Development of a Land Surface Temperature Module

A goal of this study is to incorporate spatially distributed information into the hydrologic model mHM to improve the spatial representativeness of the hydrologic fluxes and states. Herein, we will focus on the evapotranspiration since it has a high impact on the water balance. Therefore, we considered satellite derived land surface temperature fields within mHM. The spatio-temporal distribution of land surface temperature is used to constrain mHM in addition to discharge.

Since the purpose of mHM is to solve the water balance equation, land surface temperature is not required to be estimated yet. By closing the water balance, evapotranspiration (E) is estimated. The energy balance is used to simulate land

surface temperature, since the evapotranspiration is the common variable of the energy and the water balance.

The following section will introduce a parsimonious module for estimating land surface temperature based on modeled evapotranspiration. This module is called land surface temperature module in the following. It can be coupled to any hy-drologic model and will be adjoined with mHM.

On the one hand the evapotranspiration E [mm d⁻¹] is determined by closing the water balance

E =P −Q−∆S (4.1)

with mHM. Where P is precipitation [mm d⁻¹], Q is river runoff [mm d⁻¹], and

∆S [mm d⁻¹] is the change in the storages, e.g., soil moisture. On the other hand the energy balance is defined as

Rn=LE+H+G+S (4.2)

in whichR_ndenotes the net radiation [W m⁻²],LEis the latent heat flux [W m⁻²], H is the sensible heat flux [W m⁻²],Gis the soil heat flux [W m⁻²] and S are the storage terms [W m⁻²], e.g., photosynthetic or biomass heat storage. The latent heat flux LE is determined by converting the mass flux of evapotranspiration E estimated by mHM (Equation 4.1) to an energy flux. This conversion is calculated by

LE =%LE. (4.3)

In which the L is the latent heat of vaporization [kJ kg⁻¹] and % is the density of water (% = 1000kg m⁻³). The latent heat of vaporization L is approximated by L= 2501−2.37T_ausing the air temperature T_a in [^◦C] (Dyck and Peschke, 1995).

The estimation of land surface temperature is performed using the temporal res-olution of one day, because this is the temporal resres-olution of the meteorological input. For daily time steps it is assumed that the soil heat fluxG and the storage terms S are negligible (Haverd et al., 2007). Therefore Equation 4.2 simplifies to

H =Rn−LE. (4.4)

Because mHM is not estimating energy fluxes, the net radiation R_n has to be provided as an input to be able to estimate the sensible heat fluxH. Since spatially comprehensive measurements of the net radiation are not available, an alternative source of data is required. One possible source are radiation products obtained

4.4. Methodology from satellite data. A straight satellite product for net radiation is not available.

Therefore, it is determined by the single components of the radiation budget, which are derived from satellite measurements, as

R_n =Q⁽ⁱⁿ⁾_S −Q^(out)_S +Q⁽ⁱⁿ⁾_L −Q^(out)_L (4.5a)

Q^(out)_S =αQ⁽ⁱⁿ⁾_S (4.5b)

Q^(out)_L =Tb_s⁴ (4.5c)

whereQ⁽ⁱⁿ⁾_S andQ^(out)_S are the incoming and outgoing short-wave radiation [W m⁻²], respectively, and Q⁽ⁱⁿ⁾_L and Q^(out)_L are the incoming and outgoing long-wave ra-diation [W m⁻²], respectively. The outgoing short-wave Q^(out)_S radiation is es-timated using Equation 4.5b, in which α is the albedo of the land surface [−].

The outgoing long-wave radiation Q^(out)_L is approximated as emission of a gray body which can be calculated following the Stefan-Boltzmann law (Equation 4.5c).

Therein, is the emissivity [−] and σ is the Stefan-Boltzmann constant (σ = 5.67·10⁻⁸ W m⁻² K⁻⁴).

Thus, the estimation of the sensible heat fluxH [W m⁻²] (Equation 4.4) modifies to

H = (1−α)Q⁽ⁱⁿ⁾_S +Q⁽ⁱⁿ⁾_L −σTc_s⁴−LE. (4.6) Further, the thermodynamical formulation of the sensible heat H is known by

H =%acp

Tc_s−T_a

r_a (4.7)

where T_a is the air temperature [K], Tc_s the model derived land surface temper-ature [K], r_a is the aerodynamic resistance [s m⁻¹], %_a the density of air (%_a = 1.29 kg m⁻³) and c_p the specific heat capacity of air which has been assumed to be constant (c_p = 1004 J kg⁻¹ K). Combining Equation 4.6 and 4.7 leads to a polynomial of forth degree in Tc_s

(1−α)Q⁽ⁱⁿ⁾_S +Q⁽ⁱⁿ⁾_L −LE+%_ac_p

r_a T_a− %_ac_p

r_a cT_s−σcT_s⁴ = 0. (4.8) To sum up,Tb_sis the modeled variable of interest,Q⁽ⁱⁿ⁾_S ,Q⁽ⁱⁿ⁾_L ,α, andare satellite retrieved variables, %_a , σ, and c_p are constants, T_a is measured air temperature which is an input for mHM, LE is derived by closing the water balance with mHM (Equations 4.1 and 4.3), and r_a is the aerodynamic resistance which is still unknown, but will be explained in the following.

By solving Equation 4.8 four roots are obtained. The root which falls in the interval [0K,500K] is the feasible result forTc_s. During all experiments it was found that only one of the four roots does fulfill this requirement.

Still there is one unknown variable, i.e., the aerodynamic resistancer_a in [s m⁻¹], which is calculated using the equation of Allen et al. (1998):

r_a=

where z_h is the height of the humidity measurement [m], d is the zero plane dis-placement height [m], z_0m is the roughness length for momentum transfer [m], z_0h is the roughness length for heat transfer [m], k is the von Karman constant (k = 0.41), and u_z is the wind speed [m s⁻¹] at the wind speed measurement height z_m in [m]. It is assumed that the measurement heights of wind speed and humidity are equal z =z_m=z_h.

The approximations of the three variablesd= ²₃h_c,z_0m = 0.123h_c, andz_0h = 0.1z_0m are taken from Allen et al. (1998). The constants of d, z0m and z0h have been im-plemented as global parameters p₄₈, p₄₉, and p₅₀ in the land surface temperature module, respectively. These parameters need to be calibrated whereas their ranges are chosen to be between±10% of the values reported by Allen et al. (1998). Thus Equation 4.9 becomes

This shows that besides the given height z and the measured windspeed u_z, r_a is dependent on the estimation of the parameters p₄₈, p₄₉, and p₅₀ and the canopy height h_c.

Since no spatially comprehensive information about the canopy height h_c is avail-able, the Multiscale Parameter Regionalization (MPR) technique has been em-ployed to estimate hc based on the land cover information. To account for the annual development of h_c the monthly evolution of the leaf area index (LAI) is taken into consideration. The functional relationship between canopy height and the LAI, i.e., for the mixed land cover class, is assumed to be

h_c,mix(i) =p₄₇ LAI(i)

maxi LAI(i), i= 1, . . . ,12. (4.11) in whichh_c,mixis the canopy height of the mixed land cover class [m],LAI(i) is the leaf area index [m m⁻²] for monthi[−] andp₄₇is the calibration parameter [m] for

4.4. Methodology the estimation of the canopy heighth_c for the mixed land cover class (mix). The mixed land cover class is a generalized class consisting of grasslands, agricultural areas and pastures.

Both other land cover classes (urban (ur) and forest (f or)) are assumed to be constant in canopy height over the course of a year and do not depend on LAI

h_{c,f or} =p₄₅ and h_c,ur =p₄₆. (4.12)

The estimation of the canopy height is conducted on the input resolution, i.e., 100×100 m². For the upscaling to the model resolution, i.e., 4×4 km², various up-scaling operators have been tested and the arithmetic mean has proven to perform best.

To sum up, we have shown the development of a land surface temperature module which can be coupled to any environmental model. Satellite derived radiation components (Q⁽ⁱⁿ⁾_S andQ⁽ⁱⁿ⁾_L ), air temperature (T_a), wind speed (u_z), and modeled evapotranspiration (E) are used as input for the land surface temperature module.

The necessary steps for estimating land surface temperature (cT_s, Equation 4.8) are 1. The estimation of E as residual of the water balance (Equation 4.1) and 2. The calculation of Tcs (Equation 4.8) based on the aerodynamic resistance

(Equation 4.10).

To approximate the aerodynamic resistancera(Equation 4.10) the threeglobal pa-rameters p₄₈top₅₀connected to the displacement height and the roughness lengths as well as the three global parameters p₄₅top₄₇ connected to the canopy height (Equation 4.11 and Equation 4.12) are necessary. An additional parameterp51was introduced into Equation 4.8 to account for the bias, which has been observed in the satellite retrievedT_s. Finally, this parameter could be neglected because a bias insensitive error measure was designed for calibrating mHM with Ts.

These seven global parameters are estimated by an automated calibration of the model mHM. The difference between satellite derived T_s and simulated land sur-face temperature Tc_s is minimized during model calibration. The calibration pro-cedure will be explained in the following.

4.4.3 Optimization of the Coupled mHM-Land Surface