Theory and Techniques of Data Assimilation

[ra_forever8]

Suppose that we wish to estimate the true temperature in a room T_t from an unbiased background estimate T_b and an unbiased observation T_o with error variances σ^2_b and σ^2_o respectively. Suppose further that the background and observation errors are correlated, with covariance ρ. We seek an estimate T_a of the temperature of the form
T_a = α T_o + (1-α) T_b,
where α is a scalar parameter.
a) Find the value of α which gives the minimum variance estimate. Verify your answer by showing that the variance of the analysis error σ^2_a is given by
σ^2_b = (σ^2_b * σ^2_o - ρ^2) / (σ^2_b + σ^2_o - 2ρ)
b) Show that if the background and observation errors are uncorrelated (ρ= 0) then the analysis error T_a - T_b is uncorrelated with the difference between the observation and the background T_o - T_b

 

[mmwave]

a) To find the value of α that gives the minimum variance estimate, we can use the principle of minimum variance. This states that the minimum variance estimate is given by the linear combination of the unbiased background and observation estimates that minimizes the variance of the resulting estimate. In this case, we have:

T_a = α T_o + (1-α) T_b

The variance of T_a can be calculated as:

σ^2_a = α^2 σ^2_o + (1-α)^2 σ^2_b + 2α(1-α) ρ

To minimize this variance, we can take the derivative with respect to α and set it equal to 0:

d(σ^2_a)/dα = 2ασ^2_o - 2(1-α)σ^2_b + 2ρ = 0

Solving for α, we get:

α = (σ^2_b - ρ) / (σ^2_b + σ^2_o - 2ρ)

Plugging this value of α back into the equation for σ^2_a, we get:

σ^2_a = (σ^2_b * σ^2_o - ρ^2) / (σ^2_b + σ^2_o - 2ρ)

b) If ρ = 0, then the background and observation errors are uncorrelated. In this case, the covariance term in the equation for σ^2_a becomes 0, and we are left with:

σ^2_a = α^2 σ^2_o + (1-α)^2 σ^2_b

The analysis error T_a - T_b can be written as:

T_a - T_b = α(T_o - T_b)

Similarly, the difference between the observation and the background T_o - T_b can be written as:

T_o - T_b = (1-α)(T_o - T_b)

Since α and (1-α) are scalar constants, the analysis error and the difference between the observation and the background are uncorrelated. This means that the estimate T_a is independent of the difference between the observation and the background, and is solely determined by the values of T_o and T_b.