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In Inverse problem theory, what is meant by "results of the measurements of observable parameter"?

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    $\begingroup$ That's probably something that is actually measured. For example weather model may have temperature of air in different locations as model parameter, but the satellite doesn't ever observe that. It rather measures incoming radiance in many wavelengths and an algorithm deduces temperature from them. $\endgroup$ – Communisty May 9 '18 at 13:09
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Inverse problems deal with creating theories based on observed data, as opposed to making predictions about the real world based on existing theoretical models, so the "results of the measurement of observable parameter" is a confusing way of saying "the real measurements being used as data". Parameter simply refers to the value being measured.

An example would be if you have a forestry block and you want to understand why some trees are preforming better than others; you'd measure a whole group of variables that formed a set of parameters, one of which might be "tree health" based on say height, girth, leaf activity, and transpiration rate and another that's "soil fertility" based on key nutrient levels, particle size distribution, bulk density, and water holding capacity. The relationship between those two parameters then lets you formulate a theory that you can apply at other sites that you know the "soil fertility" of to predict the growth of trees there.

In the real world you have to test your theory with what in my GIS classes they called a "Ground Truth", that's where you go out and measure the parameters of a different real world site being used for the same crop and see if your theory accounts for the results at that site (we did them to confirm the relationship between satellite readings and on-the-ground data). When you've done this at a number of sites and the measurements all conform to your theory you can start applying your model as a predictive formula for sites not yet in use.

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