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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 ASABE 56(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; and Suat 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.

Abstract.  In a previous study by the same authors, a new modified Jarvis model (NMJ-model) was developed, calibrated, and validated to estimate stomatal resistance (rs) 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)-rs response subfunction developed from field measurements and a new physical term, Aexp(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 rs, 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 rs. Due to the differences in physiological and photosynthetic pathway differences between the two crops, the rs response to the same environmental variables, i.e., PPFD, vapor pressure deficit (VPD), and air temperature (Ta), 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 rs, with the coefficient of determination (r2) ranging from 0.30 to 0.38 and the root mean square difference (RMSD) between the estimated and measured rs 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 r2 (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 rs (RMSD = 38.4 s m-1). These results show the robustness and practical accuracy of the NMJ-model in estimating rs over different canopies if well calibrated for a specific crop. In terms of sensitivity and elasticity analyses, among all parameters, rs estimates were most sensitive to uncertainties introduced in parameter a1 of the PPFD subfunction due to its exponential impact on rs in the NMJ-model. Therefore, for accurate estimates of rs, uncertainties in parameter a1 should not exceed the range of -2% and 2% so that the error in estimated rs is kept between -3.5% and 3.6%. The study observed that the relative change in rs due to uncertainties in parameters a2 and a3 of the VPD subfunction was a linear function and less sensitive than the PPFD subfunction. The sensitivity of rs to uncertainties in temperature subfunction parameters (a4 and a5) was higher than that of VPD subfunction parameters, but less than that of PPFD subfunction parameters. The uncertainty in parameters a4 and a5 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 rs_min and the Aexp(1/LAI) terms, which were not accounted for in the original J-model, improved the model accuracy for estimating soybean rs.

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

Stomatal resistance (rs) 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 rs is 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 rs can be made using either steady-state or dynamic diffusion porometry instrumentation. Porometers measure rs by 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 rs and 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).

For years, strenuous research has advanced efforts to model rs from 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 rs phenomenologically 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).

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 rs data through extensive field measurements for a non-stressed maize canopy. The new re-parameterized NMJ-model was developed with a new physical term: Aexp(1/LAI), where A is the minimum leaf stomatal resistance (s m-1), and LAI is the leaf area index (unitless) to account for the influence of canopy development on rs, especially during the partial canopy cover stage. The vital rs response to light was modeled based on the photosynthetic photon flux density (PPFD) vs. rs response curves that were measured in the field. The NMJ-model demonstrated improved performance over the original Jarvis (1976) model (J-model) in estimating hourly rs, especially during the partial canopy cover stage. The calibration and the new physical term accounting for the variation in LAI improved the rs modeling performance. Irmak and Mutiibwa (2009) demonstrated that on a seasonal average basis, the J-model and the NMJ-model had similar performance in estimating rs, with a coefficient of determination (r2) of 0.74 and a root mean square difference (RMSD) between the modeled and measured rs of 48.8 s m-1 for the J-model, and r2 = 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 and rs_min term [rs_minexp(-LAI)] during the growing season in the NMJ-model improved the rs estimation, especially in the higher rs range (rs > 250 s m-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 r2 and 10% improvement in RMSD relative to the J-model (r2 = 0.64, RMSD = 35.5 s m-1 for the NMJ-model, and r2 = 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 on rs. Although Jarvis (1976) did not include variable LAI in his original rs model, 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.

The Jarvis-type models have proved to be practical in modeling rs; however, their empirical development is, in principal, a limitation in their transferability to estimate rs for 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 rs models’ performances for estimating soybean rs and potentially develop new models or re-calibrate/re-parameterize the maize rs model 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 rs models, 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 rs for soybean canopy; (2) recalibrate the models to estimate rs for 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 rs to 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 rs, climatic variables, plant physiology parameters, and soil water status in the 2007 soybean growing season in south central Nebraska.

Materials and Methods

Study Site

The 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 m3 m-3, permanent wilting point of 0.14 m3 m-3, and saturation point of 0.53 m3 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 Measurements

Measurements of surface energy fluxes (including latent heat flux (ETa), 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 (Zea mays L.), soybean [Glycine max (L.) Merr.], and winter wheat (Triticum aestivum L.) 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 dactyloides Nutt.); irrigated alfalfa (Medicago sativa L.); and rainfed switchgrass (Panicum virgatum) to riparian systems with invasive plant species [common reed (Phragmites australis), peach-leaf willow (Willow salix), and cottonwood (Populus deltoides var. 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 (Rn) 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 (Ta) 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-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 Measurements

The plant variables measured included rs, green LAI, and plant height (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 measure rs on 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 rs. The AP4 porometer has a resolution of 0.5 mol m-2 s-1 with an rs measurement speed of less than 5 s. For each rs measurement cycle, the following variables were recorded simultaneously: rs (s m-1), PPFD (mol m-2 s-1), chamber (cup) temperature (Tc, °C), and leaf-chamber temperature difference (TL- Tc, °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 rs measurements 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 rs data, PPFD vs. rs response 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 (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 rs as a function of multiplicative subfunctions of environmental and plant physiological variables without synergistic interaction. The subfunction variables include PPFD (µmol m-2 s-1), VPD (kPa), air temperature (Ta), soil water content (W,% vol), and maximum stomatal conductance (b1, m s-1). The model is expressed as:

(1)

(2)

(3)

(4)

(5)

where F is the subfunction that describes the stomatal response to a particular variable such that 0 < F < 1, although not always. F1 is the subfunction that describes the rs response to PPFD, q (s m-1) is the asymptotic value of stomatal conductance (g, m s-1) (1/rs) at infinite PPFD. Based on the asymptote from the PPFD-rs response curve of soybean presented by Mutiibwa and Irmak (2011), q was set to 0.0289 s m-1. The parameter b1 is the maximum g at full sunlight and is calculated from the relationship q = b0/(b1)2, where b0 is the nocturnal (night time) g. Parameter b2 (µmol m-2 s-1) is the slope of the PPFD vs. rs response curve at PPFD = 0. F2 is the subfunction that describes the rs response to VPD. The parameter b3 (Mg-1 s3) in equation 3 represents the slope of the relationship between rs and VPD. F3 is the subfunction that describes the rs response to Ta (in K). The subfunction F4 accounts for the effect of crop water stress on rs. It varies between 0.0 and 1.0 when soil water content (w2,% vol) varies between permanent wilting point (wwilt) and a critical value (wcr) at 0.75wsat (Thompson et al., 1981), where wsat represents the volumetric soil water content (% vol) at saturation. The term w2 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., w2 > wcr); therefore, F4 was taken as 1.0.

New Modified Jarvis (NMJ) Model

The 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-rs response 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 rs. The PPFD vs. rs response function replaced the F1 subfunction in the J-model (eq. 1). The NMJ-model has a new term extension that integrates the effect of LAI variation on rs during the growing season. The original J-model did not account for the LAI effect on rs. However, LAI has an important role in driving stomatal behavior. To account for the effect of seasonal variation of LAI on rs, an extension term of rs_min was raised to the inverse exponential function of LAI and incorporated into the NMJ-model, as shown in equation 6:

(6)

where PPFD (µmol m-2 s-1) is measured or estimated at the leaf level, VPD is in kPa, and Ta is in K. The parameter a0 is the slope of the PPFD vs. rs response subfunction (measured as 3010 s m-1 by Mutiibwa and Irmak, 2011), a1 is the exponent of the measured PPFD-rs response subfunction, parameters a2 and a3 represent the coefficients in the VPD subfunction, and parameters a4 and a5 represent the coefficients in the Ta subfunction. The term rs_min (s m-1) represents the lowest (minimum) measured rs during the growing season. Based on our extensive field measurements, rs_min for soybean canopy is 22.4 s m-1.

J-Model and NMJ-Model Calibration and Optimization

In 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 rs for soybean in the 2007 growing season. The performances of the models were evaluated using RMSD, r2, and modeling efficiency (unitless), which is expressed as:

(7)

where Oi and Pi are the observed and predicted rs, 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 rs. 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 rs measurements 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 rs by minimizing RMSD and maximizing r2. 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 r2. The RMSD evaluates the accuracy of the optimized model by measuring the deviation of the model estimates from the measured rs. During the optimization process, a constraint was added by holding the regression slope of the estimates on measured rs between 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 rs and compare the model-estimated results to measured soybean rs using the validation dataset. The validation performance statistics are presented in table 3 for the model calibration and validation.

Table 1. Daily average meteorological variables measured during the 2007 growing season when stomatal resistance (rs) measurements were made. Variables include incoming shortwave radiation (Rs), net radiation (Rn), maximum and minimum air temperature (Ta_max and Ta_min), wind speed at 3 m (u3), maximum and minimum relative humidity (RHmax and RHmin), vapor pressure deficit (VPD), and rainfall.
Date
(2007)
Meteorological Variable
Rs
(W m-2)
Rn
(W m-2)
Ta_max
(°C)
Ta_min
(°C)
RHmax
(%)
RHmin
(%)
u3
(m s-1)
VPD
(kPa)
Rainfall
(mm)
Calibration16 July30019631.516.299.155.72.42.30
20 July22314328.019.495.275.63.42.54.6
26 July29319232.819.095.048.03.02.30
31 Aug.24815727.013.4100.063.22.32.00
Validation24 July29419330.420.8100.062.32.32.70
9 Aug.29019532.519.6100.060.41.92.90.25
14 Aug.28018135.220.482.239.92.22.20
12 Sept.23713826.008.498.041.83.71.30

Sensitivity and Elasticity Analysis

The 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 rs originating 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 rs; 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 rs estimates.

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 rs by 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 rs to 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 rs relative error (%) was averaged over the calibration days. To determine the important parameters that cause the most variations in estimated rs, the elasticities, which are measures of the percent change in a dependent variable (rs) divided by the percent change in an independent variable (parameter), were calculated using equation 8:

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 (u3), maximum and minimum air temperature (Ta_max and Ta_min), relative humidity (RH), incoming shortwave radiation (Rs), and total rainfall.
PeriodMeteorological VariableMayJuneJulyAugustSeptemberOctober
2007u3 (m s-1)5.03.92.83.03.63.8
Ta_max (°C)23.527.129.329.525.119.6
Ta_min (°C)12.515.418.519.011.56.7
RH (%)71.672.977.881.373.371.5
Rs (MJ m-2 d-1)19.922.922.018.316.411.8
Rainfall (mm)1305310611767149
Long-term
(32-year)
average
u3 (m s-1)4.03.52.92.63.13.3
Ta_max (°C)22.528.130.329.225.318.3
Ta_min (°C)9.314.617.316.310.73.6
RH (%)71.370.273.274.568.867.2
Rs (MJ m-2 d-1)19.422.422.419.715.911.3
Rainfall (mm)11211093836345

(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 of rs is elastic or inelastic. An elastic response is one in which rs responds highly to small changes in the parameter, whereas an inelastic response is one in which rs does not respond much to the changes in the parameter.

Results and Discussion

Weather Conditions During Research Period

A 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). Ta_max was very close to the average, but on average Ta_min was approximately 16% higher than normal. The seasonal average relative humidity was about 5% more than long-term average.

Seasonal Trend of Measured Soybean Stomatal Resistance

Figure 2 presents the seasonal trend in measured hourly soybean rs and daily soybean LAI. For each rs measurement day, there are several hours of data points showing the daily range of rs. For example, on July 20, the measured rs ranged from 108 s m-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 and rs. Although partly obscured by the hourly fluctuations on measurement days, the rs trend depicts the theoretical parabolic variation of rs that was also observed by Monteith (1965), Monteith et al. (1965), and Irmak and Mutiibwa (2009). The rs exhibited a decreasing trend from early season toward mid-season and remained relatively constant, with later increases toward the end of the growing season. The rs values ranged from a minimum of 22.4 s m-1 to a maximum of 149 s m-1 with a seasonal average of 63 s m-1. The seasonal minimum rs (22.4 s m-1) was measured on July 26 at 4:00 p.m. The microclimatic conditions at that time were characteristic of high atmospheric evaporative demand; Rn was 577 W m-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 maximum rs value (149 s m-1) was measured on September 12 at 12:00 p.m. This was during the late growing season, and the high rs is most likely due to leaf aging and senescence. On September 12, the microclimatic conditions were Rn = 429 W m-2, air temperature = 23.5°C, wind speed = 5.1 m s-1, and VPD = 1.0 kPa. The spike in rs on 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 rs value of 149 s m-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 measure rs, and combination of all these factors.

Figure 2. Seasonal pattern of measured hourly stomatal resistance (rs) and green LAI for soybean during the 2007 growing season.

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, rs decreased to its lowest seasonal values. After late August, due to senescence and increased rs of aged leaves, rs increased rapidly as LAI also sharply decreased. Monteith et al. (1965) observed a similar seasonal trend in rs in 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 rs toward 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.

Transferability of J-Model and NMJ-Model from Maize to Soybean

In this section, the transferability of the J-model and the NMJ-model developed for maize to estimate rs for 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 rs with r2 = 0.74, RMSD = 48.8 s m-1 for the J-model, and r2 = 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 estimate rs for soybean in a different year. Figures 3a and 3b show scatter plots of the J-model and NMJ-model estimates against measured rs. Figure 3a shows that the maize-calibrated J-model substantially overestimated rs for soybean. The model r2 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 soybean rs with a low r2 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 estimating rs for 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).

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 of rs of 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.

Figure 3. Relationship between maize-calibrated stomatal resistance (rs) models used to estimate soybean rs: (a) Jarvis (1976) model (J-model), and (b) new modified Jarvis model (NMJ-model).

Calibration and Validation of Stomatal Resistance Models for Soybean

After assessing the performance of the maize-calibrated J-model and NMJ-model for estimating rs for 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., b2 and b3 coefficients), and others were substantially different (i.e., b1, b4, and b5 coefficients). The optimization of the J-model (fig. 4a) resulted in a good r2 of 0.63 and a small RMSD of 13.3 s m-1 (table 3). The model underestimated rs by 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 rs and was seemingly insensitive to changes in measured soybean rs with r2 = 0.09, RMSD = 43.5 s m-1, and EF = 0.78. Although the model r2 is low, the good EF and RMSD statistics indicate that the J-model was still able to estimate the trends in soybean rs, 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 rs of 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.

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 a2, a3, and a5 are substantially different. The NMJ-model calibrated better than the J-model, resulting in a higher r2 of 0.71, a smaller RMSD of 13.7 s m-1, and EF of 0.59. Given that the lowest measured rs was 22.4 s m-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 measured rs of soybean. The calibrated model slightly underestimated measured rs with a slope of 0.99 (fig. 4b).

Table 3. Optimized values of parameters used in equations 2, 3, 4, and 6 to estimate soybean stomatal resistance (rs) 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 rs, and EF = modeling efficiency.
Model and Parameters
after Optimization
CalibrationValidation
RMSD
(s m-1)
r2EFRMSD
(s m-1)
r2EF
J-model13.30.630.2243.50.090.78
b1 = 3.7933
b2 = 1.88 × 10-5
b3 = 0.036
b4 = 0.1061
b5 = 0.00325
b6 = 1.320
NMJ-model13.70.710.5938.40.830.75
a0 = 3010
a1 = 0.514
a2 = 0.000268
a3 = 613.4
a4 = 0.036
a5 = 0.014
Figure 4. Relationship between measured vs. estimated stomatal resistance (rs) for soybean canopy during calibration: (a) Jarvis (1976) model (J-model), and (b) new modified Jarvis model (NMJ-model).
Figure 5. Relationship between measured vs. estimated stomatal resistance (rs) 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).

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 r2 = 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 soybean rs. This is a robust performance for the model, even slightly better than the model performance over maize canopy (r2 = 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 measured rs for soybean canopy are considered (fig. 2), as well as the extremely difficult nature of modeling rs with 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 rs. 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 r2 and RMSD values demonstrate the consistency of the model’s good performance in estimating rs for different canopies if well calibrated for a specific crop.

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 rs, our validation results, as well as those observed by Irmak and Mutiibwa (2009), demonstrate the practical accuracy of estimating rs using 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 rs and re-calibration/re-parameterization of the model parameters. The enhanced modeling of rs by 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 rs during the growing season. The impact of the added LAI and rs_min term [rs_minexp(-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 rs occurred during the partial canopy phase (LAI < 3) of the growing season and late in the season during the leaf aging and leaf senescence stage.

Sensitivity Analysis of NMJ-Model

Owing 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 rs. Figures 6a through 6g show the effect of changes in the calibrated parameters of the NMJ-model on the rs estimates. 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 rs. Therefore, the response function of rs due to changes in the parameters of a given subfunction of a climatic variable is likely to be similar to changes in that climatic variable.

The sensitivity and elasticity analyses results on parameters a0 and a1 of the PPFD subfunction in the NMJ-model are presented in figures 6a and 6b. Parameter a0 (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 (%) in rs was very sensitive to relative changes (%) to a0 between 30.11 (-99%) and 4515 (50%). For greater than 50% change in a0, rs estimates became relatively insensitive. The elasticity function is constant between 30.11 (-99%) and 4515 (50%) change in parameter a0, an indication that the sensitivity of rs to a0 in this range is a linear function. The threshold is at about 4515 (50%), beyond which the sensitivity of rs to changes in parameter a0 diminishes. This could be the point at which PPFD becomes non-limiting in driving rs with 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 rs estimates using the NMJ-model. The relative error induced in rs is between -96.7% and 48.9%. Therefore for accurate estimates of rs, uncertainties in parameter a0 should not exceed -5% and 5%, in which case the relative error in rs will be within 4.5%.

The sensitivity and elasticity of analysis of rs with respect to parameter a1 are presented in figure 6b. Parameter a1, which was calibrated at a value of -0.514, was varied from -0.508 (-99%) to -1.027 (100%). Among all parameters, rs was most sensitive to relative changes in parameter a1, 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). The rs estimate was still, but moderately less, sensitive to changes in a1 above -10%. Similar to parameter a0, this could be due to PPFD becoming non-limiting, with the stomata already fully open. The elasticities show that the sensitivity function of rs to a1 is an exponential decay function. From the elasticity function, it also appears that rs becomes asymptotically insensitive to changes in a1 beyond 0.488 (-10%). In other words, rs is 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 rs to a1, the parameter uncertainty should not exceed the range of -0.508 (-1%) and -0.519 (1%), such that the error in estimated rs is 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 rs due to plausible uncertainties in parameters a0 and a1. One of the sources of uncertainties in the parameters of the PPFD subfunction may be the characteristic of hysteresis observed in rs and light functions. Other potential sources of uncertainties include measurement errors and field data inadequacy to fit the PPFD-rs response subfunction.

Because of the continuous effect of VPD, along with PPFD, on rs, these variables are considered the primary stomatal control factors for most crops (Kaufmann, 1982). The sensitivity and elasticity analysis of parameters a2 and a3 in the VPD subfunction of the NMJ-model are presented in figures 6c and 6d. Parameter a2, 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 in rs ranged from 0.046% to -0.048%, respectively. Parameter a3 (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 in rs ranged from 0.0461% to -0.0458%, respectively. Figures 6c and 6d show that the relative change in rs due to the relative change in parameters a2

Figure 6. NMJ-model output of rs sensitivity 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 (Ta) subfunction, and (g) minimum stomatal resistance (rs_min) subfunction. Squares represent elasticity, and circles represent relative variation in rs.

and a3 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 between rs and 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 rs to 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 a2 and a3 result in a relatively small decrease in rs. Normally, if soil water content is not limiting, which was the case in this experiment, then rs responds by decreasing (stomatal opening as the resistance to water vapor flow decreases) as VPD increases. The sensitivity and elasticity of rs in 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 a2 and a3. 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 of rs to relative changes in parameters a2 and a3, uncertainties in these parameters within -25% to 25% may be acceptable, with marginal impact of about -0.011% to 0.012% on rs estimates.

The effect of air temperature on rs is 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 rs is implicitly imbedded in VPD, and it is difficult to determine its impact on rs independently. The sensitivity and elasticity analysis of parameters a4 and a5 in the temperature subfunction are presented in figures 6e and 6f. Parameter a4, 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 in rs ranged from 66.5% to -32.9%, respectively. Parameter a5, 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 in rs ranged from 66.5% to -32.9%, respectively. Due to the magnitude of the relative change in rs estimates, it appears that rs is more sensitive to changes in parameters a4 and a5 than parameters a2 and a3 of the VPD subfunction. The elasticities show a negative nonlinear relationship greater than -1. This means that rs estimates 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 a4 and a5 resulted in a relative change of about 0.05% in rs. However, beyond 10% change in parameters a4 and a5, the change in rs was 5% or more. With this observation, the uncertainty in parameters a4 and a5 should range within -10% and 10%. With this range of uncertainty in the parameters, the errors in rs are 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.

The sensitivity of estimated rs to relative changes in rs_min is presented in figure 6g. The rs_min was a field-measured value of 22.4 s m-1 in this experiment and was systematically varied from 0.224 (-99%) to 44.8 (100%), and the resulting relative change in rs ranged from -0.1864% to 0.0302%, respectively. The sensitive function of relative changes in rs due to uncertainties in rs_min is non-linear. Figure 6g shows that estimates of rs are more sensitive to relative changes in rs_minbelow -50%. However, above -50%, the function tapers off and becomes almost linear as rs becomes relatively insensitive to changes in rs_min. Between -50% and 100% relative change in rs_min, the rate of change in rs (or% slope) is 0.03. Given the small errors introduced in estimated rs over a wide range of uncertainties in rs_min, estimated rs appears to be generally insensitive to rs_min. The elasticities are positive and less than 1, which is a further indication that rs is inelastic to changes in rs_min. This is an important observation because the rs_min of any given crop is an extremely difficult value to determine in the field, which requires rs measurements under a variety of weather conditions and growth stages during the entire growing season. Obtaining the rs_min of 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 rs_min for a vegetation surface as plentiful of soil water, adequate light, low humidity deficit, and optimal temperature. As mentioned earlier, in this experiment, the rs_min(22.4 s m-1) was the lowest measured soybean rs from the field measurements over the entire growing season. Although the results show that the NMJ-model is not significantly impacted by uncertainties in rs_min, effort should be taken to determine a precise rs_min to avoid cumulative uncertainties from the other calibrated parameters in the model, as discussed above.

Summary and Conclusions

This 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 (rs) 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 rs for soybean. The original maize-calibrated NMJ-model and J-model were not able to estimate soybean rs with 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 rs to 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 rs estimates. 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 rs, but had good modeling efficiency (EF = 0.78) and relatively low root mean square difference (RMSD = 43.5 s m-1) between the measured and estimated soybean rs. The NMJ-model calibrated better than the J-model, resulting in a good r2 (0.71) and a small RMSD (13.7 s m-1), slightly underestimating the measured rs with 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 rs, with an RMSD of 38.4 s m-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 rs over different canopies, if the model is well calibrated or re-parameterized for a specific crop. The enhanced modeling of rs by the NMJ-model was, in part, attributed to the integration of the term Aexp(1/LAI) to account for the effect of LAI on rs, 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 rs estimates for maize and soybean canopies and that it can be extended to other crops by field measurements of rs and re-calibration/re-parameterization of the model parameters.

Detailed sensitivity and elasticity analyses were conducted to quantify the potential relative error introduced in rs estimates due to plausible uncertainties in the NMJ-model parameters (eq. 6). The sensitivity and elasticity analysis of parameter a0 of the PPFD subfunction showed that the rs was very sensitive to relative change in a0 between -99% and 50%. For higher than 50% relative change in a0, rs became relatively insensitive. The elasticity function was constant between -99% and 50% relative changes in parameter a0, an indication that the sensitivity of rs to a0 in this range is linear. The threshold for the parameter was at about 4515 (50%), beyond which rs sensitivity to changes in parameter a0 diminished. For accurate estimates of rs, uncertainties in parameter a0 should not exceed -5% and 5% to ensure that the relative error in rs is within -4.5% and 4.5%. Among all parameters, rs estimates were most sensitive to uncertainties introduced in parameter a1 of the PPFD subfunction. For accurate estimates of rs, uncertainties in parameter a1 should not exceed the range of -2% and 2%, so that the error in estimated rs is kept between -3.5% and 3.6%. The sensitivity and elasticity of rs in response to changes in the VPD subfunction parameters were not as high as for the PPFD subfunction parameters. Therefore, uncertainties in parameters a2 and a3 between -25% and 25% may be acceptable, with marginal impact of -0.011% to 0.012% on rs estimates. The uncertainties in temperature subfunction parameters a4 and a5 had a non-linear relationship with relative change in estimated rs. The uncertainty in parameters a4 and a5 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 of rs relative changes due to uncertainties in rs_min was both non-linear and linear in two sections. In the linear section, rs was relatively insensitive to uncertainties in rs_min. In general, small errors were introduced in estimated rs over a wide range of uncertainties in rs_min. This is an important observation because rs_min is 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 rs response to a given variable under a wide range of conditions in the model calibration/re-parameterization process.

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