Short survey
Scaling-up crop models for climate variability
applications
$
J.W. Hansen
a,*, J.W. Jones
baInternational Research Institute for Climate Prediction, PO Box 1000, Palisades, NY 10964-8000, USA
bAgricultural and Biological Engineering Department, University of Florida, PO Box 110570, Gainesville, FL 32611-0570, USA
Received 9 February 2000; received in revised form 2 June 2000; accepted 6 June 2000
Abstract
Although most dynamic crop models have been developed and tested for the scale of a homogeneous plot, applications related to climate variability are often at broader spatial scales that can incorporate considerable heterogeneity. This study reviews issues and approa-ches related to applying crop models at scales larger than the plot. Perfect aggregate predic-tion at larger scales requires perfect integrapredic-tion of a perfect model across the range of variability of perfect input data. Aggregation error results from imperfect integration of het-erogeneous inputs, and includes distortions of either spatial mean values of predictions or year-to-year variability of the spatial means. Approaches for reducing aggregation error include sampling input variability in geographic or probability space, and calibration of model inputs or outputs. Implications of scale and spatial interactions for model structure and complexity are a matter of ongoing debate. Large-scale crop model applications must address limitations of soil, weather and management data. Distortion of weather sequences from spatial averaging is a particular danger. A case study of soybean in the state of Georgia, USA, illustrates several crop model scaling approaches. # 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Aggregation; GIS; Simulation; Yield forecasting; Calibration
0308-521X/00/$ - see front matter#2000 Elsevier Science Ltd. All rights reserved. P I I : S 0 3 0 8 - 5 2 1 X ( 0 0 ) 0 0 0 2 5 - 1
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$ Florida Agricultural Experiment Station, Journal Series No. R-07058.
* Correspopnding author. Tel.: +1-914-680-4410; fax: +1-914-680-4864.
Nomenclature.De®nitions of variable and subscript symbols
Variables
A Area of the region of interest
AWHC Plant-available water-holding capacity in a soil pro®le or unit soil depth
e,e, e- Particular input, vector of inputs at a point, and spatial average vector of model inputs, respectively
e,e Calibrated, eective input value and vector, respectively, of
cali-brated inputs
f(.) Model of response (e.g. crop yield) to the environment
g(.) Univariate or multivariate probability density function
p,p Proportion(s) of the area of interest in an individual and vector, respectively, of crops or land uses
R Spatial region
V Variance±covariance matrix x,z Horizontal spatial coordinates
yt,i, y-, Y- Crop yield or other response predicted at pointiin yeart, predicted
spatial average in yeart, and predicted average over space and time, respectively
a,b Intercept and slope of a least-squares linear regression y Volumetric soil water content
g Calibration multiplier for AWHC r Crop stand density
s,s Actual and predicted standard deviation, respectively, among years of actual and predicted spatial average yields
t,i,t;i;U Actual crop yield or other response at pointiin yeart, averaged over
space in yeart, and averaged over space and time, respectively Subscripts
m,i Number of spatial points or units, and an individual spatial point or unit
n,t Number of years, and an individual year
L, U, S Indicators of lower, drained upper, and saturated critical values ofy, respectively
T A time trend
1. Introduction
Dynamic, process-level crop models are playing an increasing role in translating information about climate variability into predictions and recommendations at a range of scales, tailored to the needs of agricultural decision makers. Although such models have generally been developed and tested at the scale of a homogeneous plot, decision makers often need information at broader spatial scales where the assumption of a homogeneous environment does not hold, and at higher system levels where dierent constraints operate. Growing interest in precision agriculture has brought awareness that farmers deal with spatial heterogeneity even at a ®eld scale. Policy makers are often concerned with climate impacts at district, watershed, national or broader scales. Crop model applications related to climate prediction depend critically on the assumption that the models can capture the year-to-year pattern of response to climate variability.
In this paper, we review methods for scaling up crop model predictions, and summarize challenges that arise when applying dynamic crop models developed for the plot level to broader spatial scales and higher system levels. A case study of soybean in Georgia, USA, illustrates several of these approaches. Our focus is on applications for which response to interannual climate variability is important.
2. The spatial aggregation problem
2.1. Variability in space and time
Crops are produced in an environment that varies both in space and in time. Scaling up entails applying models that assume a homogeneous environment (i.e. a point in space) to larger areas that can encompass a considerable range of spatial variability. Inputs to crop simulation models typically include daily weather, soil properties (including topography and initial conditions) and management (including cultivar characteristics); collectively these inputs de®ne the environment of the modeled system. All three types of environment inputs vary spatially. The spatial heterogeneity of a given environment input can be represented by its distribution in either geographic (e.g. in a geographical information system [GIS]) or probability (e.g. as a probability distribution) space (Band et al., 1991; King, 1991). Data col-lection methods or formats of existing data bases are likely to dictate the method of representing heterogeneity.
exhibit systematic spatial trends (i.e. aresecond-order stationary), then the variogram will reach a maximum (thesill) at a ®nite lag (therange). The range marks the limits of spatial dependence and, importantly, of spatial interpolation. The variogram often approaches the ordinate at some positive variance (thenugget) that represents small-scale, spatially independent random variation. Characteristics of a variogram depend on sample area, the minimum and maximum lag sampled, and sometimes the direction of sampling. The variogram is the basis for a family of optimal point and areal interpolation techniques known collectively askriging.
Crop yields at a particular point in space vary from season to season primarily because of the temporal variability of weather. Because of interactions between plants, soil and weather, spatial patterns of yields can also vary between years. Some aspects of variability are analogous in space and time (e.g. means, trends, auto-correlated and purely random variability). However, variability in time occurs in a single dimension, and is characterized by directional dependence; current variability can be in¯uenced by past, but not future variability.
Possible interactions of the spatial and temporal dimensions of crop model appli-cations can complicate notation, statistical description and interpretation. For example, root mean squared error (RMSE) of crop yield predictions (y) could be calculated based on deviations from observed yields () accumulated over time represented bynyears:
RMSEtime nÿ1 Xn
t1
yt;ÿt;;21=2; 1
over space represented bym®elds or measurement plots:
RMSEspace mÿ1 Xm
i1
y;iÿ;i21=2; 2
or by a combination of time and space:
RMSEcombined m nÿ1 Xn
t1
Xm
i1
yi;jÿi;j21=2: 3
2.2. Perfect aggregation
``Perfect aggregation'' (Iwasa et al., 1987) can be represented as integration of some response (e.g. crop yield) across the range of variability of the environment within some given region. Aggregation of an extensive variable (e.g. crop production) yields a spatial total, whereas aggregation of an intensive variable (e.g. crop yield) results in a spatial average. Although the discussion below focuses on crop yields, it is applicable to any model response variable.
Let the function t=f(et) represent actual crop-yield response in year tto a
par-ticular set of environmental inputs (i.e. weather, soil and management conditions) represented byet. Becauseetvaries over (x,z) space (et et x; z)),f(et) also varies
spatially. The average yieldtover a two-dimensional regionRtin a given yeartcan
be obtained by dividing the aggregate yield, integrated over (x,z) space, by the area Atof the region (adapted from King, 1991):
t Aÿt1
Rt
f et x;zdxdz: 4
Assuming a perfect modelf(et) and perfect characterization ofet(x,z), integration of
Eq. (4) will give the true spatial mean yield. We discuss the implications of model and input errors later.
Alternatively, the variability of etcan be expressed as a multivariate probability
distribution represented by the density function,g(et). Average response can then be
expressed as a statistical expectation (adapted from Rastetter et al., 1992):
t Ef et
et
f et g et det: 5
The probabilistic formulation (Eq. (5)) is equivalent to the spatial formulation (Eq. (4)) only iff(et) is a spatially independent process, meaning that interactions between
neighboring spatial units do not alter response (Band et al., 1991). We will discuss violations of this assumption later.
2.3. Bias of spatial average
Simulation studies have often inferred regional crop response to climate vari-ability based on one or a few ``representative locations''. ``Representative'' suggests that the environmentetat that location approximates the average environmentet of
the region or subregion of interest. Even if the locations are truly representative in this sense, yields simulated at representative locations will not generally represent either the spatial average or the interannual variability of regional yields because of aggregation error.
ytf et ; 6
where:
etAÿ1
R
et x;zdxdz; 7
andyt is an estimate oft. Ifetis heterogeneous andf(et) is either concave or
con-vex through the range of variability of et, then yt will be a biased estimate oft.
The degree of bias (ytÿt) is a function of the distribution ofetand the curvature
off(et).
The problem is easy to visualize for a simple case where heterogeneity of the environment is represented by two equally probable scalar values,e1ande2. Fig. 1
shows rainfed soybean (`Bragg') yield response,f , to stand density simulated by CROPGRO V.3.5 (Boote et al., 1998) using 1995 weather data for Gainesville, FL, USA, parameters for a Millhopper ®ne sand, and typical planting date (15 June) and row spacing (91 cm). To keep the example simple, we assume that stand density matched a target of 22 mÿ2(
2) on one-half of an otherwise homogeneous ®eld, but
was only 2 mÿ2(
1) on the other half due to poor germination, giving a mean
den-sity () of 12 mÿ2. The `true' mean yield () for the ®eld is the mean of yield
responses to the high (f(22)=2.76 Mg haÿ1) and low stand densities (f(2)=1.72
Mg haÿ1), or 2.24 Mg haÿ1. If we ignored stand heterogeneity and simulated yield
response to average stand density of 12 mÿ2 to obtain 2.59 Mg haÿ1, we would
underestimate by 13%. Such use of response to mean inputs to estimate mean response to heterogeneous inputs is sometimes called the ``fallacy of the averages'' (Templeton and Lawlor, 1981).
2.4. Bias of temporal variability
A second and often neglected result of using one or a few representative points to model response to a spatially heterogeneous environment is a tendency to over-estimate interannual variability. We can envision a region as having a ®nite number (m) of spatially distributed time series of annual yields for, for example, individual plants or ®elds. The interannual standard deviation of the regional average yield can be expressed in matrix form as:
pTVp1=2 8
(Helstrom, 1991), wherepis a vector of fractions of the area occupied by each of the mtime series, andVis themmvariance±covariance matrix of the individual ®elds or plants. For the simple case of two spatially segregated time series, 1 and, 2,
occurring in proportions p1and p2, with standard deviation 1 and 2 and
cross-correlationr12, the mean standard deviation of the individual time series:
sp11p22;
will overestimate the standard deviation of the aggregate time series:
ÿp2121p22222r12p1p212 1=2
;
unless,1and,2are perfectly correlated (i.e.r12=1; van Noordwijk et al., 1994).
As a simple, hypothetical illustration of the problem, we assume that the `true' average yield of maize in North Florida, USA, in any given year is the simple aver-age (i.e. p1=p2=0.5) of yields simulated with weather data from Lake City and
Ocala (Fig. 2). CERES-Maize V.3.5 (Ritchie et al., 1998) simulated 1976±1995 yields using observed precipitation and temperature data (EarthInfo, 1996), and solar irradiance generated stochastically (Hansen, 1999) using parameters calculated from
observations at Jacksonville, FL (NREL, 1992). The soil (Millhopper ®ne sand), cultivar (`McCurdy 84aa'), planting date (15 April) and planting arrangement (8 mÿ2
in 50 cm rows) are consistent with conditions and practices within the region. Simulated yields showed a slight negative correlation (r=ÿ0.109) between the two locations. Although the `true' regional mean through time (7.55 Mg haÿ1) is simply
the average of the means simulated for Lake City (7.61 Mg haÿ1) and Ocala (7.49
Mg haÿ1), the standard deviations simulated for either Lake City (1.35 Mg haÿ1) or
Ocala (1.81 Mg haÿ1) seriously overestimate the `true' standard deviation for the
region (1.07 Mg haÿ1; Fig. 3).
Many regional crop-simulation studies show a tendency to over predict observed interannual yield variability (Mearns et al., 1992; Rosenberg et al., 1992; Moen et al., 1994; Meinke and Hammer, 1995; Chipanshi et al., 1998; Rosenthal et al., 1998). Some (Mearns et al.; Meinke and Hammer; Rosenthal et al.) have attributed this
bias to aggregation error, while others (Chipanshi et al.) have explained it in terms of model errors, such as excessive sensitivity to water de®cit in dry years and pest and disease eects in wet years, that the models do not capture.
2.5. Emergent properties and processes
Moving from a homogeneous plot to a ®eld, farm, regional landscape or the bio-sphere involves more than incorporation of additional environmental heterogeneity associated with increasing spatial scale. Each of these represents a level in the hier-archy of agroecosystems. New properties and processes emerge at each system level as a result of new components (e.g. human and economic subsystems) or interac-tions among neighboring components of the system (e.g. intercrop competition). Interactions among neighboring components can violate the validity of the prob-abilistic representation of the aggregation problem (Eq. (5)). Lateral ¯ow of water, solutes and sediment emerges as a potential determinant of crop performance at the ®eld level. Interactions among intercropped species can also be important within a ®eld. Farm resource allocation, and human goals and decisions constrain crop production at a farm scale. Water allocation and competing land uses are con-straints that emerge at regional scales. Hierarchy theory (O'Neill et al., 1986; MuÈller, 1992) and Eq. (8) predict that agricultural systems at increasing scale should become less sensitive to high-frequency disturbances (e.g. interannual climate variability) in favor of lower-frequency signals (e.g. long-term climate change).
3. Aggregation approaches
Our understanding of the nature and sources of error associated with increas-ing spatial scale suggests several potential approaches for controllincreas-ing or minimizincreas-ing the eects of those errors (King, 1991; Luxmoore et al., 1991; Rastetter et al., 1992). These approaches fall under the broad categories of input sampling and calibration.
3.1. Input sampling
contribute to aggregation errors. Analyses of the linearity of response to the expec-ted range of input values (Rastetter et al., 1992) and sensitivity of mean response to the variance of inputs (Addiscott, 1993) provide an idea of the potential for aggre-gation bias due to heterogeneity of particular inputs.
3.1.1. Sampling in geographic space
Spatial dependence of variability of regionalized variables is the basis for parti-tioning a region into smaller, relatively homogeneous spatial units. Validity of the common assumption that variability within spatial units is negligible (Burke et al., 1991; Haskett et al., 1995b; de Jager et al., 1998), and therefore the utility of spatial partitioning, depends on the nugget variance and the range of spatial dependence of the particular variable relative to the size of the spatial partitions. Spatial patterns of inputs can account for much of the dependence between jointly distributed input variables. Resulting reduction of aggregation error will depend on the proportion of variability that the spatial partitions account for, and on the accuracy of the esti-mates of mean values of inputs within each unit.
GIS automate the management, analysis and display of spatial information. Vector-based GIS partition the environment into polygons of arbitrary shape representing, for example, soil map units, crop reporting districts or nearest weather station theissen polygons. Although boundaries of dierent input types (e.g. soil maps, crop reporting districts, weather station theissen polygons) generally do not coincide, GIS support overlaying of polygons of dierent input variables to create new polygons of unique input combinations. Raster-based GIS partition the envir-onment into regularly shaped and sized cells based more on convenience than on natural boundaries. The one-to-one spatial correspondence of dierent variables in a raster-based GIS simpli®es analyses. Input data formats or sampling methods (e.g. combine yield monitors, remotely sensed land use and climate data, output from dynamic atmospheric models) sometimes favor raster representation. Regional applications of crop models have used both vector (van Lanen et al., 1992; Thornton et al., 1995; Rosenthal et al., 1998) and raster (Carbone et al., 1996; Thornton et al., 1997b; de Jager et al., 1998) GIS for managing soil and weather inputs, automating spatial averaging and visualizing spatial patterns of results. The widespread applic-ability and bene®ts of GIS have prompted development of generic tools that link crop models and GIS packages (reviewed by Hartkamp et al., 1999).
3.1.2. Sampling in probability space
input distributions can require large numbers of simulation runs to achieve a given level of con®dence in output distribution parameters.
Latin hypercube sampling (McKay et al., 1979; Stein, 1987) oers a more ecient alternative to independent random sampling. Each of k input distributions is strati®ed intolequal-probability classes, which are sampled without replacement and combined randomly into l unique input vectors, requiring only l simulation runs compared tolkruns for the same intensity of independent sampling with replacement.
Methods are available to either eliminate spurious correlation among independent variables or to impose correlation among jointly distributed variables (Iman and Conover, 1982; Owen, 1994). Latin hypercube sampling has been applied to char-acterizing uncertainty of simulated crop yield and NO3ÿ-leaching (Bouma et al., 1996),
and has been proposed as a means of spatially aggregating crop and forestry yield simulations under spatial heterogeneity (Luxmoore, 1988; Luxmoore et al., 1991).
3.2. Regional calibration
Crop model predictions usually bene®t from local calibration. As discussed pre-viously, even a perfect model that is calibrated at a plot scale will yield biased aggregate predictions if the heterogeneity of the environment is not adequately characterized and sampled. If historic data are available for the response variable and region of interest, biases can be characterized and corrected through calibra-tion of model inputs or outputs. Calibracalibra-tion can correct for multiple and hidden sources of error (Rastetter et al., 1992). However, calibration precludes predictive validation using the same data set (Addiscott, 1993). A common solution is to divide the data, calibrate with one subset, then validate with a dierent subset of observed data (Power, 1993).
3.2.1. Calibration of model inputs
The planting density response example that we used to illustrate the spatial aver-aging problem (Fig. 1) can also illustrate how input calibration can correct mean aggregation bias. Observed mean aggregate yield response U replaces the model-derived mean aggregate response. In this simple case, calibration represents the entire region of interest with a single, derived value,=fÿ1(U), that has no direct
relationship to any measured values of . Using the same assumptions and the simulated aggregate yield of 2.24 Mg haÿ1 implies an eective stand density
(=6.01 mÿ2) that falls between the low (
1=2 mÿ2) and mean (=12 mÿ2) stand
densities. Simulated response to the single eective stand density will then match the aggregate response to the heterogeneous densities observed in the ®eld. Calibration should involve the mean of several years of observed yields for the region of interest. For applications involving climate variability, eective e for multiple input
vari-ables can be obtained by minimizing interannual prediction error (e.g. RMSE) using a nonlinear optimization algorithm.
argued that the use of a single eective value to represent a heterogeneous parameter is likely to invalidate both the physical interpretation of the parameter and the structure of the ®ne-scale model.
3.2.2. Calibration of model outputs
Systematic prediction errors are easily corrected by calibration of model outputs. The correction factor approach corrects mean bias by multiplying each yield pre-diction byU=Y(Haskett et al., 1995b; Russell and van Gardingen, 1997). Although the multiplicative adjustment results in a proportional change in the predicted stan-dard deviation, it does not attempt to correct interannual variability. Simulated yields adjusted by a least-squares linear correction:
yC;t yt 9
(Kunkel and Hollinger, 1991; Rosenthal et al., 1998) minimizes squared prediction error (e.g. RMSE) by removing its systematic component, leaving only unsystematic or random error. However, the standard deviation of the corrected series will generally be lower than that of the observed time series. An alternative linear correction:
yC;t tÿYs=U; 10
reproduces the mean and standard deviation of the observed series, but with higher prediction error. This correction may be preferable to Eq. (9) for risk studies where preserving interannual variability is more important than minimizing prediction error.
Historic crop yield data often display time trends in central tendency and some-times interannual variability. Time trends are usually attributed to changes in tech-nology or land-use patterns. The most frequent way to deal with yield time trends is to derive a trend,yT, t, as a parametric (Swanson and Nyankori, 1979; Carlson et
al., 1996; Mjelde and Keplinger, 1998) or smoothing (Nicholls, 1985; Hansen et al., 1998) function of the observed time series, t. Smoothing techniques separate the
relatively high-frequency response to weather variability from the lower-frequency response to technology and other factors (Hansen et al.). If appears stationary, then,tcan be detrended to a yearbbasis by an additive adjustment:
yC;t tyT;bÿyT;t: 11
Ifchanges in proportion toyT, a multiplicative adjustment:
yC;t t yT;b=yT;t; 12
Although weather is often regarded as stationary, changes in land use and crop production have been linked to time trends in precipitation (Viglizzo et al., 1995) and temperatures (Bell and Fischer, 1994). Crop models will presumably capture the eects of such climatic trends. The dierence or ratio of mean observed yields and yields simulated with ®xed management represents the component of yield trends due to factors other than weather, and may provide a superior technology trend adjustment (Bell and Fischer).
4. Dealing with imperfect models
The discussion of aggregation error holds true even when models and their inputs are perfect. However, crop models are not perfect. Increasing the spatial domain of an analysis often introduces new constraints that are controllable in plot-scale stu-dies, and new processes that result from spatial interactions or the emergence of new system components, that can invalidate regional predictions from plot-scale models. Scaling up may, at some point, involve modifying available models (``phenomenon-added modeling'' [Luxmoore et al., 1991]) to incorporate these new constraints and processes. Relevant model modi®cations could include changing model structure or embedding model code or output into a model of a larger-scale processes.
4.1. Model complexity
Two opposing schools of thought seem to exist regarding the implications of scale for model structure. The ®rst suggests that appropriate model complexity should increase with spatial scale because moving from plot to larger scales usually intro-duces additional determinants of crop production. Rabbinge (1993) classi®ed levels of crop production by the factors that limit production:potential productionlimited only by irradiance, temperature and CO2;attainable productionlimited also by water
and macronutrients; and actual production limited also by pests and toxic factors. The evolution of crop models has paralleled these levels of production. Models capable of simulating potential production processes (i.e. photosynthesis, respira-tion, partitioning, phenology) were developed ®rst, then modi®ed by the addition of models of the soil water balance, and later N dynamics and use. Recent or ongoing attempts to incorporate additional determinants of actual production include mod-els of P dynamics (Gerakis et al., 1998), drainage (Shen et al., 1998), and response to pests, diseases (Teng et al., 1998) and various soil factors that constrain root growth (Lizaso and Ritchie, 1997; Calmon et al., 1999). As models incorporate additional processes, they tend to grow in complexity and data requirements.
spatial scales eliminates the need for detailed models of ®ne-scale (e.g. cellular) processes with a small time constant.
In our opinion, a hybrid approach may often be appropriate. The physiological detail in existing process-level models captures much of the mechanism of crop response to weather variability and its interaction with management, and should not be discarded without clear justi®cation. On the other hand, incomplete under-standing of processes, and excessive model complexity and data requirements would seem to preclude the development and use of models for regional applications that simulate physiological mechanisms of all important yield determinants. As an example of the hybrid approach, R.A.C. Mitchell (IACR, Hertfordshire, UK, per-sonal communication, 1999) used dynamic models to simulate 1980±1993 winter wheat yields for 48 variety trials in the UK. Simulations explained a small portion of the interannual pattern of mean yields (Table 1). A simple linear regression function of rainfall during grain-®ll, and minimum temperatures during the coldest three consecutive days gave better predictions (r=0.59). However, simulations corrected with regression results improved predictions relative to simulations or regression alone. The post-simulation adjustment accounted for processes (presumably diseases and winter kill) that the crop models did not capture.
4.2. Spatial interactions
We can envision three processes Ð surface and subsurface hydrology, intercrop competition, and farm resource allocation Ð in which dynamic interactions in space can modify crop yields. One obvious approach to simulating each of these processes is to embed existing crop models within a model of the higher-level system. This would require the ability to iteratively model, on a daily time step, the processes (e.g. lateral water movement, intercrop competition, or farm resource allocation) of the higher-level system, then simulate resource uptake and physiological processes for the crops in each spatial unit. In the intercrop and farm examples, the overall system model must be able to handle dierent crop species with possibly dissimilar model structures (Caldwell and Hansen, 1993). Current crop models are typically struc-tured to simulate an entire growing season for one crop species. Embedding these models into models of three-dimensional hydrology, intercrop competition or farm
Table 1
Predictability of 1980±1993 winter wheat yields from UK variety trials without and with an empirical climatic correctiona
Model Without correction With correction
%RMSEb r %RMSEb r
CERES 32 0.05 7 0.68
MAFF 9 0.31 6 0.76
operations would require restructuring the models so dierent crops can be simu-lated in parallel (Caldwell and Hansen; Jones et al., 1997; Sadler and Russell, 1997; Thornton et al., 1997a). Although our experience in modeling multiple cropping systems proves that it is possible, the diculty of reorganizing model code and the need to repeat the exercise for each model revision suggests that embedded models of these higher-level systems will not be sustainable without a commitment on the part of the crop modeling community to develop and maintain an appropriate modular structure.
5. Dealing with imperfect data
The availability of input data of adequate quality and spatial coverage is perhaps the most serious practical constraint to applications of crop models at regional or larger scales (de Wit and van Keulen, 1987; Russell and van Gardingen, 1997; Heuvelink, 1998). King et al. (1997, p. 143) argued that ``upscaling to larger areas invariably means a loss in the precision and observation density of data used to parameterize a model. It also raises questions about the suitability of applying the model at a scale dierent from the one for which it was developed.'' Although we hold a more optimistic view, each type of input Ð soil, weather and management Ð presents dicult challenges. The high cost of physical measurement at the desired density for regional model applications generally necessitates interpolation from sparse measurements or estimation from more readily available surrogate data. Where existing spatial data bases are available, inconsistent spatial coverage and boundaries between soils, weather stations or climate model grid cells, and crop reporting districts present additional challenges.
5.1. Soil
Soil heterogeneity within map units has important implications for agricultural simulation applications. In a 350 kmÿ2study area in New Jersey, USA, the forest
production model, PnET, under predicted mean evapotranspiration by 16% and mean primary production by 17% using rasterized soil inputs from STATSGO relative to predictions using inputs derived from the higher resolution SSURGO spatial soil data base (Lathrop et al., 1995). In a simulation study of the hydrologic response of a grassland watershed to precipitation, mean soil parameter values gen-erated only 14% of the mean runo obtained from partitioning the watershed into 14 soil texture classes (Sharma and Luxmoore, 1979). However, using mean soil properties had little eect on simulated evapotranspiration. Other studies have shown insensitivity of mean simulated soil evaporation (Lewan and Jansson, 1996) and rice yield (Wopereis et al., 1996) to spatial heterogeneity of soil properties. Luxmoore et al. (1991, p. 286) therefore cautioned that ``in some special situations mean behavior may be representative of the whole, but this cannot be assumed.'' Fortunately, growing appreciation of the importance of soil heterogeneity for model applications and improving data-storage capabilities are prompting calls and eorts to include information about the variability of parameters within map units in soil data bases (Arnold and Wilding, 1991; Burrough, 1993; Lathrop et al.; Finke et al., 1996; Young et al., 1998).
Although higher-resolution soil data can potentially reduce aggregation bias associated with heterogeneity of soil properties, the higher-resolution data do not account for all important variability, and are often not available. When soil data bases include pro®le properties and areal proportions corresponding to multiple pedons within each map unit, parameter values for the individual soils can be used iteratively as model input. Simulation results are then aggregated by areal weighting. Alternatively, the properties can be ®t to theoretical distribu-tions for stochastic sampling (Shaer, 1988; Haskett et al., 1995a; Bouma et al., 1996).
5.2. Weather
Weather data Ð observed, estimated or predicted Ð are central to crop model applications related to climate variability and prediction. Simulations at locations far from measured data, or where essential variables or periods are missing, must rely on estimated data. The most common estimate is to simply use the nearest weather station as a proxy for unmeasured weather at the location of interest. For regional applications using spatial partitioning, theissen polygons provide an auto-matic method for identifying the nearest station to any geographical point and for obtaining areal weighting factors for each station (Carbone et al., 1996; Rosenthal et al., 1998). Remote sensing oers some promise for ®lling gaps in surface-measured weather data. Because solar irradiance data has been a problem for crop model applications due to the cost and calibration problems of sensors, the prospect of using global data bases of satellite-derived solar irradiance (Whitlock et al., 1995) directly or to parameterize stochastic generators is appealing.
Although spatial averaging and interpolation are sometimes used to estimate daily weather, spatial averaging biases the variability of daily time-series data. Because of the many nonlinear processes that they embody, crop models are sensitive not only to mean climate, but also to its variability within and between seasons (Semenov and Porter, 1995; Mearns et al., 1996; Riha et al., 1996). This is particularly important for precipitation because of its in¯uences on processes, such as solute leaching, soil erosion and crop water stress response, that depend on soil water balance dynamics. A simple example illustrates the potential problem. We estimated 1976±1995 daily weather data (i.e. observed temperatures and precipitation [EarthInfo, 1996]) and solar irradiance generated (Hansen, 1999) using parameters derived from Jackson-ville data (NREL, 1992) for GainesJackson-ville, FL, using inverse-distance-weighted avera-ges from four surrounding stations (Fig. 2). Although the interpolation procedure produced reasonable estimates of monthly total rainfall, it seriously over-predicted mean wet-day intensity and under-predicted relative frequency of wet days for all calendar months (Fig. 4).
An arti®cial increase in rainfall frequency and decrease in mean intensity due to spatial aggregation may have two contrasting eects on soil water availability and crop yield response. On the one hand, frequent low-intensity showers do not recharge soil water reserves in deeper layers, but favor increased evaporation from the soil surface, thereby increasing water stress (de Wit and van Keulen, 1987). On the other hand, increasing the frequency of rainfall events tends to reduce the duration of dry periods between rain events, thereby decreasing the probability of water stress. Simulation studies under arti®cially imposed changes in climate varia-bility suggest that conditions of low mean rainfall, high potential evaporation and high AWHC favor the ®rst mechanism (increasing soil evaporation), resulting in negative yield bias, whereas higher mean rainfall and lower AWHC favor positive bias (Carbone, 1993; Mearns et al., 1996; Riha et al., 1996).
planting date, and realistic cultivar and management inputs, the CERES-Maize V. 3.5 model predicted higher mean grain yields with less interannual variability using interpolated weather than using observed weather for any of the ®ve weather sta-tions (Table 2). Applying inverse distance interpolation to yields simulated for each station other than Gainesville resulted in more realistic mean yields, but with a low standard deviation. As our previous discussion suggests, the lower standard devia-tion is probably more representative of regional yields.
long-term change and seasonal variability. In a 1.6 million km2 region in the
Central USA, average (1953±1975) soybean yields simulated by SOYGRO using grid cell-averaged weather showed an average bias ranging from 18.5 (22 grid) to 28.0% (55 grid) relative to the simple average of yields simulated for each of >500 stations in the region (Carbone, 1993). Larger errors in individual grid cells or individual years tended to cancel each other. Easterling et al. (1998) found that errors in simulating reported (1984±1992) maize and wheat yields in a portion of the US Great Plains decreased as resolution increased from 2.82.8 to 11. Increasing resolution further to 0.50.5did not further improve predictions. These illustrations highlight the importance of spatial and temporal downscaling of climate model output in a manner that preserves both the meaningful features of the model predictions and the statistical properties of the historical daily sequences.
5.3. Management
Crop management inputs typically considered include crop species and cultivar; planting date and spatial arrangement; irrigation, fertilizer and sometimes biocide applications; and sometimes land preparation and tillage. Spatial heterogeneity of management can contribute to aggregation bias. Because management is seldom consistent from year to year, spatial representations of management variables are not generally available. Typical or recommended practices are therefore often applied uniformly within a region.
If a region includes a mixture of irrigated and rainfed production of a particular crop, knowing the relative areas of each will be important. Areas of irrigated and rainfed production are sometimes, but not always, available. In rainfed production, small changes in the timing of water shortage relative to critical periods of crop growth sometimes have profound eects on yields. Farmers therefore often diversify planting date and cultivar to reduce risk. The use of one or more representative Table 2
Simulated maize yield statistics for locations and interpolation schemes, North Florida, USA, 1976±1995a
Source Y(Mg haÿ1) s(Mg haÿ1) CV (%) Linear cross-correlation (r)
OC LC CC JA GA
Ocala (OC) 7.49 1.81 24.2 1.000
Lake City (LC) 7.61 1.35 17.7 ÿ0.109 1.000
Cross City (CC) 7.38 1.90 25.8 0.222 ÿ0.110 1.000
Jacksonville (JA) 5.76 2.10 36.4 0.310 0.208 0.452 1.000
Gainesville (GA) 6.71 1.89 28.2 0.356 0.569 0.277 0.384 1.000
Interpolated weather 8.19 0.95 11.6
Interpolated yields 7.18 1.08 15.1
a Weather and simulated yields were interpolated for Gainesville from the other locations by
cultivars is often a reasonable approximation. Alternatively, eective cultivars can be derived by calibration against observed development and yield data at the spatial scale of interest. Hodges et al. (1987) used this approach quite successfully to cali-brate nine eective maize cultivars for CERES-Maize from crop reporting district data. They then selected the best of the nine eective cultivars for each of 51 weather locations to simulate regional 1982±1985 maize yields in 14 states in the US northern Midwest. Studies have shown that using several planting dates within the reported range can improve regional yield predictions relative to using a single average planting date (Moen et al., 1994; Batchelor, Iowa State University, personal com-munication, 1999).
Moen et al. (1994) used a novel approach to incorporating the technology trend directly into simulations by changing management inputs. They varied cultivars, and planting dates and densities in each decade of 1960±1989 for a crop reporting district in eastern Illinois, USA. They also allowed N use to vary among years based on a simple economic optimization model and each year's prices.
6. County-scale soybean yields in Georgia, USA
We have discussed a number of approaches for dealing with the issues that arise when applying crop models at spatial scales that encompass substantial hetero-geneity of inputs. The case study of maize in North Florida illustrated some of the challenges related to spatial variability of weather. To illustrate possible approaches for dealing with heterogeneity of soils and management, we used the CROPGRO V. 3.5 grain legume model (Boote et al., 1998) to predict historical soybean yields in Tift County, GA, USA. The purpose is to demonstrate implications for county-scale yield predictability of heterogeneity of soil, cultivar and planting-date inputs, and of calibrating either model predictions or soil inputs.
6.1. Approach
6.1.1. Data
The National Agricultural Statistics Service provided Tift County production and yield data for 1972±1997. Observed yields were linearly detrended (Eq. (12)) and adjusted to a 1990 basis. The trend was fairly small (+11.3 kg haÿ1yearÿ1).
Because the area of irrigated soybean in the region has increased in recent years (Hoogenboom, University of Georgia, personal communication, 1999) but is not included in the county data base, we considered only 1972±1992. We selected seven random years (1977, 1978, 1979, 1981, 1984, 1989, 1990) for yield and soil calibra-tions, then calculated all goodness of ®t statistics on the remaining 14 years to maintain independence between calibration and validation data.
irradiances were generated stochastically (Hansen, 1999) based on parameters derived from the observed irradiances.
Parameters for 11 soil pro®les representing seven agriculturally important series (Table 3) were based on characterization data compiled by Perkins (1987). All seven series were Plinthic Paleudults. Areal weighting factors were derived from areas of each series under farmland (Soil Conservation Service, 1983). The lower limit L
of plant-extractable water for each horizon was set at the reported water content at 1.5 MPa. The drained upper limit U was obtained as the sum of L and AWHC
estimated from texture and organic C (Ritchie et al., 2000). Saturated hydraulic conductivity for each horizon was estimated from texture and bulk density (Jabro, 1992). The source-driven root growth model in CROPGRO uses soil-speci®c weighting factors to control the vertical distribution of new root growth. We calculated root weighting factors as a power function of depth (Jones et al., 1991), reduced for toxicity by a linear function of Al saturation between 18 and 93% based on results by Hanson and Kamprath (1979). CROPGRO uses a photosynthesis adjustment factor (SLPF) associated with each soil to adjust mean growth and yield for yield-reducing factors, such as soil fertility and pest pressure, that the model does not account for. Based on Boote et al. (1997) and our experience with experiment station data in Georgia and Florida, we set this factor to 0.92 for all soils.
Heterogeneity of planting dates was achieved by using ®ve target planting dates at 7-day intervals beginning 16 May. The model delayed planting if relative water content, (ÿL)/(UÿL), averaged to 15 cm depth fell outside 40±95%, allowing
planting dates to vary among years. We used parameters for three generic cultivars representing maturity groups V±VII.
6.1.2. Simulation and calibration
The simulation experiment consisted of six combinations of soil (single, multiple and calibrated) and management (single and multiple cultivars and planting
Table 3
Tift County, GA, soils used for soybean simulations
Series Texture No.
Tifton Loamy sand 4 221.9 49.0 21.2
Dothan Loamy sand 1 53.1 11.7 26.3
Fuquay Loamy sand 1 43.2 9.5 24.0
Ocilla Loamy sand 1 26.6 5.9 15.1
Carnegie Sandy loam 2 19.4 4.3 21.9
Lee®eld Loamy sand 1 11.0 2.4 25.0
Clarendon Loamy sand 1 10.6 2.3 25.8
Total 11 385.8 85.3
dates) and an additional post-simulation yield calibration treatment. Using single versus multiple cultivars and planting dates tested the hypothesis that heterogeneity of management reduces interannual yield variability bias. The 15 combinations of cultivar and planting date were given equal weight when averaging yields. Maturity group VI with a target planting date of 30 May was used for the homogeneous management treatments. Soil treatments included all 11 soils, a single soil (a Tifton loamy sand) without calibration, and the same soil calibrated to minimize prediction error as described below. For the yield calibration treatment, we adjusted yield predictions from the single soil, cultivar and planting date treatment used a linear correction (Eq. (9)) derived by regressing observed and predicted yields for the seven calibration years.
Under rainfed conditions, AWHC aects both mean yield and its interannual variability; low AWHC generally increases yield variability. Soil calibration entailed adjusting SLPF and AWHC to minimize squared prediction error:
SSX
n
t1
ytÿt2;
for the calibration years using a grid search. We calibrated total AWHC by adjust-ing AWHC of each soil layer by a sadjust-ingle multiplier (l), then shifting eachUto give
the target AWHC. When necessarySwas adjusted upward to prevent it from
fall-ing belowL+1.2(UÿyL).
6.2. Results and discussion
Table 4 summarizes prediction results. Without calibration, CROPGRO over-predicted the observed county mean yield (1.63 Mg haÿ1), and its standard deviation
(0.380 Mg haÿ1) and coecient of variation (0.233) among years (Fig. 5B). Using
either multiple soils or multiple cultivars and planting dates reduced but did not eliminate the standard deviation bias predicted by Eq. (8). However, the mean bias may have in¯ated the standard deviation of the predictions. The predicted CV was close to the observed when simulations incorporated heterogeneity of either soils or management. In the absence of calibration, the various goodness of ®t measures did not consistently identify a single, best approach.
As expected, soil calibration improved predictions substantially, primarily by reducing systematic error ofUand(Table 4, Fig. 5C). However, simulated results with calibrated soil parameters under-predicted observedand CV. The reductions of total error (RMSE) were accompanied by small increases of random error (RMSER). Predictions using the calibrated soil were slightly poorer when multiple
cultivars and planting dates were used. Minimum RMSE for the calibration years (0.216 Mg haÿ1 with homogeneous and 0.197 Mg haÿ1 with heterogeneous
heterogeneous management). Increasing AWHC tended to increase the mean and decrease the interannual variability of yields. Decreasing SLPF decreased mean yields. Calibrating yield predictions to observed county yields by a linear adjustment (Fig. 5D) produced the closest estimates ofUand, and the best ®t based on every other measure of goodness of ®t (Table 4).
Although results of this exercise are meant to be illustrative, they support some tentative conclusions. First, incorporating heterogeneity of soils and management reduced but did not eliminate a tendency to over-predict interannual variability of regional yields. We probably did not capture the full range of variability of soil properties or management within the county. Second, calibrating either soil inputs or simulated yields to observed regional yields improved predictability of yields outside of the calibration period. In this particular analysis, direct calibration of simulated yields proved superior to calibration of soil inputs. Finally, CROPGRO captured much but not all of the regional crop response to weather variability. This is probably due to a combination of model and input error. Using parameters that have given reasonable yield predictions at experiment stations, CROPGRO over-predicted county yields. A gap exists between experiment station and county yields of soybean in Tift County (Fig. 5A). Identifying the environmental and management factors that cause this yield gap might improve model predictions and, more importantly, identify opportunities to improve farm yields.
Table 4
Summary and goodness of ®t statisticsafor soybean yield predictions for Tift County, GA, USAb
Soils Management Y(Mg
Single Single 2.46 0.667 27.1 0.824 0.962 0.228 0.783 0.518
Heterogeneous inputs
Single Multiple 2.46 0.606 24.7 0.824 0.914 0.244 0.745 0.507
Multiple Single 2.69 0.596 22.2 1.059 1.123 0.233 0.771 0.441
Multiple Multiple 2.68 0.536 20.0 1.050 1.107 0.248 0.736 0.433
Calibrated soil parameters
Single Single 1.68 0.111 6.6 0.238 0.308 0.271 0.567 0.562
Single Multiple 1.66 0.109 6.5 0.254 0.322 0.305 0.551 0.512
Calibrated yields
Single Single 1.63 0.298 18.2 0.190 0.228 0.228 0.783 0.869
a Goodness of ®t statistics are de®ned in the Appendix.
b Observed=1.63 Mg haÿ1,=0.380 Mg haÿ1and CV=23.3%.Y,s, CV, mean, standard deviation
and coecient of variation across years of spatial average yield; MAE, mean absolute error; RMSE, root mean squared error; RMSER, random component of RMSE;r, linear cross-correlation; d, index of
7. Discussion
might give one a sense of incorporating some degree of mechanism into simula-tions, we do not have enough evidence to favor either input or output calibration. Our soybean example showed comparable improvements in predictions from both methods.
Crop models have been developed to predict yields under the intensive man-agement characteristic of experimental trials used to develop the models. Actual production is hampered by many factors that are not included in the models. Aggregating results from such models therefore usually over-estimates observed yields. Ongoing eorts will reduce the gap between simulated and actual yields. Given current model imperfections, analysis of the structure of prediction errors could identify climatic factors (e.g. excess rain during maturation or harvest) that could then be used as input to an empirical correction to model predictions. If regional predictions of crop models require empirical correction, why not replace them with simple, empirical functions of climate? Part of the answer lies in the fact that crops respond not only to seasonal climatic means, but also to the dynamics of weather events. Process-level models capture much of the eect of timing of weather, particularly rainfall, on growth and yield, and can therefore improve pre-dictions over empirical models that omit these dynamic responses. When studies are concerned with spatially aggregated response to the interactive eects of climate variability and management, the physiological detail in process-level models is essential.
Acknowledgments
We gratefully acknowledge helpful comments by W. Graham and J. Jordan and two anonymous reviewers. This work was supported in part by a grant from NOAA-Oce of Global Programs entitled ``Regional Application of ENSO-Based Climate Forecasts to Agriculture in the Americas'' and by the Global Change Sys-tem for Analysis Research and Training (START). It is adapted from a paper, ``Scaling-up Crop Models for Climate Prediction Applications'', presented at the CLIMAG Geneva Workshop, WMO Headquarters, Geneva, 27±30 September 1999.
Appendix. Model goodness of ®t statistics
Researchers use several statistics to describe the quality of model predictions relative to observations. For each of the following statistics used in this paper, t
andytrepresent observed and predicted spatial average yields for a given area for
Index Symbol Formulation Interpretation
yP,ifrom linear regression
Component of RMSE
RMSES RMSEÿRMSER Component of RMSE
that is correctable by
Addiscott, T.M., 1993. Simulation modelling and soil behaviour. Geoderma 60, 15±40.
Addiscott, T.M., 1998. Modelling concepts and their relation to the scale of the problem. Nutrient Cycling in Agroecosystems 50, 239±245.
Arnold, R.W., Wilding, L.P., 1991. The need to quantify spatial variability. In: Mausbach, M.J., Wilding, L.P. (Eds.), Spatial Variabilities of Soils and Landforms (SSSA Spec. Pub. No. 28). SSSA, Madison, WI, USA, pp. 1±8.
Band, L.E., Peterson, D.L., Running, S.W., Coughlan, J., Lammers, R., Dungan, J., Nemani, R., 1991. Forest ecosystem processes at the watershed scale: basis for distributed simulation. Ecological Model-ling 56, 171±196.
Bell, M.A., Fischer, R.A., 1994. Using yield prediction models to assess yield gains: a case study for wheat. Field Crops Research 36, 161±166.
Beven, K., 1989. Changing ideas in hydrology Ð the case of physically-based models. Journal of Hydrology 105, 157±172.
Boote, K.J., Jones, J.W., Hoogenboom, G., Pickering, N.B., 1998. The CROPGRO model for grain legumes. In: Tsuji, G.Y., Hoogenboom, G., Thornton, P.K. (Eds.), Understanding Options for Agri-cultural Production. Kluwer, Dordrecht, The Netherlands, pp. 99±128.
Bouma, J., Bootlink, H.W.G., Finke, P.A., 1996. Use of soil survey data for modeling solute transport in the vadose zone. Journal of Environmental Quality 25, 519±526.
Burke, I.C., Kittel, T.G.F., Lauenroth, W.K., Shook, P., Yonker, C.M., Parton, W.J., 1991. Regional analysis of the Central Great Plains. BioScience 41, 685±692.
Burrough, P.A., 1993. Soil variability: a late 20th century view. Soils and Fertilizers 56, 529±562. Caldwell, R.M., Hansen, J.W., 1993. Simulation of multiple cropping systems with CropSys. In: Penning
de Vries, F.W.T., Teng, P., Metselaar, K. (Eds.), Systems Approaches for Agricultural Development (Vol. 2). Kluwer, Dordrecht, The Netherlands, pp. 397±412.
Calmon, M.A., Batchelor, W.D., Jones, J.W., Ritchie, J.T., Boote, K.J., Hammond, L.C., 1999. Simu-lating soybean root growth and soil water extraction using a functional crop model. Transactions of the American Society of Agricultural Engineers 42, 1867±1877.
Carbone, G.J., 1993. Considerations of meteorological time series in estimating regional-scale crop yield. Journal of Climate 6, 1607±1615.
Carbone, G.J., Narumalani, S., King, M., 1996. Application of remote sensing and GIS technologies with physiological crop models. PE&RS 62, 171±179.
Carlson, R.E., Today, D.P., Taylor, S.E., 1996. Midwestern corn yield and weather in relation to extremes of the Southern Oscillation. Journal of Production Agriculture 9, 347±352.
Chipanshi, A.C., Ripley, E.A., Lawford, R.G., 1998. Large-scale simulation of wheat yields in a semi-arid environment using a crop-growth model. Agricultural Systems 59, 1±10.
Curran, P.J., Foody, G.M., van Gardingen, P.R., 1997. Scaling-up. In: van Gardingen, P.R., Foody, G.M., Curran, P.J. (Eds.), Scaling-up From Cell to Landscape. Cambridge University Press, Cam-bridge, pp. 1±5.
de Jager, J.M., Potgieter, A.B., van den Berg, W.J., 1998. Framework for forecasting the extent and severity of drought in maize in the Free State Province of South Africa. Agricultural Systems 57, 351±365.
de Wit, C.T., van Keulen, H., 1987. Modelling production of ®eld crops and its requirements. Geoderma 40, 253±265.
EarthInfo, 1996. Database Guide for EarthInfo CD2 NCDC Summary of the Day. EarthInfo, Inc.,
Boulder, CO, USA.
Easterling, W.E., Weiss, A., Hays, C.J., Mearns, L.O., 1998. Spatial scales of climate information for simulating wheat and maize productivity: the case of the US Great Plains. Agricultural and Forest Meteorology 90, 51±63.
Fagerberg, B., Nyman, P., Sigvald, R., Torssell, B., 1998. Estimating regional crop yields in Sweden with a simulation model and ®eld observations. Swedish Journal of Agricultural Research 28, 91±98. Finke, P.A., WoÈsten, J.H.M., Kroes, J.G., 1996. Comparing two approaches of characterizing soil map
unit behavior in solute transport. Soil Science Society of America Journal 60, 200±205.
Gerakis, A., Daroub, S., Ritchie, J.T., Friesen, D.K., Chien, S.H., 1998. Phosphorus simulation in the CERES models. In: 1998 Agronomy Abstracts. ASA, Madison, WI, USA.
Hansen, J.W., 1999. Stochastic daily solar irradiance for biological modeling applications. Agricultural and Forest Meteorology 94, 53±63.
Hansen, J.W., Hodges, A.W., Jones, J.W., 1998. ENSO in¯uences on agriculture in the southeastern US. Journal of Climate 11, 404±411.
Hanson, W.D., Kamprath, E.J., 1979. Selection for aluminum tolerance in soybeans based on seedling-root growth. Agronomy Journal 71, 581±586.
Hartkamp, A.D., White, J.W., Hoogenboom, G., 1999. Interfacing geographic information systems with agronomic modeling: a review. Agronomy Journal 91, 761±772.
Haskett, J.D., Pachepsky, Y.A., Acock, B., 1995a. Use of the beta distribution for parameterizing varia-bility of soil properties at the regional level for crop yield estimation. Agricultural Systems 48, 73±86. Haskett, J.D., Pachepsky, Y.A., Acock, B., 1995b. Estimation of soybean yields and county and state
Helstrom, C.W., 1991. Probability and Stochastic Processes for Engineers, 2nd Edition. Macmillan, New York.
Heuvelink, G.B.M., 1998. Uncertainty analysis in environmental modelling under a change of spatial scale. Nutrient Cycling in Agroecosystems 50, 255±264.
Hodges, T., Botner, D., Sakamoto, C., Hays Haug, J., 1987. Using the CERES-Maize model to estimate production for the U.S. cornbelt. Agricultural and Forest Meteorology 40, 293±303.
Iman, R.L., Conover, W.J., 1982. A distribution-free approach to inducing rank correlation among input variables for simulation studies. Communications in Statistics B11, 311±334.
Iwasa, Y., Andreasen, V., Levin, S., 1987. Aggregation in model ecosystems. I. Perfect aggregation. Eco-logical Modelling 37, 287±302.
Jabro, J.D., 1992. Estimation of saturated hydraulic conductivity of soils from particle size distribution and bulk density data. Transactions of the American Society of Agricultural Engineers 35, 557±560. Jansen, M.J.W., 1998. Prediction error through modelling concepts and uncertainty from basic data.
Nutrient Cycling in Agroecosystems 50, 247±253.
Jones, J.W., Thornton, P.K., Hansen, J.W., 1997. Opportunities for systems approaches at the farm scale. In: Teng, P.S., Krop, M.J., Ten Berge, H.F.M., Dent, J.B., Lansigan, F.P., Van Laar, H.H. (Eds.), Applications of System Approaches at the Farm and Regional Level. Kluwer, Dordrecht, The Netherlands, pp. 1±29.
Jones, C.A., Bland, W.L., Ritchie, J.T., Williams, J.R., 1991. Simulation of root growth. In: Hanks, J., Ritchie, J.T. (Eds.), Modeling Plant and Soil Systems. ASA-CSSA-SSSA, Madison, WI, USA, pp. 91±123. King, A.W., 1991. Translating models across scales in the landscape. In: Turner, M.G., Gardner, R.H. (Eds.), Quantitative Methods in Landscape Ecology: The Analysis and Interpretation of Landscape Heterogeneity. Springer-Verlag, New York, pp. 479±517.
King, D., Fox, D.M., Daroussin, J., Le Bissonnais, Y., Dannneels, V., 1997. Upscaling a simple erosion model from small areas to a large region. Nutrient Cycling in Agroecosystems 50, 143±149.
Kunkel, K.E., Hollinger, S.E., 1991. Operational large area corn and soybean yield estimation. In: Pre-prints, 20th Conference on Agricultural and Forest Meteorology, American Meteorology Society, Boston, pp. 13±16.
Lathrop Jr., R.G., Aber, J.D., Bognar, J.A., 1995. Spatial variability of digital soil maps and its impact on regional ecosystem modelling. Ecological Modelling 82, 1±10.
Lewan, E., Jansson, P., 1996. Implications of spatial variability of soil physical properties for simulation of evaporation at the ®eld scale. Water Resources Research 32, 2067±2074.
Lizaso, J.I., Ritchie, J.T., 1997. A modi®cation of CERES to predict the impact of soil water excess on maize crop growth and development. In: Krop, M.J., Teng, P.S., Aggarwal, P.K., Bouman, J., Jones, J.W., Van Laar, H.H. (Eds.), Applications of System Approaches at Field Level. Kluwer, Dordrecht, The Netherlands, pp. 153±167.
Luxmoore, R.J., 1988. Assessing the mechanisms of crop loss from air pollutants with process models. In: Heck, W.W., Taylor, O.C., Tingey, D.T. (Eds.), Assessment of Crop Loss from Air Pollutants. Elsevier, London, pp. 417±444.
Luxmoore, R.J., King, A.W., Tharp, M.L., 1991. Approaches to scaling up physiologically based soil-plant models in space and time. Tree Physiology 9, 281±292.
McKay, M.D., Beckman, R.J., Conover, W.J., 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239±245. Mearns, L.O., Rosenzweig, C., Goldberg, R., 1992. The eect of changes in interannual climatic variability on
CERES-Wheat yields: sensitivity and 2CO2studies. Agricultural and Forest Meteorology 62, 159±189.
Mearns, L.O., Rosenzweig, C., Goldberg, R., 1996. The eects of changes in daily and interannual cli-matic variability on CERES-Wheat: a sensitivity study. Clicli-matic Change 32, 257±292.
Mearns, L.O., Schneider, S.H., Thompson, S.L., McDaniel, L.R., 1990. Analysis of climate variability in general circulation models: comparison with observations and changes in variability in 2CO2
experi-ments. Journal of Geophysical Research D95, 20469±20490.
Mearns, L.O., Giorgi, F., McDaniel, L., Shields, C., 1995. Analysis of daily variability of precipitation in a nested regional climate model: comparison with observations and doubled CO2results. Global and
Meinke, H., Hammer, G.L., 1995. Climatic risk to peanut production: a simulation study for Northern Australia. Australian Journal of Agricultural Research 35, 777±780.
Mjelde, J.W., Keplinger, K., 1998. Using the southern oscillation to forecast Texas winter wheat and sorghum crop yields. Journal of Climate 11, 54±60.
Moen, T.N., Kaiser, H.M., Riha, S.J., 1994. Regional yield estimation using a crop simulation model: concepts, methods and validation. Agricultural Systems 46, 79±92.
MuÈller, F., 1992. Hierarchical approaches to ecosystem theory. Ecological Modelling 63, 215±242. Nicholls, N., 1985. Impact of the southern oscillation on Australian crops. Journal of Climatology 5, 553±
560.
NREL, 1992. User's Manual, National Solar Radiation Data Base (1961±1990). Natl. Renewable Energy Laboratory, Boulder, CO, USA.
Oliver, M.A., Webster, R., 1991. How geostatistics can help you. Soil Use and Management 7, 206±217.
O'Neill, R.V., DeAngelis, D.L., Waide, J.B., Allen, T.F.H., 1986. A Hierarchical Concept of Ecosystems. Princeton University Press, Princeton, NJ, USA.
Owen, A.B., 1994. Controlling correlations in Latin hypercube samples. Journal of the American Statis-tical Association 89, 1517±1522.
Paz, J.O., Batchelor, W.D., Colvin, T.S., Logsdon, Kaspar, T.C., Karlen, D.L., 1998. Analysis of water stress eects causing spatial yield variability in soybeans. Transactions of the American Society of Agricultural Engineers 41, 1527±1534.
Perkins, H.F., 1987. Characterization Data for Selected Georgia Soils (Georgia Agric. Exp. Stations Spec. Publ. 43). University of Georgia, Athens, GA, USA.
Power, M., 1993. The predictive validation of ecological and environmental models. Ecological Modelling 68, 33±50.
Rabbinge, R., 1993. The ecological background of food production. In: Crop Protection and Sustainable Agriculture. Ciba Foundation Symposium 177. John Wiley, Chichester, UK, pp. 2±29.
Rastetter, E.B., King, A.W., Cosby, B.J., Hornberger, G.M., O'Neill, R.V., Hobbie, J.E., 1992. Aggre-gating ®ne-scale ecological knowledge to model coarser-scale attributes of ecosystems. Ecological Applications 2, 55±70.
Reybold, W.U., TeSelle, G.W., 1989. Soil geographic data bases. Journal of Soil and Water Conservation 44, 28±29.
Riha, S.J., Wilks, D.S., Simeons, P., 1996. Impacts of temperature and precipitation variability on crop model predictions. Climatic Change 32, 293±311.
Ritchie, J.T., Crumb, J., 1989. Converting soil survey characterization data into IBSNAT crop model input. In: Bouma, J., Bregt, A.K. (Eds.), Land Qualities in Space and Time. Proceeding of the ISSS Symposium, Wageningen, 22±26 August 1998. Pudoc, Wageningen, The Netherlands, pp. 155±167. Ritchie, J.T., Gerakis, A., Suleiman, A., 1999. Simple model to estimate ®eld-measured soil water limits.
Transactions of the American Society of Agricultural Engineers 42, 1609±1613.
Ritchie, J.T., Singh, U., Godwin, D.C., Bowen, W.T., 1998. Cereal growth, development and yield. In: Tsuji, G.Y., Hoogenboom, G., Thornton, P.K. (Eds.), Understanding Options for Agricultural Pro-duction. Kluwer, Dordrecht, The Netherlands, pp. 79±98.
Rosenberg, N.J., McKenney, M.S., Easterling, W.E., Lemon, K.M., 1992. Validation of EPIC model simulation of crop responses to current climate and CO2conditions: comparisons with census, expert
judgement and experimental plot data. Agricultural and Forest Meteorology 59, 35±51.
Rosenthal, W.D., Hammer, G.L., Butler, D., 1998. Predicting regional grain sorghum production in Australia using spatial data and crop simulation modelling. Agricultural and Forest Meteorology 91, 263±274.
Russell, G., van Gardingen, P.R., 1997. Problems with using models to predict regional crop production. In: van Gardingen, P.R., Foody, G.M., Curran, P.J. (Eds.), Scaling-up from Cell to Landscape. Cam-bridge University Press, CamCam-bridge, UK, pp. 273±294.
Semenov, M.A., Porter, J.R., 1995. Climatic variability and the modelling of crop yields. Agricultural and Forest Meteorology 73, 265±283.
Shaer, M.J., 1988. Estimating con®dence bands for soil-crop simulation models. Soil Science Society of America Journal 52, 1782±1789.
Sharma, B.D., Luxmoore, R.J., 1979. Soil spatial variability and its consequences in simulated water balance. Water Resources Research 15, 1567±1573.
Shen, J., Batchelor, W.D., Jones, J.W., Ritchie, J.T., Kanwar, R., Miz, C.W., 1998. Incorporation of a subsurface tile drainage component into a soybean growth model. Transactions of the American Society of Agricultural Engineers 41, 1305±1313.
Skelly, W.C., Henderson-Sellers, A., 1996. Grid box or grid point: what type of data do GCMs deliver to climate impacts researchers. International Journal of Climatology 16, 1079±1086.
Soil Conservation Service, 1983. Soil Survey of Tift County, Georgia. US Dept. Agric., Soil Conservation Service, Washington, DC, USA.
Stein, M.L., 1987. Large sample properties of simulations using Latin hypercube sampling. Technometrics 29, 143±151.
Supit, I., 1997. Predicting national wheat yields using a crop simulation and trend models. Agricultural and Forest Meteorology 88, 199±214.
Swanson, E.R., Nyankori, J.C., 1979. In¯uence of weather and technology on corn and soybean trends. Agricultural and Forest Meteorology 20, 327±342.
Templeton, A.R., Lawlor, L.R., 1981. The fallacy of the averages in ecological optimization theory. American Naturalist 117, 390±391.
Teng, P.S., Batchelor, W.D., Pinnschmidt, H.O., Wilkerson, G.G., 1998. Simulation of pest eects on crops using coupled pest-crop models: the potential for decision support. In: Tsuji, G.Y., Hoogenboom, G., Thornton, P.K. (Eds.), Understanding Options for Agricultural Production. Kluwer, Dordrecht, The Netherlands, pp. 221±266.
Thornton, P.K., Bootlink, H.W.G., Stoorvogel, J.J., 1997a. A computer program for geostatistical and spatial analysis of crop model outputs. Agronomy Journal 89, 620±627.
Thornton, P.K., Saka, A.R., Singh, U., Kumwenda, J.D.T., Brink, J.E., Dent, J.B., 1995. Application of a maize crop simulation model in the central region of Malawi. Experimental Agriculture 31, 213±226. Thornton, P.K., Bowen, W.T., Ravelo, A.C., Wilkens, P.W., Farmer, G., Brock, J., Brink, J.E., 1997b.
Estimating millet production for famine early warning: an application of crop simulation modelling using satellite and ground-based data in Burkina Faso. Agricultural and Forest Meteorology 83, 95± 112.
Tietje, O., Tapkenhinrichs, M., 1993. Evaluation of pedo-transfer functions. Soil Science Society of America Journal 57, 1088±1095.
van Lanen, H.A.J., van Diepen, C.A., Reinds, G.J., de Koning, G.H.J., Bulens, J.D., Bregt, A.K., 1992. Physical land evaluation methods and GIS to explore the crop growth potential and its eects within the European communities. Agricultural Systems 39, 307±328.
van Noordwijk, M., Dijksterhuis, G.H., van Keulen, H., 1994. Risk management in crop production and fertilizer use with uncertain rainfall; how many eggs in which baskets? Netherlands Journal of Agri-cultural Science 42, 249±269.
Viglizzo, E.F., Roberto, Z.E., Filippin, M.C., Pordomingo, A.J., 1995. Climate variability and agroeco-logical change in the central Pampas of Argentina. Agricultural Ecosystems and Environment 55, 7±16. Warrick, A.W., 1998. Spatial variability. In: Hillel, D. (Ed.), Environmental Soil Physics. Academic Press,
San Diego, CA, USA, pp. 655±676.
Webster, R., 1985. Quantitative spatial analysis of soil in the ®eld. Advances in Soil Science 3, 1±70. Whitlock, C.H., Charlock, T.P., Staylor, W.F., Pinker, R.T., Laszzlo, I., Ohmura, A., Gilen, H.,
Konzel-man, T., DiPasquale, R.C., Moats, C.D., LeCroy, S.R., Ritchey, N.A., 1995. First global WCRP short-wave surface radiation budget dataset. Bulletin of the American Meteorological Society 76, 905±922. Wopereis, M.C.S., Stein, A., Krop, M.J., Bouma, J., 1996. Spatial interpolation of soil hydraulic
prop-erties and simulated rice yield. Soil Use and Management 12, 158±166.
