upwardspatial

Upwardspatial is a term used in theoretical and applied contexts to describe a property of spatial fields in which values increase or do not decrease along an upward direction. The concept emphasizes monotonicity with respect to a chosen vertical orientation rather than general isotropy.

Formalization: Let X be a spatial domain in R^n, and let d be a unit vector pointing upward. A scalar field f: X -> R is called upwardspatial if for all x in X and all t > 0 with x + t d in X, f(x) <= f(x + t d). In 2D, d may be (0,1) and the condition reduces to f(x, y) <= f(x, y+ t). The notion can be generalized using a convex cone C to define an order: f is C-monotone if f(x) <= f(y) whenever y - x in C.

Etymology and usage: The term is descriptive and not universally standardized, but it is used to describe vertical stratification phenomena where a quantity exhibits non-decreasing behavior with height. It is distinct from isotropy or general monotonicity, focusing on directionally dependent behavior along the upward axis.

Applications: In geography and meteorology, upwardspatial modeling helps represent vertical gradients in variables such as temperature, humidity, or pollutant concentration with altitude. In computer graphics and vision, it can describe how lighting, texture, or features vary with height in a scene. In spatial statistics and environmental science, upwardspatial assumptions can inform regression or kriging models when elevation or height is a primary predictor.

See also: monotone function, isotonic regression, spatial analysis, vertical profiling, gradient.