Usually the mapping, formula_4, 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 formula_4 or of finding the inverse model, formula_6.