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1 Introduction

Consider a physical process or system which can be described roughly by a finite set of parameters ⃗x and ⃗ b, such as the radiative transfer in the atmosphere and the optical properties in disjunct layers of the atmosphere confined into the vectors ⃗x and ⃗b. We distinguish between parameters of interest ⃗x and those ⃗b which are required to describe “the rest” of the system. The vector ⃗x is called state and shall be obtained by measurements. Sometimes there is prior knowledge about the state. Then an expectation ⃗xa (’a’ stands for a priori) and a standard deviation ⃗σa which quantifies the natural variability of ⃗x (climatology) is given. Using an adequate instrument, the outcome of the measurements will be dominated by the physics of the process. One writes

⃗y = ⃗f(⃗x,⃗b)
(1)

where ⃗y is the measurement vector with the measurement values as components and ⃗ f is the forward function which symbolises the physics of the process. In reality ⃗y often also depends on parameters of the instrument but for the present considerations it is assumed to have a perfect instrument. Besides ⃗y, a vector ⃗σɛ is obtained, which is the standard deviation of ⃗y.

1.1 Forward Model

In order to infer parts of the state from measurements the physics of the process f⃗ has to be understood. In detail, ⃗f in eq. 1 has to be approximated by a forward model ⃗F:

⃗y ≈ ⃗F (⃗x,⃗b)
(2)

The forward model should be able to simulate the outcome of the measurement at least with an accuracy better than the measurement error. In the following, the dependence of the forward model on the parameters ⃗ b is ommitted in the notation and ⃗F(⃗x) is used instead. But it is emphasized, that knowledge of ⃗b is needed to model the process correctly.

1.2 Cost Function

Assume that M measurements have been made and the state to be retrieved has N components. The cost function χ2(⃗x) is a measure of the deviation of the simulated measurement from the real measurement and is defined as

M∑ ( F (⃗x) - y )2 N∑ ( x - x )2 χ2(⃗x) = -m--------m- + -n----a,n- . m=0 σ ɛ,m n=0 σa,n
(3)

To reflect the accuracy of the measurement respectively the prior knowledge each summand is divided by the corresponding standard deviation σ. The inverse problem is then a χ2-optimisation problem, i.e. a state ˆx that minimises the cost function has to be found. If the a priori profile is considered, the resulting state is the best (optimal) compromise between the a priori knowledge and the measurement. But as will be seen later in an example the a priori information is only used where it is needed.

It is expedient to rewrite the cost function in a more compact form:

χ2(⃗x) = [⃗F(⃗x ) - ⃗y]TS -ɛ1[⃗F(⃗x) - ⃗y] + [⃗x - ⃗xa ]TS -a1[⃗x - ⃗xa].
(4)

The measurement covariance (M × M) and a priori covariance (N × N) matrices contain the squares of the standard deviations ⃗σɛ and ⃗σa on the diagonal:

Sɛ,mm = σ2ɛ,m and Sa,nn = σ2a,n
(5)

When the off-diagonal elements are filled with zeros than eqs. 3 and 4 are formally equal. There exists a class of inversion methods which fill the off-diagonal elements with non-zero values in order to get a smooth result (regularisation). The criterium which ˆx has to fullfill to minimise the cost function, i.e. to fit the simulated ⃗F(xˆ) to the measurement ⃗y is

⃗∇ χ2(ˆx) = 0 i.e. ⃗∇F⃗(⃗x)T S-1[⃗F (⃗x ) - ⃗y] + S- 1[⃗x - ⃗x ] = 0 ɛ a a
(6)

Depending on the nature of the relation eq. 1 there are two groups of inversion schemes: linear one-step and non-linear iterative retrievals.

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