# Cressman interpolation and objective analysis

I have read this question and answer – Interpolation of scattered data to a regular grid in python and I am doing something similar as I have temperature values of the atmosphere at different heights defined on a coarse grid and I want to interpolate this to a regular grid in Python and I also need to weight the matches using the Cressman weighting scheme as shown in this old paper – Cressman weighting scheme.

In this regard, I have a couple of questions. Looking at equation (1)

$$C_h = -W E_h$$

I understand what $W$ is and how to go about calculating it but it is not clear to me what the author means when he writes:

$E_h$ is the error of the interpolated value of the first guess field at the location of the observation

An error is usually a difference between two values. If that is the case which two values are being referred to here? As a first guess, I have my temperature values at irregular grid intervals and I can easily find matches within a radius of influence for a particular grid point. But which two values are contributing to the error that is being talked about here?

I have also read papers (cf. Vertical and horizontal Cressman radii) in which it is mentioned that the radius of influence that Cressman talks about in the above paper sometimes can be treated as a "deep layer" in the atmospheric sciences. In other words, the interpolated value at a grid point can include matches from not only the horizontal radius of influence on a given horizontal surface but also matches from above and below the surface in question. Is that accurate?