Filling in gaps in the observations

Historical record provides valuable information for evaluating the performance of climate models with respect to the observed changes. However, especially early historical observations are available only for sparse regions. ML can help with filling in the gaps in observations to provide a complete record for different climate variables, such as ocean carbon uptake or surface air temperature using neural networks, Kriging  , or Empirical Orthogonal Functions.

Kriging
Kriging is a statistical method of spatial prediction that (usually) relies on the second-order properties of the process analyzed. The aim of this method is to predict the value of our process at any location of interest $$s_0$$ within a domain $$D$$, using the available information of our sample. Under specific assumptions, it will be the Best Linear Unbiased Predictor (BLUP), e.g. the estimator with the minimal error variance. The main idea of Kriging is that nearby locations will tend to be similar to those more distant, which is a prediction framework it translates into a higher relevance to those nearby observations. One of the most common formulations of the Kriging estimator is:

$$\hat{Z}(s_0) = \sum_{j=1}^{n} \lambda_j Z(s_j)$$

This means that the value at the prediction location $$s_0$$ will be a linear combination of our sample. The parameters $$\lambda_j$$ aim to encode this 'relevance ' for the observation $$Z(s_j)$$ over the location prediction $$\hat{Z}(s_0)$$. Clearly, this encoding should (and can) consider the spatial dependence structure of our process, which makes this method stand out from others e.g. an inverse distance weighting approach. The assumptions and restrictions we made over our process and over the parameters $$\bold{\lambda}$$ respectively, will define different variations of Kriging. The most commons are Simple Kriging, Ordinary Kriging, and Universal Kriging.

A bit of history
This method was first employed empirically by Danie G. Krige, a South African mining engineer, on the task of determining the ore grades of panels, highly relevant since the cost of its extraction should be considerably lower than the value of the metals on the panels extracted. The work done by Krige wrapped up with his Master’ Thesis at the University of the Witwatersrand called: 'A statistical approach to some mine valuation and allied problems on the Witwatersrand ' in 1951. The generalization of this method was done by the French Engineer Georges Matheron nine years later, assigning proper weights to each sample, where these weights are determined to minimize the estimation variance under a set of constraints.