Atmospheric sounding

A limb sounder looks at the edge of the atmosphere where it is visible above the Earth. It does this in one of two ways: either it tracks the sun, moon, a star, or another transmitting satellite through the limb as the source gets occultated behind the Earth, or it looks towards empty space, collecting radiation that is scattered from one of these sources. In contrast, a nadir sounder looks down (at nadir) through the atmosphere at the surface. The SCIAMACHY instrument operates in all three of these modes. A zenith sounder looks up (at zenith) from a ground-based location.

The following applies mainly to passive sensors, but has some applicability to active sensors.

Typically, there is a vector of values of the quantity to be retrieved, ${\displaystyle {\vec {x}}}$, called the state vector and a vector of measurements, ${\displaystyle {\vec {y}}}$. The state vector could be temperatures, ozone number densities, humidities etc. The measurement vector is typically counts, radiances or brightness temperatures from a radiometer or similar detector but could include any other quantity germane to the problem. The forward model maps the state vector to the measurement vector:

${\displaystyle {\vec {y}}={\vec {f}}({\vec {x}})}$

Usually the mapping, ${\displaystyle {\vec {f}}}$, is known from physical first principles, but this may not always be the case. Instead, it may only be known empirically, by matching actual measurements with actual states. Satellite and many other remote sensing instruments do not measure the relevant physical properties, that is the state, but rather the amount of radiation emitted in a particular direction, at a particular frequency. It is usually easy to go from the state space to the measurement space—for instance with Beer's law or radiative transfer—but not the other way around, therefore we need some method of inverting ${\displaystyle {\vec {f}}}$ or of finding the inverse model, ${\displaystyle {\vec {f}}^{-1}}$.

If the problem is linear we can use some type of matrix inverse method—often the problem is ill-posed or unstable so we will need to regularize it: good, simple methods include the normal equation or singular value decomposition. If the problem is weakly nonlinear, an iterative method such Newton–Raphson may be appropriate.

Sometimes the physics is too complicated to model accurately or the forward model too slow to be used effectively in the inverse method. In this case, statistical or machine learning methods such as linear regression, neural networks, statistical classification, kernel estimation, etc. can be used to form an inverse model based on a collection of ordered pairs of samples mapping the state space to the measurement space, that is, ${\displaystyle \lbrace {\vec {x}}:{\vec {y}}\rbrace }$. These can be generated either from models—e.g. state vectors from dynamical models and measurement vectors from radiative transfer or similar forward models—or from direct, empirical measurement. Other times when a statistical method might be more appropriate include highly nonlinear problems.