Footprint parameterization derived from a graded multilayer semi-analytical model valid in homogeneously driven boundary layers described by Monin-Obukhov theory
Authors: Krapez Jean-Claude 
Published: February 2023

Micrometeorological measurements, and greenhouse-gas monitoring, performed by on-site sensors or remote sensing are not exclusively influenced by the terrain beneath the sensor position or at the optically aimed spot, sometimes they are not even influenced by it at all. The footprint function (or so-called source area) provides a description of the actual &#8220;field of view&#8221;, namely the 2D spatial distribution of the weighing function applied on the sources and sinks to yield the signal value. Given a scalar of interest, different distributions are obtained depending on whether the considered quantity is the concentration or the flux of this scalar.Footprints derived from fully analytical models are restricted due to the following hypotheses: homogeneous flow (i.e. in terms of soil-atmosphere interactions, e.g. roughness length, thermal sources driving the turbulence), the eddy diffusivity and the mean wind speed are power-law functions of height. Despite these restrictions, they are often used due to their ease and speed. The footprint functions are expressed in terms of the inverse Gamma distribution whose parameters depend on the power-law parameters. These parameters must be identified from the actual wind speed and diffusivity profiles, which are generally assimilated to profiles parameterized according to the Monin-Obukov theory. The scope of the analytical approach can be broadened by relaxing the power-law constraint regarding the profiles. In this perspective a new semi-analytical model was developed which is based on a graded multi-layer approach. A Liouville transformation is first applied which introduces a new independent variable, the Diffusion-Ascent-Associated Advection Distance (DAAAD, in place of height) and a new advection-diffusion parameter, the atmosphere inertivity, which describes the atmosphere inertia to state change at the considered position. The graded multi-layer method allows approximating the real inertivity profile by a piecewise function, with continuously joined sublayers, hence allowing a close fitting with a minimal number of sub-layers.As an example, Monin-Obukhov profiles based on the Businger-H&#246;gstr&#246;m parameterization, from unstable to stable, were considered and the footprint functions were computed for a large range of height. It was shown that the corresponding concentration and flux footprints are very accurately approximated (to less than 1-1.2% RMS error) by the functions mentioned before based on the inverse Gamma distribution. The optimal values of the two or three parameters involved therein were computed to provide a database depending on the two ratios zm/z0 and z0/L. Furthermore, an analytical parameterization of the two parameters intervening in the flux footprint has been proposed with only a slight reduction of the performance as compared to the highly accurate semi-analytical multi-layer model.A comparison to the classical Kormann & Meixner, and Hsieh et al. models, which otherwise share the same hypotheses, is finally presented. With a negligibly small increased effort in the computation of the parameterization functions, a much better rendering of the Monin-Obukhov footprints is achieved. Moreover, the Monin-Obukhov profiles are just an example, the graded multi-layer method can be applied to any pair of profiles inasmuch the K-theory can be considered valid in the studied context.