## 2 Materials and Methods
### Model Formulation
#### 2.4.3 Kcp Parameter Function  
The Kcp parameter function is defined using NDVI climatology (LSP) and published crop coefficient minimum (\(\mathrm{Kc_{min}}\)) and maximum (\(\mathrm{Kc_{max}}\)) values. The \(\mathrm{Kc}\) values provide the lower and maximum limits for the Kcp. The NDVI profile provides the temporal patterns of Kcp. \(\mathrm{Kc_{max}}\) and \(\mathrm{Kc_{min}}\) values can be defined under two conditions: 1) with the assumption that a specific crop/vegetation type will grow in a modeling cell (in this case, published values from FAO [6] or other sources can be used), 2) with the assumption that the landscape ET is based on different land cover mixtures as captured by the NDVI profile; i.e., a modeling cell will include fractions of different vegetation types and bare cover. In the second case, \(\mathrm{Kc_{max}}\) and \(\mathrm{Kc_{min}}\) can be a composite value from a weighted average of the cover composition with each modeling cell. The percent cover fractions from the MODIS VCF data will be particularly useful for this application. Equation 4 shows the formulation of the Kcp parameter as used in Equation 1 to estimate landscape ET, representing a spatial unit as determined by the resolution of the LSP data set (NDVI). \[Kcp=\frac{Kc_{\mathrm{max}}-Kc_{\mathrm{min}}}{NDVI_{\mathrm{max}}-NDVIo}*(NDVIi- NDVIo)+Kc_{\mathrm{min}} \tag{4}\]  
\[Kcp=Kc_{\mathrm{min}}\to Kcp<Kc_{\mathrm{min}} \tag{5}\]  
Where \(\mathrm{Kc_{max}}\) is the maximum (mature) \(\mathrm{Kc}\) value for a vegetation/crop type; \(\mathrm{Kc_{min}}\) is the minimum (early stage) \(\mathrm{Kc}\) value; \(\mathrm{NDVI_{max}}\) is the climatology maximum NDVI value in a year; \(\mathrm{NDVIi}\) is the climatology NDVI value for a given period \"i\" (average weekly maximum value in this case); and \(\mathrm{NDVIo}\) is the minimum reference NDVI value that is associated with the minimum \(\mathrm{Kc}\) value, which varies depending on the value-range of \(\mathrm{NDVI_{max}}\) according to Equation 6 or 7. \(\mathrm{NDVIo}\) is calculated depending on the \(\mathrm{NDVI_{max}}\) for each pixel. **CASE 1: \(\mathrm{NDVI_{max}}>\) = 0.40**  
\[NDVIo=0.3 \tag{6}\]  
**CASE II: \(\mathrm{NDVI_{max}}<\) 0.40**  
\[NDVIo=0.33*(NDVI  
fraction of the vegetation coverage is relatively high. However, in sparsely vegetated areas the magnitude of the NDVI is so low that even the maximum NDVI is sometimes less than the minimum threshold level of 0.3 that is considered an indication of the start of season in predominantly cropland areas (Figures 2 and 3). To better handle sparsely vegetated areas, the NDVIio value is defined in relation to the maximum NDVI value for such regions. By visual inspection of several LSP profiles, pixels with NDVI maximum values less than 0.4 are considered sparsely vegetated. For the conterminous United States, only 18.6% of the area had an NDVI\({}_{\text{max}}\) value lower than 0.4. These areas were mainly concentrated in the arid and semiarid southwest. For arid pixels, the NDVIio parameter will be calculated dynamically, as shown in Equation 7. The most useful information required is the timing and relative magnitude of water use. Although we move the NDVIio threshold value from 0.3 to some lower number, the timing and relative magnitude of water use will still be achieved (Figures 2 and 3). This is possible because the absolute NDVI magnitude of a pixel is a function of the percent vegetation cover, but the phenology (timing and relative magnitude of water use) is mainly captured by the temporal profile of the NDVI pattern. Assigning a Kc\({}_{\text{max}}\) value for Kcp during the peak NDVI is only appropriate if the objective is to simulate crop water use by assuming the entire modeling unit is composed of crop fields. Because crop growers make sure there is good ground coverage (closed canopy) at maturity for most crops, the Kcp function assumes that maximum water use is achieved at maximum NDVI regardless of the magnitude of the maximum NDVI.