Modeling Soil Surface Energy Fluxes from Solar Radiation, Latent Heat, and Soil plus Air Temperature Measurements Edgar G. Pavia CICESE, Ensenada, B.C., Mexico 1 Fig. 1. Experiment set-up. 2 Fig. 2. Location. Lat: 31° 52' 09" N. Long: 116° 39' 52" W. Elev: 66 m above msl. I. Experiment A bird-guarded wet-soil evaporating tray is set on an electronic balance next to a meteorological station for a 28-day period (Θ), from 11 February to 11 March 2011, at the location shown (Figures 1 and 2). II. Data We register the balance-weight (W), soil temperatures at 2 cm depth (To) and at 7 cm depth (Ts), air temperature at 2 m (Ta), solar radiation (R), precipitation (P), and other variables at Δt = 300 s intervals during Θ = N Δt, where N = 8064 is the total number of samples (Figure 3). Vienna | Austria | April N W E S Fig. 3. The recorded data. The × sign indicates evaporating-tray water recharge. III. Albedo The changing albedo of the wet-sand surface, as the sand dries up, is a major problem for modeling surface energy fluxes (Figure 4). 4a 4b 4c Fig. 4. Wet sand at different stages of wetness. IV.Empirical approach First we estimate the time rate of weight change, ΔW i = (W i-½ – W i+½ ) / Δt [kg s -1 ], the temperature differences between air and ‘surface’, ΔTo i = To i – Ta i, and between soil and ‘surface’, ΔTs i = To i – Ts i (Figure 5). Second we estimate ‘evaporation’, E i = ΔW i / A [W m -2 ], where = 2.45 x 10 6 J kg -1 is the latent heat of vaporization, and A = 0.22 m 2 is the evaporating surface area. Finally we fit E i to R i, ΔTo i, and ΔTs i to get: E’ i = a 1 R i + a 2 ΔTo i + a 3 ΔTs i ; i = 1, 2, …, n, where a 1, a 2 and a 3 are found by multiple regression. The fitting period is θ = n Δt (n<N; e.g. n = N/2). Third we test the model with independent data: n+1, n+2, …, N (Figure 6). × × ××

6 5 VI. Conclusions The empirical model works relatively well, with correlation coefficient r (E, E’ ) = 0.9, for the entire 28-d observing period (N = 8064). Obviously it cannot simulate heavy-precipitation events nor anomalously dry and windy conditions (see Figures 6 and 7). Evaporation depends strongly on solar radiation (R), since the temperature-difference terms (ΔTo and ΔTs) contribute with minimum variance explained. However these latter terms do not render the model unstable when modeling independent data (Figure 6, red line), and their coefficients, obtained by the multiple regression method, allow us to optimistically model the simplified surface energy flux balance: R n + H + G – E = 0, if the net radiation R n ~ a 1 R, the sensible heat flux H ~ a 2 ΔTo, and the surface soil heat flux G ~ a 3 ΔTs i. Vienna | Austria | April ∙ E i = ΔW i / A; i = 1, 2, …, N. ∙ E’ i = a 1 R i + a 2 ΔTo i + a 3 ΔTs i ; i = 1, 2, …, N. a 1 = 0.41 a 2 = [W m -2 C -1 ] a 3 = [W m -2 C -1 ] Fig. 6. Estimated (E) and modeled ‘evaporation’ (E’). The + sign indicates recorded precipitation (see Figure 3) ΔW (per Δt) [Kg] ΔTo [C] ΔTs [C] Kg ×× Fig. 5. Weight difference in a 5-minute interval (ΔW), air to surface temperature difference (ΔTo), and soil to surface temperature difference (ΔTs). V.Method (brief review) Make the vector: y = [E 1 E 2 … E n ] and the matrix: R 11 R 21 … R n1 X = ΔTo 12 ΔTo ΔTs 13 ΔTs 23 … ΔTs n3. With the vector of coefficients (to be found): a = [a 1 a 2 a 3 ] the problem becomes: y’ = aX. Minimizing Q = (y – aX) (y – aX) T i.e. ∂Q/∂a = 0 yields a = yX T (XX T ) -1 and thus y' = [E’ 1 E’ 2 … E’ n ]. Next, test a for stability with the independent data: (R i, ΔTo i, ΔTs i ), i = n+1, n+2, …, N, to obtain: E’ n+1, E’ n+2, … E’ N. If, in addition, a 1 < 1, a 2 < 0, and a 3 < 0, the model is stable and ‘physical’. 0 % 0 0 m/s 10 Fig. 7. Relative humidity (yellow) and wind speed (bluish). 100