Statistical errors in variational data assimilation - A theoretical one-dimensional analysis applied to Doppler wind retrieval

When the velocity field is retrieved by a variational data assimilation method from the tracer (reflectivity) pattern movement observed by a Doppler radar, the accuracy of the retrieved velocity is affected by the observational error and data resolution. The statistical aspect of this problem is examined theoretically with a one-dimensional tracer advection equation. It is found that the retrieved velocity is generally underestimated if the signal-to-noise ratio is less than 1.0 for the gradient of the observed tracer field; so there is a systematic error. This systematic error is caused by the nonlinear response of the retrieved velocity to the observed data. In addition, there is also a nonsystematic error in the retrieved velocity caused by the observational errors in the tracer field and tracer gradient field. These two types of error can be reduced by using a moving reference frame and by imposing a spatial smoothness constraint on the retrieved field.