Estimating seasonal evapotranspiration from temporal satellite images

Abstract

Estimating seasonal evapotranspiration (ET) has many applications in water resources planning and management, including hydrological and ecological modeling. Availability of satellite remote sensing images is limited due to repeat cycle of satellite or cloud cover. This study was conducted to determine the suitability of different methods namely cubic spline, fixed, and linear for estimating seasonal ET from temporal remotely sensed images. Mapping Evapotranspiration at high Resolution with Internalized Calibration (METRIC) model in conjunction with the wet METRIC (wMETRIC), a modified version of the METRIC model, was used to estimate ET on the days of satellite overpass using eight Landsat images during the 2001 crop growing season in Midwest USA. The model-estimated daily ET was in good agreement (R ² = 0.91) with the eddy covariance tower-measured daily ET. The standard error of daily ET was 0.6 mm (20%) at three validation sites in Nebraska, USA. There was no statistically significant difference (P > 0.05) among the cubic spline, fixed, and linear methods for computing seasonal (July–December) ET from temporal ET estimates. Overall, the cubic spline resulted in the lowest standard error of 6 mm (1.67%) for seasonal ET. However, further testing of this method for multiple years is necessary to determine its suitability.

Appendix: METRIC and wMETRIC models

A brief description of computational steps of Mapping Evapotranspiration at high Resolution with Internalized Calibration (METRIC) and the wet METRIC (wMETRIC) models is provided here. Readers interested in detailed process and procedures are advised to refer to Allen et al. (2007b, c) for the METRIC model and Singh and Irmak (2011) for the wMETRIC model. The computational processes are similar unless mentioned otherwise.

The net radiation (R _n) at the land surface is the difference of all the incoming and outgoing fluxes and computed as:

$$ R_{n} = R_{s}{\downarrow} - \alpha R_{s}{\downarrow} + R_{l}{\downarrow} - R_{l}{\uparrow} - (1 - \varepsilon_{o} )R_{l}{\downarrow} $$

(12)

where R _s ↓ is the incoming shortwave radiation (W m⁻²), α is the surface albedo (unitless), R _l ↓ is the incoming longwave radiation (W m⁻²), R _l ↑ is the outgoing longwave radiation (W m⁻²) and ε_o is the surface thermal emissivity (unitless). R _s ↓ is computed as a constant for the time of satellite image acquisition under the clear sky condition as:

$$ R_{s}{\downarrow} \, = \, G_{\text{sc}} \, \cos \theta \, d_{\text{r}} \, \tau_{\text{sw}} $$

(13)

where G _sc is the solar constant (W m⁻²), θ is the solar incident angle (degree), d _r is the inverse square of the relative earth–sun distance in astronomical unit, and τ_sw is the broadband atmospheric transmissivity (unitless). R _l ↓ and R _l ↑ were computed as follows:

$$ R_{l}{\downarrow} \, = \varepsilon_{a} \, \sigma \, T_{a}^{4} $$

(14)

$$ R_{l}{\uparrow} \, = \varepsilon_{o} \, \sigma \, T_{s}^{4} $$

(15)

where ε _a is the effective atmospheric emissivity (unitless), σ is the Stefan-Boltzmann constant (W m⁻² K⁻⁴), T_a is the near surface air temperature (K), ε _o is the broadband surface emissivity (unitless), and T _s is the surface temperature (K).

Soil heat flux (G) was computed as follows:

$$ G = [0.00647(T_{s} - 272.15) - 0.0955{\text{NDVI}} - 0.05]R{}_{n} $$

(16)

where NDVI is the normalized difference vegetation index (unitless).

Sensible heat flux (H) was estimated using the aerodynamic-based heat transfer equation as:

$$ H = {\frac{{\rho_{\text{a}} C_{\text{p}} {\text{d}}T}}{{r_{\text{ah}} }}} $$

(17)

where ρ_a is the air density (kg m⁻³), C _p is the specific heat of air at constant pressure (J kg⁻¹ K⁻¹), dT is the temperature difference (K) between two heights z ₁ (0.1 m) and z ₂ (2 m), and r _ah is the aerodynamic resistance to heat transfer (s m⁻¹). The dT is computed for each pixel based on linear relation between dT and T _s for the anchor (hot and cold) pixels as

$$ {\text{d}}T = aT_{\text{s}} + b $$

(18)

where a and b are the correlation coefficients for each satellite image based on reliable and accurate estimation of H at the anchor pixels. Since the stability of the atmosphere affects the aerodynamic resistance to heat transfer, stability correction was applied using Monin–Obukhov length parameter in an iterative process.

In the METRIC model, H at the cold pixel is computed based on corresponding R _n, G, and instantaneous alfalfa referenced ET (ET_r) values as follows:

$$ H = R_{\text{n}} - G - 1.05\lambda {\text{ET}}_{\text{r}} $$

(19)

The H at the hot pixel in the METRIC model is computed based on alfalfa referenced ET fraction (ET_rF) for the dry soil surface from water balance model following FAO 56 (Allen et al. 1998) as:

$$ H = R_{\text{n}} - G - {\text{ET}}_{\text{r}} {\text{F }}\lambda {\text{ET}}_{\text{r}} $$

(20)

In the wMETRIC model, H at the cold pixel was computed based on the Priestley–Taylor model (Priestley and Taylor 1972):

$$ H = R_{\text{n}} - G - \alpha {\frac{\Updelta }{\Updelta + \gamma }} \, (R_{\text{n}} - G) $$

(21)

The H at the hot pixel in the wMETRIC model was computed as:

$$ H = R_{\text{n}} - G - {\text{ET}}_{\text{r}} {\text{F }}\alpha \, {\frac{\Updelta }{\Updelta + \gamma }} \, (R_{\text{n}} - G) $$

(22)

Once the instantaneous R _n, G and H were determined, the instantaneous latent heat flux (LE, W m⁻²) was estimated using equation:

$$ {\text{LE}} = R_{\text{n}} - G - H $$

(23)

Based on the LE values, the instantaneous evapotranspiration (ET_ins, mm h⁻¹) was calculated as:

$$ {\text{ET}}_{\text{ins}} = 3,600 \, {\frac{\text{LE}}{\lambda }} $$

(24)

where λ is the latent heat of vaporization (J kg⁻¹) and computed as

$$ \lambda = \left[ {2.501 - 0.00236\left( {T_{s} - 273} \right)} \right]10^{ 6} $$

(25)

The reference ET fraction (ET_rF) was computed based on ET_ins and alfalfa referenced ET (ET_r, mm h⁻¹) from the weather data as follows:

$$ {\text{ETrF}} = {\frac{{{\text{ET}}_{\text{ins}} }}{{{\text{ET}}_{\text{r}} }}} $$

(26)

Finally, the daily ET (ET₂₄, mm day⁻¹) at each pixel within the image was computed as:

$$ {\text{ET}}_{ 2 4} = {\text{ET}}_{\text{r}} {\text{F ET}}_{\text{r24}} $$

(27)

where ET_r24 is the alfalfa referenced ET on daily basis (mm day⁻¹) based on summed up hourly ET_r.