Article Request Page ASABE Journal Article ## Transferability of Jarvis-Type Models Developed and Re-Parameterized for Maize to Estimate Stomatal Resistance of Soybean: Analyses on Model Calibration, Validation, Performance, Sensitivity, and Elasticity

D. Mutiibwa, S. Irmak## Published in

Transactions of the ASABE56(2): 409-422 (doi: ). Copyright 2013 American Society of Agricultural and Biological Engineers.Submitted for review in May 2012 as manuscript number SW 9769; approved for publication by the Soil & Water Division of ASABE in March 2013.

The mention of trade names or commercial products is for the information of the reader and does not constitute an endorsement or recommendation for use by the University of Nebraska-Lincoln or the authors.

The authors are

Denis Mutiibwa, ASABE Member,Post-Doctoral Research Associate, Department of Geography, University of Nevada, Reno, Nevada; andSuat Irmak, ASABE Member,Professor, Department of Biological Systems Engineering, University of Nebraska-Lincoln, Lincoln, Nebraska.Corresponding author:Suat Irmak, 239 L. W. Chase Hall, University of Nebraska-Lincoln, Lincoln, NE 68583, phone: 402-472-4865; e-mail: sirmak2@unl.edu.

In a previous study by the same authors, a new modified Jarvis model (NMJ-model) was developed, calibrated, and validated to estimate stomatal resistance (rAbstract._{s}) for maize canopy on an hourly time step. The NMJ-model’s unique subfunctions, different from the original Jarvis model (J-model), include a photosynthetic photon flux density (PPFD)-r_{s}response subfunction developed from field measurements and a new physical term, A^{exp(1/LAI)}, where A is the minimum stomatal resistance and LAI is the green leaf area index, to account for the influence of canopy development on r_{s}, especially during partial canopy stage in the early season and in late-season stage during leaf aging and senescence. This study evaluated the transferability of the J-model and NMJ-models that were re-parameterized and calibrated for maize canopy to estimate soybean r_{s}. Due to the differences in physiological and photosynthetic pathway differences between the two crops, the r_{s}response to the same environmental variables, i.e., PPFD, vapor pressure deficit (VPD), and air temperature (T_{a}), were substantially different. Thus, this study demonstrated the inherent limitation in applying the Jarvis-type models that were calibrated for maize to soybean without re-calibration. Maize-calibrated models performed poorly in estimating soybean r_{s}, with the coefficient of determination (r^{2}) ranging from 0.30 to 0.38 and the root mean square difference (RMSD) between the estimated and measured r_{s}ranging from 94.4 to 166 s m^{-1}. The J-model and NMJ-model were re-calibrated by parameter optimization method for soybean. The J-model calibrated well; however, the validation had poor performance results. The NMJ-model had a good calibration, resulting in a good r^{2}(0.71) and a small RMSD (13.7 s m^{-1}). The NMJ-model validation produced superior results to the J-model, explaining more than 80% of the variation in the measured r_{s}(RMSD = 38.4 s m^{-1}). These results show the robustness and practical accuracy of the NMJ-model in estimating r_{s}over different canopies if well calibrated for a specific crop. In terms of sensitivity and elasticity analyses, among all parameters, r_{s}estimates were most sensitive to uncertainties introduced in parameter a_{1}of the PPFD subfunction due to its exponential impact on r_{s}in the NMJ-model. Therefore, for accurate estimates of r_{s}, uncertainties in parameter a_{1}should not exceed the range of -2% and 2% so that the error in estimated r_{s}is kept between -3.5% and 3.6%. The study observed that the relative change in r_{s}due to uncertainties in parameters a_{2}and a_{3}of the VPD subfunction was a linear function and less sensitive than the PPFD subfunction. The sensitivity of r_{s}to uncertainties in temperature subfunction parameters (a_{4}and a_{5}) was higher than that of VPD subfunction parameters, but less than that of PPFD subfunction parameters. The uncertainty in parameters a_{4}and a_{5}should range within -10% and 10%, and the calibration of these parameters should be determined with greater precision as compared with the VPD subfunction parameters. The study confirmed that the addition of the r_{s_min}and the A^{exp(1/LAI)}terms, which were not accounted for in the original J-model, improved the model accuracy for estimating soybean r_{s}.

Elasticity analyses, Jarvis model, Maize, Optimization, Sensitivity analysis, Soybean, Stomatal resistance. Keywords.Stomatal resistance (

r) is an important and intricate phenomenon that is essential to understanding plant-soil-water-atmosphere relationships and to evaluating plant responses to various environmental variables. It is one of the drivers of photosynthesis and transpiration processes, which ultimately determine crop water productivity (also known as crop water use efficiency) and many other plant physiological functions. The phenomenon crucially regulates the biophysical link between the water source (soil moisture) and the atmospheric evaporative demand. The_{s}ris subject to independent and interactional influence of plant physiology, soil physical and chemical properties and moisture content, and atmospheric conditions. With tremendous labor and cost, continuous and viable measurements of_{s}rcan be made using either steady-state or dynamic diffusion porometry instrumentation. Porometers measure_{s}rby predicting the change in water vapor diffusion rate from a porous surface of the leaf in an enclosed chamber. The measurements are referenced to a pre-calibration fit of a standard porous surface of know diffusion resistances. The_{s}rand stomatal aperture and behavior can also be measured and monitored using observations under a microscope, using cobalt-chloride paper, the leaf-chamber (cuvette) transpiration method, and mass-flow porometry methods (Kirkham, 2005, pp. 392-393)._{s}For years, strenuous research has advanced efforts to model

rfrom soil moisture, carbon dioxide concentrations, and climatic variables. However, the physiological knowledge about stomatal functioning may not be adequate to provide a mechanistic model linking stomatal resistance to all driving variables. The alternative approach is to estimate_{s}rphenomenologically from environmental variables. Jarvis (1976) developed a descriptive and multiplicative model to estimate stomatal resistance as a function of environmental variables, soil moisture, and plant water status. The model predicted the response of stomata to environmental variables operating as resistance stress functions without synergy. Jarvis (1976) described the model (J-model) as a useful interim way of using field measurements to describe very complex and changing properties of the stomata. Since its development, the model has been extended, recalibrated, and/or re-parameterized into various Jarvis-type models for different vegetation surfaces and environmental conditions (Farquhar, 1978; Farquhar et al., 1980; Lohammar et al., 1980; Kauffman, 1982; Jarvis and McNaughton, 1986; Ball et al., 1987; Noilhan and Planton, 1989; Massman and Kaufmann, 1991; Pleim and Xiu, 1995; Viterbo and Beljaars, 1995; Niyogi and Raman, 1997; Green and McNaughton, 1997; Thomas et al., 1999; Irmak and Mutiibwa, 2009, 2010)._{s}In the Irmak and Mutiibwa (2009) study, a new modified Jarvis model (NMJ-model) was presented along with the original Jarvis (1976) and Green and McNaughton (1997) models, and Irmak and Mutiibwa (2009) calibrated/re-parameterized and validated the NMJ-model against porometer-measured

rdata through extensive field measurements for a non-stressed maize canopy. The new re-parameterized NMJ-model was developed with a new physical term:_{s}A^{exp(1/LAI)}, whereAis the minimum leaf stomatal resistance (s m^{-1}), and LAI is the leaf area index (unitless) to account for the influence of canopy development onr, especially during the partial canopy cover stage. The vital_{s}rresponse to light was modeled based on the photosynthetic photon flux density (PPFD) vs._{s}rresponse curves that were measured in the field. The NMJ-model demonstrated improved performance over the original Jarvis (1976) model (J-model) in estimating hourly_{s}r, especially during the partial canopy cover stage. The calibration and the new physical term accounting for the variation in LAI improved the_{s}rmodeling performance. Irmak and Mutiibwa (2009) demonstrated that on a seasonal average basis, the J-model and the NMJ-model had similar performance in estimating_{s}r, with a coefficient of determination (r_{s}^{2}) of 0.74 and a root mean square difference (RMSD) between the modeled and measuredrof 48.8 s m_{s}^{-1}for the J-model, and r^{2}= 0.74 and RMSD = 50.1 s m^{-1}for the NMJ-model, for a non-stressed, subsurface drip-irrigated maize canopy. The inclusion of the variation in the LAI andrterm [_{s_min}rexp(-LAI)] during the growing season in the NMJ-model improved the_{s_min}restimation, especially in the higher_{s}rrange (_{s}r> 250 s m_{s}^{-1}), as compared to the J-model. When the period of partial canopy cover was considered separately (LAI range from 1.20 to approximately 2.5), the addition of the LAI term in the NMJ-model resulted in 8% improvement in r^{2}and 10% improvement in RMSD relative to the J-model (r^{2}= 0.64, RMSD = 35.5 s m^{-1}for the NMJ-model, and r^{2}= 0.59, RMSD = 39.0 s m^{-1}for the J-model). The enhanced performance of the NMJ-model was attributed to the calibration of the model to a specific crop and environmental conditions. In another study by Noilhan and Planton (1989), the J-model was extended and improved through environmental modulation of minimum stomatal resistance and by the integration of the inverse LAI effect onr. Although Jarvis (1976) did not include variable LAI in his original_{s}rmodel, the variability of LAI is known to control the amount of light scattered and absorbed by the plant canopy, impacting stomatal functions and responses. Finnigan and Raupach (1987) linked LAI, theoretically, to the vegetation’s diffusive source/sink capacity, which regulates the mass and energy exchange rate of the plant canopy via stomatal regulation._{s}The Jarvis-type models have proved to be practical in modeling

r; however, their empirical development is, in principal, a limitation in their transferability to estimate_{s}rfor different crops. Since different plant species (e.g., maize vs. soybean) have different stomatal response to the same environmental variable, a model that was developed for maize canopy may not accurately represent stomatal behavior of soybean canopy. Because maize and soybean are the dominant agronomic crops produced in Nebraska and many other Midwestern U.S. states, and in many other countries, there is a need to test the maize_{s}rmodels’ performances for estimating soybean_{s}rand potentially develop new models or re-calibrate/re-parameterize the maize_{s}rmodel for soybean canopy. In the research conducted by Irmak and Mutiibwa (2009), the NMJ-model was developed for maize canopy under south central Nebraska environmental and climatic conditions and management practices. The three main objectives of this research were to: (1) evaluate the transferability of the two_{s}rmodels, the J-model (Jarvis, 1976) and the NMJ-model (Irmak and Mutiibwa, 2009), that were calibrated and re-parameterized for non-stressed maize canopy to estimate_{s}rfor soybean canopy; (2) recalibrate the models to estimate_{s}rfor soybean, using extensive datasets measured through an independent field campaign for soybean, by applying the same approach presented by Irmak and Mutiibwa (2009); and (3) investigate the sensitivity of_{s}rto the NMJ-model coefficients in the subfunctions of environmental variables in the model. The models were recalibrated and validated using data from extensive field measurements of_{s}r, climatic variables, plant physiology parameters, and soil water status in the 2007 soybean growing season in south central Nebraska._{s}

Materials and Methods

Study SiteThe field measurements for this study were conducted in 2007 at the University of Nebraska-Lincoln, South Central Agricultural Laboratory (SCAL) near Clay Center, Nebraska. The site is located in Clay County in the south central part of the state at 40° 34' N and 98° 8' W at an elevation of 552 m above mean sea level (Irmak, 2010). The soil at the site is a Hastings silt loam (fine, montmorillonitic, mesic Udic Argiustoll), with 0.5% slope, which is a well-drained soil on uplands, with field capacity of 0.34 m

^{3}m^{-3}, permanent wilting point of 0.14 m^{3}m^{-3}, and saturation point of 0.53 m^{3}m^{-3}. The particle size distribution is 15% sand, 65% silt, and 20% clay, with 2.5% organic matter content in the topsoil (Irmak, 2010). The experimental field is 13 ha in size and irrigated with a subsurface drip irrigation system. The drip lines were installed at about 0.40 m below the soil surface with 0.45 m emitter spacing on the drip lines and 1 LT h^{-1}flow rate with pressure-compensating drip emitters (Netafim-USA, Fresno, Cal.). The field was irrigated two or three times per week to meet plant water requirement. The soil water content was measured using a neutron probe soil moisture meter (model 4302, Troxler Electronics Laboratories, Inc., N.C.) at 0.30, 0.60, 0.90, and 1.20 m soil depths twice a week throughout the season. For each irrigation application, the soil water deficit was replenished to approximately 90% of the field capacity in the top 0.90 m soil profile to maintain non-stressed plant conditions and to reserve storage in the soil profile for potential rainfall. The effective rooting depth for soybean in the experimental region is 0.90 m. The total available water holding capacity of the top 0.90 m soil profile is approximately 175 mm. The maximum allowable depletion was set to approx.imately 40% to 45% of the total available water. A total of seven irrigations were applied during the 2007 growing season [July 23 (9 mm), July 26 (13 mm), August 7 (17 mm), August 10 (13 mm), August 13 (26 mm), August 16 (21 mm), and August 20 (11 mm)] with a seasonal total of 110 mm. The total rainfall from emergence until physiological maturity (May 26 to September 30) measured in the experimental field was 354 mm. Plants were maintained with regular pest and disease control practices when needed (Irmak, 2010). The soybean [Glycine max(L.) Merr.] crop was planted on May 21 with a planting density of approximately 188,000 plants ha^{-1}. The planting row spacing was 0.762 m with a west-east planting direction. Plants emerged on May 26, reached flowering stage around July 14-15, reached pod formation stage (R3) around July 20, reached complete canopy closure around August 2 (73 days after planting), fully matured on September 30, and were harvested on October 24, 2007 (Irmak, 2010).

Micrometeorology MeasurementsMeasurements of surface energy fluxes (including latent heat flux (ET

), sensible heat flux, soil heat flux, and net radiation) and other climatic variables (air temperature, relative humidity, wind speed and direction, and precipitation) were made using a Bowen ratio energy balance system (BREBS) (Radiation and Energy Balance Systems (REBS), Inc., Bellevue, Wash.), which was stationed in the middle of the experimental field (Irmak, 2010). The site and the BREBS are part of the Nebraska Water and Energy Flux Measurement, Modeling and Research Network (NEBFLUX; Irmak, 2010), which is a network of 11 flux towers that are installed and operated on an hourly basis in various parts of Nebraska on vegetation surfaces ranging from irrigated and rainfed croplands, including maize (_{a}Zea maysL.), soybean [Glycine max(L.) Merr.], and winter wheat (Triticum aestivumL.) under different tillage and irrigation practices; irrigated and natural grasslands, including mixture of tall fescue (Festuca arundinacea), Kentucky bluegrass (Poa pratensis), smooth bromegrass (Bromus inermis), creeping foxtail (Alopecurus arundinacea), and buffalograss (Bouteloua dactyloidesNutt.); irrigated alfalfa (Medicago sativaL.); and rainfed switchgrass (Panicum virgatum) to riparian systems with invasive plant species [common reed (Phragmites australis), peach-leaf willow (Willow salix), and cottonwood (Populus deltoidesvar.occidentalis), etc.). The NEBFLUX towers measure all surface energy flux variables, meteorological variables, plant physiological parameters, soil water content (every 0.30 m up to 1.80 m on an hourly basis), soil characteristics, and agronomical components, including biomass production and/or yield, for a significant number of different vegetation surfaces. For this study, net radiation (R) was measured using a REBS model Q*7.1 net radiometer. Incoming and outgoing shortwave and longwave radiation were measured simultaneously using a REBS model THRDS7.1 double-sided total hemispherical radiometer that is sensitive to wavelengths from 0.25 to 60 µm (Irmak, 2010). Air temperature (_{n}T) and relative humidity (RH) gradients were measured using two platinum resistance thermometers and monolithic capacitive humidity sensors (REBS models THP04015 and THP04016, respectively) with resolutions of 0.0055°C for temperature and 0.033% for relative humidity. The measured temperature and relative humidity gradients were used to calculate vapor pressure deficit (VPD). Precipitation was recorded using a sensor (model TR-525, Texas Electronics, Inc., Dallas, Tex.). Wind speed and direction at 3 m height were monitored using a cup anemometer (model 034B, Met One Instruments, Grant Pass, Ore.). The anemometer had a wind speed range of 0 to 44.7 m s_{a}^{-1}and threshold wind velocity of 0.28 m s^{-1}. The BREBS used an automatic exchange mechanism that physically exchanged the temperature and humidity sensors every 15 min at two heights above the canopy. All variables were sampled every 60 s, averaged, and recorded on an hourly basis using a CR10X datalogger and AM416 relay multiplexer (Campbell Scientific, Inc., Logan, Utah) (Irmak, 2010). Extensive maintenance procedures that were described by Irmak (2010) were followed weekly to ensure continuous and good quality data collection throughout the year. Additional detailed description of the BREBS setup and instrumentation are provided by Irmak (2010).

Stomatal Resistance, Plant Green LAI, and Plant Height MeasurementsThe plant variables measured included

r, green LAI, and plant height (_{s}h). A dynamic diffusion porometer (model AP4, Delta-T Devices, Ltd., Cambridge, U.K.) equipped with an unfiltered GaAsP photodiode light sensor with a spectral response similar to photosynthetically active radiation response (Irmak and Mutiibwa, 2009; Mutiibwa and Irmak, 2011) was used to measureron randomly selected green and healthy soybean leaves. Before taking readings, the porometer was calibrated based on the manufacturer’s recommendations. The unit was recalibrated every time RH changed by ±10% from the previously set value and whenever air temperature changed by ±4°C from the temperature at the time of previous calibration. The porometer head unit contained fast-response sensors to measure cup and leaf temperatures, allowing automatic temperature compensation to be applied when measuring_{s}r. The AP4 porometer has a resolution of 0.5 mol m_{s}^{-2}s^{-1}with anrmeasurement speed of less than 5 s. For each_{s}rmeasurement cycle, the following variables were recorded simultaneously:_{s}r(s m_{s}^{-1}), PPFD (mol m^{-2}s^{-1}), chamber (cup) temperature (T, °C), and leaf-chamber temperature difference (_{c}T-_{L}T, °C). On a given field measurement day, on average, three readings from each leaf, six leaves from each plant, and fifteen to thirty plants were sampled for_{c}rmeasurements and averaged per hour. Each reading corresponded to one complete diffusion cycle in which the sensor and leaf reached equilibrium with the RH in the chamber. This study uses the_{s}rdata, PPFD vs._{s}rresponse curves, and other supporting field data that were measured by Mutiibwa and Irmak (2011). The reader is referred to Mutiibwa and Irmak (2011) for more detailed description of the field measurement. LAI was measured using a plant canopy analyzer (model LAI-2000, Li-Cor Biosciences, Lincoln, Neb.) once a week during the growing season. On average, a total of 60 LAI measurements were taken across the field on each field measurement day and averaged for that day. LAI measurements were started at 32 days after planting (DAP) (June 22) when LAI was approximately 1.10. On the same days of LAI field measurements, plant height (_{s}h) measurements were taken by measuring soybean plants from the soil surface to the tip of the tallest leaf for 14 to 17 randomly selected plants, and the values were averaged for that week.

Jarvis Model (J-Model)The J-model (Jarvis, 1976) estimates

ras a function of multiplicative subfunctions of environmental and plant physiological variables without synergistic interaction. The subfunction variables include PPFD (µmol m_{s}^{-2}s^{-1}), VPD (kPa), air temperature (T), soil water content (_{a}W,% vol), and maximum stomatal conductance (b_{1}, m s^{-1}). The model is expressed as:(1)

(2)

(3)

(4)

(5)

where

Fis the subfunction that describes the stomatal response to a particular variable such that 0 <F< 1, although not always.F_{1}is the subfunction that describes therresponse to PPFD,_{s}q(s m^{-1}) is the asymptotic value of stomatal conductance (g, m s^{-1}) (1/r) at infinite PPFD. Based on the asymptote from the PPFD-_{s}rresponse curve of soybean presented by Mutiibwa and Irmak (2011),_{s}qwas set to 0.0289 s m^{-1}. The parameterb_{1}is the maximumgat full sunlight and is calculated from the relationshipq=b_{0}/(b_{1})^{2}, whereb_{0}is the nocturnal (night time)g. Parameterb_{2}(µmol m^{-2}s^{-1}) is the slope of the PPFD vs.rresponse curve at PPFD = 0._{s}F_{2}is the subfunction that describes therresponse to VPD. The parameter_{s}b_{3}(Mg^{-1}s^{3}) in equation 3 represents the slope of the relationship betweenrand VPD._{s}F_{3}is the subfunction that describes therresponse to_{s}T(in K). The subfunction_{a}F_{4}accounts for the effect of crop water stress onr. It varies between 0.0 and 1.0 when soil water content (_{s}w_{2},% vol) varies between permanent wilting point (w) and a critical value (_{wilt}w) at 0.75_{cr}w(Thompson et al., 1981), where_{sat}wrepresents the volumetric soil water content (% vol) at saturation. The term_{sat}w_{2}represents the deep soil profile moisture (volumetric water content at 1 m below the soil surface). In this study, the soil moisture at the effective plant root zone depth was maintained at optimum level (i.e.,w_{2}>w); therefore,_{cr}F_{4}was taken as 1.0.

New Modified Jarvis (NMJ) ModelThe NMJ-model that was developed for maize canopy was presented by Irmak and Mutiibwa (2009). The important features of the new model included the PPFD-

rresponse function for the crop canopy, which was measured and constructed in the field to account for the effect of different ranges of canopy light distribution on_{s}r. The PPFD vs._{s}rresponse function replaced the_{s}F_{1}subfunction in the J-model (eq. 1). The NMJ-model has a new term extension that integrates the effect of LAI variation onrduring the growing season. The original J-model did not account for the LAI effect on_{s}r. However, LAI has an important role in driving stomatal behavior. To account for the effect of seasonal variation of LAI on_{s}r, an extension term of_{s}rwas raised to the inverse exponential function of LAI and incorporated into the NMJ-model, as shown in equation 6:_{s_min}(6)

where PPFD (µmol m

^{-2}s^{-1}) is measured or estimated at the leaf level, VPD is in kPa, andTis in K. The parameter_{a}a_{0}is the slope of the PPFD vs.rresponse subfunction (measured as 3010 s m_{s}^{-1}by Mutiibwa and Irmak, 2011),a_{1}is the exponent of the measured PPFD-rresponse subfunction, parameters_{s}a_{2}anda_{3}represent the coefficients in the VPD subfunction, and parametersa_{4}anda_{5}represent the coefficients in theTsubfunction. The term_{a}r(s m_{s_min}^{-1}) represents the lowest (minimum) measuredrduring the growing season. Based on our extensive field measurements,_{s}rfor soybean canopy is 22.4 s m_{s_min}^{-1}.

J-Model and NMJ-Model Calibration and OptimizationIn the Irmak and Mutiibwa (2009) study, the J-model and NMJ-model were calibrated for maize for the 2006 growing season. Therefore, we first evaluated the transferability of the maize-calibrated models to estimate

rfor soybean in the 2007 growing season. The performances of the models were evaluated using RMSD, r_{s}^{2}, and modeling efficiency (unitless), which is expressed as:(7)

where

Oand_{i}Pare the observed and predicted_{i}r, respectively, and is the mean of observed data. The models were re-calibrated and re-parameterized for soybean by applying the parameter optimization procedure and then validated using porometer-measured soybean_{s}r. The days for the soybean stomatal resistance measurements were randomly and evenly divided to create two datasets: one for calibration, and one for validation. The dates and meteorological conditions on the days when the_{s}rmeasurements were made for model calibration and validation are presented in table 1. Using the Solver tool in Microsoft Excel 2010, the parameters of the models were optimized for the best-fit model on measured_{s}rby minimizing RMSD and maximizing r_{s}^{2}. The procedure of optimization involves searching the parameter space for a parameter value that is optimal with respect to the specified objective conditions, such as minimizing RMSD and maximizing r^{2}. The RMSD evaluates the accuracy of the optimized model by measuring the deviation of the model estimates from the measuredr. During the optimization process, a constraint was added by holding the regression slope of the estimates on measured_{s}rbetween 0.90 and 1. A constraint is a logical condition that an optimized model must satisfy. The calibrated model performance was evaluated using EF (eq. 7), which assesses the fraction of the variance of the measured values that is explained by the model. The EF ranges between one and negative infinity, and values close to unity are an indication of good performance of the model. The validation of the models was implemented by using the models to estimate soybean_{s}rand compare the model-estimated results to measured soybean_{s}rusing the validation dataset. The validation performance statistics are presented in table 3 for the model calibration and validation._{s}

Table 1. Daily average meteorological variables measured during the 2007 growing season when stomatal resistance ( r) measurements were made. Variables include incoming shortwave radiation (_{s}R), net radiation (_{s}R), maximum and minimum air temperature (_{n}Tand_{a_max}T), wind speed at 3 m (_{a_min}u_{3}), maximum and minimum relative humidity (RHand RH_{max}), vapor pressure deficit (VPD), and rainfall._{min}Date

(2007)Meteorological Variable R_{s}

(W m^{-2})R_{n}

(W m^{-2})T_{a_max}

(°C)T_{a_min}

(°C)RH _{max}

(%)RH _{min}

(%)u_{3}

(m s^{-1})VPD

(kPa)Rainfall

(mm)Calibration 16 July 300 196 31.5 16.2 99.1 55.7 2.4 2.3 0 20 July 223 143 28.0 19.4 95.2 75.6 3.4 2.5 4.6 26 July 293 192 32.8 19.0 95.0 48.0 3.0 2.3 0 31 Aug. 248 157 27.0 13.4 100.0 63.2 2.3 2.0 0 Validation 24 July 294 193 30.4 20.8 100.0 62.3 2.3 2.7 0 9 Aug. 290 195 32.5 19.6 100.0 60.4 1.9 2.9 0.25 14 Aug. 280 181 35.2 20.4 82.2 39.9 2.2 2.2 0 12 Sept. 237 138 26.0 08.4 98.0 41.8 3.7 1.3 0

Sensitivity and Elasticity AnalysisThe technique of parameter optimization is implemented by searching, under specified conditions, the parameter space to find the optimal parameters for the best fit in a model. Therefore, by applying sensitivity and elasticity analysis on the NMJ-model parameters, the study analyzes the potential variations and errors in the estimated

roriginating from the potential parameter uncertainties over the parameter space. The analysis objectives are to determine the important parameters, which cause the most significant variations in estimated_{s}r; determine the type of sensitivity and elasticity functions; identify threshold values where the optimal strategy changes; and determine the relative variation (uncertainty percentages) in each parameter for practically accurate_{s}restimates._{s}For most sensitivity analysis studies, the focus usually is to evaluate the relative error induced in the model output due to potential uncertainties and errors and the magnitude of changes in the input variables. In the NMJ-model sensitivity and elasticity analysis, the relative error introduced in estimated

rby the input parameter’s potential relative variation over the parameter space during the optimization process was evaluated. By holding all variables and parameters constant, the sensitivity of_{s}rto the variation in a given parameter was investigated by systematically varying the calibrated value of the investigated parameter within the conceivable range. This technique of sensitivity analysis is referred to as one-at-a-time sensitivity analysis (Hamby, 1994). Each parameter was deviated from the baseline value (optimized or calibrated value) by -99%, -75%, -50%, -25%, -10%, -5%, -2%, -1%, 1%, 2%, 5%, 10%, 25%, 50%, 75%, and 100%. The sensitivity analysis was carried out on field measurement days when data were collected for calibration, as shown in table 1. For each parameter value’s deviation from the baseline, the_{s}rrelative error (%) was averaged over the calibration days. To determine the important parameters that cause the most variations in estimated_{s}r, the elasticities, which are measures of the percent change in a dependent variable (_{s}r) divided by the percent change in an independent variable (parameter), were calculated using equation 8:_{s}

Figure 1. Daily rainfall events and cumulative seasonal rainfall measured in the experimental field from May 1 through October 31, 2007.

Table 2. Daily average meteorological variables measured from May to October 2007 and long-term averages at Clay Center, Nebraska. Variables include wind speed at 3 m ( u_{3}), maximum and minimum air temperature (Tand_{a_max}T), relative humidity (RH), incoming shortwave radiation (_{a_min}R), and total rainfall._{s}Period Meteorological Variable May June July August September October 2007 u_{3}(m s^{-1})5.0 3.9 2.8 3.0 3.6 3.8 T(°C)_{a_max}23.5 27.1 29.3 29.5 25.1 19.6 T(°C)_{a_min}12.5 15.4 18.5 19.0 11.5 6.7 RH (%) 71.6 72.9 77.8 81.3 73.3 71.5 R(MJ m_{s}^{-2}d^{-1})19.9 22.9 22.0 18.3 16.4 11.8 Rainfall (mm) 130 53 106 117 67 149 Long-term

(32-year)

averageu_{3}(m s^{-1})4.0 3.5 2.9 2.6 3.1 3.3 T(°C)_{a_max}22.5 28.1 30.3 29.2 25.3 18.3 T(°C)_{a_min}9.3 14.6 17.3 16.3 10.7 3.6 RH (%) 71.3 70.2 73.2 74.5 68.8 67.2 R(MJ m_{s}^{-2}d^{-1})19.4 22.4 22.4 19.7 15.9 11.3 Rainfall (mm) 112 110 93 83 63 45 (8)

Essentially, the elasticities (

E) of parameters are the first derivatives of the sensitivity function. For linear sensitivity functions, the elasticities are constant values; however, for non-linear sensitivity functions, the elasticities are first-derivative functions of the sensitivity functions. This analysis identifies parameter spaces where the response ofris elastic or inelastic. An elastic response is one in which_{s}rresponds highly to small changes in the parameter, whereas an inelastic response is one in which_{s}rdoes not respond much to the changes in the parameter._{s}

Results and Discussion

Weather Conditions During Research PeriodA summary of the daily average meteorological conditions for the 2007 growing season and their comparisons relative to the long-term (32-year) averages are presented in table 2, and daily rainfall events and seasonal cumulative values are presented in figure 1. During the growing season, the total rainfall from May through end of September (473 mm) was very close to the long-term growing season average value (461 mm). The largest rainfall event occurred on August 22 as 80 mm. Another large rainfall was recorded on July 9 as 56 mm. Although July and August were wetter than the long-term average, the rainfall amount from after July 9 until August 22 was not enough to meet crop water requirement. The seasonal average wind speed was 13% greater than the long-term average, with the highest monthly average wind speed occurring in May (5.0 m s

^{-1}).Twas very close to the average, but on average_{a_max}Twas approximately 16% higher than normal. The seasonal average relative humidity was about 5% more than long-term average._{a_min}

Seasonal Trend of Measured Soybean Stomatal ResistanceFigure 2 presents the seasonal trend in measured hourly soybean

rand daily soybean LAI. For each_{s}rmeasurement day, there are several hours of data points showing the daily range of_{s}r. For example, on July 20, the measured_{s}rranged from 108 s m_{s}^{-1}at 9:00 a.m. to the lowest value of 24 s m^{-1}at solar noon at 3:00 p.m., with a large diurnal range of 84 s m^{-1}. There was an opposite trend between LAI andr. Although partly obscured by the hourly fluctuations on measurement days, the_{s}rtrend depicts the theoretical parabolic variation of_{s}rthat was also observed by Monteith (1965), Monteith et al. (1965), and Irmak and Mutiibwa (2009). The_{s}rexhibited a decreasing trend from early season toward mid-season and remained relatively constant, with later increases toward the end of the growing season. The_{s}rvalues ranged from a minimum of 22.4 s m_{s}^{-1}to a maximum of 149 s m^{-1}with a seasonal average of 63 s m^{-1}. The seasonal minimumr(22.4 s m_{s}^{-1}) was measured on July 26 at 4:00 p.m. The microclimatic conditions at that time were characteristic of high atmospheric evaporative demand;Rwas 577 W m_{n}^{-2}, air temperature was extremely high as 31.6°C, wind speed was 3.4 m s^{-1}, and VPD was 1.8 kPa. The seasonal maximumrvalue (149 s m_{s}^{-1}) was measured on September 12 at 12:00 p.m. This was during the late growing season, and the highris most likely due to leaf aging and senescence. On September 12, the microclimatic conditions were_{s}R= 429 W m_{n}^{-2}, air temperature = 23.5°C, wind speed = 5.1 m s^{-1}, and VPD = 1.0 kPa. The spike inron July 20 at 9:00 a.m. (fig. 2) was due to cloudy conditions and the typical coolness and low VPD (0.34 kPa) of morning hours. The highest_{s}rvalue of 149 s m_{s}^{-1}that was measured in this research, potentially lower than the values reported in the literature for soybean, was due to a combination of several factors, such as the well-watered conditions of soybean under subsurface drip irrigation, lower than long-term average air temperatures in June and July, considerably greater than long-term average relative humidity throughout the growing season, greater rainfall amounts during the peak atmospheric demand periods in July and August, the physiological differences between the soybean variety grown in this study and those studied in the literature, differences in performance between the instruments used to measurer, and combination of all these factors._{s}

Figure 2. Seasonal pattern of measured hourly stomatal resistance ( r) and green LAI for soybean during the 2007 growing season._{s}Soybean LAI (fig. 2) was 1.1 in late June, peaked at 5.1 in August, and the end-season value in late September was 1.9, with a seasonal average of 3.9 (from June 22 to September 20). As LAI increased to the maximum during the mid-growing season,

rdecreased to its lowest seasonal values. After late August, due to senescence and increased_{s}rof aged leaves,_{s}rincreased rapidly as LAI also sharply decreased. Monteith et al. (1965) observed a similar seasonal trend in_{s}rin the late growing season and related it to increased epidermal resistance as leaves aged. Slatyer and Bierhuizen (1964) and Brown and Pratt (1965) discussed the increase in_{s}rtoward the end of the season and observed that the stomata of older leaves became less responsive and remained only partly open, even at midday with sufficient sunshine._{s}

Transferability of J-Model and NMJ-Model from Maize to SoybeanIn this section, the transferability of the J-model and the NMJ-model developed for maize to estimate

rfor soybean is evaluated. In Irmak and Mutiibwa (2009), the two models were calibrated for maize, and the models produced very good results, estimating field-measured_{s}rwith r_{s}^{2}= 0.74, RMSD = 48.8 s m^{-1}for the J-model, and r^{2}= 0.74, RMSD = 50.1 s m^{-1}for the NMJ-model. In this study, first without recalibration, the maize-calibrated models were used to estimaterfor soybean in a different year. Figures 3a and 3b show scatter plots of the J-model and NMJ-model estimates against measured_{s}r. Figure 3a shows that the maize-calibrated J-model substantially overestimated_{s}rfor soybean. The model r_{s}^{2}was 0.38 and RMSD was 94.4 s m^{-1}, a significant deviation in the model’s performance for soybean relative to its performance for maize. Before calibration, the NMJ-model overestimated measured soybeanrwith a low r_{s}^{2}of 0.30 and a very high RMSD of 166 s m^{-1}, a significant lapse in the performance of the model for soybean relative to maize. These results indicate the inherent limitation of J-type models that are calibrated for maize in estimatingrfor soybean due to differences in the two crops’ basic physiologic functions and their different photosynthetic pathways, which are implicitly imbedded in the model coefficients during calibration. These results also indicate that re-parameterization of J-type models results in crop-specific parameter (variable) coefficients, limiting their transferability to other crops and/or different environments. These models are empirical representations of the behavior of a very complex biological system, rather than representtations of simple physics of the system. Nevertheless, the models are applicable and valuable within their limits of validity (Raupach and Finnigan, 1988)._{s}Jarvis (1976) studied mainly forest canopies [Douglas fir (

Pseudotsuga menziesii), Sitka spruce (Picea sitchensis), and Scots pine (Pinus sylvestris)] to develop his original model to estimate stomatal conductance. Our results demonstrate the need for re-parameterization of J-type models for accurate estimation ofrof a specific crop. The physiological structure differences between maize and soybean are primary reasons for the non-transferability of the maize model to soybean (Kirkham, 2005). The physiological differences between maize and soybean result in two different photosynthetic systems, and this results in their having two different stomatal resistances under the same conditions. Maize is a C4 plant and soybean is a C3 plant, and their stomata are anatomically different. C4 plants have dumbbell-shaped stomata, whereas C3 plants have bean-shaped stomata that function differently under the same environmental conditions. In general, the stomata of C4 plants have higher stomatal resistance than those of C3 plants. Differences in the shape, size, and distribution and changes in the growth of stomata with the development of the plant also contribute to the differences in stomatal response between maize and soybean (Kirkham, 2005, pp. 382-382) to the same environmental conditions. Kirkham (2011, pp. 172-174) provides an excellent and in-depth comparisons of the differences in stomatal functions between C4 and C3 plants._{s}

Figure 3. Relationship between maize-calibrated stomatal resistance ( r) models used to estimate soybean_{s}r: (a) Jarvis (1976) model (J-model), and (b) new modified Jarvis model (NMJ-model)._{s}

Calibration and Validation of Stomatal Resistance Models for SoybeanAfter assessing the performance of the maize-calibrated J-model and NMJ-model for estimating

rfor soybean (with poor performance), the models were re-calibrated and re-parameterized for soybean by parameter optimization. The results are presented in table 3 and figures 4a and 4b for calibration and in figures 5a and 5b for validation. Compared to the maize-optimized parameters of the J-model, as presented by Irmak and Mutiibwa (2009), some of the soybean-optimized parameters were very similar (i.e.,_{s}b_{2}andb_{3}coefficients), and others were substantially different (i.e.,b_{1},b_{4}, andb_{5}coefficients). The optimization of the J-model (fig. 4a) resulted in a good r^{2}of 0.63 and a small RMSD of 13.3 s m^{-1}(table 3). The model underestimatedrby less than 30% and had a modeling efficiency (EF) of 0.22 (table 3). Overall, the calibration of the J-model yielded moderate results. However, for the validation (fig. 5a), the model performance was poor. The model substantially underestimated_{s}rand was seemingly insensitive to changes in measured soybean_{s}rwith r_{s}^{2}= 0.09, RMSD = 43.5 s m^{-1}, and EF = 0.78. Although the model r^{2}is low, the good EF and RMSD statistics indicate that the J-model was still able to estimate the trends in soybeanr, but not the magnitudes. When Jarvis (1976) presented the original model, his results showed that the model accounted for 51% and 73% of the variation in_{s}rof two different datasets, and some of the variations in performance results in his study were attributed to inadequacies in the distribution of the data rather than to inadequacies in the model. Stomatal resistance is an extremely intermittent process, variant on leaf, plant, and field-level conditions. Thus, characterizing a single average value over a field for a given hour is a simple and useful concept; however, the depiction is a coarse assumption._{s}The NMJ-model calibration results for soybean are presented in figure 4b and table 3. Compared to the maize-optimized parameters of the NMJ-model presented by Irmak and Mutiibwa (2009), the soybean-optimized parameters

a_{2},a_{3}, anda_{5}are substantially different. The NMJ-model calibrated better than the J-model, resulting in a higher r^{2}of 0.71, a smaller RMSD of 13.7 s m^{-1}, and EF of 0.59. Given that the lowest measuredrwas 22.4 s m_{s}^{-1}, an RMSD of 13.7 s m^{-1}was an indication of a small calibration error, but is also an indication that the model’s predictive ability was within acceptable limits. The model was able to account for 71% of the variation in measuredrof soybean. The calibrated model slightly underestimated measured_{s}rwith a slope of 0.99 (fig. 4b)._{s}

Table 3. Optimized values of parameters used in equations 2, 3, 4, and 6 to estimate soybean stomatal resistance ( r) using the Jarvis (1976) model (J-model) and the new modified Jarvis (1976) model (NMJ-model): RMSD = root mean square difference between measured and estimated_{s}r, and EF = modeling efficiency._{s}Model and Parameters

after OptimizationCalibration Validation RMSD

(s m^{-1})r ^{2}EF RMSD

(s m^{-1})r ^{2}EF J-model 13.3 0.63 0.22 43.5 0.09 0.78 b_{1}= 3.7933b_{2}= 1.88 × 10^{-5}b_{3}= 0.036b_{4}= 0.1061b_{5}= 0.00325b_{6}= 1.320NMJ-model 13.7 0.71 0.59 38.4 0.83 0.75 a_{0}= 3010a_{1}= 0.514a_{2}= 0.000268a_{3}= 613.4a_{4}= 0.036a_{5}= 0.014

Figure 4. Relationship between measured vs. estimated stomatal resistance ( r) for soybean canopy during calibration: (a) Jarvis (1976) model (J-model), and (b) new modified Jarvis model (NMJ-model)._{s}

Figure 5. Relationship between measured vs. estimated stomatal resistance ( r) for soybean canopy using models re-parameterized for soybean during validation: (a) Jarvis (1976) model (J-model), and (b) new modified Jarvis model (NMJ-model)._{s}In general, model validation helps to establish a confidence in the calibration. A model is considered to be verified if its accuracy and predictive capability have been proven to be within acceptable limits of error by testing the independency of the calibration data (Konikow, 1978). The NMJ-model validation on an independent dataset produced superior results of r

^{2}= 0.83, RMSD = 38.4 s m^{-1}, and EF = 0.75. The model was able to explain more than 80% of the variation in the measured soybeanr. This is a robust performance for the model, even slightly better than the model performance over maize canopy (r_{s}^{2}= 0.74, RMSD = 50.1 s m^{-1}) observed by Irmak and Mutiibwa (2009). The RMSD of 38.4 s m^{-1}is within the acceptable limits of error when the ranges of measuredrfor soybean canopy are considered (fig. 2), as well as the extremely difficult nature of modeling_{s}rwith great accuracy, especially on a short (hourly) time step. The EF being close to 1 is an indication of the model’s ability to explain the variance of the measured_{s}r. The EF value of 0.75 for the soybean crop and 0.73 for the maize crop (Irmak and Mutiibwa, 2009) and similarities in the r_{s}^{2}and RMSD values demonstrate the consistency of the model’s good performance in estimatingrfor different canopies if well calibrated for a specific crop._{s}Similar to other Jarvis-type models, the NMJ-model performance can be further improved with integration of the complex interaction effect of the variables. Nonetheless, using the multiple regression procedure on the primary independent variables that drive

r, our validation results, as well as those observed by Irmak and Mutiibwa (2009), demonstrate the practical accuracy of estimating_{s}rusing the NMJ-model for soybean and maize crops, with crop-specific calibration parameters, and that NMJ-model can be extended to other crops by field measurements of_{s}rand re-calibration/re-parameterization of the model parameters. The enhanced modeling of_{s}rby the NMJ-model is attributed to the integration of the effect of LAI variation into the model. As such, the model accounts for the effect of plant development on_{s}rduring the growing season. The impact of the added LAI and_{s}rterm [_{s_min}rexp(-LAI)] was assessed in figure 8 of Irmak and Mutiibwa (2009). Their results showed that most of the contribution of the LAI term to the modeling of_{s_min}roccurred during the partial canopy phase (LAI < 3) of the growing season and late in the season during the leaf aging and leaf senescence stage._{s}

Sensitivity Analysis of NMJ-ModelOwing to the uncertainties in the calibration procedure, the parameter values used in the calibrated model may not be very precise. Consequently, the calibrated parameters may not accurately represent the biological system under a different set of conditions or environmental stresses (Anderson and Woessner, 1992). Therefore, the primary purpose of sensitivity analysis in this study is to evaluate the uncertainties in the calibrated parameters of the NMJ-model and quantify the plausible relative error introduced in the estimated soybean

r. Figures 6a through 6g show the effect of changes in the calibrated parameters of the NMJ-model on the_{s}restimates. The parameters in the subfunctions of each climatic variable and the climatic variables themselves (because they are independent) were varied to quantify the respective relative error in_{s}r. Therefore, the response function of_{s}rdue to changes in the parameters of a given subfunction of a climatic variable is likely to be similar to changes in that climatic variable._{s}The sensitivity and elasticity analyses results on parameters

a_{0}anda_{1}of the PPFD subfunction in the NMJ-model are presented in figures 6a and 6b. Parametera_{0}(fig. 6a), which was calibrated at a value of 3010 s m^{-1}(table 3), was systematically varied from 30.11 (-99%) to 6000 (100%). Figure 6a shows that the relative change (%) inrwas very sensitive to relative changes (%) to_{s}a_{0}between 30.11 (-99%) and 4515 (50%). For greater than 50% change ina_{0},restimates became relatively insensitive. The elasticity function is constant between 30.11 (-99%) and 4515 (50%) change in parameter_{s}a_{0}, an indication that the sensitivity ofrto_{s}a_{0}in this range is a linear function. The threshold is at about 4515 (50%), beyond which the sensitivity ofrto changes in parameter_{s}a_{0}diminishes. This could be the point at which PPFD becomes non-limiting in drivingrwith the stomata fully open. These results indicate that any uncertainties introducing errors between 30.11 (-99%) and 4515 (50%) will linearly, and probably significantly, affect the_{s}restimates using the NMJ-model. The relative error induced in_{s}ris between -96.7% and 48.9%. Therefore for accurate estimates of_{s}r, uncertainties in parameter_{s}a_{0}should not exceed -5% and 5%, in which case the relative error inrwill be within 4.5%._{s}The sensitivity and elasticity of analysis of

rwith respect to parameter_{s}a_{1}are presented in figure 6b. Parametera_{1}, which was calibrated at a value of -0.514, was varied from -0.508 (-99%) to -1.027 (100%). Among all parameters,rwas most sensitive to relative changes in parameter_{s}a_{1}, especially between 0.005135 (-99%) and 0.488 (-10%). This could be attributed to the exponential nature of the parameter in the NMJ-model (eq. 6). Therestimate was still, but moderately less, sensitive to changes in_{s}a_{1}above -10%. Similar to parametera_{0}, this could be due to PPFD becoming non-limiting, with the stomata already fully open. The elasticities show that the sensitivity function ofrto_{s}a_{1}is an exponential decay function. From the elasticity function, it also appears thatrbecomes asymptotically insensitive to changes in_{s}a_{1}beyond 0.488 (-10%). In other words,ris very elastic between 0.005135 (-9%) and 0.488 (-10%), beyond which point the response becomes inelastic. Due to this apparent high and non-linear sensitivity of_{s}rto_{s}a_{1}, the parameter uncertainty should not exceed the range of -0.508 (-1%) and -0.519 (1%), such that the error in estimatedris kept between -3.5% and 3.6%. This sensitivity and elasticity analysis of the PPFD subfunction depicts the plausible extent of errors introduced in modeling_{s}rdue to plausible uncertainties in parameters_{s}a_{0}anda_{1}. One of the sources of uncertainties in the parameters of the PPFD subfunction may be the characteristic of hysteresis observed inrand light functions. Other potential sources of uncertainties include measurement errors and field data inadequacy to fit the PPFD-_{s}rresponse subfunction._{s}Because of the continuous effect of VPD, along with PPFD, on

r, these variables are considered the primary stomatal control factors for most crops (Kaufmann, 1982). The sensitivity and elasticity analysis of parameters_{s}a_{2}anda_{3}in the VPD subfunction of the NMJ-model are presented in figures 6c and 6d. Parametera_{2}, which was calibrated at a value of 0.036, was systematically varied from 0.00036 (-99%) to 0.072 (100%), and the resulting relative error inrranged from 0.046% to -0.048%, respectively. Parameter_{s}a_{3}(fig. 6d), which was calibrated at a value of 0.0141, was systematically varied from 0.000141 (-99%) to 0.0283 (100%), and the resulting relative error inrranged from 0.0461% to -0.0458%, respectively. Figures 6c and 6d show that the relative change in_{s}rdue to the relative change in parameters_{s}a_{2}

Figure 6. NMJ-model output of rsensitivity to the parameters of (a and b) photosynthetic photon flux density (PPFD) subfunction, (c and d) vapor pressure deficit (VPD) subfunction, (e and f) air temperature (_{s}T) subfunction, and (g) minimum stomatal resistance (_{a}r) subfunction. Squares represent elasticity, and circles represent relative variation in_{s_min}r._{s}and

a_{3}is a linear function. The linear relationships of both parameters have a percentage slope of -0.05, which is equivalent to the constant value of the elasticity function shown in the figures. Thorpe et al. (1980) and Monteith (1965) also observed a linear relationship betweenrand VPD from field measurements. As mentioned earlier, varying the parameters in a subfunction effectively varies the microclimatic variable independently, in this case VPD, thus producing a similar response function of_{s}rto the changes in the microclimatic variable. The elasticity values reported in figures 6c and 6d are negative and less than -1, which means that increases in parameters_{s}a_{2}anda_{3}result in a relatively small decrease inr. Normally, if soil water content is not limiting, which was the case in this experiment, then_{s}rresponds by decreasing (stomatal opening as the resistance to water vapor flow decreases) as VPD increases. The sensitivity and elasticity of_{s}rin response to changes in VPD subfunction parameters is not as drastic as that of PPFD; this could be due to generally gradual changes in relative humidity in the natural environment. In contrast, PPFD is usually intermittent due to atmospheric and cloudiness variations, in addition to the distribution variability of PPFD in the canopy, especially during the windy conditions. The elasticity function is a constant function at -0.05, apart from between -10% and 10% of relative change in parameters_{s}a_{2}anda_{3}. Within this range (-10% and 10%), the function asymptotically approaches the nominal value (0%) from both sides of the plot, as shown in figures 6c and 6d. In general, due to the low sensitivity ofrto relative changes in parameters_{s}a_{2}anda_{3}, uncertainties in these parameters within -25% to 25% may be acceptable, with marginal impact of about -0.011% to 0.012% onrestimates._{s}The effect of air temperature on

ris considered secondary (Kaufmann, 1982) because its effect is limited during extreme conditions. In addition, due to the strong correlation between temperature and VPD, the effect of temperature on_{s}ris implicitly imbedded in VPD, and it is difficult to determine its impact on_{s}rindependently. The sensitivity and elasticity analysis of parameters_{s}a_{4}anda_{5}in the temperature subfunction are presented in figures 6e and 6f. Parametera_{4}, which was calibrated at a value of 0.000268, was systematically varied from 0.00000268 (-99%) to 0.000535 (100%), and the resulting relative change inrranged from 66.5% to -32.9%, respectively. Parameter_{s}a_{5}, which was calibrated at a value of 613.356, was systematically varied from 6.134 (-99%) to 1226.712 (100%), and the resulting relative change inrranged from 66.5% to -32.9%, respectively. Due to the magnitude of the relative change in_{s}restimates, it appears that_{s}ris more sensitive to changes in parameters_{s}a_{4}anda_{5}than parametersa_{2}anda_{3}of the VPD subfunction. The elasticities show a negative nonlinear relationship greater than -1. This means thatrestimates are highly sensitive and dynamic to uncertainties in the parameters of the temperature subfunction, as compared with the VPD subfunction. An uncertainty of 1% change in parameters_{s}a_{4}anda_{5}resulted in a relative change of about 0.05% inr. However, beyond 10% change in parameters_{s}a_{4}anda_{5}, the change inrwas 5% or more. With this observation, the uncertainty in parameters_{s}a_{4}anda_{5}should range within -10% and 10%. With this range of uncertainty in the parameters, the errors inrare kept within 4.7% and -4.4%. Therefore, the calibration of temperature subfunction parameters should be determined with greater precision than the VPD subfunction parameters._{s}The sensitivity of estimated

rto relative changes in_{s}ris presented in figure 6g. The_{s_min}rwas a field-measured value of 22.4 s m_{s_min}^{-1}in this experiment and was systematically varied from 0.224 (-99%) to 44.8 (100%), and the resulting relative change inrranged from -0.1864% to 0.0302%, respectively. The sensitive function of relative changes in_{s}rdue to uncertainties in_{s}ris non-linear. Figure 6g shows that estimates of_{s_min}rare more sensitive to relative changes in_{s}rbelow -50%. However, above -50%, the function tapers off and becomes almost linear as_{s_min}rbecomes relatively insensitive to changes in_{s}r. Between -50% and 100% relative change in_{s_min}r, the rate of change in_{s_min}r(or% slope) is 0.03. Given the small errors introduced in estimated_{s}rover a wide range of uncertainties in_{s}r, estimated_{s_min}rappears to be generally insensitive to_{s}r. The elasticities are positive and less than 1, which is a further indication that_{s_min}ris inelastic to changes in_{s}r. This is an important observation because the_{s_min}rof any given crop is an extremely difficult value to determine in the field, which requires_{s_min}rmeasurements under a variety of weather conditions and growth stages during the entire growing season. Obtaining the_{s}rof a crop requires having many ideal conditions in place, including optimal atmospheric evaporative demand, optimal soil water content, and a healthy crop at an optimal development stage. Kelliher et al. (1995) described the optimal environmental conditions required to achieve_{s_min}rfor a vegetation surface as plentiful of soil water, adequate light, low humidity deficit, and optimal temperature. As mentioned earlier, in this experiment, the_{s_min}r(22.4 s m_{s_min}^{-1}) was the lowest measured soybeanrfrom the field measurements over the entire growing season. Although the results show that the NMJ-model is not significantly impacted by uncertainties in_{s}r, effort should be taken to determine a precise_{s_min}rto avoid cumulative uncertainties from the other calibrated parameters in the model, as discussed above._{s_min}

Summary and ConclusionsThis progression study of the new modified Jarvis model (NMJ-model), developed for maize by Irmak and Mutiibwa (2009), extends the model to soybean and presents an analysis of model performance, calibration, validation, sensitivity, and elasticity of leaf stomatal resistance (

r) estimates to uncertainties in the calibrated model parameters. The study evaluated the transferability of the original Jarvis (1976) model (J-model) and the new modified Jarvis model (NMJ-model) that were calibrated/parameterized for maize to estimate_{s}rfor soybean. The original maize-calibrated NMJ-model and J-model were not able to estimate soybean_{s}rwith a reasonable accuracy. The inherent limitation in the transferability of Jarvis-type models that are calibrated/re-parameterized for a specific crop to another crop is due to the differences in the crops’ phenomenological development and physiological functions and differences in the response of_{s}rto the same environmental variables between different crops. These differences justify the need for re-parameterization of models for specific crops for more accurate and robust_{s}restimates. The J-model and NMJ-model were re-calibrated by parameter optimization for soybean. The J-model calibrated well for soybean. However, the validation had mixed results. The model underestimated_{s}r, but had good modeling efficiency (EF = 0.78) and relatively low root mean square difference (RMSD = 43.5 s m_{s}^{-1}) between the measured and estimated soybeanr. The NMJ-model calibrated better than the J-model, resulting in a good r_{s}^{2}(0.71) and a small RMSD (13.7 s m^{-1}), slightly underestimating the measuredrwith a slope of 0.99. The NMJ-model validation on an independent dataset produced superior results. The NMJ-model was able to explain more than 80% of the variation in the measured_{s}r, with an RMSD of 38.4 s m_{s}^{-1}and EF of 0.75. The results were slightly better than the performance of the model observed over non-stressed maize canopy by Irmak and Mutiibwa (2009).These results show the robustness and practical accuracy of the NMJ-model in estimating

rover different canopies, if the model is well calibrated or re-parameterized for a specific crop. The enhanced modeling of_{s}rby the NMJ-model was, in part, attributed to the integration of the term_{s}A^{exp(1/LAI)}to account for the effect of LAI onr, especially during partial canopy (early season) and leaf aging and/or senescence (late season). Overall, the results of this study and the observations by Irmak and Mutiibwa (2009) confirmed that the NMJ-model can provide robust and accurate_{s}restimates for maize and soybean canopies and that it can be extended to other crops by field measurements of_{s}rand re-calibration/re-parameterization of the model parameters._{s}Detailed sensitivity and elasticity analyses were conducted to quantify the potential relative error introduced in

restimates due to plausible uncertainties in the NMJ-model parameters (eq. 6). The sensitivity and elasticity analysis of parameter_{s}a_{0}of the PPFD subfunction showed that therwas very sensitive to relative change in_{s}a_{0}between -99% and 50%. For higher than 50% relative change ina_{0},rbecame relatively insensitive. The elasticity function was constant between -99% and 50% relative changes in parameter_{s}a_{0}, an indication that the sensitivity ofrto_{s}a_{0}in this range is linear. The threshold for the parameter was at about 4515 (50%), beyond whichrsensitivity to changes in parameter_{s}a_{0}diminished. For accurate estimates ofr, uncertainties in parameter_{s}a_{0}should not exceed -5% and 5% to ensure that the relative error inris within -4.5% and 4.5%. Among all parameters,_{s}restimates were most sensitive to uncertainties introduced in parameter_{s}a_{1}of the PPFD subfunction. For accurate estimates ofr, uncertainties in parameter_{s}a_{1}should not exceed the range of -2% and 2%, so that the error in estimatedris kept between -3.5% and 3.6%. The sensitivity and elasticity of_{s}rin response to changes in the VPD subfunction parameters were not as high as for the PPFD subfunction parameters. Therefore, uncertainties in parameters_{s}a_{2}anda_{3}between -25% and 25% may be acceptable, with marginal impact of -0.011% to 0.012% onrestimates. The uncertainties in temperature subfunction parameters_{s}a_{4}anda_{5}had a non-linear relationship with relative change in estimatedr. The uncertainty in parameters_{s}a_{4}anda_{5}should range within -10% and 10%, and the calibration of these parameters should be determined with greater precision as compared with the VPD subfunction parameters. The sensitive function ofrrelative changes due to uncertainties in_{s}rwas both non-linear and linear in two sections. In the linear section,_{s_min}rwas relatively insensitive to uncertainties in_{s}r. In general, small errors were introduced in estimated_{s_min}rover a wide range of uncertainties in_{s}r. This is an important observation because_{s_min}ris a difficult value to determine in field conditions. The uncertainties introduced with the aforementioned parameters into the NMJ-model can be controlled or reduced, to a degree, by substantially increasing the number of stomatal resistance measurements to account for the_{s_min}rresponse to a given variable under a wide range of conditions in the model calibration/re-parameterization process._{s}## References

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