31 Definition 531 Let f be a totally differentiable function on D R 2 with

31 definition 531 let f be a totally differentiable

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observations in the following Definition 5.3.1. Definition 5.3.1. Let f be a totally differentiable function on D R 2 with gradient 109
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f. The aspect direction corresponding to f at s , denoted by U asp p s q : D B , where B is the unit circle, is defined as asp p q } f p s q}q U s . (5.4) f p s The angle associated with U asp p s q is denoted by θ asp p s q and is defined by asp p D e 2 f p s q tan θ s q e 1 p q . (5.5) D f s The (steepest) slope associated with f at s , denoted by R p s q : D R is defined as R p s q } f p s q} . (5.6) Again, as an angle, tan γ asp p s q R p s q . (5.7) Figure 5.1 provides both 3-D and 2-D mesh plots of a fixed function with additional layers of arrow plots representing its corresponding aspect surface (arrow directions) and associated slopes (length of arrows). The function evaluated here is f p s q pp cos p s 1 π { 180 qq 2 p cos p s 2 π { 180 qq 2 q 2 , where s P r 0 , 90 s r 0 , 90 s . The slopes and aspects can be calculated analytically. 5.3.2 Adding stochasticity; directional derivative processes Gradients and more generally linear functionals of stochastic processes have been studied extensively; refer to Adler (2009) for a comprehensive review of the literature. Banerjee et al. (2003) laid down an inferential framework for directional gradients on a spatial surface in a fully Bayesian framework, enabling inference on 110
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(possibly non-linear) functionals over the entire spatial domain with associated uncertainty quantification through the posterior distribution. See also Banerjee and Gelfand (2003) for detailed distribution theory on Gaussian processes and its gradients. We now set up the basic distributional theoretical background to define and study aspect and slope processes as continuous stochastic processes over a spatial domain. First, we review stochastic extensions of the notions of directional derivative and total derivative discussed in the previous subsection from Banerjee et al. (2003). We restrict our attention to Gaussian processes with isotropic covariance kernels, noting that many of the definitions and results can be extended more generally. (a) 3-dimensional view 111
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(b) 2-dimensional view Figure 5.1 : Display of the surface of a f p s q with associated gridded slope and aspect surface. The arrow indicates the aspect direction, the length of the arrow indicates the associated slope Let Y p s q be a mean zero stationary Gaussian process (GP) with isotropic covariance function of the form C p s , s 1 q K p|| s s 1 ||q on a domain D R 2 . Two commonly used choices are the power exponential family K p t q σ 2 exp p t φ ν q , 0 ν 2 and the Mat´ern family K p t q σ 2 p t φ q ν K ν p t φ q , with K ν being the modified Bessel function of the second kind of order ν . The stationarity and isotropy assumptions are used mainly to simplify expressions for the induced covariance structure for the gradients and the definitions below hold more broadly.
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