When I learned “Kalman filter” for the first time, I was not sure how to distinguish it from “Yule-Walker equation” (time series), “Pade approximant, (unfortunately, the wiki page does not have its matrix form). Wiener Filter” (signal processing), etc. Here are those publications, specifically mentioned the name Kalman filter in their abstracts found from ADS.

• Application of Data Assimilation Method for Predicting Solar Cycles (2008) in ApJ
• Time series analysis in astronomy: Limits and potentialities (2005) in A&A
  From the abstract: Only techniques of data aalysis developed in a specific physical context can be expected to provide useful results. The filed of stochastic dynamics appears to be an interesting framework for such an approach.
• Determination of the Mass of Jupiter Using the Motion of Its Ninth Satellite and a Kaiman-Bucy Filter (1972) in A&A

The motivation of introducing Kalman filter although it is a very well known term is the recent Fisher Lecture given by Noel Cressie at the JSM 2009. He is the leading expert in spatial statistics. He is the author of a very famous book in Spatial Statistics. During his presentation, he described challenges from satellite data and how Kalman filter accelerated computing a gigantic covariance matrix in kriging. Satellite data of meteorology and geosciences may not exactly match with astronomical satellite data but from statistical modeling perspective, the challenges are similar. Namely, massive data, streaming data, multi dimensional, temporal, missing observations in certain areas, different exposure time, estimation and prediction, interpolation and extrapoloation, large image size, and so on. It’s not just focusing denoising/cleaning images. Statisticians want to find the driving force of certain features by modeling and to perform statistical inference. (They do not mind parametrization of interesting metric/measure/quantity for modeling or they approach the problem in a nonparametric fashion). I understood the use of Kalman filter for a fast solution to inverse problems for inference.

There are three distinctive subjects in spatial statistics: geostatistics, lattice data analysis, and spatial point pattern analysis. Because of the resemblance between the spatial distribution of observations in coordinates and the notion of spatially random points, spatial statistics in astronomy has leaned more toward the spatial point pattern analysis than the other subjects. In other fields from immunology to forestry to geology whose data are associated spatial coordinates of underlying geometric structures or whose data were sampled from lattices, observations depend on these spatial structures and scientists enjoy various applications from geostatistics and lattice data analysis. Particularly, kriging is the fundamental notion in geostatistics whose application is found many fields.

Hitherto, I expected that the term kriging can be found rather frequently in analyzing cosmic micro-wave background (CMB) data or large extended sources, wide enough to assign some statistical models for understanding the expected geometric structure and its uncertainty (or interpolating observations via BLUP, best linear unbiased prediction). Against my anticipation, only one referred paper from ADS emerged:

Topography of the Galactic disk – Z-structure and large-scale star formation
by Alfaro, E. J., Cabrera-Cano, J., and Delgado (1991)
in ApJ, 378, pp. 106-118

I attribute this shortage of applying kriging in astronomy to missing data and differential exposure time across the sky. Both require underlying modeling to fill the gap or to convolve with observed data to compensate this unequal sky coverage. Traditionally the kriging analysis is only applied to localized geological areas where missing and unequal coverage is no concern. As many survey and probing missions describe the wide sky coverage, we always see some gaps and selection biases in telescope pointing directions. So, once this characteristics of missing is understood and incorporated into models of spatial statistics, I believe statistical methods for spatial data could reveal more information of our Galaxy and universe.

A good news for astronomers is that nowadays more statisticians and geo-scientists working on spatial data, particularly from satellites. These data are not much different compared to traditional astronomical data except the direction to which a satellite aims (inward or outward). Therefore, data of these scientists has typical properties of astronomical data: missing, unequal sky coverage or exposure and sparse but gigantic images. Due to the increment of computational power and the developments in hierarchical modeling, techniques in geostatistics are being developed to handle these massive, but sparse images for statistical inference. Not only denoising images but they also aim to produce a measure of uncertainty associated with complex spatial data.

For those who are interested in what spatial statistics does, there are a few books I’d like to recommend.

• Cressie, N (1993) Statistics for spatial data
  (the bible of statistical statistics)
• Stein, M.L. (2002) Interpolation of Spatial Data: Some Theory for Kriging
  (it’s about Kriging and written by one of scholarly pinnacles in spatial statistics)
• Banerjee, Carlin, and Gelfand (2004) Hierarchical Modeling and Analysis for Spatial Data
  (Bayesian hierarchical modeling is explained. Very pragmatic but could give an impression that it’s somewhat limited for applications in astronomy)
• Illian et al (2008) Statistical Analysis and Modelling of Spatial Point Patterns
  (Well, I still think spatial point pattern analysis is more dominant in astronomy than geostatistics. So… I feel obliged to throw a book for that. If so, I must mention Peter Diggle’s books too.)
• Diggle (2004) Statistical Analysis of Spatial Point Patterns
  Diggle and Ribeiro (2007) Model-based Geostatistics