Directory UMM :Data Elmu:jurnal:A:Agricultural & Forest Meterology:Vol101Issue2-3Maret2000:

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A parsimonious, multiple-regression model of wheat yield

response to environment

S. Landau

a,∗

_{, R.A.C. Mitchell}

_{, V. Barnett}

_{, J.J. Colls}

_{, J. Craigon}

_{, R.W. Payne}

a_{Department of Biostatistics and Computing, Institute of Psychiatry (King’s College), London SE5 8AF, UK} b_{Biochemistry and Physiology Department, IACR-Rothamsted, Harpenden, Hertfordshire AL5 2JQ, UK}

c_{Department of Mathematics, University of Nottingham, Nottingham NG7 2RD, UK}

d_{Environmental Science Division, University of Nottingham, Sutton Bonington, Loughborough, Leicestershire LE12 5RD, UK} e_{Statistics Department, IACR-Rothamsted, Harpenden, Hertfordshire AL5 2JQ, UK}

Received 8 February 1999; received in revised form 20 November 1999; accepted 23 November 1999

Abstract

A database of nearly 2000 yield observations from winter wheat crops grown in UK trials between 1976 and 1993 was used to develop a new model of effects of weather on wheat yield. The intention was to build a model which was parsimonious (i.e., has the minimum number of parameters and maximum predictive power), but in which every parameter reflected a known climate effect on the UK crop-environment system to allow mechanistic interpretation. To this end, the model divided the effects of weather into phases which were predicted by a phenology model. A maximum set of possible weather effects in different phenological phases on yield was defined from prior knowledge. Two-thirds of the database was used to select which effects were necessary to include in the model and to estimate parameter values. The final model was tested against the independent data in the remaining third of the data set (246 aggregated yield observations) and showed predictive power (r=0.41), which was improved when comparing against mean annual yields (r=0.77). The final model allowed the relative importance of the 17 explanatory variables, and the weather effects they represent (defined before fitting), to be assessed. The most important weather effects were found to be: (1) negative effects of rainfall on agronomy before and during anthesis, during grain-filling and in the spring (2) winter frost damage (3) a positive effect of the temperature-driven duration of grain-filling and (4) a positive effect of radiation around anthesis, probably due to increased photosynthesis. The model developed here cannot be applied outside the UK, but the same approach could be employed for applications elsewhere, using appropriate yield, weather and management data. ©2000 Elsevier Science B.V. All rights reserved.

Keywords:Winter wheat; Grain yields; Weather; Prediction; Parsimony

1. Introduction

A variety of mathematical models relating en-vironmental and management factors to crop yield have been proposed throughout this century. Most of

∗_{Corresponding author. Tel.:}_{+44-0171-919-3313;} fax:+44-0171-919-3304.

E-mail address:[email protected] (S. Landau).

these models can be broadly classified as empirical regression-type models, derived from large amounts of yield data, or deterministic crop simulation mod-els, based on experiments on crops and incorporating knowledge of processes. Early investigations into the effects of climate on crops used regression-type mod-els. Work on wheat yields in the UK was dominated by analyses of the relationship between climate variables and crop yields from a long-term experiment (the 0168-1923/00/$ – see front matter ©2000 Elsevier Science B.V. All rights reserved.

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Broadbalk wheat experiment) at Rothamsted (Fisher, 1924; Tippett, 1926; Alumnus, 1932; Cochran, 1935; Buck, 1961; Thorne et al., 1988 and Chmielewski and Potts, 1995, amongst others). With these single-site models, workers were able to find correlations be-tween observed and predicted yields in the range 0.45–0.6, thus explaining 20–35% of the variability in grain yields. Further wheat/climate investigations in the UK include the early work of Lawes and Gilbert (1880); Hooker (1907) and Barnard (1936) and the relatively recent work of Spence (1989).

Regression models have been criticised, since un-derlying mechanisms which transform climatic input into yield are not explicitly described and the hierar-chical structure of the underlying physiological pro-cesses is not taken into account (Katz, 1977; Monteith, 1981; France and Thornley, 1984; Touré et al., 1994). For example, monthly climatic effects predicted by a regression model are not easily interpreted from a physiological background because the model can only be an approximation of the underlying processes, and may fail to include some of them. Because of their empirical nature, regression models are restricted to the range of climate data from which they are developed.

Advances in scientific understanding of the plant’s growth processes led to the formulation of determin-istic growth models. They attempt to simulate the growth processes throughout the year by modelling relevant plant processes. The Global Climate and Terrestrial Ecosystems (GCTE) group recognises that there are at least 14 wheat models which attempt to account for physiological processes that gov-ern wheat growth and development (GCTE, 1992). Wheat simulation models applicable to UK climatic conditions include AFRCWHEAT2 (Porter, 1993), CERES-wheat (Ritchie and Otter, 1985) and SIRIUS (Jamieson et al., 1998). Such models are widely ap-plied in decision support and studies of the impact of climate change on wheat production.

Crop simulation models assume that the dynamic mechanistic process formulations can be represented accurately, and that the model parameters can be correctly determined. However, of necessity, most of the important processes within simulation models are described by empirical functions, since no models exist for the enormously complex (and poorly un-derstood) mechanisms underlying phenology, canopy

development and senescence, partitioning etc. Thus, simulation models also cannot be used outside the re-gion they were developed for with confidence. Young et al. (1996) argued that “a data-based mechanistic modelling” philosophy is the way forward. This sug-gests a model category — a mechanistic and statistical model — between the extreme modelling approaches outlined earlier which assume that either nothing or everything is known about the crop-climate system. The means to achieve this is the derivation of the most parsimonious model based on knowledge of the system; i.e. one that uses the minimum number of parameters without losing predictive power.

In the present study, we employ a large yield data set from agricultural experiments on winter wheat in the UK to develop a new model for predicting well-managed wheat grain yields from climatic in-put. The objective of this work was to develop, as an example of the new methodology described ear-lier, a parsimonious, empirically-based model which takes into account mechanisms of wheat growth and development. In order to establish how well such a hybrid-model behaves in the multi-site, multi-year UK environment, we also perform an extensive inde-pendent test of the suggested model.

2. Data sources and methods

2.1. Winter wheat trials database and weather data

An electronic database was established consist-ing of wheat trials undertaken durconsist-ing the period September 1975–August 1993. The database con-tains grain yields at 85% dry matter together with additional information such as treatments, experi-mental design, grid reference, altitude, sowing date, cultivar and type of trial (e.g. variety trial, nitrogen response trial). Trials from most UK agricultural in-stitutes were included (details in Landau, 1998). Data were restricted to autumn-sown, fungicide-treated trials of cultivars of bread-making varieties that had been on the UK recommended list. For each variety the treatment combination that produced the highest average plot yield was taken to reflect the trial’s best managed yield. This procedure implicitly assumed that a high-yielding treatment had been applied and that the crops were overall well-managed in terms

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Fig. 1. Locations of crop trials included in the well-managed yield data set (840 trials). The plotting symbol indicates the yield sample to which a trial was allocated: the first (O) or second development sample (1) or the test sample (×).

of factors such as cultivation, spraying regime and nutrient supply.

The well-managed yield data set consisted of 840 trials (site × year combinations) and 1992 average yield observations from major wheat-growing areas in the UK (Fig. 1). The trial set was split into three samples. We refer to these samples as the ‘first de-velopment sample’, the ‘second dede-velopment sample’ and the ‘test sample’. The development samples were used to develop the new yield-climate model, while the yield data in the test sample were reserved for in-dependent testing of the new model. Two-thirds of the field trials were assigned at random to a development sample and the remaining third to the test sample. Due to time constraints on the project, the develop-ment sample was further divided (again at random). A

test of the predictive accuracy of existing crop models (Landau et al., 1998) and the early model development stage was carried out only with the first development sample.

The allocation of the mainland trials to the three samples was performed by stratified random sampling from the set of trials. Strata were defined by regions within harvest years. Ten homogeneous weather zones were used to define regions (UK Meteorological Of-fice, 1985). The sizes of the subsamples from each stratum were chosen to be proportional to the stra-tum sizes. Due to late availability all Northern Ireland trials were allocated to the test sample. Fig. 1 shows that all three samples were representative of main-land wheat growing regions and Table 1 demonstrates the size of the sampling variation in annual average yields.

Daily climate data were considered (as in sim-ulation models) as our starting point in the search for a more empirically-based statistical model. We considered minimum and maximum daily tempera-tures and daily radiation and rainfall totals to be the essential climatic variables affecting wheat yields. Daily weather data for the period covered by the winter wheat trials database were retrieved for 234 meteorological stations within the UK. Radiation records were sparse (only 19 meteorological sta-tions recorded radiation throughout the period of interest) but sunshine durations were available. Be-cause meteorological stations do not necessarily coincide with the crop sites, methods to interpo-late yearly series of daily weather data for arbi-trary sites in the UK were developed (Landau and Barnett, 1996) and used to generate daily weather data at the crop sites. The selected interpolation method took account of the spatial and temporal variation in the weather data and explained vari-ation in observed weather well for a set of dates at randomly selected sites (94, 97, 84 and 74% of the variation in minimum and maximum temper-ature, sunshine hours and rainfall were explained, respectively). Because radiation records were sparse, radiation was estimated from the interpolated sun-shine durations by a standard linear relationship (Rietfield, 1978). The latter method was able to ex-plain 95% of the variation in daily radiation mea-surements over 2 years at the available recording stations.

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Table 1

Descriptive statistics for the three yield samples (1=first development sample, 2=second development sample, 3=test sample) by harvest year

Yield sample 1 2 3 1 2 3 1 2 3 1 2 3

Harvest year Number of trials Number of yield values Mean yield (t ha−1₎ _{Standard deviation (t ha}−1₎

1976 3 2 1 4 3 1 7.51 7.17 6.21 0.55 0.71 –

1977 2 1 1 4 2 2 6.82 7.50 8.30 0.67 0.43 0.78

1978 2 3 2 3 6 4 7.11 6.47 6.45 0.15 0.94 0.25

1979 3 3 1 9 9 4 7.67 6.60 6.04 0.86 0.61 0.57

1980 4 4 1 10 6 1 8.24 8.12 5.13 0.86 1.74 –

1981 35 20 22 45 22 25 8.35 8.09 8.58 1.40 1.54 1.47

1982 41 26 33 90 47 69 8.29 8.03 7.82 1.19 1.37 1.70

1983 39 28 29 81 57 59 8.84 8.04 8.69 1.38 1.43 0.84

1984 28 20 29 63 40 75 9.77 9.52 9.12 1.52 1.19 1.40

1985 26 22 21 79 54 57 7.95 7.85 7.43 1.03 0.98 1.10

1986 35 19 30 95 58 93 8.65 8.27 7.94 1.09 0.90 1.12

1987 30 16 23 53 40 62 7.44 7.21 7.57 1.14 1.01 1.09

1988 26 16 18 50 25 29 8.12 8.21 7.89 0.91 1.09 1.11

1989 12 9 7 29 21 17 8.11 8.36 7.71 0.86 0.90 1.20

1990 11 8 18 35 18 44 8.85 9.01 8.34 1.20 1.11 1.23

1991 13 12 17 29 28 39 8.36 8.94 8.01 1.18 0.90 1.54

1992 14 6 10 56 22 42 8.66 8.62 8.59 1.25 1.97 1.06

1993 17 7 14 80 25 71 9.23 8.87 8.80 1.70 1.42 1.58

1976–1993 341 222 277 815 483 694 8.52 8.23 8.19 1.37 1.36 1.40

2.2. Strategies adopted for developing the new hybrid-model

It was envisaged in the initial stages of this project that an existing crop model could be identified which offered the closest prediction of grain yield, and a par-simonious model would then have been developed on the basis of the selected model. However, we found that none of the crop models considered (AFRC-WHEAT2, CERES-wheat, modified version 3.0 and SIRIUS, version 3) was able to explain the variation in grain yields in the first development sample of well-managed yields (Landau et al., 1998), the rea-sons for which are discussed in Jamieson et al. (1999) and Landau et al. (1999). Hence, rather than reducing the complexity of a simulation model, the develop-ment of a parsimonious model was approached from a minimalist viewpoint, introducing effects one by one and allowing inclusion in the final model only if they improved prediction.

The first and second development samples of wheat yields represented the empirical basis upon which to build the model. The hybrid-model was developed by employing elements of both empirical and mechanistic modelling. Knowledge of wheat physiology was used

to suggest simple expressions of climate effects on yields. Then this large set of observed yields was used to assess the empirical importance of the suggested climate variables. Only empirically important climate variables were included in the final model.

The climate response sub-model was developed in two stages — a variable selection process and a model fitting process. Firstly, during the variable selection process, a set of physiologically meaningful climate variables, which could potentially explain variations in the observed yields, was established. This set of potential explanatory variables constitutes the maxi-mum model. Secondly, during the model fitting pro-cess, this model was reduced to contain only statisti-cally significant terms to form a parsimonious model. The variable selection process was based on yields from the first development sample alone, whereas the final model was fitted to yields from both the first and second development sample.

2.3. The variable selection process

A pool of climatic explanatory variables to be considered during the variable selection process was created using physiological/agronomical guidance as

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follows. Firstly, in order to reduce the number of in-puts greatly, summaries of the climatic effects over physiologically-meaningful periods were investigated rather than the effects of daily climatic inputs. The crop year was split into five phases: a vegetative phase, an early-reproductive phase, an anthesis phase, a grain-filling phase and the remaining pre-harvest phase. Details concerning the choice of these phases are given in Section 2.4. Secondly, knowledge of wheat physiology and agronomy was employed to suggest simple expressions for expected climate ef-fects within these phases. An expression could contain a single variable, or of a set of variables (principally two variables) whose added effects would reflect the expected climate effect. For example, one possible expression of an expected (disease-related) rainfall effect during a phase could be the additive effects of mean and maximum rainfall during the phase. Knowledge was also used to determine the type of dependence expected for variables involved in each expression — either positively or negatively related to grain yield.

A cumulative procedure using regression methods was adopted in the variable selection to establish the maximum model. During the procedure expressions of expected climate effects were assessed for their poten-tial in explaining variation in observed yields. Within each step of the procedure alternative expressions of a single expected climate effect were compared and the explanatory variables involved in the best expres-sion, if any, were added to the set of potential ex-planatory variables. The percentage of yield variation explained by a regression model was measured by the adjusted coefficient of determination (R2ad, for defini-tion see Payne et al., 1993). In each step the expres-sion which increased the R2ad most, relative to the

R2adof the best model in the previous step, and whose coefficients estimate(s) showed the expected sign(s), was selected as a potential candidate for the final model.

We considered climatic effects in the order of their anticipated importance to grain yield. The importance of each effect was evaluated in terms of our knowl-edge of wheat physiology and agronomy combined with results from an explorative analysis (details in Landau, 1998). When selecting variables, yield ob-servations were used in aggregated form. As in the test of the crop models (Landau et al., 1998), the

ob-servations were averaged for each year within 1 km squares, which represent the precision of the available grid references. The ‘replicates’ within the squares re-flect trials where several valid cultivars were tested together, or where several trials were situated in the same square. Aggregation of the observed data set sim-plified the variable selection process by reducing the variation in grain yield to a component potentially ex-plainable by climate variables. To take account of the fact that means based on many observations are less variable than means calculated from few observations, weights reflecting the number of replicates were em-ployed in all the regressions.

2.4. Initial choice of phenological stages

We aimed to define periods within which to aggre-gate the climate data that reflect key periods of plant development. Although the general dependence of de-velopment stages on thermal time is well established, differences exist in the approaches taken to model the relationship. Date of anthesis has been shown to be the most crucial developmental date for predict-ing grain yields from climate. Grain yields have been found to be most sensitive to changes in radiation lev-els around anthesis (Fischer, 1975, 1985; Willington and Biscoe, 1985; Savin and Slafer, 1991; Tribo¨ı and Ntonga, 1993; Mitchell et al., 1996) and the date of an-thesis determines the onset of the grain-filling phase. The latter is important in that climatic conditions dur-ing this phase, especially those determindur-ing the dura-tion of grain-filling itself, have been shown to affect grain yields (Monteith and Scott, 1982; Fischer, 1983; Spiertz and Vos, 1985; Moot et al., 1996).

We therefore identified the best existing predictor of anthesis dates by comparing anthesis date predictions of crop models applicable to the UK (AFRCWHEAT2, CERES-wheat and SIRIUS; run for cultivar Avalon) with a set of observed anthesis dates from 57 UK trials (described in Landau, 1998). This showed that CERES-wheat best predicted the date of anthesis (Table 2). CERES-wheat and AFRCWHEAT2 showed almost identical correlations with the observed dates but CERES-wheat predictions were less biased. SIR-IUS widely overpredicted the anthesis dates due to a fault which has been corrected in later versions of the model (M.A. Semenov, personal communication).

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Table 2

Correlation between observed and predicted anthesis dates (r), bias of predictions (bias) and root mean square error (RMSE) of predictions from crop models for the set of 57 observed anthesis dates

Crop model CERES-wheat AFRCWHEAT2 SIRIUS

r 0.61 0.62 0.18

Bias (days) 3.11 −5.32 9.02

RMSE (days) 7.41 8.57 14.51

CERES-wheat was therefore used to define a 21-day anthesis phase by allowing a 10-day window on either side of the predicted anthesis date.

The crop year was then split into five phases (Phases I–V) by estimating an early-reproductive phase to occur before and a grain-filling phase to oc-cur after the predicted anthesis phase. The remaining days at the beginning and end of the of the crop year were assigned to a vegetative and pre-harvest phase, respectively. Fig. 2 illustrates alternative definitions for the phases before (Phase II) and after (Phase IV)

Fig. 2. Alternative phase definitions used during the variable selection process. Phase I starts with sowing (S) and Phase V ends with harvest (H). Phase III is the 21-day window around the anthesis date predicted by CERES-wheat (ACE). There are three options for the

start of Phase II:SIIa(18th April) andSIIb(10th May) are fixed dates while the date ofSIIIc depends on thermal time (500 degree days

before the start of Phase III). There are two options for the end of Phase IV:EIVa is 600 degree days (above base temperature of 0◦C)

andEIVb is 470 degree days (above a base temperature of 1◦C and below an upper limit of 26◦C) after the end of Phase III.

the anthesis phase (Phase III).SIIa andSIIb represent an early and a late estimate of terminal spikelet. The fixed day predictors were derived from the idea that sensitivity to radiation peaks at anthesis and dimin-ishes towards terminal spikelet, so that variations in terminal spikelet dates would be unimportant as long as the area of low sensitivity was identified.SIIc measures the distance between anthesis and terminal spikelet in thermal time using an approximation of the ARCWHEAT1 thermal time requirement (Weir et al., 1984). All formulations used to predict the end of grain-filling were based on thermal time.E_IVa

reflects an approach to estimating the grain-filling phase which is intermediate between AFRCWHEAT2 and SIRIUS, whereas EIVb is defined according to the CERES-wheat formula (all for cultivar Avalon). The latter generally results in a shorter grain-filling duration than the other two models. During the vari-able selection process, expressions of climate effects as well as these alternative phase definitions were tested.

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2.5. The model fitting process

At the end of the variable selection process the best definitions of the phenological phases were iden-tified. The next step was the translation of these def-initions into a phenology sub-model and the genera-tion of climate input required in the maximum model using this sub-routine. Then during the model fitting process formal inference was used to assess the sig-nificance of terms included in the maximum model on the basis of both development samples. A parsi-monious yield response sub-model was determined by step-wise dropping of variables from the maxi-mum model. At each step the explanatory variable which tested as insignificant at the 5%-level using the

F-test and gave the smallest variance ratio, or a vari-able which showed an unexpected sign for its coef-ficient estimate, was dropped. All dummy variables were kept in the model as adjustment factors. Vari-ables reflecting main effects were only dropped, once any interaction term involving them had disappeared. Also, non-linear threshold parameters were retained as they were believed to be needed to ensure physiologi-cal meaningfulness. The significance testing assumed independently distributed normal errors with expecta-tion zero and unknown but constant variance. Residual diagnostics were employed throughout to check the distributional assumptions. In contrast to the variable selection process the original non-aggregated yields were used during the fitting process.

The relative importance of each term in the final cli-mate response sub-model was assessed by decompos-ing the model’s regression sum of squares into com-ponents due to each term. Because the explanatory variables were not orthogonal to each other, the order in which the terms were added into/dropped out of the model affected the part of the sum of squares that was attributed to them. Therefore, for comparison pur-poses, a forward and a backward selection procedure were employed to achieve a decomposition.

2.6. Testing the new hybrid-model

The new parsimonious hybrid-model was tested with independent observed yields in the test sample to provide an assessment of the predictive accuracy of the new hybrid-model in practice. This also allowed

comparison of the predictive accuracy of the new hybrid-model with that of the mechanistic crop mod-els which had already undergone independent testing for UK well-managed yields (Landau et al., 1998).

As in Landau et al. (1998), observed yields were av-eraged within 1 km squares within each year to match the precision of the interpolated weather variables. The root mean square error (RMSE) of differences be-tween observed and predicted yields and correlations were employed to measure the accuracy of the new hybrid-model for predicting temporally and spatially distributed UK yields. The new model’s accuracy for predicting annual average yields was also measured in order to assess the model’s ability to predict purely temporal variation in UK well-managed yields. To take account of the fact that the variance of average yields is inversely proportional to the number of originally available yields all accuracy measures were weighted accordingly.

3. Results

3.1. Simplified CERES-wheat phenology sub-model

A parsimonious phenology model which splits the crop year into the five phases during which climate data is aggregated was defined by simplifying the CERES-wheat phenology sub-routine (Hodges and Ritchie, 1991). The structure and parameters of this sub-routine were retained because firstly, this phe-nology model had performed best in explaining the variation in a set of observed anthesis dates (Table 2) and it was used to define the initial anthesis phase (Fig. 2). Secondly, during the variable selection pro-cess the CERES-wheat formula for the duration of grain-filling (EIVb, Fig. 2) was identified as the one which explained most yield variation by aggregated climate input during a grain-filling phase. The defi-nition identified to aggregate climate input during an early-reproductive phase (S_IIc, Fig. 2) had not orig-inated from CERES-wheat but was easily redefined within the framework of this sub-routine (Table 3).

The phenological sub-model requires specification of the cultivar-specific parameters sensitivity to ver-nalisationυ, sensitivity to photoperiodρ, phyllochron intervalφ and crop genetic coefficient ψ. Parameter settings were taken from the CERES-wheat model

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Table 3

Phases defined by parsimonious phenology sub-model and their approximate interpretation according to CERES-wheata

Phase Start (day of crop year) End (day of crop year) Interpretation

I S SII−1 Sowing to early terminal spikelet

II SII A−11 Early terminal spikelet to start of ear growth

III A−10 A+10 Start of ear growth to start of grain-filling

IV A+11 EIV Grain-filling

V EIV+1 H End of grain-filling to harvest

a_{The simplified CERES-wheat phenology subroutine is utilised to predict the terminal spikelet stage (}_S

II), the date of anthesis (A) and

the end of grain-filling (EIV). The crop year is defined as the period between 1st September and 31st August the following year.

as those supplied for cultivar Avalon (υ=0.033, ρ=0.008,φ=95 degree days,ψ=470 degree days).

The parsimonious phenology sub-model requires fewer inputs and is of less complexity than the CERES-wheat phenology sub-routine. Simplifications of the CERES-wheat routine were achieved by omit-ting adjustments believed to be of little effect in the UK and by using simpler formulations. Specifically, the simplified routine (i) bases all degree day criteria on mean daily temperatures within a range rather than employing a complicated mechanism involving min-imum and maxmin-imum temperatures within different temperature ranges, (ii) omits an adjustment for snow cover, (iii) does not model a delay in germination due to insufficient soil water content, (iv) uses simpler formulae for calculating daily vernalisation units and (v) does not allow for vernalisation to be reversed under warm conditions.

The simplified phenology model was tested against the set of observed anthesis dates. The predic-tive power of the simpler model (r=0.62, bias 2.9 days, RMSE 7 days) was almost identical to that of CERES-wheat (cf. Table 2) with the anthesis date pre-dictions of the two models differing by a maximum of 2 days (correlation between predictions,r=0.995). To ensure that the simpler phenology model had not omitted features important under wider climatic con-ditions than were present in the 57 trials where anthe-sis dates were observed, the model’s predictions were also compared with CERES-wheat predictions for the first development sample. Again, all differences be-tween predictions were found to be less than 3 days.

3.2. Parsimonious yield response sub-model

During the variable selection process the maxi-mum model was established from the pool of climatic

explanatory variables (based on the n=303 aggre-gated yield observations in development, Sample 1). The pool of climatic explanatory variables reflected direct and indirect effects (i.e. via effects on pests, diseases, agronomy) of climate on well-managed yields. Emphasis was placed on climatic effects likely to dominate in the UK. A summary of the climate effect considered during the variable selection pro-cess is given in Table 4. More details can be found in Landau (1998).

Table 5 lists the 22 explanatory variables contained in the maximum model for climate effects on grain yield. The model was described by the basic linear relationship

Yi =β1+ 23 X j=2

βjXij+εi

whereYi, i=1, . . . ndenotes the i-th grain yield

ob-servation, Xij is the respective value of the j-th

ex-planatory variable Xj as defined in Table 5 and βj,

j=1. . .23 denote the linear parameters. The model also contained two non-linear parameters (α,γ) em-ployed in the definitions ofX15 andX16, respectively. To ease interpretation of the model’s constantβ1, ex-planatory variables were centred (Table 5). Finally, the error termεi reflects factors not accounted for in the

model, for example management factors or mere ran-dom (intrinsic) variability.

During the variable selection process, the relatively few yields observed during the early harvest years 1976–1980 (14 out of 303 aggregated values) were consistently over-estimated, presumably due to ma-jor changes in technology in the late 1970s. Hence in all subsequent analyses yields from these years were excluded to ensure availability of more recent technology.

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Table 4

Summary of climate effects assessed during the variable selection processa

Climate effect category Number of steps in Phase Climate effect Expected direction of

cumulative procedure climate effect on yield

Effects of rainfall 1 IV Disease −

Drought +

Lodging −

Sprouting −

2 II Disease −

Drought +

3 III Disease −

Drought +

Effect of radiation interception

4 III Carbon assimilation: amount

light energy available

5 IV Carbon assimilation: amount

light energy available

and temperature driven dura-tion of phase

−

6 II Carbon assimilation: amount

light energy available

and temperature driven dura-tion of phase

−

(Radiation damage) (−) Interaction between water

and radiation levels

7 II Radiation damage under

drought conditions

−

Yield loss under extreme 8 I–II Frost damage +

temperatures 9 III–IV Heat damage −

10 Meiosis Cold damage +

11 Anthesis Cold damage +

Yield loss due to harvest 12 V Shedding of over-ripe grain −

conditions Wetness at harvest −

Yield loss due to drilling 13 I Delay in sowing date −

conditions Wetness at sowing −

Effects during vegetative growth

14 I Varied temperature effects

(early canopy development, encouragement of aphid pop-ulation growth)

15 I Carbon assimilation +

16 I Disease, nitrogen leaching. . . −

a_{In each step of the cumulative procedure a set of alternative expressions for the respective climate effect(s) were investigated. The}

phases refer to the vegetative Phase (I), the early-reproductive Phase (II), the anthesis Phase (III), the grain-filling Phase (IV) and the pre-harvest Phase (V).

After the end of the selection process climatic ex-planatory variables could be defined over phases re-turned by the phenology sub-model. The first and sec-ond development samples of grain yields were then used during the model fitting process to select empir-ically important variables from this maximum model and to estimate their effects. A total ofn=1242 grain yield observations during the period 1981–1993 was analysed (see Table 1). The total variance of these ob-served grain yields amounted to 1.88 t2ha−2of which

R2ad=26.3% was accounted for by fitting the maxi-mum model.

Table 6 demonstrates the effect of the step-wise dropping of the unwanted explanatory variables. The effect of total radiation during Phase IV (X11) remained negative after dropping the interaction be-tween radiation and trial type (X13). Therefore, X11 was excluded from the model to ensure interpretabil-ity of model terms. Table 7 lists the constant term and the 17 explanatory variables included in the

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Table 5

Set of explanatory variables constituting the maximum modela

Explanatory Model term Interpretation (indicating the period over which

sum-variable maries were calculated)

X1 β1 Mean yield for non-variety trials for which sowing date

and harvest date were known and harvest took place before 31st August

X2 β2 (ln(P(mean)+0.1)−0.33) Main effect of mean and maximum daily rainfall in Phase

X3 β3 (ln(P(max)−P(mean)_+0.1)_−2.22)

X4 β4 V Main effect of variety trials

X5 β5 (ln(P(mean)+0.1)−0.33)V Interaction effect of mean daily rainfall in Phase IV and

the variety trial factor

X6 β6 (ln(P(prop)/(1−P(prop)))−0.69) Main effect of proportion of days in Phase II when rain

occurred

X7 β7 (ln(P(mean)+0.1)−0.12) Effect of mean and maximum daily rainfall in Phase III

X8 β8 (ln(P(max)−P(mean)+0.1)−1.87)

X9 β9 (R(tot)−351) Effect of total radiation in Phase III

X10 β10(D−31.98) Main effect of duration of Phase IV

X11 β11(R(tot)−517.6) Main effect of total radiation in Phase IV

X12 β12(D−31.98)V Interaction effect of duration of Phase IV and the variety

trial factor

X13 β13(R(tot)−517.6)V Interaction effect of total radiation in Phase IV and the

variety trial factor

X14 β14(R(mean)−15.49) Main effect of mean daily radiation in Phase II

X15 β15(((1−P(prop)−α)I[α,1](1−P(prop)))1/2 R(mean) −6.26) Interaction effect of the proportion of days in Phase II

when rain occurred and the mean daily radiation in Phase II

X16 β16((T(min3)−γ)I(−∞,γ](T(min3))+3.63) Effect of minimum daily temperatures throughout the

crop year

X17 β17(ln(P(mean)+0.1)+0.304) (1−B) (1−C) Effect of mean daily rainfall in the week before harvest

X18 β18B Effect of unknown harvest date

X19 β19C Effect of harvest later than 31st August

X20 β20(S−43.87) (1−F) Effect of delay in sowing date

X21 β21F Effect of unknown sowing dates

X22 β22(ln(P(mean)+0.1)−0.298) Effect of rainfall during February/March

X23 β23(ln(P(mean)+0.1)−0.267)(1−F) Effect of rainfall during the first 2 weeks after sowing a_{All explanatory variables are defined over phases returned by the phenology sub-model in Section 3.1 (Table 3). Linear parameters}

are denoted byβi; non-linear parameters byαandγ. For specified periods the mean (P(mean)) and maximum daily total rainfall (P(max)) measured in mm, the proportion of days when rain occurred (P(prop)_{), the total radiation (}_R(tot) _{in MJ m}−2_{), the mean daily radiation}

(R(mean) _{in MJ m}−2 _{per day), the minimum mean daily minimum temperature of three consecutive days (}_T(min3)_in◦_{C) and the duration} of the period (Dmeasured in days) itself are employed as climate input summaries. The dummy variableVreflects a variety trial factor. It takes the value one if the respective yield value has been obtained from a variety trial and the value zero otherwise. VariablesB,Cand

Fdefine dummy variables employed to take account of conditions at harvest and sowing. Variable Btakes value one when the harvest date is unknown;Ctakes value one when the harvest date was known but harvest occurred after the 31st August. VariableFtakes the value one when the sowing date was unknown.Sis the sowing date measured in days since 1st September.I[a,b](x) denotes the indicator

function which takes the value one whenxlies within the interval [a,b] and takes the value zero otherwise.

selected yield response sub-model and their estimated effects. This final parsimonious yield response model was able to explain R2_ad=26.1% of the variance in grain yields. It predicted grain yields in development Samples 1 and 2 with a RMSE of 1.17 t ha−1 _and

achieved a correlation between yield observations and predictions ofr=0.52.

The relative importance of each term in the yield response sub-model was assessed by decomposing the model’s regression sum of squares into components

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Table 6

Sums of squares (SS) and variance ratios (VR) for terms dropped from the maximum modela

Model (d.f.) Explanatory SS VR

variable(s)

Null model (1239) X1 2328 –

Maximum model (22) X1,. . .,X23 640 20.96

Drop variable (1) X23 0.004 <0.001

Drop variable (1) X13 0.66 0.48

Drop variable (1) X11 2.97 2.15

Drop variable (1) X12 1.38 1.00

Drop variable (1) X8 2.86 2.06

Selected model (17) X1,. . .,X7,X9,

X10,X14,. . .,X22

632 26.78

a_{A single term significantly improved the model fit at}

the 5%-level if its variance ratio exceeded the 95%-quantile of an F-distribution with 1 and 1200 degrees of freedom (F1,1200;0.95=3.84).

due to each term using a forward and a backward se-lection procedure. Table 8 shows that both decompo-sitions attributed similar relative importances to the model terms. A major change in ranking was only found for the sowing date (X20), the effect of the du-ration of Phase IV (X10) and the mean rainfall effect during Phase IV (X2).

Table 7

Parameter estimates and their standard errors (s.e.) for the parsi-monious yield response sub-model (n=1240)

Explanatory Estimate of Estimate of variable linear parameter non-linear

βj (s.e.) parameter (s.e.)

X1 8.649 (0.164) –

X2 −1.346 (0.14) –

X3 0.5731 (0.0893) –

X4 −0.1466 (0.0991) –

X5 0.518 (0.139) –

X6 −2.06 (0.29) –

X7 −0.2237 (0.0703) –

X9 0.002021 (0.000851) –

X10 0.0622 (0.0121) –

X14 0.1483 (0.0407) –

X15 −0.3766 (0.0674) 0.155 (0.0065)

X16 0.0636 (0.0142) −3.21 (2.29)

X17 −0.1360 (0.0564) –

X18 −0.1080 (0.0938) –

X19 −0.522 (0.130) –

X20 −0.00822 (0.00392) –

X21 0.10 (0.117) –

X22 −0.6628 (0.0984) –

Table 8

Sums of squares (SS) and variance ratios (VR) for each term included in the parsimonious yield response sub-modela

Step Forward selection Backward selection Variable SS VR Variable SS VR 1 X16 107.54 59.98 X18 1.88 1.35

2 X6 109.26 64.04 X21 1.47 1.06

3 X22 83.67 51.02 X20 5.86 4.21

4 X7 62.43 39.24 X9 8.16 5.84

5 X19 32.24 20.59 X17 7.62 5.43

6 X20 28.13 18.21 X4 16.18 11.44

7 X2 21.02 13.75 X14 22.73 15.88

8 X3 56.45 38.04 X15 17.20 11.91

9 X5 36.11 24.81 X19 24.94 17.04

10 X10 26.16 18.22 X5 30.17 20.29

11 X15 16.06 11.28 X10 44.97 29.54

12 X14 22.30 15.85 X7 49.05 31.43

13 X4 12.81 9.17 X3 58.73 36.54

14 X17 6.87 4.94 X2 42.25 25.76

15 X9 7.27 5.24 X22 83.67 49.04

16 X21 1.47 1.06 X6 109.26 60.93

17 X18 1.88 1.35 X16 107.54 57.25 a_{A step-wise forward and backward variable selection}

proce-dure were employed. A term significantly improved the model fit at the 5%-level if its variance ratio exceeded 3.84.

3.3. Independent testing

Evaluation of the predictive accuracy of the parsi-monious hybrid-model on the basis of development samples 1 and 2 was expected to be over-optimistic since these yield data represent the empirical basis from which the model was developed. Independent testing of the new model was based on yield data from the test sample. This sample containedn=246 1 km square yield values during the time period for which the new model was applicable (1981–1993, see Table 1).

Fig. 3 demonstrates the performance of the new par-simonious hybrid-model for predicting the temporally and spatially distributed UK well-managed yields in the test sample. The parsimonious hybrid-model was almost unbiased (bias=0.078 t ha−1), the correlation between observed and predicted yields was r=0.41 (significantly different from zero at the 5% level) and the RMSE when predicting yields with the new model was 1.21 t ha−1_.

Fig. 4 shows the temporal trend in annual average UK well-managed yields and their predictions from the hybrid-model. The model predictions followed the

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Fig. 3. Yields predicted by the parsimonious hybrid-model plot-ted against observed grain yields in the independent test sample (n=246). The 1:1 line is shown representing perfect agreement. observed annual average yields well. The correlation between annual average predicted and observed yields was r=0.77 (again significant at the 5% level). This

Fig. 4. Annual average observed yields in the independent test sample (closed symbols) and annual average predictions from the parsimonious hybrid-model (open symbols) plotted against year.

indicated that purely temporal variation in annual av-erage yields was more easily accounted for by cli-matic differences according to the new hybrid-model than the combined spatial-temporal variation in UK well-managed yields.

4. Discussion

We have developed a new model for wheat grain yield response to environment. The model is ap-plicable to spatially and temporally distributed UK well-managed yields. This new parsimonious hybrid-model represents an attempt to bring together empirically-based statistical modelling with mecha-nistic modelling. The hybrid-model consists of simple expressions of climate-yield mechanisms for which empirical evidence exists, thus ensuring empirical importance and interpretability.

The hybrid-model adheres to the principle of par-simony. The complexity of the phenology sub-model is much reduced relative to that of the CERES-wheat sub-routine, and it is driven by daily mean tem-peratures alone. Although daily series of minimum temperatures, daily radiation levels and daily rainfall

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values remain the initial inputs to the yield response sub-model, its structure is very simple. The rela-tionship between the aggregated (and possibly trans-formed) climatic explanatory variables is assumed to be linear, with individual climate effects simply adding to each other.

We have demonstrated the ability of the parsimo-nious hybrid-model to predict temporally and spatially distributed UK well-managed yields in a large inde-pendent test (246 aggregated yields, correlation be-tween observed and predicted yields 0.41, Fig. 3). Predictive power of the developed model on the tem-poral as well as on the spatial scale is evident be-cause the sample of observed yields was constructed to cover a range of years (1981–1993) and all major UK wheat-growing regions. In fact, the hybrid-model pre-dicted temporal differences in annual average yields more accurately (correlation between observed and predicted annual average yield 0.77) than differences in the original spatially and temporally distributed yields. The latter suggests that within-year variation in yields is more affected by factors not included in the model — for example differences in sub-optimal management and physical site characteristics.

However, the predictive accuracy achieved was consistent with results in the literature for models for a single site. The most relevant finding was that of Chmielewski and Potts (1995) whose single-site yield-climate regression model achieved a corre-lation of 0.44 when predicting a long-term series (1854–1967) of farm yard manure treated grain yields from rainfall and temperature data in an independent test. The success of the hybrid-model is in sharp con-trast to that shown by the crop models in the previous validation study (Landau et al., 1998). However, the lack of success of the crop models may result from the fact that these cater for optimally managed yields which are not affected indirectly by weather (i.e. through sub-optimal management). Indirect weather effects, for example negative rainfall effects were found to play a major role even in well-managed UK yields and explain part of the improved predictive accuracy of the new model.

4.1. Empirical importance of climate effects

The yield response sub-model (Tables 5 and 7) serves to identify physiologically and agronomically

justified climate effects which are of empirical impor-tance for well-managed UK wheat trials. These can be ranked in order of importance (Table 8, backward selection procedure):

• Negative effects of rainfall during the estimated early-reproductive phase, the estimated anthesis phase, the estimated grain-filling phase and the February/March period are the most dominant ef-fects in the climate response sub-model (explaining 54% of the grain yield variation accounted for by the model).

• Yield loss due to extreme frosts accounts for a further 17% of the explained variation in grain yields.

• Yield is affected by the type of trial (variety or non-variety). In absence of climate effects, smaller yields are predicted for variety trials. But variety trials are found to be less sensitive to rainfall during grain-filling. Trial type differences amount to 7% of the explained yield variation.

• The model predicts that the longer the duration of the grain-filling phase, the higher the yields. This climate effect accounts for a further 7% of the vari-ation explained by the model.

• Damage due to adverse harvest conditions (high rainfall in the week preceding harvest or late har-vesting) accounts for 6% of the variation.

• Positive radiation effects during the early-repro-ductive phase and the anthesis phase contribute 5% of the variation explained by the model.

• The model predicts that rainfall and radiation levels in the early-reproductive phase interact with each other. Once a drought threshold is reached, radia-tion has a negative effect, with the strength of the effect depending on the drought level. This interac-tion accounts for 3% of the variance explained by the model.

• Finally, delaying the sowing date has a negative effect on yield. This contributes the remaining 1% to the explained variation. However, the forward selection procedure assigned a higher contribution (Table 8).

Negative rainfall effects account for the major-ity of the yield variation explained. Consistent with an assessment by Monteith and Scott (1982) UK well-managed grain yields are shown to be affected more by indirect climate effects than by direct climate effects.

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4.2. Physiological/agronomical interpretation

The dominant negative effect of rainfall during grain-filling is consistent with expectation from the viewpoint of ripening diseases. Amount and duration both have a detrimental effect on yield by providing favourable conditions for diseases, denying access of machinery to the land, washing off sprays. Possibly lodging and sprouting may contribute to the effect, but these would be expected to occur during specific parts of grain-filling, and lodging would be associated with the intensity of rainfall rather than the average amount. Relevant expressions for the latter effects were considered, but did not perform as well as the expressions consistent with a disease effect. On the basis of these arguments, negative rainfall effects dur-ing the anthesis phase are also attributed mostly to increased incidence of disease.

Apart from incomplete disease control and water-logging, the negative rainfall effects during February/March may be due to washing away of sprays, or leaching of fertiliser treatments since a small, 40 kg ha−1 nitrogen is recommended at this time (UK Ministry of Agriculture Fisheries and Food, 1985). The main nitrogen application is recommended for the early stem extension stage (usually April), corresponding approximately to the start of the early reproductive phase. It therefore seems likely that the negative effects of rainfall during the early reproduc-tive phase was related to nitrogen leaching, consistent with the finding that rainfall during the March to May period has a strong negative effect on crude protein concentration (Smith and Gooding, 1996).

The finding of yield differences between variety and non-variety trials is attributed to differences in management. However, confounding with a technol-ogy trend or with geographical characteristics cannot be ruled out because the major trial series contributing non-variety trials (UK Agricultural Development Ad-visory Service trials) were carried out in England and Wales before 1989. The dependence of sensitivity to rainfall on trial type may be explained if variety trials have a higher level of disease control than non-variety trials.

The positive effect of winter minimum temperatures below a threshold, the positive effect of the duration of grain-filling and the positive effect of radiation lev-els during the early reproductive and anthesis phases

are in line with physiological expectations of win-ter kill, beneficial effects of prolonged grain-filling and radiation-driven growth, respectively. In fact, the model assigns more importance to the duration of grain-filling than to radiation levels, supporting the assessment by Monteith and Scott (1982). However, the threshold of −3.2◦_{C below which winter kill} is estimated to occur is higher than the threshold value of −20◦_{C suggested by Petr (1991) and the} range of threshold values for tiller damage used in CERES-wheat (−18◦_{C to}₋₆◦_C).

The model predicts yield loss due to wet conditions in the week before harvest. It also includes a penalty for late (after August) harvesting, which may have a detrimental effect because the risk of shedding in the meantime and of harvesting under wet conditions is increased (Cranstoun, 1996). Finally, the date of sowing itself is modelled to affect yield. Consistent with the findings of Green and Ivins (1985) the model assumes that later sowing date reduces yield.

Consistent with the finding of Baier and Robert-son (1968) that yield was most closely related with soil moisture before anthesis, empirical evidence of drought effects was found only during the early repro-ductive phase. During this phase, yield loss depend-ing on drought and radiation levels is predicted when a threshold of 15% drought days during the phase is exceeded. No empirical evidence of such effects was found during anthesis and grain-filling phases, possi-bly due to the overpowering negative effects of rain-fall. Drought during later phases may be less damag-ing to wheat crops because of their ability to use stem reserves (Fischer, 1983).

Sensitivity estimates of climate variables showed smaller effects than expected (Table 7). The model es-timates a loss of 0.062 t ha−1 for each day by which the duration of grain-filling is reduced compared to 0.175 t ha−1 per day estimated from the findings of Vos (1981). The sensitivity to the radiation total dur-ing the anthesis phase (0.002 t ha−1MJ−1m2) is also much smaller than sensitivities derived from the litera-ture (0.0094–0.0144 t ha−1_MJ−1_m2_{). Finally, the} lit-erature estimate of a loss of 0.028 t ha−1_{for each day} delay in sowing after mid-September (Green and Ivins, 1985) greatly exceeds the model’s sensitivity estimate of 0.0082 t ha−1 _{per day. These findings suggest that} experiments where one factor is varied under condi-tions of optimal management may exaggerate effects

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compared to that effect averaged over many different sites, years, varieties and where there are interacting stresses.

5. Concluding remarks

The particular model developed here could not be applied outside the UK. However, with yield, weather and management data from a wider range of climates, the approach used could be used to extend the model and many more of the effects suggested by knowl-edge would become significant. Many of the features included in simulation models might well be incorpo-rated, always justified as improving prediction. The ex-ception to this rule would be for climate change appli-cations, where it would be necessary to include terms in the model which do not improve prediction with current data (e.g. to describe responses to CO2), and in this case long-term projects gathering new yield data to continually check predictions would be needed. As all crop models contain empiricisms, we suggest that it is essential to test all models used to predict yields with such data to identify the optimum approach. It is therefore important that statisticians, crop physiolo-gists/modellers and agronomists collaborate for future work in developing tools to predict yield.

Acknowledgements

This work was funded by the Biotechnological and Biological Sciences Research Council. Funding for R.A.C. Mitchell from the UK Ministry of Agriculture, Fisheries & Food is acknowledged. We thank the Agri-cultural Development Advisory Service, the National Institute of Agricultural Botany, Biomathematics and Statistics Scotland, the Scottish Agricultural College, the Department of Agriculture for Northern Ireland, the Morley Research Centre and the Harper Adams College for providing data.

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Table 6

Sums of squares (SS) and variance ratios (VR) for terms dropped from the maximum modela

Model (d.f.) Explanatory SS VR

variable(s)

Null model (1239) X1 2328 –

Maximum model (22) X1,. . .,X23 640 20.96 Drop variable (1) X23 0.004 <0.001

Drop variable (1) X13 0.66 0.48

Drop variable (1) X11 2.97 2.15

Drop variable (1) X12 1.38 1.00

Drop variable (1) X8 2.86 2.06

Selected model (17) X1,. . .,X7,X9,

X10,X14,. . .,X22

632 26.78 a_{A single term significantly improved the model fit at} the 5%-level if its variance ratio exceeded the 95%-quantile of an F-distribution with 1 and 1200 degrees of freedom (F1,1200;0.95=3.84).

Table 7

Parameter estimates and their standard errors (s.e.) for the parsi-monious yield response sub-model (n=1240)

Explanatory Estimate of Estimate of

variable linear parameter non-linear

βj (s.e.) parameter (s.e.)

X1 8.649 (0.164) –

X2 −1.346 (0.14) –

X3 0.5731 (0.0893) –

X4 −0.1466 (0.0991) –

X5 0.518 (0.139) –

X6 −2.06 (0.29) –

X7 −0.2237 (0.0703) –

X9 0.002021 (0.000851) –

X10 0.0622 (0.0121) –

X14 0.1483 (0.0407) –

X15 −0.3766 (0.0674) 0.155 (0.0065)

X16 0.0636 (0.0142) −3.21 (2.29)

X17 −0.1360 (0.0564) –

X18 −0.1080 (0.0938) –

X19 −0.522 (0.130) –

X20 −0.00822 (0.00392) –

X21 0.10 (0.117) –

X22 −0.6628 (0.0984) –

Table 8

Sums of squares (SS) and variance ratios (VR) for each term included in the parsimonious yield response sub-modela Step Forward selection Backward selection

Variable SS VR Variable SS VR

1 X16 107.54 59.98 X18 1.88 1.35 2 X6 109.26 64.04 X21 1.47 1.06 3 X22 83.67 51.02 X20 5.86 4.21 4 X7 62.43 39.24 X9 8.16 5.84 5 X19 32.24 20.59 X17 7.62 5.43 6 X20 28.13 18.21 X4 16.18 11.44 7 X2 21.02 13.75 X14 22.73 15.88 8 X3 56.45 38.04 X15 17.20 11.91 9 X5 36.11 24.81 X19 24.94 17.04 10 X10 26.16 18.22 X5 30.17 20.29 11 X15 16.06 11.28 X10 44.97 29.54 12 X14 22.30 15.85 X7 49.05 31.43 13 X4 12.81 9.17 X3 58.73 36.54 14 X17 6.87 4.94 X2 42.25 25.76 15 X9 7.27 5.24 X22 83.67 49.04 16 X21 1.47 1.06 X6 109.26 60.93 17 X18 1.88 1.35 X16 107.54 57.25 a_{A step-wise forward and backward variable selection} proce-dure were employed. A term significantly improved the model fit at the 5%-level if its variance ratio exceeded 3.84.

3.3. Independent testing

Fig. 3 demonstrates the performance of the new par-simonious hybrid-model for predicting the temporally and spatially distributed UK well-managed yields in the test sample. The parsimonious hybrid-model was almost unbiased (bias=0.078 t ha−1_{), the correlation} between observed and predicted yields was r=0.41 (significantly different from zero at the 5% level) and the RMSE when predicting yields with the new model was 1.21 t ha−1_.

Fig. 4 shows the temporal trend in annual average UK well-managed yields and their predictions from the hybrid-model. The model predictions followed the

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Fig. 4. Annual average observed yields in the independent test sample (closed symbols) and annual average predictions from the parsimonious hybrid-model (open symbols) plotted against year.

4. Discussion

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4.1. Empirical importance of climate effects

The yield response sub-model (Tables 5 and 7) serves to identify physiologically and agronomically

justified climate effects which are of empirical impor-tance for well-managed UK wheat trials. These can be ranked in order of importance (Table 8, backward selection procedure):

• Yield loss due to extreme frosts accounts for a further 17% of the explained variation in grain yields.

• The model predicts that the longer the duration of the grain-filling phase, the higher the yields. This climate effect accounts for a further 7% of the vari-ation explained by the model.

• Damage due to adverse harvest conditions (high rainfall in the week preceding harvest or late har-vesting) accounts for 6% of the variation.

• Positive radiation effects during the early-repro-ductive phase and the anthesis phase contribute 5% of the variation explained by the model.

• The model predicts that rainfall and radiation levels

in the early-reproductive phase interact with each other. Once a drought threshold is reached, radia-tion has a negative effect, with the strength of the effect depending on the drought level. This interac-tion accounts for 3% of the variance explained by the model.

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4.2. Physiological/agronomical interpretation

Apart from incomplete disease control and water-logging, the negative rainfall effects during February/March may be due to washing away of sprays, or leaching of fertiliser treatments since a small, 40 kg ha−1 _{nitrogen is recommended at this} time (UK Ministry of Agriculture Fisheries and Food, 1985). The main nitrogen application is recommended for the early stem extension stage (usually April), corresponding approximately to the start of the early reproductive phase. It therefore seems likely that the negative effects of rainfall during the early reproduc-tive phase was related to nitrogen leaching, consistent with the finding that rainfall during the March to May period has a strong negative effect on crude protein concentration (Smith and Gooding, 1996).

is estimated to occur is higher than the threshold value of −20◦_{C suggested by Petr (1991) and the}

range of threshold values for tiller damage used in CERES-wheat (−18◦_{C to}₋₆◦_C).

Sensitivity estimates of climate variables showed smaller effects than expected (Table 7). The model es-timates a loss of 0.062 t ha−1 _{for each day by which} the duration of grain-filling is reduced compared to 0.175 t ha−1 _{per day estimated from the findings of} Vos (1981). The sensitivity to the radiation total dur-ing the anthesis phase (0.002 t ha−1_MJ−1_m2_{) is also} much smaller than sensitivities derived from the litera-ture (0.0094–0.0144 t ha−1_MJ−1_m2_{). Finally, the} lit-erature estimate of a loss of 0.028 t ha−1_{for each day} delay in sowing after mid-September (Green and Ivins, 1985) greatly exceeds the model’s sensitivity estimate of 0.0082 t ha−1 _{per day. These findings suggest that} experiments where one factor is varied under condi-tions of optimal management may exaggerate effects