Conventional footprint models cannot account for the heterogeneity of the urban landscape imposing a pronounced uncertainty on the spatial interpretation of eddy-covariance (EC) flux measurements in urban studies. This work introduces a computational methodology that enables the generation of detailed footprints in arbitrarily complex urban flux measurements sites. The methodology is based on conducting high-resolution large-eddy simulation (LES) and Lagrangian stochastic (LS) particle analysis on a model that features a detailed topographic description of a real urban environment. The approach utilizes an arbitrarily sized target volume set around the sensor in the LES domain, to collect a dataset of LS particles which are seeded from the potential source area of the measurement and captured at the sensor site. The urban footprint is generated from this dataset through a piecewise postprocessing procedure, which divides the footprint evaluation into multiple independent processes that each yield an intermediate result. These results are ultimately selectively combined to produce the final footprint. The strategy reduces the computational cost of the LES–LS simulation and incorporates techniques to account for the complications that arise when the EC sensor is mounted on a building instead of a conventional flux tower. The presented computational framework also introduces a result assessment strategy which utilizes the obtained urban footprint together with a detailed land cover type dataset to estimate the potential error that may arise if analytically derived footprint models were employed instead. The methodology is demonstrated with a case study that concentrates on generating the footprint for a building-mounted EC measurement station in downtown Helsinki, Finland, under the neutrally stratified atmospheric boundary layer.

Micrometeorological measurements in densely built city environments pose an antipodal problem: they are essential in establishing the fundamental basis for the study of urban microclimates, but these measurements are endowed with pronounced uncertainties, which mainly originate from the topographic and elemental complexity of the urban landscape. The resulting noncompliance between the theory and practice in urban micrometeorological measurements undermines the study of how our cities interact with the surrounding atmosphere. At the very heart of this discord lies the problem concerning the determination of effective source areas, or footprints, of urban flux or concentration measurements.

The footprint is a concept used to describe the surface area that contains
the sources and sinks which contribute to the measured quantity obtained by a
sensor

The underlying assumptions are often acceptable in measurement sites where the sensors are mounted on towers that have been appropriately placed above homogeneous forested landscapes and well above the surface roughness sublayer height where the effects of the individual roughness elements disappear. However, due to practical regulations constraining measurement campaigns in densely populated cities, sufficiently tall flux towers cannot be erected above the skyline of central urban areas. It is often inevitable that if the urban microclimate is to be studied experimentally, the measurements must be obtained near the border of the roughness sublayer by sensors that are mounted either on low-rise towers or on top of tall buildings. In these suboptimal conditions, assumption (2) becomes strictly invalid and assumption (3) highly questionable because urban boundary layer (UBL) flows are typically characterized by developing and strongly heterogeneous flow conditions, particularly at lower elevations where individual buildings influence the turbulence.

Considering that the analytical footprint models effectively provide
ellipse-shaped probability distributions for the source contributions without
any regard to topographic heterogeneities, it becomes clear that the use of
such source-area models becomes highly suspect in real urban conditions. This
is an unacceptable state of affairs in the urban micrometeorology research
and immediately calls for targeted efforts to alleviate the uncertainties
associated with the invaluable urban flux-measurement data. Although, the
first efforts by

As a response, this works introduces a new numerical methodology to construct detailed topography-sensitive footprints for complex urban flux measurement sites by the means of pre- and postprocessing developments and a large-eddy simulation (LES) solver suite that features an embedded Lagrangian stochastic (LS) particle model. This coupled model will be referred to with the acronym LES–LS. The proposed methodology is designed to be first and foremost a postprocessing procedure, which exploits the current state-of-the-art LES–LS modeling framework in an urban setting with a minimal investment in the initial setup.

The principal objective is to provide a reliable computational framework, founded on a high-resolution LES–LS analysis, to generate the most accurate footprint estimates feasible without the need to conduct tracer gas experiments, which are nearly impossible to arrange in residential areas. These computationally generated footprints open up the possibility to study the appropriate placement of new measurement stations and to assess the magnitude of the potential misinterpretation which may arise from the application of closed-form footprint models to urban flux or concentration measurements. The proposed framework is also supplemented by a convenient technique to approximate this error with the assistance of a land cover classification dataset.

The methodology is demonstrated with a numerical case study, which is staged
in Helsinki, the coastal capital city of Finland, and focuses on the
eddy-covariance (EC) measurement site mounted on the roof of Hotel Torni

This study employs the PArallelised LES Model PALM

The PALM model utilized in this study is an open-source numerical
solver for atmospheric and oceanic flow simulations. The software has been
carefully designed to run efficiently on massively parallel supercomputer
architectures and it is therefore exceptionally well suited for
high-resolution UBL simulations considered herein. The LES model employs
finite-difference discretization on staggered Cartesian grid and utilizes an
explicit Runge–Kutta time-stepping scheme to solve the evolution of velocity
vector

The embedded Lagrangian particle model in PALM implements the
time-accurate evolution of discrete particles (either with or without mass)
through a technique that conforms to the LES approach: the trajectories are
integrated in time such that the transporting velocity field is decomposed
into deterministic (i.e., resolved) and stochastic (i.e., subgrid-scale)
contributions. The deterministic velocity components are directly obtained
from the LES solution, while the random components are evaluated according to

While the LES–LS simulations are carried out in large supercomputing facilities, the preprocessing of the urban topography model and the postprocessing of the final footprint from raw data is performed on a personal workstation utilizing freely available numerical scripting and data visualization technologies. See the paragraph on code availability at the end of this paper.

The urban topography model, used in describing the bottom wall boundary of the
LES domain, is prepared from a detailed 2 m resolution laser-scanned dataset
of the Helsinki area

Raster maps of topography height

The horizontal domain for the LES analysis extends

The first half of the topography model (where

The lateral sides were made identical for cyclic boundary condition treatment by applying a zero-height margin that smoothly blends toward the values in the interior.

Immediately upstream of the outlet boundary, a margin with sloping terrain height is applied to force the highly turbulent flow (caused by the buildings near the end of the domain) to slightly accelerate before reaching the outlet boundary where reversed flow causes numerical difficulties.

Visualization of the topography height distribution underlying the
LES domain. The particle release area is enveloped by a white dashed line.
The size of the precursor domain is outlined in the top left corner. The
location of the turbulence recycling plane is marked by a black dotted line
at

The meteorological conditions for the simulation are adopted from 9 September
in 2012 when near-neutral ABL conditions were recorded with the EC
measurements made on top of the Torni building. Lidar measurements

For the precursor simulation the solver was run with an option that
explicitly conserves the initial mass flow rate across the system, which was
specified by initializing the velocity field with a constant value

The precursor LES solution was computed on a grid that has the same
resolution and vertical dimension as the principal urban LES grid, but its
lateral dimensions are smaller by an integer division. Table

The precursor simulation generates a highly resolved ABL solution that will
be utilized, first, in a recursive manner to initialize the entire urban LES
flow field with turbulence and, second, to aid construction of appropriate inlet
boundary conditions though a technique labeled

Denoting prognostic field variables by

In this study, the recycling plane is situated, as shown in
Fig.

Computational grid specifications.

The embedded LS particle model is employed such that, after the initial
transients in the LES solution have subdued (after approximately 5 min of
simulation), the release of particles is activated within the region outlined
in Fig.

Denoting the Lagrangian coordinate vector of the

The raw particle data for constructing footprints through LES–LS modeling in
an arbitrarily heterogeneous environment are obtained by setting a target
volume around the specified sensor location

Consider the problem of strictly concentrating on the exact location

Adopting this strategy reduces the level of rigor required at the setup
stage of the LES–LS analysis and simplifies the guidelines for the particle
acquisition: the target volume should be centered at

A three-dimensional rendering of the urban topography near Hotel
Torni

The precursor simulation is run for 1.5 h physical time to develop the
desired ABL profile. The initialization of the primary LES–LS computation
with this precursor solution expectedly results in short-lived unphysical
fluctuations around the urban topography, but after

During the LES–LS simulation, the sampling of particle hits at the target
volume

According to the issues discussed in Sect.

Example discretization of target volume

Each target subvolume now yields an associated subset

The individual sectional footprints are typically evaluated from subsets that
contain an insufficient number of particle data entries needed to obtain a
converged footprint distribution.

The piecewise processing of the footprint carries an inherent difficulty that
arises in situations where the mean flow displays strong gradients within the
target volume. This is evidently present in the considered case study
featuring an EC sensor mounted close to the top of a building. The difficulty
relates to the evaluation of

Unfortunately, at the required level of target volume discretization, the
excessive number of individual

Crosswind integrated distributions of piecewise postprocessed
footprints obtained with different levels of target volume discretizations
using

The method has a prerequisite that the deficiently obtained footprint (for
instance, obtained via insufficient target box discretization) must exhibit a
properly leveled off far field, because the approach fundamentally relies on
the following simple assertion: if the footprint distribution plateaus in the
far field, this asymptote can be amended to become the zero reference level,
which deviates from the “correct” asymptote by a negligibly small offset.
Accepting this assertion and the associated approximation paves the way for a
corrective coordinate rotation scheme which can be laid out by first
classifying the data contributing to the far-field footprint via subsets

Table

This realignment of the coordinate rotation plane within a larger subvolume, when examined in contrast with the reference technique where the coordinate rotation is performed at full LES resolution, alters how some of the individual particles contribute to the footprint. However, this discrepancy gives rise to an error that is distributed throughout the footprint domain. Therefore, the validity of the far-field correction approach hinges upon the magnitude of this distributed error and its sensitivity to the target box discretization. The sensitivity can be established by carrying out the selective assembly of the footprint result for different levels of target box discretizations.

Since its conception it has been clear that the piecewise postprocessing
approach must be endowed with the capacity to incorporate a sensitivity
analysis phase into the final assembly of the footprint result. One of the
driving motivators for developing the piecewise approach arose from the need
to reduce the computational cost of collecting a large number of particle
hits by an arbitrarily sized target volume around

Diagnostic data from the application of far-field correction in the
coordinate rotation. The farthest 15 % of the source area in the LES domain
is considered (i.e.,

Thus, the process of selectively assembling the final footprint result begins
by first defining an inadequately converged initial footprint, which
represents the desired preform at

The process begins by setting at the 0th iteration

The iterative process continues such that new candidate contributions

The obtained final result, which combines the earlier accepted additions,
features

Illustration of the selective assembly of the final footprint for

As long as the individual subsets contain a sufficient number of particle
data entries (

Taking into account the far-field correction procedure, the postprocessing
procedure for evaluating a footprint from a LES–LS obtained dataset can be
described in the following steps.

Split the original dataset

Evaluate an approximate footprint in a piecewise manner by applying
Eq. (

Inspect the approximate footprint result to identify the extent of the far field (by specifying

Evaluate the sectional footprints

select initial guess for

perturb the coefficient

exit the loop if

compute derivative

set

Select the appropriate set

Assemble the final footprint via Eq. (

Comparison of identically normalized footprint distributions

It is noteworthy that in step 2 for the approximate footprint evaluation and
in step 4a for the initialization of the optimization loop, the values for
the mean vertical velocities

The proposed methodology, founded on high-resolution LES–LS analysis and a piecewise postprocessing approach, has been shown to be a reliable, robust, and accessible, although computationally expensive, approach to generating topography-sensitive footprints in real urban applications. Since the underlying motivation for this development effort sprung from the need to evaluate the potential error that may arise when analytical, closed-form footprint models are applied to urban flux measurements, this work also proposes a technique to approximate the magnitude of this error in the absence of field validation studies. This approach hinges on the assumption that, in a real urban application, a topography-sensitive footprint obtained through a highly resolved LES–LS analysis features a higher level of accuracy and a lower level of uncertainty than any available closed-form footprint model.

The proposed assessment technique compares the obtained LES–LS footprint
result to an analytical model, which belongs to the group of closed-form
models that would otherwise be employed in similar studies, by applying the
footprint distributions to the land cover classification (

Parameters used in the Korman and Meixner footprint model.

A preliminary comparison between the obtained LES–LS and KM footprint
distributions,

Comparison of identically normalized LES–LS

Comparison of normalized, crosswind integrated LES–LS and KM footprints. A light blue dashed line indicates the start of urban topography and the gray dashed line marks the location of the EC sensor.

The comparative technique proposed for assessing the potential error, that may
arise if urban measurements are interpreted with closed-form footprint
models, exploits the land cover dataset under the assumption that the

The comparison is carried out by extracting the area corresponding to the LES
domain from the

Raster map of land cover types,

A pie-chart of source-area fractions

Comparison of source-area fractions

Repeating the introduced assessment technique for multiple representative
meteorological conditions paves the way for a numerical approach that allows
the obtained urban flux measurements to be interpreted either differently or
with improved confidence. Naturally, having access to real source strength
distributions opens up the ability to utilize LES–LS footprints (or
positively assessed analytical footprints) to carry out detailed emission
inventories

The utility of the eddy-covariance method in measuring the exchanges of mass, heat, and momentum between the urban landscape and the overlying atmosphere largely depends on the ability to determine the effective source area, or footprint, of the measurement. In situations where the heterogeneity of the surface becomes relevant, like for urban landscapes, and the structures surrounding the measurement site can no longer be considered as a homogeneous layer of roughness elements, the use of analytical footprint models becomes highly suspect. In order to diminish the resulting uncertainties and to obtain the ability to assess the applicability of analytical models, the ability to evaluate complex footprints with high resolution becomes essential.

This work presents a numerical methodology to generate topography-sensitive footprints for real urban EC flux measurement sites. This methodology is based on high-resolution LES–LS analysis where the simulation domain features a detailed description of the urban topography and accounts for the entire vertical extent of the atmospheric boundary layer. The online-coupled LS model within the LES solver is employed to simulate a constant release of inert gas emissions from the potential upwind source area of the considered EC sensor. The necessary data for the footprint generation are obtained from the LES–LS analysis by setting up a finite target volume around the sensor location and, over a sufficiently long simulation period, gathering a record of particles that hit this target. To generate an estimate for the flux footprint, this dataset is subjected to a postprocessing procedure that involves a coordinate rotation step, which eliminates the effect of the mean flow on the flux evaluation. But, if the considered EC sensor is mounted on a building (instead of a conventional tower-like structure) in the vicinity of which strong mean flow gradients occur, standard postprocessing techniques fail to produce physically meaningful footprints unless the target volume size is reduced to correspond with the LES grid spacing. This inevitably leads to prohibitive computational costs. Therefore, this work introduces a robust piecewise postprocessing strategy, which facilitates the evaluation of footprints despite the added complexity. The piecewise approach involves splitting the original dataset into a series of subsets which are all independently postprocessed to yield incompletely converged intermediate footprint estimates. The splitting is done by applying Cartesian discretization to the target volume in order to generate a series of subvolumes that correspond to the subsets. However, to facilitate a sufficiently accurate coordinate rotation treatment in the presence of strong gradients, the size of these subvolumes must also be reduced to match the resolution of the LES grid. This causes their number, and hence the number of intermediate sectional footprints, to become excessive, motivating the development of a new approximate scheme labeled far-field correction, which enables the subvolume size to be increased and the postprocessing effort to be reduced significantly. In the piecewise postprocessing approach, the final, completely converged, footprint is eventually selectively assembled from the obtained set of intermediates.

The methodology is demonstrated in a real urban application where the
objective is to compute a highly resolved topography-sensitive footprint for
the Hotel Torni EC flux measurement sensor mounted on the roof of a tall
building situated in the downtown area of Helsinki, Finland. The EC sensor's
measurement height is 60 m above the ground level and 36 m above the
surrounding mean building height (24 m). The meteorological conditions for
the LES simulation were adopted from measurements on 9 September 2012 when
southwesterly winds and a neutrally stratified boundary layer of 300 m
height were recorded. A detailed topography map of Helsinki at 2 m
resolution from

This paper also introduces an accessible technique to employ the obtained
high-resolution topography-sensitive urban footprint in estimating the
potential error that may arise when an analytical footprint model is used to
interpret urban EC measurements. The underlying stipulation for this method
is that it does not require knowledge of real source strength distributions.
Thus, it is proposed that a detailed land cover type classification (

The context of this paper is limited to laying out the new methodology for generating urban footprints and exploiting them in the assessment of analytical models. It is evident that changes in the meteorological and anthropogenic conditions will influence the results and a proper assessment of the applicability of analytical models at a given EC measurement site will require that these conditions are varied, necessitating numerous footprint evaluations. This paper lays the numerical groundwork for such future investigations.

PALM is open-source software
released under GNU General Public License (v3) and freely available upon
registration at

The authors declare that they have no conflict of interest.

This study was supported by Academy of Finland (grant no. 284701, 1281255, 277664, and 281255). The computing resources were provided by CSC – IT Center for Science Ltd., Finland (grand challenge project gc2618). The authors would like to sincerely acknowledge Curtis Wood for the meteorological data acquisition and Tiina Markkanen, Siegfried Raasch, Andrey Glazynov, and Juha Lento for the help and advice they provided. The authors also wish to express their gratitude to the peer reviewers whose comments and feedback helped to improve the paper.Edited by: Simon Unterstrasser Reviewed by: two anonymous referees