Flux footprint models are often used for interpretation of flux tower measurements, to estimate position and size of surface source areas, and the relative contribution of passive scalar sources to measured fluxes. Accurate knowledge of footprints is of crucial importance for any upscaling exercises from single site flux measurements to local or regional scale. Hence, footprint models are ultimately also of considerable importance for improved greenhouse gas budgeting. With increasing numbers of flux towers within large monitoring networks such as FluxNet, ICOS (Integrated Carbon Observation System), NEON (National Ecological Observatory Network), or AmeriFlux, and with increasing temporal range of observations from such towers (of the order of decades) and availability of airborne flux measurements, there has been an increasing demand for reliable footprint estimation. Even though several sophisticated footprint models have been developed in recent years, most are still not suitable for application to long time series, due to their high computational demands. Existing fast footprint models, on the other hand, are based on surface layer theory and hence are of restricted validity for real-case applications.

To remedy such shortcomings, we present the two-dimensional parameterisation for Flux
Footprint Prediction (FFP), based on a novel scaling approach for the
crosswind distribution of the flux footprint and on an improved version of
the footprint parameterisation of

The new footprint parameterisation requires input that can be easily determined from, for example, flux tower measurements or airborne flux data. FFP can be applied to data of long-term monitoring programmes as well as be used for quick footprint estimates in the field, or for designing new sites.

Flux footprint models are used to describe the spatial extent and position of
the surface area that is contributing to a turbulent flux measurement at a
specific point in time, for specific atmospheric conditions and surface
characteristics. They are hence very important tools when it comes to
interpretation of flux measurements of passive scalars, such as the
greenhouse gases carbon dioxide (

In recent years, the application of footprint models has become a standard
task in analysis of measurements from flux towers

Long-term and short-term flux observations are exposed to widely varying
atmospheric conditions and their interpretation therefore involves an
enormous amount of footprint calculations. Despite the widespread use of
footprint models, the selection of a suitable model still poses a major
challenge. Complex footprint models based on large-eddy simulation

Existing footprint modelling studies offer the potential for simple
parameterisations as, for example, proposed by

To fill this gap,

Recently, footprint model outputs have frequently been combined with surface
information, such as remote sensing data

This study addresses the issues and shortcomings mentioned above. We present
the new parameterisation for Flux Footprint Prediction
(FFP), with improved footprint predictions for elevated measurement heights
in stable stratifications. The influence of the surface roughness has been
implemented into the scaling approach explicitly. Further and most
importantly, the new parameterisation also describes the crosswind spread of
the footprint and hence, it is suitable for many practical applications. Like
all footprint models that do not simulate the full time- and space-explicit
flow, FFP implicitly assumes stationarity over the eddy-covariance
integration period (typically 30 min) and horizontal homogeneity of the flow
(but not of the scalar source/sink distribution). As in

Mathematically, the flux footprint,

Derivation and evaluation of the footprint parameterisation are based on
footprint calculations using LPDM-B

Compared to the original parameterisation of

Velocity scales (friction velocity,

As expected for such a broad range of scenarios, the resulting footprints of
LPDM-B simulations show a vast range of extents and sizes.
Figure

Range of peak locations (

An additional set of 27 LPDM-B simulations was run for independent evaluation
of the footprint parameterisation. Measurement heights that are typical for
flux tower sites were selected for this evaluation set, with boundary layer
conditions again ranging from convective to stable.
Table

The vast range in footprint sizes presented above clearly manifests that it
is not practical to fit a single footprint parameterisation to all real-scale
footprints. An additional step of footprint scaling is hence needed, with the
goal of deriving a universal non-dimensional footprint. Ideally, such
dimensionless footprints collapse to a single shape or narrow ensemble of
curves. We follow a method that borrows from Buckingham

As in

Velocity scales (

With the above scaling parameters, we form four dimensionless

The non-dimensional form of the crosswind-integrated footprint,

As a next step, the above scaling procedure is applied to all footprints of
Scenarios 1 to 8 (Table

Density plot of scaled crosswind-integrated footprints of LPDM-B
simulations as in Table

Density plot of real-scale (left panel) and scaled (right panel)
lateral dispersion of LPDM-B simulations as in Table

Crosswind dispersion can be described by a Gaussian distribution function
with

Similar to the crosswind-integrated footprint, we aim to derive a scaling
approach of the lateral footprint distribution. We choose

The successful scaling of both along-wind and crosswind shapes of the footprint into narrow ensembles within a non-dimensional framework provides the basis for fitting a parameterisation curve to the ensemble of scaled LPDM-B results. Like for the scaling approach, the footprint parameterisation is set up in two separate steps, the crosswind-integrated footprint, and its crosswind dispersion.

The ensemble of scaled crosswind-integrated footprints

The footprint parameterisation (Eq.

Figure

Performance of the footprint parameterisation evaluated against all
scaled footprints of LPDM-B simulations as in Table

A single function can also be fitted to the scaled crosswind dispersion. In
conformity with

Typically, users of footprint models are interested in footprints given in a
real-scale framework, such that distances (e.g. between the receptor and
maximum contribution to the measured flux) are given in metres or kilometres.
Depending on the availability of observed parameters, the conversion from the
non-dimensional (parameterised) footprints to real-scale dimensions can be
based on either Eqs. (

Example footprint estimate for the convective
Scenario 1 of Table

The distance between the receptor and the maximum contribution to the
measured flux can be approximated by the peak location of the
crosswind-integrated footprint. The maximum's position can be deduced from
the derivative of Eq. (

The two-dimensional footprint function can be calculated by applying the
crosswind dispersion (Eq.

evaluate

derive

invert Eqs. (

evaluate

Often, the interest lies in the extent and location of the area contributing
to, for example, 80 % of the measured flux. For such applications, there
are two approaches: (i) the crosswind-integrated footprint function,

For case (i) starting at the receptor location, we denote

There is no near-analytical solution for the description of the source area,
the extent of the fraction

If the size and position of the two-dimensional

The presented footprint model is computationally inexpensive and hence can be
run easily for several years of data in, for example, half-hourly time steps.
Each single data point can be associated with its source area by converting
the footprint coordinate system to geographical coordinates, and positioning
a discretised spatial array containing the footprint function onto a map or
aerial image surrounding the receptor position. In many cases, an aggregated
footprint, a so-called footprint climatology, is of more interest to the user
than a series of footprint estimates. The aggregated footprint can be
normalised and presented for several levels of relative contribution to the
total aggregated footprint. Figure

Example footprint climatology for the ICOS flux tower Norunda,
Sweden, for 1–31 May 2011. The red dot depicts the tower location with a
receptor mounted at

Combined with remotely sensed data, a footprint climatology provides
spatially explicit information on vegetation structure, topography, and
possible source/sink influences on the measured fluxes. This additional
information has proven to be beneficial for analysis and interpretation of
flux data

Certain remotely sensed data, for example airborne lidar data, allow for
approximate derivation of the zero-plane displacement height and the surface
roughness length

Performance of the footprint parameterisation evaluated against the
second set of LPDM-B footprints of Table

Exhaustive evaluation of footprint models is still a difficult task, and,
clearly, tracer-flux field experiments would be very helpful. We are aware
that in reality such experiments are both challenging and expensive to run.
However, the aim of the present study is not to present a new footprint
model, but to provide a simple and easily accessible parameterisation or
“shortcut” for the much more sophisticated, but highly resource intensive,
Lagrangian stochastic particle dispersion footprint model LPDM-B of

The capability of the footprint parameterisation to reproduce the real-scale
footprint of LPDM-B simulations is tested by means of the full extent of the
footprint, its peak location, peak value, and its crosswind dispersion.
Performance metrics show that for all stability classes (convective, neutral,
and stable scenarios), the footprint parameterisation is able to predict the
footprints simulated by the much more sophisticated Lagrangian stochastic
particle dispersion model very accurately
(Table

Results shown here clearly demonstrate that our objective of providing a
shortcut to LPDM-B has been achieved. The full model was tested successfully
against wind tunnel data

Sensitivity of footprint peak location (

For the calculation of footprints with FFP, the values of the input
parameters

The sensitivity of the FFP derived footprint estimate to changes in

Comparison of FFP simulations for scenarios of Table

Since FFP is based on LPDM-B simulations, LPDM-B's application limits are
also applicable to FFP. As for most footprint models, these include the
requirements of stationarity and horizontal homogeneity of the flow over time
periods that are typical for flux calculations (e.g. 30–60 min). If applied
outside these restrictions, FFP will still provide footprint estimates, but
their interpretation becomes difficult and unreliable. Similarly, LPDM-B does
not include roughness sublayer dispersion near the ground, nor dispersion
within the entrainment layer at the top of the convective boundary layer.
Hence, we suggest limiting FFP simulations to measurement heights above the
roughness sublayer and below the entrainment layer (e.g. for airborne flux
measurements). The

The requirements and limits of FFP for the measurement height and stability
mentioned above can be summarised as follows:

Same as Fig.

In the following, we compare footprints of three of the a most commonly used
models with results of FFP: the parameterisation of

For the comparison, the three above models and FFP were run for all scenarios
listed in Table

Figure

The along-wind extents of the footprint predictions of KM01 are very similar
to HKC00's results, and hence the comparison of KM01 against FFP is similar
as well: larger footprint extents resulting from KM01 than from FFP in most
cases except for free convection and mixed layer scenarios, where FFP's
footprints extend further (Fig.

KRC04 and FFP were both developed on the basis of LPDM-B simulations. Hence,
as expected, the results of these two footprint parameterisations agree quite
well (Fig.

To date, the availability of observational data suitable for direct
evaluation of footprint models is very limited, and hence the performance of
footprint models cannot be tested against “the truth”. Nevertheless, as
stated in Sect.

Flux footprint models describe the area of influence of a turbulent flux measurement. They are typically used for the design of flux tower sites, and for the interpretation of flux measurements. Over the last decades, large monitoring networks of flux tower sites have been set up to study greenhouse gas exchanges between the vegetated surface and the lower atmosphere. These networks have created a great demand for footprint modelling of long-term data sets. However, to date available footprint models are either too slow to process such large data sets, or are based on too restrictive assumptions to be valid for many real-case conditions (e.g. large measurement heights or turbulence conditions outside Monin–Obukhov scaling).

In this study, we present a novel scaling approach for real-scale two-dimensional footprint data from complex models. The approach was applied to results of the backward Lagrangian stochastic particle dispersion model LPDM-B. This model is one of only few that have been tested against wind tunnel experimental data. LPDM-B's dispersion core was specifically designed to include the range from convective to stable conditions and was evaluated successfully using wind tunnel and water tank data, large-eddy simulation and a field tracer experiment.

The scaling approach forms the basis for the two-dimensional flux footprint parameterisation FFP, as a simple and accessible shortcut to the complex model. FFP can reproduce simulations of LPDM-B for a wide range of boundary layer conditions from convective to stable, for surfaces from very smooth to very rough, and for measurement heights from very close to the ground to high up in the boundary layer. Unlike any other current fast footprint model, FFP is hence applicable for daytime and night-time measurements, for measurements throughout the year, and for measurements from small towers over grassland to tall towers over mature forests, and even for airborne surveys.

Comparison of FFP simulations for scenarios of Table

There may be situations where footprint estimates are needed for only one
specific stability regime, for example, when footprints are calculated for
only a short period of time, or for a certain daytime over several days. For
such cases, it may be beneficial to use footprint parameterisation settings
optimised for this stability regime only. While scaled footprint estimates
for neutral and stable conditions collapse to a very narrow ensemble of
curves, footprints for strongly convective situations may also include
contributions from downwind of the receptor location. For neutral and stable
conditions, a specific set of fitting parameters for FFP has been derived
using the LPDM-B simulations of Scenarios 4 to 6 (Table

Please note that when applying these fitting parameters, the footprint functions for convective and neutral/stable conditions will not be continuous. We hence suggest to use the universal fitting parameters of Table A1 for cases where a transition between stability regimes may occur.

Fitting parameters of the crosswind-integrated footprint
parameterisation and of the crosswind footprint extent for a “universal”
regime (Scenarios 1 to 9), and for specifically convective (Scenarios 1* and 1
to 3) or neutral and stable regimes (Scenarios 4 to 9). For each scenario,
all measurement heights and roughness lengths were included. See Table

Determination of the boundary layer height,

For stable and neutral conditions there are simple diagnostic relations with
which the boundary layer height can be estimated.

For convective conditions, the boundary layer height cannot be diagnosed due
to the nearly symmetric diurnal cycle of the surface heat flux. It must
therefore be integrated employing a prognostic expression, and starting at
sunrise (before the surface heat flux first becomes positive) when the
initial height is diagnosed using one of the above expressions. The
slab model of

For the parameterisation of the scaled footprints, using continuous
functions, we need to address an issue that arises from the discrete nature
of LPDM-B. Like in all stochastic particle dispersion models, the number of
particles (

For the crosswind-integrated footprint, mean advection of the particle plume
over time

For the integration of the footprint parameterisation

The code of the presented two-dimensional flux footprint parameterisation FFP
and the crosswind-integrated footprint can be obtained from

The authors would like to thank Anders Lindroth, Michal Heliasz, and Meelis Mölder for the Norunda flux tower data. We also thank three anonymous reviewers for their helpful suggestions. This research was supported by the UK Natural Environment Research Council (NERC, NE/G000360/1), the Commonwealth Scientific and Industrial Research Organisation Australia (CSIRO, OCE2012), by the LUCCI Linnaeus center of Lund University, funded by the Swedish Research Council, Vetenskapsrådet, the Austrian Research Community (OeFG), the Royal Society UK (IE110132), and by the German Helmholtz programme ATMO and the Helmholtz climate initiative for regional climate change research, REKLIM. Edited by: J. Kala