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2 Linear Retrieval

In the case of a linear forward function (approximated by a linear forward model), one may write

⃗F (⃗x) = K ⃗x thus ⃗∇ ⃗F(⃗x ) = K.
(7)

The minimisation problem then is:

KT S-ɛ1[K ⃗x - ⃗y] + S -a1[⃗x - ⃗xa] = 0
(8)

and is readily solved for ⃗x:

T -1 - 1-1 T -1 -1 ˆx = [K S ɛ K + Sa ] [K S ɛ ⃗y + Sa ⃗xa].
(9)

2.1 Example

To demonstrate the method, the linear retrieval of a NO2 vertical profile performed using simulated data (fig. 2.1). The measurement vector consists of NO2 SCD’s which would be measured during the ascent of a ballon with an instrument looking horizontally into the layer of the ballon height. The forward model was used to simulate the measurement. Then the profile was altered and the inversion scheme was tested using the altered profile as a priori input. As shown in the figure, the algorithm reproduces the real profile which was used to simulate the measurement. If the tropospheric region is magnified, it is observed that the algorithm tends to return the a priori profile information in regions where the overlap of the real profile and the a priori expectation is small.

PIC PIC
Figure 1: Simulated NO2 vertical profile retrieval. The right panel shows the troposphere only.

2.2 Averaging Kernel

For a sensitivity analysis, two questions may be asked: How much does the result ˆx depend on the a priori state and on the real state of the atmosphere? The former part of the question is answered by the derivative

∂xˆ-= [KT S -1K + S-1]-1S- 1 ∂⃗xa ɛ a a
(10)

The trace of the resulting matrix provides the number of parameters which are introduced by the a priori state vector. It is eligible, that this number tends to zero. Vice versa, and answering the latter question the averaging kernel matrix is obtained by assuming ⃗y = K⃗xreal in eq. 9

∂ ˆx ----- = [KT S -ɛ1K + S-a1]-1KT S -ɛ1K ∂ ⃗xreal
(11)

The trace of the averaging kernel matrix quantifies the information content of the measurement and is the number of components of ˆx which could be retrieved independently from the a priori state. A more detailed discussion on averaging kernels and information content can be found in [Rodgers 2000].

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