# Theory and Techniques of Data Assimilation

#### grandy

##### Member
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