This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross-covariance functions on S2, referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms.

Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O di + dj and |x - y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x - y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

/pdf/construction-of-correlation-functions-in-two-and-three-5v9yuj9d1h.pdf

Construction of correlation functions in two and three dimensions

Gravitation, Gauge Theories and Differential Geometry

The problem of deriving approximations for multinormal integrals is examined using results of asymptotic analysis. The boundary of the integration domain given by g(x¯)=0 is simplified by replacing g(x¯) by its Taylor expansion at the points on the boundary with minimal distance to the origin. Two approximations which are obtained by using a linear or quadratic Taylor expansion are compared. It is shown that, applying a quadratic Taylor expansion, an asymptotic approximation for multinormal integrals can be obtained, whereas using linear approximations large relative errors may occur.

Asymptotic approximations for multinormal integrals

Estimating principal curvatures and principal directions of a surface from a polyhedral approximation with a large number of small faces, such as those produced by iso-surface construction algorithms, has become a basic step in many computer vision algorithms, particularly in those targeted at medical applications. We describe a method to estimate the tensor of curvature of a surface at the vertices of a polyhedral approximation. Principal curvatures and principal directions are obtained by computing in closed form the eigenvalues and eigenvectors of certain 3/spl times/3 symmetric matrices defined by integral formulas, and closely related to the matrix representation of the tensor of curvature. The resulting algorithm is linear, both in time and in space, as a function of the number of vertices and faces of the polyhedral surface. >

/pdf/estimating-the-tensor-of-curvature-of-a-surface-from-a-2k4m2vorqz.pdf

Estimating the tensor of curvature of a surface from a polyhedral approximation

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

System Theory on Group Manifolds and Coset Spaces

Let S be an n-surfacc in ℝ n +1, oriented by the unit normal vcctor field N, and let p ∈ S. The Weingarten map L p : S p → S p , defined by L p (v) = − ∇vN for v ∈ S p , measures the turning of the normal as one moves in S through p with various velocities v. Thus L p measures the way S curves in ℝ n +1 at p. For n = 1, we have seen that L p is just multiplication by a number K(p) the curvature of S at p. We shall now analyze L p when n > 1.

Curvature of Surfaces

In this chapter we shall develop some machinery which we shall need in the next chapter to prove one of the most celebrated theorems in differential geometry, the Gauss-Bonnet theorem. We shall first discuss n-surfaces-with-boundary. Then we shall develop a little bit of the differential calculus of forms.

Surfaces with Boundary

Let C = f−1(c), where f: U → ℝ, be a plane curve in the open set U ⊂ ℝ2, oriented by N = ∇f/∥∇f∥. Then, for each p ∈ C. the Weingartcn map Lp, is a linear transformation on the 1-dimensional spacc Cp. Sincc every linear transformation from a 1-dimensional space to itself is multiplication by a real number, there exists, for each p ∈ C, a real number K(p) such that 

$$ {L_p}\left( v \right)\, = \,\kappa \left( p \right)\nu \,for\,all\,\nu \, \in \,{C_p} $$

. K(p) is called the curvature of C at p.

Curvature of Plane Curves

Associated with each real valued function of several real variables is a collection of sets, callcd level sets, which are useful in studying qualitative properties of the function. Given a function f: U → ℝ. where U ⊂ ℝ n +1,its level sets are the sets f-1 (c) defined, for each real number c, by 

$$ {f^{ - 1}}\,\left( c \right)\, = \,\left\{ {\left( {{X_1},...,\,{X_{n + 1}}} \right)\, \in U:\,f\,\left( {{X_1},...,\,{X_{n + 1}}} \right)\, = \;c} \right\} $$

. The number c is called the height of the level set, and f -1 (c) is called the level set at height c. Since f-l(c) is the solution set of the equation f(x1…, x n +1) = c, the level set f -1 (c) is often described as “the set f(x1,…, x n +1) = c.”

Graphs and Level Sets

Contents: Graphs and Level Sets- Vector Fields- The Tangent Space- Surfaces- Vector Fields on Surfaces Orientation- The Gauss Map- Geodesics- Parallel Transport- The Weingarten Map- Curvature of Plane Curves- Arc Length and Line Integrals- Curvature of Surfaces- Convex Surfaces- Parametrized Surfaces- Local Equivalence of Surfaces and Parametrized Surfaces- Focal Points- Surface Area and Volume- Minimal Surfaces- The Exponential Map- Surfaces with Boundary- The Gauss-Bonnet Theorem- Rigid Motions and Congruence- Isometries- Riemannian Metrics

Elementary Topics in Differential Geometry

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