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This article is an open-access article distributed under the terms and conditions of the CreativeCommons Attribution license (http://creativecommons.org/licenses/by/3.0/).

In order to make the prediction of land surface heat fluxes more robust, two improvements were made to an operational two-layer model proposed previously by Zhang. These improvements are: 1) a surface energy balance method is used to determine the theoretical boundary lines (namely ‘true wet/cool edge’ and ‘true dry/warm edge’ in the trapezoid) in the scatter plot for the surface temperature _{m} – ^{2}∼50 w/m^{2}, which is consistent with the site scale measurement of latent heat flux. Because soil evaporation and vegetation transpiration are not measured separately from the field site, _{2} flux is used to examine the modeled _{veg}. Similar trends of seasonal variations of vegetation were found for the canopy transpiration retrievals and _{2} flux measurements. The above differences are mainly caused by 1) the scale disparity between the field measurement and the MODIS observation; 2) the non-closure problem of the surface energy balance from the surface fluxes observations themselves. The improved method was successfully used to predict the component surface heat fluxes from the soil and vegetation and it provides a promising approach to study the canopy transpiration and the soil evaporation quantitatively during the rapid growing season of winter wheat in northern China.

Evapotranspiration (

By treating the soil-vegetation system as a single uniform leaf, the big-leaf model simplified the mechanism of the energy exchange between the surface and the atmosphere, and therefore the regional scale evapotranspiration simulation is made. This category of models is simple and convenient to use, but the limitation is that this big-leaf approximation in the model is not applicable to surfaces with highly spatial heterogeneity due to large differences of surface energy exchange between soil and vegetation, such as in arid or semi-arid areas. Therefore, a two-layer model is proposed and the surface available energy is partitioned between soil and vegetation to overcome the limitation of the big-leaf model. These models have an improvement over the big-leaf models when applied to sparsely vegetated surfaces [

In the existing two-layer models, the cores of the algorithm primarily lie in two aspects: (1) accurately decomposing surface temperature of mixed pixel (_{m}) into soil temperature (_{soil}) and vegetation temperature (_{veg}); (2) obtaining accurate surface resistances, such as aerodynamic resistance, canopy resistance, residual resistance. In recent years, many attempts had been made to investigate the two issues. For instance, Norman and Kustas [_{n}_{a}); Zhang _{m}) and net radiation of mixed pixel, and finally to estimate soil evaporation (_{soil}) and vegetation transpiration (_{veg}). Because the multi-angle satellite data is not always available, multi-angle method of surface temperature decomposing is limited for applications, In contrast, PCACA algorithm is more convenient because only single angle remote sensed data are required and it can be provided from most of the satellite data. Additionally, by using the layered energy-separating algorithm the core of which is Bowen-ratio energy balance method, the uncertainties in surface energy partitioning based on the Beer's law are reduced.

PCACA algorithm and layered energy-separating algorithm utilize the scatter plot of the surface temperature against vegetation fraction cover (_{m} – _{m} – _{a}, aerodynamic resistance _{a}, surface reflectivity _{m} – _{soil} values for all pixels at an iso-line are equivalent, so are for _{veg} values, which is just like the case that while measuring the same area constituted by soil and vegetation at varying view angles, _{soil} and _{veg} are invariable and thus _{m} observations only vary with vegetation fraction cover. Under this condition, _{soil} and _{veg} could be obtained by calculating the slopes of these iso-lines of equal soil water availability (d_{m}/d_{m} – _{soil} and _{veg} calculations, The temperature separation finally influences

In this paper, to improve the accuracy of _{m} –

The two-layer model used in this study was presented by Zhang

_{m}_{smin}).

Soil water content equals to field capacity; ‘true dry/warm edge’ represents zero evapotranspiration and has maximum surface resistance to evapotranspiration (_{smax}). If the positions of the two edges are determined, the shape and the structure of the trapezoid can be fixed and the consequent calculations of the surface heat fluxes could be done. To make the illustration easier to follow, four points are defined: _{sd} and _{sw} represent the points of true dry bare soil and true water saturated bare soil, respectively. _{vd} represents true dry full-cover vegetation and _{vw} represents true water saturated, full-cover vegetation. The above definitions all correspond to the ideal surface conditions, namely there exist driest and wettest bare soil and full-cover vegetation. In reality there are always insufficient number of pixels that can cover all kinds of soil wetness and vegetation fraction cover within the study area, which leads to a difficulty in determining “true wet/cool edge” and “true dry/warm edge”, as a result, “observed wet/cool edge” and “observed dry/warm edge” (dashed lines) are often defined according to the envelop shape of the actual scatter plot to represent actual two extreme soil moisture conditions and are used to replace “true wet/cool edge” and “true dry/warm edge” in most applications, although some errors would be introduced. Iso-line of equal vegetation fraction cover intersects “true dry edge” and “true wet edge” at true maximum temperature and true minimum temperature denoted as _{mi,max} and _{mi,min}, respectively. It has to be noted that for the trapezoid constructed by data with coarser pixel resolution (eg. 1km), surface temperature at “true dry edge” generally is higher than that at “observed dry edge”, contrarily surface temperature at “true wet edge” is lower than that at “observed wet edge” according to the findings of Carlson [

The PCACA algorithm is a method to decompose the mixed surface temperature. As mentioned above, its physical basis is the fact that soil water status can be represented by the configuration of _{m}-

The basic formulations used to decompose mixed surface temperature are shown in

_{m}, _{veg} and _{soil} are the broadband emissivities of mixed pixel, vegetation and bare soil, respectively; σ is the Stefan-Boltzmann constant; d_{m}/d_{veg} of 0.97 and _{soil} of 0.95 were used. By simply weighting the fractional cover for vegetation and bare soil, mixed pixel emissivity _{m} for each pixel can be computed following

From the above three equations, we can see that to solve _{soil} and _{veg},

According to _{u} is for upper bound, _{L} is for lower bound). A detailed method for this will be illustrated in section 3; (2) linearly interpolating

Considering that the precise formulation of the relationship is actually unknown, linear interpolation is a reasonable approximation according to the study on the configuration of _{m}

Essentially, the aim of the Layered Energy-separating algorithm is to calculate Bowen-ratio (_{soil} and _{veg}, respectively. By using the relationship between Water Deficit Index _{0}) illustrated by Moran

After _{s}, _{v}, _{s} and _{v} are obtained using the above methods, net radiation at the soil surface and at the vegetation surface (_{n,soil}, _{n,veg}) can be calculated following

_{0} is the solar incident total radiation and is regarded as spatially uniform for clear sky conditions at the regional scale, usually obtained from standard meteorological station; _{soil} and _{veg} are the albedo of bare soil and vegetation, also calculated by the PCACA algorithm (seen from _{sky} is the average sky emissivity and is approximately set to 1.0 in the study. _{sky} is average sky temperature and usually approximates to the temperature at 37° view sky angle [

In terms of the Bowen-ratio energy balance method, soil evaporation (_{soil}) and vegetation transpiration (_{veg}) can be retrieved based on _{soil} and _{veg}, expressed as

As mentioned above, the locations of true dry edge and the true wet edge are crucial in the application of PCACA algorithm. The previously used method to determine the trapezoid boundary often leads to uncertainties because of subjective judgement. To reduce the errors from this respect, a physically based method, which takes account of the surface energy balance, is presented in this study.

According to _{sd}, _{sw}, _{vd} and _{vw} can determine the envelop shape of the trapezoid, that is to say, as long as their values are obtained. Consequently, the true dry edge and the true wet edge can be determined. In the study, surface energy balance method was adopted to compute their values. For pixels at the true dry edge, _{n}_{sd}. In the same way, _{vd} is formulated as

_{p}_{a}_{sd} is the albedo of dry bare soil.

From these two equations, we can see that _{sd} or _{vd} could not be iteratively computed until parameters of _{sd}, _{vd}, _{sky}, _{sda}, _{vda}, _{sda}, _{vda} are acquired. From the _{m}_{sd}, _{vd} points under the conditions that the variations in atmospheric conditions are very small due to the spatially uniformity. In the study, we chose 50 pixels around the upper-left corner in the trapezoid representing the observed driest bare soil and 50 pixels around upper-right corner in the trapezoid representing the observed driest full-cover vegetation, and the highest _{a} and _{a} in each 50 pixels were selected as _{sda} and _{sda}, _{vda} and _{vda}, respectively. The method of retrieving the spatial distribution of _{a} and _{a} will be described in the following. The calculation of _{a} requires _{a}_{s}, the retrievals about which will also be described. As for _{sd} and _{vd}, we selected the lowest albedo values of bare soil and full-cover vegetation within the whole scene of remotely sensed image (pixels) because low albedo would result in high surface temperature at the same soil water content, judging from _{a}, _{a}, _{s} and _{a}:

Surface temperature, as a heat or cold source, influences the variations of air temperature by heating or cooling near-surface atmosphere. In most cases, high surface temperature is accompanied by high air temperature and low surface temperature is accompanied by low air temperature. By using this relationship between them and assuming the following ratio expressed in

_{a}

Like the relationship between air temperature and surface temperature, there are also strong interactions between near-surface actual vapor pressure and soil moisture. If there is no horizontal and vertical advection, vapor in the atmosphere mainly comes from soil water by an evapotranspiration process. Contrarily, the vapor gradient between surface and air influences the intensity of evapotranspiration. By assuming the following ratio relationship between near-surface actual vapor pressure and soil water, we interpolated _{a}, where soil moisture status was characterized quantitatively by the soil apparent thermal inertia:
_{i}, _{s}_{min} is the minimum surface temperature during the daytime which usually occurs before sunrise when _{n}=0 and for all pixels it can be assumed to be the same value [

In the retrieval of evapotranspiration, _{s} is used to correct the difference between the vapor pressure at the surface (_{s}) and the saturated vapor pressure at the evaporating front (_{s}*). In theory, _{s} ranges from ∞ to 0 corresponding to the surface conditions of potential evapotranspiration and zero evapotranspiration, respectively, namely the true dry edge and the true wet edge. However, in reality the condition of zero evapotranspiration (r_{smax} = ∞) rarely occurs for vegetated surface even in semi-arid environment primary due to root zone soil water uptake, consequently, we selected a pixel closest to the observed driest bare soil where a meteorological station is located to calculate _{smax} by _{a}, _{smax} is about 1000 (s m^{-1}) according to the calculation, which is in agreement with Qiu's observations [_{smin} is set to 0 in the study.

After the upper and lower limits of _{s} are determined, _{s} is interpolated linearly within each _{si} for a pixel at (_{i} , _{i}) equals:

Besides air temperature, aerodynamic resistance is a site-specific variable and can not be retrieved directly by remote sensing. Although Monin-Obukhov similarity theory has been widely used to estimate it, the accurate calculations for the spatial distributions of roughness length (_{0}), wind speed (

An expression of _{a} is obtained on the basis of the energy-balance by substituting _{s}* is the saturated vapor pressure at _{m} estimated using the classic formulation with regard to surface temperature, as _{a}, _{a} and _{s} are estimated by the approaches mentioned previously.

Above all, all necessary parameters to calculate _{sd}, _{vd} can be retrieved according to the _{m} and _{m} is obtained from MODIS (Moderate Resolution Imaging Spectroradiometer) standard land data products, MOD11 and MOD02 [

Many studies [_{m}– _{sw} and _{vw} also can be parameterized on the basis of energy balance equation like _{sd} and _{vd}, the inputs (_{sd}, _{vd}, _{sda}, _{vda}, _{sda}, _{vda}) cannot be obtained by the above methods. Taking account of the fact that the surface radiant temperature of dense vegetation is very close to the ambient air temperature [_{vw}. As for _{sw}, we adopted an approximation of using the surface temperature of standing water body (such lake) as the surface temperature of true wet bare soil _{sw}, that is to say, standing water body is regarded as the surface of potential evapotranspiration. In fact, it is not uncommon to find some patches of standing water body in a remote sensed image. In this study, the surface temperature of Dongping Lake (35.965°, 116.81°; water area: 209 km^{2}) was used as _{sw}. In applications, we found that the pixels with mixed surface temperatures below the true wet edge all scattered around the cloud pixels and the coastline pixels.

Using the above methods, the positions of the true dry edge and the true wet edge in the triangle can be located. When surface radiant temperature is mainly dominated by surface soil water content, the following relationship between surface temperature and soil water content is tenable (_{sd}, _{sw}, _{vd} and _{vw} due to the constant surface temperature.

Same as the triangle method, an important assumption for the PCACA algorithm is that the surface evapotranspiration is primarily constrained by soil water availability, based on which _{soil} values for all pixels with an equal water availability are identical, so are for _{veg} values, that is to say, the configuration of _{s} and _{s}

In the study, the effects of four controlling factors (_{a}, _{a}, _{a} and albedo) on surface temperature were eliminated. According to the assumption, _{a}, _{a}, _{a} and albedo values of each pixel should be equal, namely they are spatially homogeneous, to ensure that only water availability controls surface temperature. To meet this requirement, the average values of _{a}, _{a}, _{a} and albedo in the image are assigned to each pixel and a new mixed surface temperature (_{mi}’ ) is calculated based on energy balance equation. _{mi}’ is the assumed temperature controlled only by soil water availability and therefore the new trapezoid constructed by _{m} and _{m}’ – _{m} – _{m}’, _{veg}’ (vegetation temperature controlled only by soil water availability) and _{soil}’ (soil temperature controlled only by soil water availability) can be obtained. Now the problem is that the calculated _{veg}’ and _{soil}’ cannot represent the actual temperature of vegetation and soil after the above transformation., Therefore they can't be directly used in the consequent calculations. To estimate the true vegetation temperature and true soil temperature (_{veg} and _{soil}) from _{veg}’ and _{soil}’, the following method is used.

Assuming that the thermal energy fraction assigned to the vegetation and the soil are constants, the following expression [_{soil}, _{veg} and _{m} values would all increase with the increase of solar radiation intensity and all influenced by wind speed. It is not likely to happen that the _{soil} increases, but _{veg} decreases in the same atmospheric conditions. Therefore, the assumption is reasonable in practice and it is different from Beer's law using fractional vegetation cover as the weight to partition thermal energy. Using this equation, _{soil}^{4} - _{veg}^{4} can be solved as

Combined with _{soil} and _{veg} can be solved. _{soil} and _{veg} can be directly obtained from _{soil} and _{veg} is used instead of the absolute temperatures and _{s}_{oil} and _{veg} values.

The study area is located in the North China Plain and ranges from 35.2N to 40.84N in latitude, from 113.68E to 119.54E in longitude. The land use in the area is dominated by the rotating cropping of winter wheat and summer maize. Millet, soybean and cotton are also scattered planted in summer [

In the study, field measurements from 135 standard meteorological stations were used. The measurements include air temperature and actual vapor pressure at 2 m height above the surface, solar incoming radiation, surface radiative temperature, wind speed, upward longwave radiation, upward shortwave radiation, downward longwave radiation and downward shortwave radiation. _{2}/H_{2}O analyzer since 2002. _{2} fluxes were originally sampled at 10 Hz and values averaged over 30 min were used in the study to validate the above mentioned methods. The rectangle symbol (■) shows the location of Dongping Lake. Water surface temperature has been measured for seven years since 2001 and was used to validate the MODIS land surface temperature products used in this study, and thereby to ensure the accuracy of _{sw} values.

MODIS land data products, including MOD11 (Land Surface Temperature), MOD03 (Geolocation Data Set), MOD05 (Total Precipitable Water), MOD02 (Calibrated Geolocated Radiance) and MOD35 (Cloud Mask), for clear sky during springtime between Mar and June when winter wheat is the dominant crop were used in the study. The visible channels of MOD02 were processed with atmospheric correction using the SMAC algorithm [_{n}, PCACA algorithm and Layered Energy-separating Algorithm were applied.

The instantaneous surface albedo was obtained by averaging reflectance values for several visible and near-infrared channels with the wavelength as the weight. This would introduce some errors because channel reflectance values observed by a satellite are reflectance at only one Sun-target-sensor configurations and it is generally different from the hemispherical reflectance. But we assumed its influence to be small as pointed by Nishida [

_{max} and _{min} are _{min}=0.09 and _{max}=0.78. Other variables, such as _{n}, _{a}, _{a}, _{s}, were obtained using the above mentioned methods.

_{soil}, _{veg} obtained from the original model and _{soil}’, _{veg}’ obtained from the improved model at an iso-line of equal water availability.

Apparently, the results of _{soil}’, _{veg}’ are in closer agreement with the above assumptions that soil surface temperature for all pixels at an iso-line are identical, so are the vegetative surface temperature. It provides a better physical foundation for the model.

Due to the absence separate measurement of the soil evaporation, soil heat flux, vegetation transpiration and vegetation heat flux from the field sit, estimates of _{n}-_{n}-

Because direct comparison of _{soil} and _{veg} cannot be performed, the relationship between modeled _{veg} and measured CO_{2} were used to indirectly validate the two-layer model since CO_{2} fluxes are closely related to _{veg}[_{veg} and measured CO_{2} flux. Note that the minus sign (-) for CO_{2} fluxes means that the flux transferring direction is from up to down. The performance of the model was evaluated using the root mean squares difference (RMSD) and the mean absolute difference (MAD), which are defined as

The figures demonstrate that (1) no significant bias is found between the modeled and measured _{n}-^{2}=0.79) suggests that _{n, soil} and _{n, veg} calculated from _{soil} and _{veg} are reasonable because _{n}-_{veg} and measured CO_{2} flux shows good agreement though few points in 2006 showed different variations perhaps influenced by horizontal/vertical advection or other factors. Vegetation transpiration generally increases with the crop growth, at the same time, since vegetation absorbs more CO_{2} due to the more active photosynthesis in the daytime, CO_{2} fluxes also increases. On the contrary, _{veg} and CO_{2} fluxes both became very small after crop harvest or in winter. In the study, winter wheat was harvest in June, therefore _{veg} and CO_{2} fluxes rapidly decreased on June 15 and June 19 in 2005.

In fact, there are two other factors which contribute to the uncertainty in the above validation. One is that point measurements usually can not represent the whole MODIS pixel (1 km^{2}) because of the large scale disparity between them. The other is that the observed _{n}, ^{2}) for modeled

_{veg} maps on May 2, 2005 for the North China Plain retrieved by the improved model and the original model, respectively. Differences were found between them. The _{veg} value from the improved model varies from 222 (w/m^{2}) to 456 (w/m^{2}), while the original _{veg} values varied from 50 (w/m^{2}) to 560 (w/m^{2}). In terms of the experiences and the knowledge on the vegetation transpiration, a small _{veg} dynamical range is more reasonable in the rapid growing season of winter wheat, that is to say, the results from the improved model is better than that of the original model although it is difficult to quantitatively evaluate the two results due to the absence of field measurements of the vegetation transpiration and soil evaporation at the satellite pixel scale. Furthermore, according to the above illustrations, the physical basis of the improved model is strengthened against the original model.

This paper presented two improvements of a two-layer model for the estimation of the land surface heat fluxes. The weakness in the original model was identified: (1) a subjective method to determine the true boundary lines from the scatter plot for the surface temperature of mixed pixel versus the fractional vegetation cover; (2) the assumption that the configuration of _{m} – _{a}, _{a}, _{a} and albedo) on surface temperature solves the problem (2). At the same time, the interpolation methods of _{a} and _{a}, the methods for acquiring spatial distributions of _{a} and _{s} were also investigated. Finally, the improved model was applied to the North China Plain. The results showed good agreement with _{2} flux, and the vegetation transpirations retrieved by the improved model and the original model, the effectiveness of the improved two-layer model for estimating soil evaporation and vegetation transpiration were indirectly proved.

This work was supported jointly by the State Key Development Program for Basic Research of China with grant number 2007CB714401-3, the Knowledge Innovation Project of IGSNRR, CAS (grant No. GXIOG-A05-11), the Knowledge Innovation Program of CAS (grant No. KZCX2-YW-326), the Young Talents Forefront Project of IGSNRR, CAS (grant No.07v70050SZ), the National Key Project of Scientific and Technical Supporting Programs Funded by Ministry of Science & Technology of China (No. 2006BAC08B0407) and the Program of “One Hundred Talented People” of the Chinese Academy of Sciences (CAS)

The relationship between the temperature of mixed pixel and vegetative fractional cover for 8∼14_{m}, _{veg} and _{soil} are respectively the surface temperature of mixed pixel, vegetation canopy and soil surface. _{m}, _{veg} and _{soil} are the emissivities of mixed pixel, vegetation and bare soil; σ is the Stefan-Boltzmann constant. _{veg} and _{soil} cannot be retrieved only using this equation. Computing the differential coefficient of _{m} to

Multiplying

Integrating _{soil} can be expressed as:
_{veg} can be expressed as:

According to _{veg}=0.97 and _{soil}=0.95. In terms of the numerical simulations, the small
_{soil} and _{veg}, therefore we ignored the difference between _{veg} and _{soil}. This simplification means that

Furthermore, using

In fact,

Albedo of mixed pixel is the weighed-average value of soil albedo and vegetation albedo, showed as _{m} to

Integrating _{soil} and _{veg} can be expressed as:

Evidently, as the method of decomposing mixed surface temperature _{m}, PCACA also can be applied to separate albedo of mixed pixel using the configuration of _{m}-

^{-1}for sparse vegetation

_{2}flux for wheat and studying of regional distribution

_{2}Transfer Dynamics

Scatter plot of surface temperature against vegetation fraction cover □ true dry edge □ observed dry edge □ observed wet edge □ true wet edge □ iso-line of equal vegetation fraction cover □ iso-line of equal soil water availability.

Relationship between surface temperature and soil water content.

Spatial distribution of standard meteorological stations in the study area.

Comparisons between _{soil}, _{veg} obtained from the original model and _{soil}’, _{veg}’ obtained from the improved model at iso-line of equal water availability.

Modeled versus measured available energy, _{n}-

Modeled versus measured latent heat flux,

Seasonal variation of the modeled _{veg} and measured CO_{2} flux during winter wheat growing period 2005 and 2006 (from March to June).

_{veg} maps on May 2, 2005 in North China Plain retrieved by the improved model (left image) and the original model (right image).