To avoid unnecessary data traffic it is sometimes desirable to apply mean averaging kernels to mean profiles of atmospheric state variables. Unfortunately, application of averaging kernels and averaging are not commutative in cases when averaging kernels and state variables are correlated. That is to say, the application of individual averaging kernels to individual profiles and subsequent averaging will, in general, lead to different results than averaging of the original profiles prior to the application of the mean averaging kernels unless profiles and averaging kernels are fully independent. The resulting error, however, can be corrected by subtraction of the covariance between the averaging kernel and the vertical profile. Thus it is recommended to calculate the covariance profile along with the mean profile and the mean averaging kernel
