Rossby Waves and Frontal Waves

Objective analysis [scalar fields] We would like to estimate a property S(x) at a point x given a set of observations s(xi ) (containing errors) of the property at other spatial points. In the absence of other information (climatology, for example), we assume that the data mean represents the true mean and use a linear estimator for the deviations � s
(xi )a(xi , x) S̃
(x) = or, using summation convention, S̃
(x) = s
(xi )a(xi , x) (1) The problem now becomes the choice of a. We form an error estimate 1

[S̃ (x) − S
(x)]2  2 1 1 =
s
(xi )s
(xj )a(xi , x)a(xj , x) −
S
(x)s
(xi )a(xi , x) +
S
(x)S
(x) 2 2 = (2) We seek the minimum error with respect to the values of the coefficients a(xi , x) ∂ =0 ∂a(xi , x) which implies
s
(xi )s
(xj )a(xj , x) =
S
(x)s
(xi ) (3) The symmetry of
s
(xi )s
(xj ) has been used. We write this in terms of the covariance for the field C(x − x
) =
S
(x)S
(x
) assuming that the measurement noise is uncorrelated and has variance σ 2 [C(xi − xj ) + σ 2 δij ]a(xj , x) = C(xi − x) If we know or can approximate the covariance function, we can set up and solve this linear system to give a(xi , x) for any target point x a(xj , x) = [C(xi − xj ) + σ 2 δij ]−1 C(xi − x) (4) Not only can we substitute this in (1) to find the estimated field at x, we can also get an estimate of the error by using (3) and (4) in (2) = 1 1 C(0) − C(x − xj )[C(xi − xj ) + σ 2 δij ]−1 C(xi − x) 2 2 Note that the errors depend only on the sampling positions and C, σ. Therefore we can design sampling strategies given an estimate of the covariance and the noise.