Determining the ground heat flux using a variational technique

In micrometeorology, historically relative little attention was paid to measurement of the ground heat flux at the surface. The reason was that it was considered a small term comprising typically about one tenth of the net radiation for vegetated surfaces (De Bruin and Holtslag 1988). Recently, however, attention for ground heat flux has increased, because of two reasons. First, micrometeorological research has shifted towards drier areas where the ground heat flux is a more important component of the surface energy budget (Verhoef et al. 1995; Passerat de Silans 1997; Heusinkveld et al. 2004). Second, the ground heat flux is acknowledged as an important component of the surface energy balance in stable nocturnal conditions (Van de Wiel et al. 2002).
In most application, the ground heat flux at the surface is not measured directly, but derived from measurements of the soil heat flux and soil temperatures at certain depths below the surface. Usually, this is done by the method of De Vries and Van Wijk (1966), where it is assumed that time series of ground heat flux and soil temperature can be analyzed using harmonic analysis. Inserting the harmonics in the diffusion equation gives then a set of ordinary differential equations that can be solved yielding the vertical distribution of the amplitudes of the harmonics with depth. Extrapolation to the surface gives the harmonic of the surface ground heat flux. Disadvantages of this method are that it assumes parameters as the heat capacity of the soil and the conductivity to independent of time and invariable with depth, that it neglects slow cycles ( ~ > 1 day), that in many application it neglects useful available information on soil temperatures, while it is smoothing high frequencies, present at the surface.
The aim of the presentation is to show results of a novel method for determining the surface ground heat flux for the Cabauw site (52N, 5E) in the Netherlands. The new method derives the ground heat flux by applying the adjoint technique i.e. a technique where observations are statistically optimally assimilated into a model that describes in detail the transport of heat from the grass vegetation into the soil and within the soil column. As such, a framework is established in which surface ground heat flux estimates are obtained that are both dynamically consistent and are optimal with respect to the different measurements taken at the soil site of the Cabauw site. Our approach can be summarized as follows:
1. the heat transport model is run forward using the temperature of the vegetation and the temperature at 12 cm as boundary conditions;
2. the adjoint of the heat transport model is run backward, driven by the discrepancies between modeled and measured soil temperatures at 0.04 cm, 2 cm, 4 cm, and 8 cm and ground heat fluxes at 5 cm and 10 cm;
3. the control variables, the skin conductivity describing the transport of heat between the vegetation and the upper soil, the soil conductivity of the soil and the volumetric heat capacity of the soil, are updated;
4. decision is made whether steps 1 to 3 are repeated or the values of the control variables have converged sufficiently.
The results indicate that this approach gives optimal estimates of the ground heat flux at the surface that are based on all available information. The method facilitates that the control variables vary on scales longer than one day, it takes account of the impact of slow cycles via the prescribed boundary conditions, and it takes account of high-frequency variations in the ground heat flux at the surface. Furthermore, the method can be extended in a straightforward way to account for non-homogeneities in the soil column.