It is generally accepted that the ground heat flux accounts for a significant fraction of the surface energy balance. In land surface models, the ground heat flux is typically estimated through a numerical solution of the heat conduction equation. Recent research has shown that this approach introduces errors in the estimation of the energy balance. In this paper, we calibrate a land surface model using a numerical solution of the heat conduction equation with four different vertical spatial resolutions. It is found that the thermal conductivity is the most sensitive parameter to the spatial resolution. More importantly, the thermal conductivity values are directly related to the spatial resolution, thus rendering any physical interpretation of this value irrelevant. The numerical solution is then replaced by an analytical solution. The results of the numerical and analytical solutions are identical when fine spatial and temporal resolutions are used. However, when using resolutions that are typical of land surface models, significant differences are found. When using the analytical solution, the ground heat flux is directly calculated without calculating the soil temperature profile. The calculation of the temperature at each node in the soil profile is thus no longer required, unless the model contains parameters that depend on the soil temperature, which in this study is not the case. The calibration is repeated, and thermal conductivity values independent of the vertical spatial resolution are obtained. The main conclusion of this study is that care must be taken when interpreting land surface model results that have been obtained using numerical ground heat flux estimates. The use of exact analytical solutions, when available, is recommended.
An accurate estimate of the surface energy balance is very important for
climate modeling and numerical weather prediction
However,
The problem with the numerical estimation of the ground heat flux in land surface models is that their vertical spatial resolution is too coarse to accurately estimate the soil temperature gradient. This gradient can be very steep near the soil surface, and errors in its estimation are compensated for by adopting fictitious values for the soil thermal parameters (the thermal conductivity and heat capacity). The use of analytical solutions of the heat conduction equation can be expected to partially solve this problem.
A few attempts have been undertaken to derive analytical solutions of the
heat conduction equation that can easily be implemented in land surface
models. A number of solutions can be found in
This paper focuses on the estimation of ground heat fluxes and soil thermal properties using a land surface model. It is first examined whether or not calibrated soil thermal properties are independent of the vertical spatial resolution of the model, if the heat conduction equation is solved numerically. An analytical solution of the heat conduction equation is then derived, with temporally varying boundary conditions, which can be applied in the model. Using this analytical solution instead of the numerical approximation, the dependence of the obtained soil thermal properties on the model spatial resolution is then investigated.
The model parameters that need to be estimated.
The data used in this study have been acquired in the framework of the
AgriSAR (AGRIcultural bio/geophysical retrieval from frequent
repeat pass SAR (Synthetic Aperture Radar) and optical imaging) 2006 campaign, for which the test site was located in
Mecklenburg-Vorpommern in northeastern Germany, approximately 150 km North of
Berlin. More specifically, time-domain reflectometry (TDR)-based soil
moisture observations and Bowen ratio-energy balance (BREB)-based
observations of the energy balance components in a large winter wheat field
were available from 20 April to 5 July 2006, with the Bowen ratio data
containing a number of gaps. The soil moisture was measured at a depth of 5,
9, 15, and 25 cm. Meteorologic data from the weather station at Görmin
were available as well and can be used as model forcing from 2005 onwards.
All observations were converted to an hourly time step by averaging the
10 min observations. For this study, all model simulations were performed
from 1 April 2006 to 5 July 2006, with an hourly time step, unless
differently stated. A detailed description of this data set is given in
For the purpose of this study, the water and energy balance model developed
in
The model couples three physical equations. The movement of soil water in the
unsaturated zone is modeled using a numerical solution of the Richards
equation
Table
The model is applied with four different uniform vertical spatial resolutions, namely, 0.01, 0.025, 0.05, and 0.1 m.
The parameter estimation algorithm used in this paper, particle swarm optimization (PSO), is based on the collective behavior of individuals in
decentralized, self-organizing systems. These systems are created through a
population of individuals that interact locally with each other and with the
community. These interactions lead to global behavior, which can result in
the achievement of certain objectives. Examples of such systems in nature are
abundant: ant colonies, swarms of birds, and schools of fish
Comparison between the modeled and the observed energy balance terms
for the simulation with
Comparison between the modeled and the observed soil moisture values for the four different spatial resolutions, and a numerical solution of the heat conduction equation. The model results for all four spatial resolutions are very similar and therefore difficult to distinguish.
In order to estimate the model parameters, observations of the net radiation (
Results of the linear regressions between the energy balance
observations (
The model simulations resulting in the lowest objective function values for
the different spatial resolutions will be analyzed in this section.
Figure
Table
This result can be explained by the independence of the albedo (
In order to solve the issue related to the dependence on the grid resolution
in the use of a numerical solution of the heat conduction equation, we
propose the use of an analytical solution. First, the steady-state temperature
profile for a constant temperature at the bottom (
It should be noted that with this analytical solution it is no longer
necessary to calculate the soil temperature profile in order to calculate the
ground heat flux. In the original model formulation, the heat conduction
equation needed to be solved numerically, using the surface skin temperature
as a boundary condition, so the temperature of the first soil layer could be
calculated, and the ground heat flux could be computed. However, Eq. (
A synthetic test case is used to intercompare the analytical and the
numerical solutions. Equations (
Figure
Comparison between soil temperature profiles obtained using the
numerical (
Averages of the eight lowest objective function values and the corresponding parameter values.
Figure
Figure
Comparison between soil temperature profiles obtained using the
numerical (
Comparison of the resulting ground heat fluxes from the fine and coarse spatial and temporal resolutions to the analytical solution.
Comparison between the modeled and the observed energy balance terms
for the simulation where
Comparison between the modeled and the observed soil moisture values for the four different spatial resolutions, and an analytical solution of the heat conduction equation.
Results of the linear regressions between the energy balance
observations (
Table
A pooled variance
The key conclusion from these simulations is that the overall model performance is independent of the type of calculation of the ground heat flux (analytically or numerically), but that the results of the model calibration are more robust (i.e., independent of the spatial resolution) if an analytical solution of the ground heat flux equation is used.
A water and energy balance model, using a numerical solution of the heat conduction equation, has been calibrated against energy balance and soil moisture observations, for four different vertical spatial resolutions (0.01, 0.025, 0.05, and 0.1 m). It has been found that a number of parameters are dependent on this resolution, with the soil thermal conductivity values showing the largest dependence. An analytical solution of the heat conduction equation has then been derived, allowing for the bottom and top boundary conditions (i.e., the bottom and surface skin temperatures) to vary over time. Using this analytical solution has the advantage that the soil temperature profile no longer needs to be computed. For fine spatial and temporal resolutions the analytical and numerical solutions cannot be distinguished, but different results are obtained for resolutions typically used in land surface models. When the ground heat flux is calculated using this analytical solution, and the model is calibrated, the obtained soil thermal conductivity is no longer dependent on the model spatial resolution. Furthermore, the variability in the obtained soil-heat capacity is also strongly reduced.
The results in this paper indicate that a similar model performance is obtained when the ground heat flux is calculated analytically or numerically. However, the calibration is more robust, and the parameter values more physically interpretable, if the analytical solution is used. One must thus be careful when using numerical solutions of the heat conduction equation in land surface models, and preference should be given to the use of analytical solutions. The solution derived in this paper does not allow for temporally varying soil thermal properties, and ongoing research is focusing on the derivation of an analytical solution that is straightforward to apply in land surface models in these conditions.
The data can be accessed by contacting the AgriSAR team:
A solution of the heat conduction–convection equation is derived first, since
this equation is analytically more straightforward to solve than the heat
conduction equation because of the easier inversion from the Laplace domain.
Furthermore, this general solution can be used for purposes outside the scope
of this paper. The limit case with zero convection is then calculated. The
governing equation is
In order to obtain a realistic initial condition, we will calculate the
steady-state solution. For example, the profile at the end of a very long,
hot day. The equation becomes
We will use a constant time steps
We will calculate the temperature profile at
Through again calculating the temperature profile at time
For the temperature profile, we define two variables
In this case
Edoardo Daly and Valentijn R. N. Pauwels developed the idea to use an analytical solution to model the ground heat flux in land surface models, instead of numerical solutions. Valentijn R. N. Pauwels solved the equation, prepared the manuscript, and performed the model simulations. The results were extensively discussed and analyzed by both authors.
Valentijn R. N. Pauwels is funded through Australian Research Council Future Fellowship grant number FT130100545. Edited by: P. Gentine Reviewed by: J. Wang and two anonymous referees