Isoline retrieval is a remote sensing inverse method that retrieves one or more isolines of a trace atmospheric constituent or variable. When used to validate another contour, it is the most accurate method possible for the task. When used to retrieve a whole field, it is a general, nonlinear inverse method and a robust estimator.
For validating advected contours
Rationale Suppose we have, as in contour advection, inferred knowledge of a single contour or isoline of an atmospheric constituent, q and we wish to validate this against satellite remote-sensing data. Since satellite instruments cannot measure the constituent directly, we need to perform some sort of inversion. In order to validate the contour, it is not necessary to know, at any given point, the exact value of the constituent. We only need to know whether it falls inside or outside, that is, is it greater than or less than the value of the contour, q0. This is a classification problem. Let:
j = { 1 ; q < q 0 2 ; q ≥ q 0 {\displaystyle j={\begin{cases}1;&q<q_{0}\\2;&q\geq q_{0}\end{cases}}}
be the discretized variable. This will be related to the satellite measurement vector, y → {\displaystyle {\vec {y}}} , by some conditional probability, P ( y → | j ) {\displaystyle P({\vec {y}}|j)} , which we approximate by collecting samples, called training data, of both the measurement vector and the state variable, q. By generating classification results over the region of interest and using any contouring algorithm to separate the two classes, the isoline will have been "retrieved." The accuracy of a retrieval will be given by integrating the conditional probability over the area of interest, A:
a = 1 A ∫ A P [ c ( r → ) | y → ( r → ) ] d r → {\displaystyle a={\frac {1}{A}}\int _{A}P\left[c({\vec {r}})|{\vec {y}}({\vec {r}})\right]\,d{\vec {r}}}
where c is the retrieved class at position, r → {\displaystyle {\vec {r}}} . We can maximize this quantity by maximizing the value of the integrand at each point:
max ( a ) = 1 A ∫ A { max j P [ j | y → ( r → ) ] } d r → {\displaystyle \max(a)={\frac {1}{A}}\int _{A}\left\lbrace \max _{j}P\left[j|{\vec {y}}({\vec {r}})\right]\right\rbrace \,d{\vec {r}}}
Since this is the definition of maximum likelihood, a classification algorithm based on maximum likelihood is the most accurate method possible of validating an advected contour. A good method for performing maximum likelihood classification from a set of training data is variable kernel density estimation.
Training data There are two methods of generating the training data. The most obvious is empirically, by simply matching measurements of the variable, q, with collocated measurements from the satellite instrument. In this case, no knowledge of the actual physics that produce the measurement is required and the retrieval algorithm is purely statistical. The second is with a forward model:
y → = f → ( x → ) {\displaystyle {\vec {y}}={\vec {f}}({\vec {x}})\,}
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