{x})\mid\mathbf{U} ]$ does not depend on site x under the stationarity of SRF Z(x); thus, we have in particular $\mathbb{E} [ Y_{v}\mid\mathbf{U} ] = \mathbb {E} [ Y(\mathbf{x}^{\ast})\mid\mathbf{U} ]$, and $\operatorname{Var} ( \mathbb{E} [Y_{v}\mid\mathbf{U} ] ) = \operatorname{Var} ( \mathbb{E} [Y(\mathbf{x}^{\ast})\mid \mathbf{U} ] )$. Combining these expressions with Eq. (14) yields

$$ \frac{S_{Z}(v)}{S_{\mathbf{U}}(v)} = \frac{ \sigma^2_{v}}{\operatorname{Var}(\mathbb{E} [ Y(\mathbf {x}^\ast) \mid\mathbf{U} ])}. $$

(15)

The ratio of site sensitivity indices gives (Eq. (4))

$$ \frac{S_{Z}}{S_{\mathbf{U}}} = \frac{ \operatorname{Var}(\mathbb{E} [ Y(\mathbf{x}^\ast)\mid Z(\mathbf{x}) ])}{\operatorname{Var}(\mathbb{E} [ Y(\mathbf{x}^\ast)\bigm| \mathbf{U} ])}. $$

(16)

We notice that for point-based models $\operatorname{Var} [ \mathbb {E}(Y(\mathbf{x}^{\ast})\mid Z(\mathbf{x})) ] = \operatorname{Var} [ \mathbb{E}_{Z}Y(\mathbf{x}^{\ast}) ] = \sigma^{2}$ (definition of $\mathbb{E}_{Z}Y(\mathbf{x})$ (Eq. (7))). Finally, it follows from Eqs. (15) and (16) that

$$ \frac{S_{Z}(v)}{S_{\mathbf{U}}(v)} =\frac{S_{Z}}{S_{\mathbf{U}}} \cdot\frac{\sigma^2_{v}}{\sigma^2}. $$

Appendix B: Hermitian Expansion of Random Field $\mathbb {E}_{Z}Y(\mathbf {x})$

The random field $\mathbb{E}_{Z}Y(\mathbf{x})$ can be written (Eqs. (2), (7)) as a transformation of the Gaussian random field Z(x) through the function $\bar{\psi}: z \mapsto\int_{\mathbb{R}^{n}} \psi (\mathbf{u},z) \cdot f_{U}(\mathbf{u}) \, d\mathbf{u}$

$$\mathbb{E}_{Z}Y= \bar{\psi} ( Z ), $$

where f _U (⋅) is the multivariate pdf of random vector U. Under the hypothesis that the first two moments of Y(x) exist, random field $\mathbb{E}_{Z}Y(\mathbf{x})$ has finite expected value and finite variance. Thus, $\bar{\psi}$ belongs to the Hilbert space $L^{2}(\mathcal {G})$ of functions ϕ:ℝ→ℝ, which are square-integrable with respect to Gaussian density g(.). Hence, $\bar {\psi}$ can be expanded on the sequence of Hermite polynomials (χ _k ) _k∈ℕ, which forms an orthonormal basis of $L^{2}(\mathcal{G})$ (Chilès and Delfiner 1999)

$$\bar{\psi} = \sum _{k=0}^{\infty} \alpha_k \cdot\chi_k \quad \mbox{with~} \chi_k(z) = \frac{1}{\sqrt{k!} } \cdot\frac {1}{g(z)} \cdot \frac{\partial^k }{\partial z^k}g(z), $$

where coefficients α _k are given by: $\alpha_{k} = \int_{\mathbb{R}} \chi_{k}(z) \bar{\psi}(z) g(z) \, dz$. It follows that $\mathbb{E}_{Z}Y (\mathbf{x})$ can be written as an infinite expansion of polynomials of Z(x)

$$\forall\mathbf{x}\in\mathcal{D}, \quad\mathbb{E}_{Z}Y(\mathbf {x})= \sum _{k=0}^{\infty} \alpha_k \cdot\chi_k \bigl[Z(\mathbf{x}) \bigr]. $$

Its covariance then gives (Chilès and Delfiner 1999)

$$\operatorname{Cov} \bigl(\mathbb{E}_{Z}Y(\mathbf{x}), \mathbb {E}_{Z}Y(\mathbf{x}+ \mathbf {h}) \bigr) = \sum _{k=0}^{\infty} \alpha_k^2 \cdot \biggl[\frac{ C(\mathbf{h})}{C(0)} \biggr]^k = \sum _{k=0}^{\infty} \lambda_k^2 \cdot \bigl[ C(\mathbf{h} ) \bigr]^k, $$

where C(h) is the covariance function of GRF Z(x) and λ _k =α _k ⋅C(0)^−k/2.