6 P. Lagacherie et al. Agriculture, Ecosystems and Environment 81 2000 5–16
simulations can then be undertaken by using these input data, if necessary, in combination with pedo-
transfer functions Bouma and van Lanen, 1987. The so obtained results are assumed to be valid for the
whole mapping unit, thus allowing easy mapping by classical GIS procedures.
This straightforward approach becomes question- able when yield must be mapped over large areas.
Generally, small scale soil maps 1:250,000 are the only available soil information over such areas.
At this scale the mapping units often include several taxonomic units, each of them characterised by a rep-
resentative profile. The common practice is to select the representative profile of the dominant taxonomic
units which can lead to neglect a substantial part of the mapping unit area Le Bas et al., 1998. Further-
more the taxonomic units of a small scale soil map cannot be reduced to the description of a unique set
of parameters measured at a given site because of its high within-unit variability. Resuming, the exclusive
use of quantitative information derived from a small scale soil map leads to a misrepresentation of the soil
variability of the region. This may have consequences on yield mapping and further decision making.
A possible alternative is to use the qualitative de- scription of the soil taxonomic units STUs of a
small scale map as input data for crop models. This description provides a more synthetic understanding
of the taxonomic units taking into account the entire information collected by the soil surveyor in the field,
i.e. soil profiles, auger hole observations and surface features observations. The description of a STU also
includes a description of environmental attributes e.g. geology, land use, slope. If maps of these at-
tributes exist for the studied region, the environmental descriptions can be used for estimating the location
of each STU within the complex soil mapping units of the 1:250,000 scale soil map.
All this qualitative information, generally based on well-established codification systems FAO-UNESCO,
1981; Baize and Jabiol, 1995, is now easily avail- able in soil databases Oldeman and van Engelen,
1993; Bornand et al., 1994; King et al., 1994. This information is, however, still underexploited in yield
predictions. The approach presented below estimates crop yields over a region by coupling simulations
with the qualitative descriptions of taxonomic soil units of a 1:250,000 soil database. Such coupling is
made through agrotransfer functions derived from a set of crop model simulations undertaken over a set
of representative soil climate situations. The proposed approach uses possibility theory for representing and
handling the required qualitative information. It was applied within the framework of the EC funded re-
search project IMPEL Rounsevell et al., 1998 for mapping hard wheat yields evolutions with changing
climate conditions over a region of 1200 km
2
located in the Languedoc plain southern France. This paper
focuses on the estimation of wheat yield under actual climate conditions. The climate changes issues dealt
in the IMPEL project are detailed in Wassenaar et al. 1999.
2. The spatial approach
2.1. Representing qualitative descriptions of soil taxonomic units by possibility distributions
Fig. 1 shows an example of description of STU in a regional soil database. STUs are non-geographic
components of soil landscape units King et al., 1994 delineated in small scale soil maps. The first part of
STU description Fig. 1, in italics deals with the STU environment whereas the second part is a description
of the soil properties which characterise the STU.
A soil or environment attribute qualifying a STU is expressed by a categorical variable corresponding to
a range of values according to a coding system. For example, the soil texture class ‘clay loam’, defined
in the texture triangle, corresponds to three intervals of values for the clay, silt and sand content. This
means precisely that if a STU is described as ‘clay loam’ then, 1 all the clay silt and sand content val-
ues included in the intervals defining ‘clay loam’ can be encountered within the STU, 2 nothing can be
said about the respective chances of occurrence of any of the individual values included in the intervals.
However, clay, silt and sand content values slightly inferior to the lower clay loam intervals limits, or
slightly superior to the upper ones, cannot be totally rejected since textural classes are subjective estima-
tions prone to errors and imprecisions. The qualitative description of a STU should, therefore, be expressed
by a set of intervals of values bounded by fuzzy limits.
P. Lagacherie et al. Agriculture, Ecosystems and Environment 81 2000 5–16 7
Fig. 1. An example of qualitative description of STU in a regional soil database. STU: soil taxonomic units, SMU: soil mapping unit, italic: description of STU environment, normal: description of STU soil properties.
A faithful representation of the information avail- able in soil database is of major concern in this study.
As intervals of values do not provide any information about the chances of occurrence of each soil prop-
erty values within the STUs, they cannot be translated into probability distributions without implicitly intro-
ducing a hypothesis which can influence final results Cazemier, 1999. Possibility theory provides a more
appropriate data representation which overcomes this problem Martin-Clouaire et al., 2000. In this frame-
work, knowledge about a variable X is represented by a possibility distribution π
X
Zadeh, 1978; Dubois and Prade, 1988 mapping the domain of X into the
[0, 1] interval. The distribution π
X
can be viewed as the membership function of the fuzzy set of possible
values of X. For any element s, π
X
s is the degree of possibility that X=s, with the convention that π
X
s=0 indicates that X cannot take the value s and π
X
s=1 means that nothing prevents X to take the value s. In
the transition area with possibilities strictly between 0 and 1, π
X
s
1
π
X
s
2
expresses that s
1
is a more plausible value than s
2
. The set of values s such that π
X
s=1 is called the core of the distribution; the set of values s such that π
X
s0 is called the support of the distribution.
Fig. 2 gives an example of trapezoidal possibility distribution of clay content π derived from a field
determination of a textural class clay loam. The core of the description corresponds to the interval of values
provided by the codification system used, here the FAO texture triangle. The lower and the upper boundaries of
the support of the distribution are defined, respectively, by subtracting to the core lower boundary and adding
to the core upper boundary a maximal admitted error which is fixed here at 10.
2.2. Mapping of soil hydraulic properties over the region
The general procedure used for mapping soil hy- draulic properties from qualitative descriptions of
STUs of a small scale soil map is illustrated in Fig. 3. This approach includes three steps which are
Fig. 2. possibility distribution of clay content p for the FAO textural class ‘clay loam’.
8 P. Lagacherie et al. Agriculture, Ecosystems and Environment 81 2000 5–16
Fig. 3. A general procedure for mapping soil hydraulic property from a regional soil database.
summarised below. More details can be found in Martin-Clouaire et al. 2000 and Cazemier 1999.
The first step Fig. 3 consists in predicting the soil hydraulic properties of interest for each STU. This
amounts to solving a system of equalities and inequal- ities as the following which estimates the available
water capacity awc.
w 10 = 31.2 − 0.185 ∗ sand + 0.0675 ∗ clay − 7.63 ∗ bulk density + I
err1
w 1500 = 39.9 − 0.094 ∗ silt − 0.204 ∗ sand − 12.3 ∗ bulk density + I
err2
awc = w10 − w1500 ∗ bulk density ∗ depth ∗ 100 − stone
100 clay + silt + sand = 100
w 10 w1500
w 10 = 4.66 + 0.914 ∗ w1500 + I
err3
1
The first two equations are pedotransfer functions pro- viding an estimate of the water retention properties to
determine awc. The third equation is the awc formula Baize and Jabiol, 1995. The last three mathemati-
cal expressions are constraints linking the variables of the first three equations. These relations express gen-
eral physical rules or statistical knowledge from soil databases.
In the example shown, the variables clay, silt, sand, bulk density, depth and stone are imprecisely known
and have values given by possibility distributions see Section 2.1 representing fuzzy intervals. The error
terms I
err1
, I
err2
and I
err3
are initially distributions of regression residuals which are traduced into fuzzy in-
tervals Martin-Clouaire et al., 2000 for being com- patible with the other variables. The computation of
the fuzzy value of the awc is therefore an arithmetic calculus over non-independent fuzzy quantities which
can be formulated as a fuzzy constraint satisfaction problem CSP Dubois et al., 1993 without requir-
ing any simplification hypothesis. In order to solve it efficiently, the constraint solver CON’FLEX Reiller
and Vardon, 1996 was used.
The second step of the spatial procedure is the map- ping of the STUs Fig. 3. This involves examining
each location in turn to find out the STUs that might be present on the basis of the environmental charac-
teristics of the STUs and of the site. The estimation of the possibility that a particular STU is present at a
particular site is done in two operations •
A fuzzy pattern matching Badia and Martin Clouaire, 1989 for calculating for each environ-
mental attribute the compatibility between the set of values characterising the STU as given by its
qualitative description Fig. 1 and the value taken by this attribute at the considered site as provided
by the environment maps geology map, DTM, . . . included in the regional database. The matching
may be partial due to the fact that the set may be
fuzzy and the retrieved value may be fuzzy too Cazemier, 1999.
• A conjunctive combination Dubois and Prade,
1988 which summarises the possibility values ob- tained for each individual attribute into a single
possibility degree.
The last step is the mapping of the variable awc over the entire region Fig. 3. This is done by pooling
at every location the fuzzy estimates of awc of each possible STU obtained at step 1, taking into account
the possibilities of presence of the STU as determined at step 2. This step involves a disjunctive combina-
tion operation Dubois and Prade, 1988. Altogether this three-step procedure returns a possibility dis-
tribution of the awc variable for each point of the region.
P. Lagacherie et al. Agriculture, Ecosystems and Environment 81 2000 5–16 9
2.3. Estimating crop yields over the region Once the techniques presented above are applied,
estimation of crop yields over the region can be done, using the possibility distribution of soil parameters and
climate parameters as inputs in a crop model. The selection of the crop model is important. Obv-
iously, it must be simple enough for being applica- ble over vast areas. However, a simple model has also
its limitations in spatial applications Leenhardt et al., 1995. A simple model generally neglects a lot of
physical processes that limits its application to areas and topics in which these processes play a minor role.
Furthermore, the cost reduction involved by limiting the measured physical parameters of complex models
is often counteracted by the need of more local ex- perimental data for calibrating the global parameters
considered in simple models.
A hybrid modelling approach containing two steps was applied in this study Wassenaar et al., 1999:
1 running a mechanistic crop model over a set of soil–climate situations including the soil and climate
variability of the studied region and 2 deriving a set of simple agrotransfer functions — linear or logarith-
mic expressions — by regression analysis from the dataset obtained in the previous step. These agrotrans-
fer functions can be considered as a very simplified crop model that can be used over vast areas. The main
advantage of this method is that it integrates a mecha- nistic crop model for representing accurately the phys-
ical process involved with the necessity of handling a very simple crop yield estimate, because of the limited
data availability cited before. However, these agro- transfer functions have a limited sphere of validity.
First, they are specific to a particular output of the crop model, e.g. the average hard wheat yield calculated
over the 1984–1991 period for a wheat–maize Zea mays L. rotation. Second, their spatial validity must
clearly be defined as demonstrated for similar mod- els like pedotransfer functions Bastet et al., 1997. In
the technique presented here, a reasoned sampling of soil–climate situations Wassenaar et al., 1999 was
necessary for obtaining an effective validity of the agrotransfer functions over the whole region.
The use of fuzzy estimates of soil parameter in the agrotransfer functions and the systematic presence of
associated error terms expressed as fuzzy subsets leads to a fuzzy estimate of crop yield. This fuzzy estimate
of crop yield can be determined by arithmetic calculus with fuzzy quantities Dubois and Prade, 1988. The
use of CSP technique as the one involved for calculat- ing awc was not required here since no statistical rela-
tions between agrotransfer functions parameters were considered.
2.4. Mapping fuzzy estimates of crop yields Mapping of crop yields over the region is of major
concern for further decision making. A possibility distribution is a very complex information which
cannot be summarised by a single value attached to a pixel. The proposed solution, as suggested by
Hootsmans 1996 was two-fold: extracting synthetic indicators from the possibility distributions and using
an interactive display environment.
The synthetic indicators were derived in view to an- swer the following practical question: ‘Is the predicted
wheat yield higher or lower than the given threshold value Z
?’ This type of question occurs frequently since decision makers are very often interested in
the position of this value relatively to a meaningful threshold value e.g. the lowest value for economical
viable cultivation of a specific crop. Fig. 4 illustrates how the results for each situation can be interpreted.
Given a possibility distribution expressing the esti- mated crop yield at a point x Mg ha
− 1
, the value of Z
in relation to the distribution results in one of five possible modalities, i.e. certainly above the
threshold; possibly above the threshold; certainly be- low the threshold; possibly below the threshold; and
undecided. The modality certainly above the thresh-
Fig. 4. Interpretation of possibility distributions of yield values p in order to map yield predictions. Z
is a user-fixed threshold of yield value.
10 P. Lagacherie et al. Agriculture, Ecosystems and Environment 81 2000 5–16
old means that the smallest value of the support of the possibility distribution representing the hydro-
logic property at point x is above the threshold. The modality possibly above the threshold means that the
smallest value of the core of the possibility distribu- tion is above the threshold. The next two modalities
were defined similarly. Undecided was used when the threshold falls within the core of the distribution the
true value can be above or below the threshold.
A Graphic User Interface for Windows written in Delphi 3 Borland
TM
using MapObjects
TM
ESRI
TM
was build for exploring interactively different values of the threshold Z
.
3. Application to the mapping of wheat yields over the Hérault-Libron-Orb valleys
