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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 R2 with gradient109
∇f. The aspect direction corresponding to f at s, denoted by Uasppsq : D B, whereB is the unit circle, is defined asaspp q}∇fpsq}qUs.(5.4)∇fpsThe angle associated with Uasppsq is denoted by θasppsq and is defined byasppDe2fpsqtanθsqe1 p q.(5.5)D f sThe (steepest) slope associated with f at s, denoted by Rpsq : D R is defined asRpsq}∇fpsq}.(5.6)Again, as an angle,tanγasppsqRpsq.(5.7)Figure 5.1 provides both 3-D and 2-D mesh plots of a fixed function withadditional layers of arrow plots representing its corresponding aspect surface(arrow directions) and associated slopes (length of arrows). The function evaluatedhere is fpsq ppcosps1π{180qq2 pcosps2π{180qq2q2, where s P r0,90s r0,90s. Theslopes and aspects can be calculated analytically.5.3.2Adding stochasticity; directional derivative processesGradients and more generally linear functionals of stochastic processes have beenstudied extensively; refer to Adler (2009) for a comprehensive review of theliterature. Banerjee et al. (2003) laid down an inferential framework for directionalgradients on a spatial surface in a fully Bayesian framework, enabling inference on110
(possibly non-linear) functionals over the entire spatial domain with associateduncertainty quantification through the posterior distribution. See also Banerjee andGelfand (2003) for detailed distribution theory on Gaussian processes and itsgradients.We now set up the basic distributional theoretical background to define andstudy aspect and slope processes as continuous stochastic processes over a spatialdomain. First, we review stochastic extensions of the notions of directionalderivative and total derivative discussed in the previous subsection from Banerjeeet al. (2003). We restrict our attention to Gaussian processes with isotropiccovariance kernels, noting that many of the definitions and results can be extendedmore generally.(a)3-dimensional view111
(b)2-dimensional viewFigure 5.1: Display of the surface of a fpsq with associated gridded slope and aspectsurface. The arrow indicates the aspect direction, the length of the arrow indicatesthe associated slopeLet Y psq be a mean zero stationary Gaussian process (GP) with isotropiccovariance function of the form Cps,s1q Kp||s s1||q on a domain D R2. Twocommonly used choices are the power exponential family Kptq σ2 expp tφνq, 0 ⁄ ν⁄2 and the Mat´ern family Kptq σ2ptφqνKνptφq, with Kνbeing the modified Besselfunction of the second kind of order ν. The stationarity and isotropy assumptionsare used mainly to simplify expressions for the induced covariance structure for thegradients and the definitions below hold more broadly.