Fórmulas METRIC

Objetivo

Presentar las fórmulas del modelo METRIC usando el motor de visualización de JavaScript llamado MathJax en una página web.

1. Descripción del modelo METRIC

Mapping evapotranspiration at high resolution with internalized calibration (METRIC) is a satellite-based image-processing model for calculating evapotranspiration (ET) as a residual of the surface energy balance. METRIC uses as its foundation the pioneering SEBAL energy balance process developed in The Netherlands by Bastiaanssen, where the near-surface temperature gradients are an indexed function of radiometric surface temperature, thereby eliminating the need for absolutely accurate surface temperature and the need for air-temperature measurements (Allen et al., 2007).

2. Balance de energía

Actual net radiation flux at the surface (Rn) represents the radiant energy at the surface that is partitioned into H, G, and LE. Rn is computed by subtracting all outgoing radiant fluxes from all incoming radiant fluxes and includes solar and thermal radiation.

2.1. Radiación Neta

The net radiation flux at the surface ($R_n$) represents the actual radiant energy available at the surface.

$$ R_n = R_{S\downarrow} - \alpha R_{S\downarrow} + R_{L\downarrow} - R_{L\uparrow} - (1- \varepsilon_0)R_{L\downarrow} $$ $$ R_n = (1- \alpha)R_{S\downarrow} + (R_{L\downarrow} - R_{L\uparrow}) - (1- \varepsilon_0)R_{L\downarrow} $$ $$ \text{Net surface radiation} = \text{gains} - \text{losses} $$

Dónde:

• $R_n \;$ : Flujo de radiación neta $[W/m^2]$
• $\alpha \;\;\;$ : Albedo de superficie $[-]$
• $R_{S\downarrow}$ : Radiación de onda corta entrante $[W/m^2]$
• $R_{L\downarrow}$ : Radiación de onda larga entrante $[W/m^2]$
• $R_{L\uparrow}$ : Radiación de onda larga saliente $[W/m^2]$
• $\epsilon_0 \;\;\;$ : Emisividad del ancho de banda en la superficie / broad-band surface thermal emissivity

El término $(1- \epsilon_0)R_{L\downarrow}$ representa la fracción de radiación entrante de onda larga (incoming long-wave radiation) reflejada desde la superficie.

2.2. Flujo del calor del suelo

Soil heat flux is the rate of heat storage in the soil and vegetation due to conduction. General METRIC applications compute G as a ratio $G/R_n$ using an empirical equation by Bastiaanssen (2000) representing values near midday

$$ \frac{G}{R_n} = ( T_s - 273.15 ) (0.0038 + 0.0074 \alpha ) (1-0.98 \; \text{NDVI}^4) \; \; \; \; \text{(26)} $$

Dónde

• $T_s$ : surface temperature $[K]$
• $\alpha$ : surface albedo $[-]$

G is then calculated by multiplying $G/R_n$ by $R_n$.

2.3. Flujo de calor sensible

Sensible heat flux is the rate of heat loss to the air by convection and conduction, due to a temperature difference. It is computed using the following one-dimensional, aerodynamic, temperature gradient based equation for heat transport: $$ H = \rho_{\text{air}} C_p \frac{dT}{ r_{\text{ah}} } $$ Dónde:

• $\rho_{\text{air}}$: air density $[kg/m^{3}]$
• $C_p$: air specific heat $[1004 \; J/kg/K]$
• $dT$: temperature difference (T1 - T2) between two heights (z1 and z2) $[K]$
• $r_{\text{ah}}$: aerodynamic resistance to heat transport $[s/m]$

Referencias

Libros

• Allen, R. G., Pereira, L. S., Raes, D., & Smith, M. (2006). Evapotranspiración del cultivo: guías para la determinación de los requerimientos de agua de los cultivos. Roma: FAO, 298(0).
  https://www.fao.org/3/x0490s/x0490s00.htm

Artículos

1. Allen, R. G., Tasumi, M., & Trezza, R. (2007). Satellite-based energy balance for mapping evapotranspiration with internalized calibration (METRIC)—Model. Journal of irrigation and drainage engineering, 133(4), 380-394. https://doi.org/10.1061/(ASCE)0733-9437(2007)133:4(395))

Muchas gracias por leer. Saludos!

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