remote Estimation of Actual Crop Coefficients Using Remotely Sensed Vegetation Indices and Soil Water Balance Modelled Data

: A new procedure is proposed for estimating actual basal crop coefficients from vegetation indices (K cb VI ) considering a density coefficient (K d ) and a crop coefficient for bare soil. K d is computed using the fraction of ground cover by vegetation (f c VI ), which is also estimated from vegetation indices derived from remote sensing. A combined approach for estimating actual crop coefficients from vegetation indices (K c VI ) is also proposed by integrating the K cb VI with the soil evaporation coefficient (K e ) derived from the soil water balance model SIMDualKc. Results for maize, barley and an olive orchard have shown that the approaches for estimating both f c VI and K cb VI compared well with results obtained using the SIMDualKc model after calibration with ground observation data. For the crops studied, the correlation coefficients relative to comparing the actual K cb VI and K c VI with actual K cb and K c obtained with SIMDualKc were larger than 0.73 and 0.71, respectively. The corresponding regression coefficients were close to 1.0. The methodology herein presented and discussed allowed for obtaining information for the whole crop season, including periods when vegetation cover is incomplete, as the initial and development stages. Results show that the proposed methods are adequate for supporting irrigation management.


Introduction
The accurate estimation of crop water requirements plays an important role in the improvement of crops water use and irrigation performance. This issue is particularly relevant considering the need for intensification of irrigated agriculture and increased water scarcity in several regions of the world.
A common approach for the estimation of crop water requirements is the Kc-ETo approach adopted by FAO56 [1] where a reference crop evapotranspiration (ETo) is multiplied by a crop coefficient (Kc) for estimating crop evapotranspiration, ETc. The ETo represents the climatic demand of the atmosphere while the Kc represents the differences distinguishing the reference crop and the considered crop in terms of ground cover, canopy properties and aerodynamic resistance, thus in terms of crop ET. Alternatively, crop evapotranspiration may be computed directly from ground observations using a combination equation such as the Penman-Monteith equation [2,3]. However, this approach is more demanding than the Kc-ETo approach and is not used for operational purposes but limited to research.
Another approach for estimating crop ET is based on surface energy balance models that use remote sensing thermal infrared data (e.g., [4][5][6]). These models estimate crop ET by subtracting the soil heat flux (G) and sensible heat flux (H) from the net radiation (Rn) at the surface. Such models allow to directly integrating the effects related with soil water deficit or water vapor pressure in the crop ET estimation [7]. Crop coefficients can be further derived considering the ETo [8][9][10]. However, these models present greater complexity and larger amount of input data than the Kc-ETo approach [7]. The Kc-ETo approach is commonly accepted for operational and research objectives [11]. A single or dual Kc approach may be used [1]. In the single approach both crop transpiration and soil evaporation are timely averaged into a single coefficient (Kc), whereas in the dual approach a daily basal crop coefficient (Kcb), representing primarily the plant transpiration, and a daily soil evaporation coefficient (Ke) are considered separately, i.e., Kc = Kcb + Ke.
For transferability purposes, Kc, Kcb and ETc in FAO56 represent ET rates under optimal, well-watered conditions [1]. However, in the field and in common practice, crop conditions are often not optimal due to insufficient or non-uniform irrigation, crop density, soil salinity and/or agronomic management. Potential ETc must then be replaced by the actual ETc (ETc act), and the resulting Kc is renamed Kc act and Kcb is also renamed Kcb act [11]. The term actual is adopted in this study. As commented by Pereira et al. [11], this concept better supports estimation and transferability avoiding the need to define multiple Kc values for the same crop as has occurred in the past for several crops, e.g., vines and orchards. Moreover, adopting Kc act and Kcb act rather than using the potential crop coefficients allows to perform irrigation management closer to the actual conditions, which is a main motivation of the present study.
Standard, potential Kc and Kcb values are defined and tabulated for a wide range of agricultural crops [1,12], but appropriate corrections may be required adopting a stress coefficient (Ks) to obtain the actual Kc (Kc act = Ks Kc or Kcb act = Ks Kcb). In addition, the transferability of Kc and Kcb values requires appropriate adjustment to local climate [1]. The adjustment of tabulated Kc and Kcb values for local conditions is also necessary when differences occur in planting density and geometry, vegetation height, and canopy architecture particularly in case of tree and vine crops [1,12].
In the last two decades, remote sensing (RS) data has increasingly been used for monitoring and mapping the spatial and temporal variation of ET and thus for computing crop water requirements (e.g., [13][14][15]). One of the most widely used approaches considers the relationship between vegetation indices (VI) derived from RS reflectance data and (actual) crop coefficients, either Kc or Kcb. The basis of this approach relies on the close correlation of several VI and various biophysical characteristics of the plants, e.g., leaf area index (LAI), ground cover fraction (fc), biomass, and physiological processes depending on light absorption by the canopy, including ET (e.g., [16][17][18][19]). With the VI approach, because these indices reflect the actual vegetation cover conditions, the estimated Kc or Kcb represent actual rather than potential Kc or Kcb. As proposed by Pereira et al. [11], these coefficients should then be referred to as Kc act and Kcb act despite many authors have not adopted this conceptual difference and that most VI-based methods are unable to accurately observe reductions in Kc and Kcb caused by acute water or salinity stress.
One advantage of using VI-based actual crop coefficients is the ability to account for variations in plant growth due to abnormal weather conditions, e.g., the impact of a frost occurrence [20]. Using VI-based crop coefficients also allows obtaining the spatial variation of actual Kc (or Kcb) within fields. In addition they provide for a field-to-field description of the variation of actual Kc (or Kcb) due to variations in planting dates, plant spacing and cultivars [11]. A review of the advantages and disadvantages of VI-based crop coefficients was presented by Allen et al. [21].
The Normalized Difference Vegetation Index (NDVI; [22]) and the Soil Adjusted Vegetation Index (SAVI; [23]) are the most commonly VI used to estimate actual Kc and Kcb. The formulation of both VI combines the reflected light in the red and near infrared (NIR) bands, thus providing an indirect measure of the absorption of red light by chlorophylls (a and b) and reflectance of NIR by the mesophyll structure in leaves [17,24]. Choudhury et al. [25] refer that using SAVI instead of NDVI allows extending the range over which the VI respond to the increase in vegetation amount/density beyond a LAI around 3, which is related with NDVI saturation problems for high LAI values. In addition, the NDVI is considered more sensitive than SAVI to soil background reflectance changes due to the moisture of soil surface [7,13,26].
Several relationships between Kcb (or Kc) and VI are reported in the literature for different vegetation types, with the most recent studies focusing the Kcb estimation because plant transpiration is more directly related with the VI. Some examples of this approach consider a linear relationship between Kcb and NDVI [13,20,25,27,28], and between Kcb and SAVI [13,25,26] but an exponential relationship between NDVI and Kc was also considered [29]. A few studies report a Kcb estimation based on fc or LAI obtained through VI, including for field and tree crops [7,30,31]. However, in most studies, the relationship between Kcb and VI was established for non-stressed conditions and for conditions of dry soil surface, which often do not represent the actual conditions of crop management. VI derived crop coefficients often do not account for the reduction in Kcb due to water or salinity stress [7,11]. Then the above referred stress coefficient Ks should be considered to obtain an actual Kc or Kcb, i.e., Kc act and Kcb act [1]. Most of those studies were established for a single crop and its application for other crops, particularly for discontinuous tree crops, is still limited. Differently, the approach proposed by Mateos et al. [7], which approximates Kcb using fc and VI, refers to several crops and shows good results. However, this application does not always allow estimating Kcb for periods when the soil is not yet fully covered by the crop.
Although several equations were proposed for approaching Kcb via VI, a consensual equation that can be consistently used for different vegetation types and conditions is still lacking. In such context, a density coefficient (Kd) [12] may be used in the formulation of Kcb through VI to help incorporating the impact of vegetation density and height. Such approach would allow Kcb to be computed for a range of row crops, orchards and vines, thus including conditions when soil is not fully covered by the crop. For further estimating Kc, a water balance modeling approach adopting the dual crop coefficient approach [1] is required to estimate soil evaporation and thus to estimate the evaporation coefficient Ke that can be combined with the VI based Kcb, i.e., Kc = Kcb + Ke. The SIMDualKc model [32], which performs a daily water balance of the surface soil layer, provides for a daily Ke. Thus, by applying the dual coefficient approach and performing the soil water balance, the SIMDualKc model provides information on the stress coefficient Ks, on potential and actual Kcb, on Ke and ultimately on potential and actual Kc.
Considering the advances and limitations discussed above, particularly the need to represent actual rather than potential crop and evapotranspiration conditions for irrigation management purposes, the main objectives of this study are: (i) developing and testing a new equation for estimating actual Kcb based on VI and considering the stress coefficient Ks and the density coefficient Kd computed with the fraction of ground cover fc estimated with VI; (ii) developing and testing a combined approach for estimation of the actual Kc using the VI-based Kcb and the evaporation coefficient Ke obtained with a daily soil water balance model; and (iii) testing the adequacy of the approach to different crop types, namely maize, barley, and an olive orchard.

Study Areas
Three crops were considered to test the approach of actual Kc estimate based on reflectance VI in combination with the SIMDualKc soil water balance model: a super high density olive orchard, maize, and barley.
The super high density olive orchard test site is located in Viana do Alentejo, in South of Portugal (38°24′46′′ N, 7°43′38′′ W, 143 m a.s.l.; Figure 1). This olive orchard occupies a total area of 78 ha in an undulating terrain. The olive orchard was planted in 2006 in a hedgerow system with 1.35 m × 3.75 m spacing (1975 trees ha −1 ), and an orientation South-North. The olive trees are of cultivar Arbequina. The fraction of ground covered by the vegetation (fc) was approximately 0.35 and tree height was around 3.5 m. The procedure to obtain fc was based upon the measurements of the projection of crown diameters in row direction and perpendicular to it in 51 trees. Soils are sandy loam, with soil water content averaging 0.24 cm 3 ·cm −3 at field capacity and 0.12 cm 3 ·cm −3 at the wilting point. Ground data from two consecutive years (2011 and 2012) were collected and used to validate information of several agronomic and biophysical parameters as detailed by Paço et al. [8] and Pôças et al. [33]. In 2012, between February 20th and 25th, an extremely heavy frost affected the orchard causing a strong leaf fall. A severe pruning of the trees was therefore applied following the frost occurrence. Some areas of the olive orchard were more affected by the frost, and consequently the pruning was not uniform along the orchard, which increased the variability of the vegetation conditions within the orchard in 2012.  Table 1. The crop growth stages were defined according to Allen et al. [1]: (i) initial stage, from the sowing/planting until 10% of ground cover (Ini); (ii) development stage, from 10% of ground cover until maximum ground cover by vegetation (Dev); (iii) mid-season stage, from full cover until maturity (Mid), and (iv) late-season stage, from the beginning of senescence and leaves yellowing until harvesting (Late). The observations of fc were visually performed and adjusted with the help of photographs of the ground shadow by the crop near solar noon [34].
Barley was sown in field 1; two seasons were studied: (i) 2012, a dry year, and (ii) 2012/2013, a wet year [35]. The barley field was cropped with 200 kg·ha −1 of malting barley (Cv. Publican) seeds using an inter-row spacing of 0.15 m. The plants density was measured before tillering, when the crop attained tree leaves, averaging 342 and 319 plants per m 2 respectively in 2012 and 2013. The lengths of the crop growth stages are given in Table 1. The barley fc observations were performed similarly to the maize crop; furthermore, LAI measurements were also used to estimate fc and therefore to confirm related observations as referred by Pereira et al. [35]. Soils in fields 1 and 2 are loamy sand soils, with total available water TAW = 171 and 149 mm·m −1 respectively; field 3 is a silty-loam soil, with TAW = 209 mm·m −1 . Ground data collected in maize and barley study areas included LAI and fc, whose measurements were used to calibrate the SIMDualKc model as detailed by Paredes et al. [34] and Pereira et al. [35].

Satellite Imagery
Vegetation indices used for estimating the basal crop coefficients were obtained from Landsat 5 TM and Landsat 7 ETM+ satellite images. Landsat images of Path 203/Row 033 were considered for the olive orchard study area, while for the other study areas the Path 204/Row 033 was used. Table 2 presents the summary of the cloud-free satellite image dates considered for each study area and crop. The satellite image of 03 January 2013 was eliminated from the set of considered images due to the unfavorable light conditions during this period of the year, which affected the VI values. The images were geometrically and atmospherically corrected using the procedure described by Allen et al. [15] and Tasumi et al. [36]. The procedure included image calibration using the coefficients proposed in the literature for Landsat 5 TM and Landsat 7 ETM+ [37,38]. The VI were calculated on a pixel-by-pixel basis and averaged for the entire study plot in each study area. Pixels from the edges of the fields and from roads within the olive orchard were not considered. The VI computed for the study were the Normalized Difference Vegetation Index (NDVI; [22]) and the Soil Adjusted Vegetation Index (SAVI; [23]).

SIMDualKc
The SIMDualKc is a soil water balance model that applies the dual crop coefficient approach, thus separately computing the daily soil evaporation and crop transpiration [32,39]. This model has been successfully applied to estimate ET and assess Kc of a wide range of field crops [34,35,39], including tree crops [8,40]. Data from SIMDualKc were used as benchmark, i.e., a standard against which comparisons can be done, mainly for the comparison of Kc and Kcb results. SIMDualKc was previously calibrated using ground data for the three crops, as described later in this section.
In SIMDualKc, Kcb is computed with the equation below [12,32] where impacts of plant density and/or leaf area are taken into consideration by a density coefficient: where Kd is the crop density coefficient, Kcb full is the estimated basal Kcb for peak plant growth conditions having nearly full ground cover (or LAI > 3), and Kc min is the minimum Kc for bare soil (in the absence of vegetation). The Kc min value is about 0.15 under typical agricultural conditions and ranges 0.0-0.15 for native vegetation depending on rainfall frequency. Kcb is adjusted by the model for local climatic conditions where the average minimum relative humidity differs from 45% and/or the average wind speed is different from 2 m·s −1 [1,12,32]. Kd is computed with the equation proposed by Allen and Pereira [12]: where fc eff is the effective fraction of ground covered or shaded by vegetation near solar noon [ ], ML is a multiplier [ ] on fc eff describing the effect of canopy density on shading and on maximum relative ET per fraction of ground shaded (to simulate the physical limits imposed on water flux through the plant root, stem and leaf systems), and h is the mean height of the vegetation [m]. Therefore, the use of Kd allows incorporating the impact of vegetation density and height in the estimation of Kcb. Kd has also been used in other studies to adjust Kcb to actual density conditions for full cover crops [41] and in partial cover crops using both ground [42] or remote sensing data [43]. When soil water deficit occurs, the stress coefficient Ks [ ] is computed by the model using a daily soil water balance for the entire root zone. Ks is expressed as a linear function of the root zone depletion Dr [1,32]: where TAW and RAW are, respectively, the total and readily available soil water [mm], Dr is the root zone depletion [mm], and p is the depletion fraction for no stress [ ]. Kcb is multiplied by Ks to account for the effects of soil water stress and thus obtaining Kcb act. The calculation of the soil evaporation coefficient Ke [ ] follows the procedures proposed by [1,44] which include the daily water balance of the evaporation soil surface layer that allows to compute the evaporation reduction coefficient Kr [ ] from the cumulative depth of water depleted (evaporated) from the topsoil [1]. It results that Ke is then computed daily with: as a function of Kc max, the maximum value of Kc following rain or an irrigation event and Kcb in that day (Equation (1)) after adjustment with Kr. Ke is limited to few Kc max that represents the fraction of Kc max that refers to the fraction few of the soil that is both exposed and wetted, i.e., the fraction of soil surface from which most evaporation occurs. The background and computation of Kcb, Ks and Ke in SIMDualKc is described in detail by Rosa et al. [32].
The calibration of SIMDualKc aims at optimizing the crop parameters Kcb and p relative to the various crop growth stages, as well as the soil evaporation parameters, the deep percolation parameters, and the runoff curve number (CN) using trial and error procedures until small errors were achieved. Further information is provided by [8,34,35]. For the olive orchard, the calibration was performed comparing daily transpiration data simulated with those obtained with sapflow measurements (and also eddy covariance and energy balance measurements as auxiliary data for calibration and data quality screening), as described in Paço et al. [8]. For the maize and barley crops, the calibration was performed by minimizing the differences between observed and simulated soil water content, as described in Paredes et al. [34] and Pereira et al. [35]. In all cases, the results of goodness of fit indicators showed good model performance and small errors of estimates [8,34,35]. Furthermore, ET partitioning was performed in other studies, e.g., testing the soil evaporation algorithm through comparing simulated with microlysimeter observations [45,46], crop transpiration against sap flow measurements [8,40] and crop ET compared with eddy covariance ET data [47]. An example of the results of the model calibration for barley is presented in Figure 2, which shows that the simulated soil water content accurately follows the dynamics of observations throughout the barley crop season [35]. ; θSat, θFC, θp and θWP are respectively the SWC at saturation, field capacity, wilting point and at non-stressed depletion fraction p). Precipitation is represented by dark bars and irrigation by the light ones (source: [35]).

Basal Crop Coefficients Derived from Reflectance Vegetation Indices
The proposed methodology is based on the dual crop coefficient approach, thus integrating daily Kcb and Ke, and considering Ks when water stress occurs [1,12]. Considering this approach, a new equation was developed to estimate Kcb from the density coefficient Kd and Kc min as proposed by Allen and Pereira [12] and from vegetation indices derived from satellite images, Kcb VI:  (6) where VIi corresponds to the VI for a specific date and pixel, VImax is the VI for maximum vegetation cover and VImin is VI for minimum vegetation cover (bare soil). This equation adapts to crops with continuous ground cover and to tree crops with bare soil by including Kd. For tree crops having an active ground cover, Kc min is replaced by a Kcb cover, corresponding to the Kcb of the ground cover in the absence of tree foliage [12]. Kd in Equation (6) is computed with Equation (2) using the effective fraction of ground covered estimated with the Equation (7) presented below. Table 3 presents Kd ranges for the three crops during the mid-season stage. (1) K d computed using effective fraction of ground covered or shaded by vegetation derived from ground data; (2) K d computed using effective fraction of ground covered or shaded by vegetation estimated with vegetation indices using Equation (7); (3) A single value is presented because a single satellite image was available for the mid-season stage.
Both NDVI and SAVI vegetation indices may be used. The SAVI index is considered less sensitive to soil background reflectance variability than other VI, thus with higher potential for application in tree crops with discontinuous ground cover. NDVImax was set to 0.75-0.85 for the three crops, and the value 0.10 was set for NDVImin, both based on reference values for, respectively, an irrigated agricultural field with full cover and a bare soil [48][49][50]. When SAVI was considered, the SAVImin was set to 0.09 while the SAVImax was adjusted to 0.75 according to Mateos et al. [7].
The Kc min was set to a value between 0.10 and 0.15 (0.13 was considered in this study) according to values proposed by Allen and Pereira [12]. For the olive orchard, Kcb cover was considered instead of Kc min and set to a value between 0.15 and 0.18 (0.17 was considered in this study) due to the presence of sparse vegetation cover between tree rows. Reference values for several parameters considered in the computation of Equation (6) can be found in Allen and Pereira [12]. Kd was computed with Equation (2) where, in a first approach, the fc eff was the same considered in SIMDualKc, which was based in field observations (fc field). In a second approach, fc eff was computed using an equation based on vegetation indices derived from remote sensing reflectance data (fc VI): (7) where β1 is an empirical coefficient (ranging between 0 and 1) depending upon the maximum VI value in each crop stage. VIi is the average VI of the study area pixels for each date, VImax and VImin correspond to the VI respectively for maximum and minimum vegetation cover. Both NDVI and SAVI may be used, with the latter producing better results for incomplete cover crops, e.g., olive orchards. NDVImax and NDVImin were set according to references in the literature [48][49][50] (Table 4). For the super high density olive orchard, SAVImax and SAVImin were considered following values proposed by Mateos et al. [7] (Table 4). The parameter β2 corresponds to an adjustment coefficient associated with crop senescence and leaves yellowing. Both NDVI and SAVI are sensitive to leaf senescence thus yielding smaller values at the end of the growing cycle than before peak development was reached [51]. Therefore, the β2 adjustment is considered to compensate the decrease in VI due to the senescence and/or yellowing of plants, which is independent from fc. Values of β2 are presented in Table 4.
(1) The values were defined according to stages of the crop growth. For the development stage were considered sub-stages by dividing the growth stage in two (spring-summer crops; maize) or three (winter-spring crops; barley) periods; (2) 0 is considered when the crop is in its maximum development and 0.5 close to the harvest (a value of 0.2 was set in the first half of the late stage).
The four main stages of the crop growth, described by Allen et al. [1], were considered in the definition of β1: (i) Ini, (ii) Dev, (iii) Mid, and (iv) Late. Due to the great variability of the ground cover by the vegetation throughout Dev, a differentiation of β1 values according to sub-stages was considered: (i) two sub-stages in spring-summer crops (maize); (ii) three sub-stages in winter-spring crops (barley), where the Dev is longer. For the olive orchard, a crop with persistent leaves and where the soil cover by the vegetation is incomplete, the β1 value was set to 1 throughout the year, except for the period of pruning and in the subsequent weeks, when it was slightly decreased to account for the decrease in canopies vegetation. Values of β1 are presented in Table 4.
Most of the optical VI track the effects of long-term water stress on plants but do not allow the early detection of water stress [52]. Therefore, Equation (6) expresses the potential Kcb, thus transpiration occurring at maximum rate. However, the occurrence of water stress cause a decrease of the potential value and Ks must be applied to obtain actual Kcb VI. In the current approach, Ks computed with SIMDualKc was used. The dates when water stress was detected through the daily soil water balance are given in Table 2. Therefore, hereafter, the actual Kcb VI refers to the Kcb VI computed by Equation (6)  Because Ks and Ke used in the estimation of actual Kc VI are computed with SIMDualKc, the results of comparing actual Kc VI with Kc SIMDualKc are somehow influenced by the fact that values of Ks and Ke used in both cases are the same. Nevertheless, this fact does not impede the assessment of the computation of the actual Kcb VI because this is the procedure that gives the possibility to compare the actual crop coefficients from both approaches and retrieves useful information from satellite imagery aiming at irrigation management and farmer advising.
A linear regression forced through the origin was used for the comparisons and the corresponding regression and determination coefficients were consequently used as indicators. Additionally, the root-mean-square deviation (RMSD; Equation (8)) and the relative mean deviation (RMD; Equation (9)) were used to compare two sets of data, with Xi relative to estimations with SIMDualKc, having mean , and Yi relative to the VI estimates (i = 1,2, ..., n).
Since the database available for this study only comprises 15 VI observations for maize and olive and seven for barley (Table 2), the validation of Kcb estimated based on VI was performed using the "leave-one-out" (LOO) cross-validation [53]. Applications of the LOO cross-validation in remote sensing studies are provided by [54,55]. The LOO cross-validation evaluates the model performance for observations not considered in the estimation step, thus providing independent estimates of the predictive capability of the selected models. This technique consists in the removal of one observation from the dataset used and the estimation of a new regression model with the remaining observations. This new regression model is used to estimate the Kcb VI of the observation withdrawn (Kcb VI LOO). Equations (8) and (9) were also used to estimate the goodness of cross-validation results, with Xi and Yi representing respectively Kcb SimDualKc and Kcb VI LOO data. A similar approach was considered for the Kc and fc estimated based on VI.

Estimation of the Fraction of Ground Cover from VI
The results of fc estimated from NDVI (fc VI; Equation (7)) were compared with fc based on field observations (fc field) for the maize and barley crops (Figure 3). The comparison of fc VI and fc field have shown a good agreement for both crops, with R 2 ≥ 0.81 and the regression coefficient b close to 1.0 ( Figure 3). For maize, RMSD was 0.10 and RMD 10.6%, while for barley RMSD was 0.06 and RMD was 8.1%. For the olive crop, due to the low variability of fc field values, a regression for comparing fc VI and fc field was not considered. The fraction of ground cover for this tree crop with persistent leaves is nearly constant throughout the year, except for the pruning period. As such, for the analysis of the fc VI results, only the indicators of residual estimation deviations were used: RMSD = 0.22 and RMD = 69.2%. When SAVI was used for the estimation of fc VI results of all indicators improved much, with RMSD= 0.02 and RMD = 5.5%. The absolute mean differences between the fc VI and the fc field for the olive orchard were small, ranging from 0.003 to 0.046. Contrarily, for the annual crops (maize and barley), the use of SAVI in the estimation of fc VI did not improve the residual estimation deviations.
Results of cross validation show a good performance of the fc VI estimation, with RMSD of 0.06. 0.04, and 0.01 respectively for maize, barley and olive. RMD results for the cross validation varied between 2.6% for olive and 7.5% for maize.

Estimation of Kcb from Reflectance-Based Vegetation Indices (Kcb VI)
The Kcb VI (Equation (6)) was adjusted using the Ks derived from SIMDualKc computations for the dates when stress occurred ( Table 2) to obtain the actual Kcb VI. The values of Kcb SIMDualKc used in comparisons with Kcb VI were also adjusted to water stress using Ks. Both NDVI and SAVI were tested for the estimation of Kcb VI and the results obtained with the two VI were compared.
The comparison of actual Kcb VI with actual Kcb SIMDualKc for the studied crops is presented in Figure 4. The Kcb VI values in Figure 4 were estimated using fc NDVI (Equation (7)) with the objective of fully estimating actual Kcb from reflectance-based vegetation indices. Figure 4a shows Kcb VI for maize when using NDVI, which has a determination coefficient (R 2 ) lower than using SAVI (Figure 4b) but has a higher regression coefficient (b = 0.98) relative to SAVI (Table 5). Thus, Kcb SAVI underestimates Kcb values. For barley (Figure 4c,d) both b and R 2 are better when using NDVI (Table 5). For the olive orchard estimating Kcb VI with SAVI leads to b equal to 1.22 and a R 2 equal to 0.83; when using NDVI there is an overestimation of the Kcb values, b = 1.8 (Figure 4e,f).  The statistical indicators relative to the comparison between Kcb VI and Kcb SIMDualKc for all crops are presented in Table 5. The RMSD between Kcb VI and Kcb SIMDualKc for the set of image dates considered for maize was equal to 0.16 when using NDVI for the computation of Kcb VI and 0.22 when SAVI was adopted. RMD was also higher when adopting SAVI. The regression coefficient b = 0.98 when NDVI is used indicates a statistical similarity between Kcb VI and Kcb field. For barley, b and R 2 are higher for Kcb NDVI and RMSD and RMD are smaller than for Kcb SAVI. For the olive crop there is a clear   Table 5). For the olive crop, however, estimating Kcb VI using fc SAVI (see Equations (6) and (7)) provided much better results ( Figure 5 and Table 5; Kcb SAVI *) than using Kcb VI with fc NDVI (Table 5). Comparing Kcb SAVI with Kcb SIMDualKc it results a higher R 2 = 0.88 with b of 1.03 (Table 5). In addition, RMSD decreased from 0.08 to 0.02 and RMD from 21.7 to 5.6% (Table 5). RMSD and RMD obtained in the cross validation procedure for the three crops are also presented in Table 5. The results obtained for cross validation are consistent with those obtained for the estimation procedure.  Table 6 presents the average and standard error of the mean or the range of the Kcb VI obtained for all fields and all crops. Results refer to the use of both NDVI and SAVI with Equation (6). As for the previous analysis, the best results refer to Kcb NDVI in case of maize and barley and Kcb SAVI in case of the olive orchard. Comparing results for both VIs in this Table 6 it becomes apparent that estimations using NDVI lead to higher Kcb VI results, which relates to the nature of the VI index.   (1) 0.18 ( ) (2) 0.78 (±0.13) Development (4) [0. 44 (5) 0.87 ( ) (2) 0.61 (±0.05) (6) Late-season (4) [ Note: f c VI were obtained from NDVI for maize and barley and from SAVI for olive. (1) No Landsat images available for this crop growth stage; (2) Only one Landsat image available for this crop growth stage; (3) Period after pruning; (4) Range of values for this crop growth stage; (5) Stress occurred in three out of the eight image dates considered; (6) Stress occurred in four out of the eight image dates considered. Figure 6 shows selected examples of the seasonal variation of the daily Kcb, Kcb act, Ke and Kc act obtained with SIMDualKc for maize [34], barley [35] and olive [8] and the fitting of actual Kcb VI obtained for those sets of data. Kcb VI used in Figure 6 refer to Kcb NDVI in case of maize and barley and to Kcb SAVI in case of olive. For the olive orchard, the fc SAVI (Equation (7)) were used in the computation of Kd while fc NDVI were used for maize and barley. Unfortunately, there were few images for maize crop in the year when water stress occurred. Results show that actual Kcb VI match quite well the actual Kcb curve for the three crops. The variability of weather conditions in those years are well apparent through the numerous peaks of the evaporation coefficient Ke due to both rainfall and irrigation events. The effect of water stress on Kcb is well apparent by the deviations of Kcb act from the potential Kcb, thus when the Kcb act curve lays below the Kcb curve. Summarizing, Figure 6 demonstrates the goodness of the approach used to estimate Kcb act from a vegetation index.

Estimation of Actual Kc from Reflectance-Based Vegetation Indices (Kc VI)
The actual Kc VI values were obtained by combining the actual Kcb VI with Ke derived from SIMDualKc. The results for Kc VI presented in Figure 7 and Table 7 were obtained using the Kcb VI that provided the best estimation for each crop as presented in Section 3.2. Thus, Kcb NDVI was considered for the computation of Kc VI for maize and barley crops, while Kcb SAVI was adopted for olive. Figure 6 shows the Kc VI corresponding to the same dates when the Kcb VI were obtained for selected data sets referred in Section 3.2. It may be observed that despite the enormous variability of Ke due to rainfall and irrigation wetting events the computed Kc VI match well the actual Kc curves for the three crops analyzed.  When comparing Kc VI and Kc SIMDualKc a high determination coefficient was obtained (R 2 ≥ 0.72) together with a regression coefficient very close to 1.0 for the three crops studied (Figure 7 and Table 7). The best statistical indicators were obtained for the olive orchard (Table 7), particularly with quite small RMSD and RMS. RMSD and RMD obtained in the cross validation procedure for the three crops show results very similar to those obtained for the estimation procedure ( Table 7).
The average and range of the actual crop coefficients estimated with vegetation indices combined with the computed Ke (Kc VI) are given in Table 8 for the various crops. Note: f c VI were obtained from NDVI for maize and barley and from SAVI for olive. (1) No Landsat images available for this crop growth stage; (2) Only one Landsat image available for this crop growth stage; (3) High evaporation occurrence in one of the image dates considered in the initial stage due to precipitation occurrence. (4) Range of daily Kc values (not time-averaged); (5) High evaporation in two of the dates considered in this crop growth stage; (6) Period after pruning; (7) Stress occurred in three out of the eight image dates considered; (8) Stress occurred in four out of the eight image dates considered.

Estimation of the Fraction of Ground Cover from Vegetation Indices
The statistical results relative to the estimation of fc VI indicate a good performance of Equation (7) for the annual crops studied (maize and barley). The R 2 values observed when comparing fc VI with fc field are highly significant for both crops (Figure 3). These R 2 values are within the range of the values reported by Trout et al. [56] for the estimation of canopy cover using NDVI derived from a multispectral camera for a set of 11 annual and perennial horticultural crops. Purevdorj et al. [57] reported similar R 2 results when estimating the percentage of vegetation cover in grasslands and lawn grass using NDVI derived from AVHRR satellite images, thus with a coarser spatial resolution. Estimated deviations are small, for both maize and barley, with RMSD of 0.10 and 0.06, respectively.
The estimation of fc in the olive orchard also showed good estimation, with low RMD < 10%. When the two years studied (2011 and 2012) were analyzed separately, the average absolute deviations between fc VI and fc field were slightly higher in 2012 than in 2011. These results are likely due to the larger variability in the vegetation conditions within the orchard in 2012 because a severe frost occurred in February which was followed by an heavy and non-uniform pruning that increased the variability of fc within the orchard as discussed by Paço et al. [8].
The better performance of SAVI for the estimation of fc VI in the high density olive orchard, having a large fraction of soil exposed, is likely due to the lower sensitivity of this VI to soil background influence [23]. Similar conclusions were reported by Purevdorj et al. [57] relative to the better performance of SAVI to estimate the vegetation cover for low crop densities. Differently, in a study with savannah-type open woodlands dominated by evergreen oak species (montados or dehesa), the NDVI performed slightly better than SAVI for the estimation of tree canopy cover [58] but the study areas were characterized by an extremely variable understory including bare soil, dry grass and a few evergreen shrubs that impacted the VIs values as discussed by the authors.

Estimation of Actual Kcb from Reflectance-Based Vegetation Indices (Kcb VI)
The R 2 results relative to the regression comparing the actual Kcb VI with Kcb SIMDualKc for maize were highly significant using NDVI (Table 5). Although using NDVI led to a lower R 2 than SAVI, the regression coefficient comparing Kcb SAVI with Kcb SIMDualKc (b = 0.79; Table 5) indicates an underestimation of Kcb SAVI values. The regression coefficient obtained when comparing Kcb NDVI with Kcb SIMDualKc is much better (b = 0.98; Table 5), which may be related with the tendency of NDVI reaching a maximum value at about the same time as Kcb, i.e., at a LAI around 3 [14,59]. Contrarily, SAVI tends to continue increasing for LAI above 3 [21,25,60]. Thus, according to Allen et al. [21], the NDVI supports better than SAVI the estimation of Kc and Kcb in crops reaching high LAI values as for the full cover crops considered in the current study.
The base data on maize refers to three different seasons and three different fields, which results in a large variability of soil, climate and crop management conditions. A large variability of Kcb act was therefore observed despite the potential/standard Kcb was the same after appropriate calibration of SIMDualKc [34]. This variability contributed for a larger variance of the actual Kcb VI and Kcb SIMDualKc and made it more difficult to parameterize a single value for Kc min and β1 (Equations (6) and (7), respectively) when considering all three maize seasons and three cropped fields. The variability conditions observed in the three different fields and seasons during the initial and development stages may justify why comparing Kcb NDVI with Kcb SIMDualKc leads to a smaller R 2 ( Table 5).
The actual Kcb NDVI obtained for the maize mid-season stage averaged 0.92 (Table 6). This value relates with the water stress that occurred during the mid-season stage of 2010 in both surveyed fields which was detected in two of the images, particularly that of 30 July 2010. The SIMDualKc computed Ks values for this date were 0.55 and 0.29 respectively for field 1 and field 2. Such low Ks values highly impacted actual Kcb for this stage. Figure 6a shows well the impact of water stress during that period, with the Kcb act curve largely below the Kcb curve. The actual Kcb NDVI are within the range of values presented by Padilla et al. [31]. Relative to the late season, the range of Kcb NDVI values in Table 6 refer to three images available for this period. One of these images was captured the day prior to harvest (19 September 2011 in field 1) and the Kcb NDVI was then equal to 0.48, therefore similar to the Kcb end proposed by Allen et al. [1], Kcb end = 0.5. However, values for Kcb end relate to crop management and therefore are difficult to be compared with other values in literature.
For the regression comparing the actual Kcb VI and Kcb SIMDualKc for barley the statistical indicators show that NDVI performed better, both in terms of the regression and determination coefficients and RMSD and RMD (Table 5 and Figure 4c,d). The actual Kcb NDVI obtained for the initial crop growth stage (actual Kcb ini NDVI = 0.18; Table 6) is just slightly higher than the value presented by Allen et al. [1], Kcb ini = 0.15, which relates with the fact that no water stress occurred in that stage (Figure 6b). Kcb NDVI values for the development growth stage ranged between 0.25 and 0.91 (Table 6), which refer to values commonly accepted [1]. The actual Kcb mid NDVI = 0.87 (Table 6) is lower than the standard Kcb = 1.10 proposed by Allen et al. [1] and values obtained for crops with similar canopy [30,31,61]. This is likely due to different vegetation conditions in the 2012/2013 crop season and to the adjustment to climate [35]. The Kcb NDVI presented in Table 6 for the late season ranged between 0.86 and 0.69 and refer to two images available for that stage. These images provided for likely appropriate actual Kcb values and compared well with values obtained with SIMDualKc (Figure 6b, [35]).
For the olive orchard, results obtained with SAVI are good ( Figure 5) and in agreement with the above mentioned lower sensitivity of this VI to changes in soil background [23], which is particularly relevant for tree crops with discontinuous ground cover. Furthermore, SAVI improves the linearity between the VI and biophysical parameters as observed by other researchers [25,56]. Mateos et al. [7] also obtained good results when using SAVI for estimating Kcb in peach and mandarin orchards. Contrarily, in a study with grapevine, Campos et al. [13] did not find significant improvements in the accuracy when using SAVI but the application excluded days with wet soil. Accurate results for olive in the current study are likely due to the use of Kd in Equation (6).
The average Kcb SAVI value obtained for the mid-season stage in olive (0.37; Table 6 and Figure 5c) was affected by the occurrence of water stress in four out of the eight dates considered in this stage, with Ks < 0.77 in two of these dates. The impacts of water stress on Kcb is clearly shown in the example presented in Figure 6c, with the Kcb act curve laying bellow the Kcb curve during the mid-season. The actual Kcb SAVI for this stage would increase to 0.42 when only dates without water stress or with very mild stress (Ks > 0.95) are considered. The actual Kcb SAVI values for olive orchards for the mid-and late-seasons, respectively 0.37 and ranging 0.40 to 0.33 (Table 6), are within the range of values for olive orchards with an effective fraction of ground cover (fc eff) between 0.25 and 0.5, with a Kcb mid ranging between 0.35 and 0.55 and Kcb end ranging from 0.30 to 0.50 [12]. During the initial stage, some sparse vegetation was growing in the inter-rows, thus Kcb was only slightly influenced by this sparse vegetation. Consequently, Kcb ini SAVI = 0.33 was an appropriate value for this orchard.
Overall results obtained have shown a good performance of the Kcb VI Equation (6) for all three crops, particularly integrating results of fc VI when computing the density coefficient Kd used in Equation (6). It must be stressed that the use of Kd, which integrates the information on fc and crop height, and of Kc min provided good results in modeling the actual Kcb VI (Equation (6)) for crops with very different canopy structures and ground cover conditions as those used in this study (continuous versus discontinuous ground cover) for the whole crop season including when water stress was observed.

Estimation of Actual Kc by Combining Kcb VI with Ke from SIMDualKc (Kc VI)
The statistical indicators from comparing the actual Kc VI with Kc SIMDualKc show a good performance of the model for the three crops (Table 7 and Figure 7). The deviations of estimate are very low for olive and barley and slightly higher for maize. Results are in agreement with those discussed for Kcb VI, which is a component of Kc VI. For the maize crop, the actual Kc NDVI values obtained for the development stage ranged between 0.90 and 1.30, which relate with a large variability of Ke values, as for the example in Figure 6a. The average Kc NDVI = 0.98 ± 0.11 for the mid-season (Table 8) reflects also a large variability as exemplified in Figure 6a, where one value exceeds 1.20 and the other, due to water stress, is as low as 0.40. When only cases without water stress are considered, Kc VI increases to 1.14, a value close to the Kc reported by Gonzalez-Piqueras et al. [26]. Regarding the Kc NDVI values for the late-season, values obtained are likely to occur because they are within the interval limited by the Kc values at mid-season and those expected for end season.
The actual Kc NDVI estimated for the initial and development stages for barley (0.78 and 0.75-1.18; Table 8) reflects the abundance of rainfall during these stages, with various Ke peaks occurring then (Figure 6b). For the mid-season stage the value obtained is lower than the values estimated with SIMDualKc (Kc SIMDualKc = 0.99), thus indicating an underestimation of Kc. The average value obtained for the late-season stage (0.88) is close to that obtained by Liu et al. [62] for the period between wax ripeness and harvesting (equal to 0.83).
The results obtained for the olive orchard show a very good performance of the model for estimating Kc SAVI (Figure 7c). The average results of Kc SAVI (Table 8) are within the range of those proposed by Allen and Pereira [12] for an fc eff between 0.25 and 0.50, which vary between 0.80 for the Kc ini (considering an active ground cover), 0.40-0.60 for the Kc mid (no active ground cover), and 0.35-0.75 for the Kc end (depending on the ground cover). The results obtained throughout the year follow a pattern similar to results reported by other authors [43,63], as discussed by Paço et al. [8].
It is important to highlight that results presented herein correspond to actual Kc, thus reflecting the management options observed in the study areas.

Conclusions
The overall results for the three crops-Maize, barley and olive-indicate that the newly proposed Equation (7) adequately models the fraction of soil covered by vegetation throughout the crop season. The results obtained indicate a better performance of SAVI for the computation of the fraction of ground cover for the incomplete cover crop, olive, and of the NDVI for the annual crops, maize and barley.
Using the novel Equation (6), which applies a density coefficient, the estimation of actual basal crop coefficients through reflectance-based vegetation indices was successful, with good accuracy. The use of NDVI was preferred for maize and barley and SAVI was more accurate for olive. These results are consistent with the lower sensitivity of SAVI to changes of reflectance in soil background due to soil moisture. This approach allowed considering the whole crop season, including periods when the soil cover by vegetation is low, as in the initial and development growth stages, which was not considered in several studies previously published in the literature. The use of the density coefficient allowed considering crops with different canopy structures and incorporating the impact of vegetation density and height. The use of the stress coefficient computed with the soil water balance SIMDualKc made it possible to apply the methodology not only in days when soil water was adequate to sustain full plant transpiration but also in days when heavy water stress conditions occurred, contrarily to most of the approaches previously published that do not account for the crop water stress in their formulation.
The proposed methodology for estimating actual crop coefficients follows a dual crop coefficient approach by combining a remotely sensed basal crop coefficient and an evaporation coefficient computed with the soil water balance SIMDualKc. Therefore, the approach allows integrating the effect of plant transpiration with that of soil evaporation.
The approaches proposed in this study, combining VI data with soil water balance data, provide for an operational methodology to support irrigation management considering the actual cropping conditions and associating real time satellite information with irrigation scheduling models potentially applicable to diverse crop types. Nevertheless, further studies are desirable to better test and implement the methodology, particularly for other crops and using a larger number of image dates.
writing the present manuscript, also in collaboration with the other authors. The second and third authors contributed with field work and SIMDualKc modeling relative to the crops studied as well as contributing to test the approaches. The fourth author was responsible for the remote sensing study while the fifth and senior author was responsible for aspects related with evapotranspiration and water balance and for the revision of the manuscript. Adjustment coefficient associated with crop senescence and leaves yellowing