In the context of flux and concentration measurements in micrometeorology, the footprint is a concept used to describe the surface area that contains the sources (and sinks) which contribute to the measured signal. In another words, it is the 'field of view' of a given sensor whose identification is essential in interpreting the obtained flux or concentration values in their correct spatial extent. Technically, the footprint is a transfer function f, which relates the value of a measurement η at location x^{*} to the spatial distribution of flux or concentration sources Q from a volumetric domain Ω of interest:

Thus, the footprint acts as a spatial weighting function for the sources. Explicit analytical expressions have been derived for the footprint functions (Schmid, 2002), but only under the assumptions that (1) steady-state conditions prevail during the analyzed period, (2) turbulent fluctuations in the atmospheric boundary layer (ABL) are horizontally homogeneous, and (3) there is no vertical advection. In urban environments, where the ABL flows are characterized by strong heterogeneity, assumption (2) becomes strictly invalid and assumption (3) highly questionable. For these reasons, analytical footprint models cannot be utilized reliably in urban studies motivating the search for new, generally applicable techniques to produce footprints.

The most promising approach applicable to heterogeneous sites combines Lagrangian stochastic (LS) particle model with large-eddy simulation (LES), which offers the most suitable numerical strategy to study ABL flows (Steinfeld et. al., 2008; Hellsten et. al., 2015). The coupling of LS with LES (LES-LS) refers to an approach where trajectories of passive weightless particles are simultaneously solved with the flow field. To account for the stochastic nature of the subgrid-scale turbulence, the time evolution of the particles is solved utilizing velocity fields that contain both resolved and stochastic contributions.

The objective of this investigation is to develop a reliable numerical framework for constructing footprints for real urban measurement sites, which may be situated suboptimally on top of buildings instead of towers. The case study is staged around the downtown region of Helsinki with a focus on the eddy-covariance (EC) sensor mounted on the roof of a centrally situated Hotel Torni building.

In this study the Parallelized LArge-eddy Model, PALM (Maronga et. al., 2015), is used to run the coupled LES-LS simulations. PALM implements the non-hydrostatic, filtered, incompressible Navier-Stokes equations together with a subgrid-scale model according to Deardorff (1980).

The topography for the Helsinki LES model has been constructed from detailed 2 m resolution laser-scanned data (Nordbo et. al., 2015) and the computational domain comprises of a 4 km by 2 km land area, which is oriented to place Hotel Torni in a pivotal location. See Figure 1.

To ensure that the relevant turbulent structures are accurately resolved even between the buildings and within the street canyons, a spatial resolution of Δx = Δy = Δz = 1 m was specified for the LES grid. The meteorological conditions for the simulation were adopted from September 9th in 2012 when south-westerly wind and near neutral ABL conditions were recorded above the urban internal boundary layer. The boundary layer height (obtained from Lidar data) of 300 m was fixed by specifying a mean potential temperature profile with a strong inversion layer. Physically meaningful inlet boundary conditions were generated by recycling the solution from yz-plane at back to the inlet. For this reason, the first half of the computation domain (0 < x < 2km) features flat terrain, which conveniently coninsides with the natural sea surface. To reduce the computational time required to reach statistically stationary conditions, the urban simulation was initialized with a pre-computed ABL solution over a flat surface.

The particles for the LS model are seeded 1 m above the topography and released at 10 s intervals. Those particles that hit a target volume V , set around Hotel Torni's measurement site, are sampled and their coordinates of origin and incident velocity data recorded. The target volume has to be larger than one grid cell in order to accumulate a sufficiently large number of particle hits (~ 10^{7}). The recorded hits yield a large dataset S, which is subsequently divided into a series of subsets s_{ijk} (s.t. S = ∑ _{k} ∑ _{j} ∑ _{i}s_{ijk}) each containing the record obtained from a target subvolume ΔV _{ijk}. The division of the target volume into N_{tot} = N_{i}N_{j}N_{k} subvolumes follows a structured cartesian discretization. This procedure enables a systematic piecewise post-processing of independently mean-flow corrected footprint functions f_{ijk}, which locally adapt to the variable wind conditions just above the Hotel Torni building. The final footprint is assembled by combining the individual contributions

where n_{ijk} are the normalization factors of the individual footprints.

Figure 2 illustrates a comparison between an analytical (Korman and Meixner, 2001) and LES-LS generated footprint functions for Hotel Torni's EC sensor. The LES-LS footprint reflects in unprecedented detail the complex dispersion characteristics of the flow, which interacts with the densely built urban environment featuring perimeter blocks with courtyards. Further analysis of the urban footprint lays bare the importance of accounting for the sudden variations in source weights, which occur between roof-tops and street canyons, when interpreting e.g. measurements of pollutant emissions that primarily originate from traffic.