Revisiting the Desroziers Diagnostic Across Gaussian and SOAR-Type Spatial Correlation Regimes

The Desroziers diagnostic (D05) is widely used to estimate observation error covariances, but its behavior under different spatial correlation structures remains insufficiently understood. Through twin experiments with a modified shallow water model, this study evaluates D05 across Gaspari-Cohn, uncorrelated, SOAR-type, and mixed Gaspari-Cohn-SOAR spatial correlation regimes. D05 reproduces the prescribed correlation shape well in the Gaspari-Cohn and uncorrelated cases, while small variance biases remain when the assimilation system uses a diagonal observation-error covariance matrix. Under the SOAR-type correlation settings considered here, D05 produces longer diagnosed correlation scales than the prescribed curves, and this positive length-scale bias increases in the mixed Gaspari-Cohn-SOAR experiments. These results indicate that D05 is sensitive to the functional form of the prescribed spatial correlation and to covariance misspecification in the assimilation system. The findings provide guidance for interpreting D05-derived observation-error correlations in atmospheric data assimilation, especially when the true observation-error dependence structure differs from conventional compact-support correlation models.