The kriging update model and recursive space-time function estimation

We present a method for efficiently fitting a time series of spatial functions to observed data. The method is closely related to kriging, which is an interpolation method based on a stochastic data model. While kriging is effective and versatile for estimating individual functions from observed data, it must be extended to incorporate temporal correlation. In this paper, we introduce temporal correlation to kriging in the form of a stochastic state equation representation—the kriging update model. This permits a recursive solution that is akin to Kalman filtering to estimate time series of functions that avoids growing data problems associated with other spacetime extensions of kriging. The state equation representation incorporates the principle assumption of universal kriging: that the mean is deterministic but unknown. We derive the estimate using best linear unbiased estimation and state the result in a concise algorithm for general use in arbitrary spatial dimensions. To demonstrate the algorithm, we apply it to two sets of test functions and provide an example application in estimating heart motion from medical images.