The contribution and weighting functions of radiative transfer - theory and application to the retrieval of upper-tropospheric humidity

Abstract

Several interesting problems in remote sensing can be traced back to the question of the origin along the line of sight of the registered photons. In this paper we revive old concepts that directly follow from the equation of radiative transfer, namely the contribution and weighting functions. We give them, however, a new mathematical form by transforming them into a pair of probability density functions which have the advantage that they can be used in a more flexible manner. We derive these functions, demonstrate a simple relation between them and show how they can be used in principle. Then we proceed with simple applications to a case of upper-tropospheric humidity (UTH) retrieval. In particular, we show how the mean emission pressure level and mean emission temperature change with increasing UTH. We show that the mean emission pressure increases with increasing humidity and remains almost unchanged for UTH values greater than 50 %. The mean emission temperature is decreasing exponentially as UTH increases. The sensitivities of the mean emission pressure to various quantities, e.g. the temperature lapse rate, or retrieval situations, e.g. whether UTH or UTH with respect to ice is considered or which of two different versions of a receiver is used, is generally small compared to the 2σp-width of the layer. The relation of the contribution and weighting functions to Jacobians is discussed as well. We note that the dependence of the mean emission pressure level and other statistical quantities can be formulated using the radiances or brightness temperatures directly. The new method thus offers additional possibilities for interpretation of data from passive remote sensing, and examples are given. In addition of deriving the desired product (for instance, UTH) one can derive and map the mean emission location, its width, and other physical properties like mean temperature of the emission layer. The necessary probability density functions are contained in the solution of the radiative transfer equation and can thus be obtained from runs of the corresponding models. We recommend that radiative transfer models be equipped with facilities to compute and output the contribution and weighting functions.