Forward Modeling and the Inverse Problem

Let us say that we are going to build a satellite instrument that will measure ultraviolet light scattered back from the Earth's atmosphere, and from those measurements we will measure the profiles of atmospheric constituents like ozone. We know enough about the structure of the atmosphere to make a reasonable computer model of "typical" profiles of chemical concentrations, temperature, and pressure. And, because we have a reasonably good understanding of the way light and our atmospheric constituents interact, we can write a program to solve the radiative transfer problem. That is, we put in photons at the top of the atmosphere, from the direction of the sun, over a certain location on the Earth, and, using scattering cross sections and phase functions, we can calculate the fraction of those incoming photons that will wind up coming toward the satellite instrument. This process is called Forward Modeling, and can be done very well even on PC-class computers.

Of course, the point of remote sensing is to infer from the space-based measurements what the profile is. In the forward modeling problem, you have to assume you know what it is. So you can do some experiments, systematically tinkering with the profiles, to build up a kind of library of profiles (you make them up) and the amount of backscattered radiation they give rise to (from the radiative transfer calculation). Now, you want to use this library to be able to solve the inverse problem: Given satellite-measured, backscattered spectral radiances, can you infer the profile? This process is called Retrieval.

There's a problem. If you look at a single wavelength, you will find that many profiles can give you the same backscattered spectral radiance. If you measure radiances at several different wavelengths, you can narrow the choices down quite a bit. But, in general, it is always true that there are many atmospheric profiles that can explain the measured radiances. At that point, you have to apply some mathematical techniques to select a profile that is in some way better than all the others. These techniques also need to take into account the fact that every measured radiance has a certain amount of uncertainty.