M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55 45
Results of the assessment must be interpreted in re- lation to the studied situation and the crop model. Un-
certainties that strongly affect results certainly need consideration: more or better basic data are necessary
or the model needs improvement, or both. Uncertain- ties that contribute little to the variation of calculated
results suggest that, for the studied conditions, there is little need for better data. However, the possibility
that the factor considered is not well accounted for in the model should not be excluded, e.g. a model that
does not take account of possible chemical root barri- ers will obviously not be sensitive to their occurrence.
Note further that this assessment considers only part of all possible errors, i.e. the results indicate a mini-
mum level of uncertainty.
2. Methods
2.1. Research strategy •
Water-limited sugarcane yield potentials were cal- culated with a crop model, using soil data inferred
from reference profiles of the study region. Results were compared with on-farm yield records to indi-
cate yield gaps, and yield variation among neigh- bouring plots was determined to assess uncertainty
in actual yield data.
• Probability density functions determined for RDM
in dependence of Al saturation and for the volume fraction of soil water held at potential greater than
− 1.5 MPa, 2 days after field saturation θ
2d−1.5 MPa
, cm
3
cm
− 3
; often referred to as available water ca- pacity were used in a Monte Carlo method, gen-
erating realisations of RDM andor θ
2d−1.5 MPa
as input in the crop growth model.
• Comparative crop growth calculations were done
with two alternative mechanisms of soil water up- take by the crop. The first option assumes that water
uptake under drought stress is determined by the part of the soil in which water is most readily avail-
able. The second option assumes that water uptake is a function of the total amount of available soil
water. Option 2 is commonly applied by irrigation engineers, but option 1 seems more appropriate for
application in dry land cropping, when dry spells precede rains that are insufficient to wet the entire
root zone. 2.2. Study locations
The study fields are located in the surroundings of Araras, São Paulo State, Brazil Fig. 1. Monthly
average climatic data of Limeira station are given in Table 1. Fifteen sugarcane fields ±100 ha each,
with each field comprising two to five adjacent plots were analysed. The soil in each field was sampled at
two depths 0–20 and 60–80 cm by auger at 7–16 sites and in 13 fields, one soil profile was examined
in detail, as described by van den Berg and Oliveira 2000a,b. The profiles were used as reference profile
for the field in which they were sited. For the two fields without profile, data were used from profiles of
other fields with similar soil characteristics as judged by soil texture and chemical properties of the auger
samples.
2.3. Crop data The ‘Usina São João’ sugarcane mill provided data
on planting, harvesting and fresh cane yields of 81 crops of sugarcane cv. NA5679 on the study fields.
Data on cane composition sugars and non-soluble solid contents of fresh cane stalks were available
for 51 of these crops. These data pertain to samples taken randomly from lorries entering the mill after
harvest.
2.4. The simulation model 2.4.1. General outlines
The crop growth simulation model that was used for this study is derived from PS123 Driessen and
Konijn, 1992; Driessen, 1997. Major modifications are: a priority rules were established for assimilate
partitioning over crop organs to mimic the availabil- ity of sinks and their demand for assimilates; b a
storage pool sucrose was introduced that acts as source and sink for carbohydrates and c the model
offers several options to calculate soil water uptake by plants. An overview of the procedures used in the
model is given in Table 2. A listing with detailed explanation of the modifications can be provided by
the corresponding author, on request. A short descrip- tion of the approaches for crop water uptake is given
below.
46 M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55
Fig. 1. Location of study region in Brazil and study fields in study region.
2.4.2. Representation of the rooted soil and initial settings
The soil as considered in the model is divided in compartments. Each compartment has a thickness of
25 cm, except the deepest one. The lower boundary of that deepest compartment represents the lower bound-
ary of the rooted soil zone until it attains a thickness of 25 cm, when a new lower compartment is initialised.
Rooting depth at emergence was set to 40 cm for plant cane and to RDM for ratoon cane, i.e. it is as-
sumed that the root system remains intact after harvest c.f. Ball-Coelho et al., 1992. Downward extension
Table 1 Average monthly climate data at Limeira 22
◦
32
′
S, 47
◦
27
′
WG; 639 m a.m.s.l., 1940–1990
a
January February March April May June
July August September October November December
Average mean t
◦
C 22.8
22.7 22.2
20.4 18.3
17.0 16.8
18.7 20.0
20.9 21.5
22.0 Average maximum t
◦
C 29.1
29.2 28.9
27.5 25.3
24.4 24.7
27.2 27.7
28.2 28.5
28.5 Average minimum t
◦
C 18.0
18.1 17.2
14.8 12.4
11.1 10.6
12.0 13.3
14.9 15.9
17.0 Precipitation mm
236 192
165 68
56 41
28 31
63 128
153 231
No. of rainy days 18
16 14
7 6
5 4
3 6
11 12
17 Bright sunshine h
194 181
214 213
208 195
219 226
202 208
212 178
a
Source: Instituto Agronômico de Campinas.
of the rooted zone is assumed to proceed at a constant rate of 1 cm per day, until it reaches depth RDM.
Initial soil water contents of each compartment were set at θ
2d
soil water content 2 days after field satu- ration for plant cane and at the final value under the
previous crop for ratoon cane. 2.4.3. Transpiration and water withdrawal by roots
Maximum transpiration rate T
m
, cm per day, i.e. transpiration of a well-watered crop, is calculated ac-
cording to the original PS123 setting see Table 2. The assessment of how transpiration is affected by
M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55 47
Table 2 Principal features of the crop growth model
Feature Description
Crop growth Gross photosynthesis rate
Driessen and Konijn 1992; function of intercepted radiation and light use efficiency; levels off at temperature dependent light saturation plateau
Maintenance respiration rate Driessen 1997; temperature dependent mass fraction for each organ;
maintenance respiration is discounted after assimilate partitioning Conversion efficiency of primary
assimilates to plant material Penning de Vries et al., 1989; organ specific parameters, based on
biomass composition Leaf mass and area dynamics
Assimilate allocation to leaves occurs at maximum rate whenever net marginal photosynthesis gains exceed 0, and stops when gains are ≤0. An
average value of specific leaf area area per unit mass, is used to calculate leaf area index LAI. Leaves die due to physiologic ageing. Leaf area
also decreases when respiration exceeds assimilate allocation to leaves
Stem growth Basic assimilate allocation to stems is proportional to assimilate allocation
to leaves. More assimilates are allocated to stems when leaf growth stops while soil water is readily available T
a
T
m
≥ 0.8
Root growth Basic assimilate allocation to roots is proportional to S
root,max
− S
root
, where S
root,max
is a target value for root biomass S
root
kg ha
− 1
. More assimilates can be allocated to roots when leaf growth stops and soil
water status limits transpiration T
a
T
m
0.8 Formation and remobilization of stored reserves
Reserves are formed when assimilates remain available after allocation for structural growth. Remobilization is a simple function of temperature
and water availability. Stored reserves cannot exceed 0.6 of total stem biomass
Links between crop growth and water balance Gross photosynthesis rate is assumed to be linearly related to T
a
T
m
Assimilate partitioning is affected see above Canopy temperature rises. A linear relation is assumed between T
m
− T
a
and increase in maximum temperature. This affects phenological devel- opment, leaf ageing and maintenance respiration
Water balance Potential evapotranspiration rate E
Penman method as implemented by Driessen 1997 Maximum transpiration rate T
m
Driessen and Konijn 1992; function of E , leaf area index LAI,
extinction coefficient for global radiation k
e
and a wind turbulence factor F: T
m
= E
1 − e
k
e
LAI
F Actual transpiration rate T
a
See text Maximum evaporation rate E
m
from wet cropped soil surface
Driessen and Konijn 1992; function of E , LAI including death leaves
and extinction coefficient for global radiation: E
m
= E
e
k
e
LAI
Actual evaporation rate E
a
Based on Supit et al. 1994, after Ritchie 1972, as function of number of days after last rainE
. Extraction from surface layer only Infiltration
Gauged rainfall+irrigation; not corrected for leaf interception or runoff
a
Percolation and deep drainage Simple ‘tipping bucket’ approach, using field capacity concept
a a
The study soils are very permeable and there is no groundwater influence. Tensiometer readings on covered soils of the study fields suggest that water content decreases some 0.01 cm cm
− 3
between 2 and 5 days after field saturation van den Berg, 1996.
soil water status, is based on a combination of the ap- proach of Allen et al. 1998 and a free interpretation
of a large number of field trials, reviewed by Gard- ner 1983. It is assumed that, from an initially wet
soil, a fraction p
soil
of total available soil water can be removed before actual transpiration rate T
a
, cm per day drops below T
m
. When the fraction of extracted available water exceeds p
soil
, T
a
T
m
decreases linearly with the amount of available water left in the soil. For
this study, p
soil
was determined as a function of at- mospheric demand according to Allen et al. 1998.
In the model, p
soil
is assumed to be distributed over the rooted soil depth as indicated in Fig. 2, in analogy
with the cumulative soil water uptake distribution at
48 M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55
Fig. 2. Schematic representation of depth dependent soil water depletion fractions.
first wilt noted by Gardner 1983. Each compartment i
has a specific soil water depletion fraction p
i
. Water uptake by roots during time interval 1 1 day
is accomplished in up to N+1 steps, where N is the number of rooted compartments. For each step j, the
compartment in which water is most readily available is presumed to be the one with the greatest value of a
dimensionless variable f
r,i,j
, which is calculated as f
r,i,j
= θ
i,j
− θ
− 1.5 MPa
1 − p
i
θ
2d−1.5 MPa
1 where θ
i,j
is the water content of compartment i, just before step j, and θ
− 1.5 MPa
the soil water retained at −
1.5 MPa matric potential. The rate of water withdrawal U
i,j
cm per day from this compartment is calculated as
U
i,j
= T
m
f
r,i,j
2 U
i,j
was bounded by the conditions that U
i,j
≤ T
m
and that the amount of water withdrawn from compart-
ment i may not exceed the amount that is actually available. Further water uptake from compartment i
is blocked when its water content during 1 decreases ≥
0.02 cm
3
cm
− 3
. This maximum sink term Hoogland et al., 1981 was introduced to avoid excessive water
uptake from a small compartment. Subsequently, θ
i,j
is adjusted and the procedure is repeated for step j+1, etc.
The model offers two options to calculate T
a
: 1. T
a
is calculated after completion of step N+1 by integrating U
i,j
of Eq. 2. In this representation, T
a
is determined by the water content of the soil compartments in which water is most readily
available. 2. Alternatively, T
a
is calculated previously as T
a
= T
m
θ
soil
− θ
− 1.5 MPa
1 − p
soil
θ
2d−1.5 MPa
3 bounded by the condition T
a
≤ T
m
, where θ
soil
is the water content averaged over the rooted part of
the soil. Eqs. 1 and 2 are used in this option but the ‘loop’ is left as soon as cumulative water
withdrawal during 1 reaches T
a
. When soil water distribution is homogeneous, both
options yield practically identical results as, e.g. WOFOST Boogaard et al., 1998 and SUCROS van
Laar et al., 1997. However, option 1 yields con- siderably larger T
a
s when wet and dry layers occur immediately above or below each other.
2.5. Assessment of uncertainties in actual yield estimates
Cane dry matter yields kg ha
− 1
were calculated for each field as the product of fresh cane yield and
content of sugar+insoluble solids. Standard deviations of yields from individual plots within a field were
estimated as well. If only data on fresh cane weights were available 30 cases, average dry matter yields
and standard deviations of yields among adjacent plots for each harvest were approximated as
Y
k
= P
i= 1,n
CF
i,k
A
i
P
i= 1,n
A
i
4a S
2 Y
k
= C
2
S
2 F
k
+ F
2 k
S
2 C
+ 2Y
k
S
C
S
F
k
r
C,F
4b where Y
k
is the estimated dry cane yield kg ha
− 1
of a field at harvest k, C the average dry matter sugar+insoluble solids content of fresh cane stalks
at harvest 0.29 kg kg
− 1
, n the number of plots in the field, F
i,k
the mass of fresh cane stalks at plot i at harvest k kg ha
− 1
, A
i
the surface area of plot i ha, S
Y
k
the estimated standard deviation of dry cane yields among plots within a field at harvest k kg ha
− 1
, S
F
k
the standard deviation of fresh cane yields among
M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55 49
plots within the field at harvest k kg ha
− 1
, S
C
the estimated standard deviation of dry matter content of
fresh cane calculated from data provided by the mill 0.029 kg kg
− 1
and r
C,F
the correlation coefficient of fresh cane yield and cane dry matter content −0.491.
2.6. Maximum rooting depths The determination of values of RDM as input in the
simulation model, is based on the following reasoning, in analogy with Jones et al. 1991.
For the reference profiles, it is assumed that ‘rootability’ of a soil horizon is unaffected i.e. 1.0
if Al
3+
saturation is less than 40, and nil i.e. 0.0 if Al
3+
saturation exceeds 80. An intermediate value is taken when Al
3+
saturation is between 40 and 80. The rootability indices of each soil horizon
are multiplied by their thickness down to a depth of 200 cm and summed to obtain the value of the
‘effective’ RDM.
Data from the auger samples and a pedo-transfer function were used to determine a probability density
function for the assessment of errors in the represen- tativeness of calculated effective RDM values. The
derivation of the pedo-transfer function is illustrated in Fig. 3, which shows effective RDM and Al
3+
sat- uration of the 60–80 cm layer Al
60
–
80
of 31 profiles of strongly weathered soils in south and south-east
Fig. 3. Maximum effective rootable depth RDM as a function of the Al
3+
saturation at 60–80 cm depth Al
60
–
80
, .
Brazil including those from the study fields. RDM is not affected by the Al
3+
saturation of the 0–20 cm layer, which, modified by liming, never exceeded the
lower threshold value of 40. The following relation was derived from these data to generate RDM values:
RDM
i,j
= a
Al
j, 60
–
80
+ b + ε
5 bounded by the limits 30≤RDM
i,j
≤ 200, where
RDM
i,j
is the ith generated value of RDM for field j
cm, Al
j , 60
–
80
the average value of Al
60
–
80
in field j , a, b the regression coefficients calculated
from the data in Fig. 3 excluding data pairs with Al
60
–
80
25: a=−2.565 cm and b=265.9 cm and ε
a stochastic variable cm representing the deviation from the mean value of RDM.
van den Berg and Oliveira 2000a showed that Al
60
–
80
presents considerable within-field variation in the study region, but spatial autocorrelation is weak
for the sampling intervals they considered. This, and the small number of sites in each field prompted us to
assume a Gaussian probability distribution for RDM, with average ε=0 and standard deviation S
RDM
. Stan- dard statistical procedures see e.g. Helstrom, 1991
were used to estimate S
RDM
from the standard devia- tion of Al
60
–
80
in field j S
Al
j, 60
–
80
, the standard de- viations of regression coefficients a and b S
a
= 1.181,
S
b
= 70.14 of Eq. 5, and the correlations between a
and b r
a,b
=− .955
S
2 RDM
= Al
2 j,
60
–
80
S
2 a
+ a
2
S
2 Al
j, 60
–
80
+ S
2 b
+ 2Al
j, 60
–
80
S
a
S
b
r
a,b
6 S
Al
j, 60
–
80
was approximated by the root pooled within field variance of Al
60
–
80
=14.98. Preliminary analysis suggested that within-field values of Al
60
–
80
have approximately normal distribution. The same was implicitly assumed for the residuals of the reg-
ression equation derived from Fig. 3. Note that Eq. 6 only considers uncertainties in es-
timated rooting depths as related to the lateral variabil- ity of Al
3+
saturation and the use of the pedo-transfer function of Eq. 5. It ignores uncertainties incurred in
the rather speculative estimation of RDM as a function of vertical changes in Al
3+
saturation of the soil. Root distribution recorded in soil profile descriptions is not
of much help in this case because crops on different soils were also in different stages of development.
50 M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55
Besides, it is difficult to quantify the implications of such general statements as ‘few fine roots’.
2.7. Generating water retention data For a data set including the study soils, van den
Berg 1996 showed that θ
2d−1.5 MPa
as inferred from tensiometer readings in the field and water retention
curves in the lab, is correlated with the soil’s dry bulk density and total iron content. van den Berg 1997
suggested that if these data are not available, as is the case for the auger samples in this study, an av-
erage value of 0.17 cm
3
cm
− 3
can be substituted for θ
2d−1.5 MPa
. He found a standard deviation S
θ
2d−1.5 MPa
of 0.03 cm
3
cm
− 3
. However, comparing gravimetri- cally determined field water contents with tensiome-
ter readings and lab data, suggested that the average value may be overestimated by up to 0.04 cm
3
cm
− 3
. The model used requires input of the upper limit of
available water water stored in the soil after excess water is removed by drainage; in our case represented
by θ
2d
cm
3
cm
− 3
, the lower limit of available water estimated at θ
− 1.5 MPa
, cm
3
cm
− 3
, and an assessment of the water content of ‘air-dry soil’ θ
AD
cm
3
cm
− 3
. These data were generated as follows:
1. θ
2d−1.5 MPa
values are obtained by assuming a normal distribution with average=either 0.17 or
0.13 cm
3
cm
− 3
and S
θ
2d−1.5 MPa
= 0.03 cm
3
cm
− 3
. To avoid senseless data, generated values were boun-
ded by 0.07 cm
3
cm
− 3
6 θ
2d−1.5 MPa
6 0.26 cm
3
cm
− 3
. 2. θ
− 1.5 MPa
, i.e. the presumed lower boundary of available water cm
3
cm
− 3
is calculated using a transfer function derived by van den Berg 1996
for a data set including the same soils θ
− 1.5 MPa
= 0.02 + 0.27clay + silt
7a where clay+silt is the average content of clay+
silt, kg kg
− 1
of the soil in the field. 3. θ
2d
is calculated as θ
2d
= θ
2d−1.5 MPa
+ θ
− 1.5 MPa
7b 4. For the assessment of air-dry water contents θ
AD
, a ‘residual soil water content’ θ
r
cf. van Genuchten, 1980 was calculated according to van den Berg
1997, as θ
r
= 0.064+0.19clay+silt
2
− 2.7 × 10
2
C
2 org
7c where C
org
is the organic carbon content kg kg
− 1
. The calculated value was bound by the limits
0≤θ
r
≤ θ
− 1.5 MPa
− 0.005. θ
r
was assumed to be retained at a soil water potential of −10 MPa and
0.25θ
− 1.5 MPa
was assumed to correspond with a soil water potential of −100 MPa and less.
5. Intermediate values were obtained by loglinear in- terpolation where necessary. In the current version
of the model this is only used for the assessment of water contents of air-dry soil material, i.e. in equi-
librium with the atmosphere.
2.8. Independent realisations from normal distributions
Independent random samples from normal distri- butions to generate values for RDM and θ
2d−1.5 MPa
were simulated by first generating uniformly dis- tributed random numbers with the FORTRAN RAN
function and the RANØ subroutine described by Press et al. 1986. Transformation to a normal distribution
was done according to Casella and Berger 1990.
2.9. Model runs The crop model was used to calculate water-limited
yield potentials of sugarcane for the following config- urations:
Effective RDM and θ
2d−1.5 MPa
obtained from reference profiles
1 RDM calculated stochastically according to
Eq. 5; θ
2d−1.5 MPa
set to an average value of 0.17 cm
3
cm
− 3
; method 1 for water uptake 2
Water retention data generated stochastically; RDM set to the field average Eq. 5; ε set
to 0; method 1 for water uptake 3
As 2, but with average θ
2d−1.5 MPa
set to 0.13 instead of 0.17
4 Both water retention data average θ
2d−1.5 MPa
set to 0.17 and RDM are generated stochastically; method 1 for water uptake
5 As 4, but using method 2 for water uptake
6 As 4, but with average θ
2d−1.5 MPa
set to 0.13 instead of 0.17
7 As 4, but disregarding uncertainties related to
the use of the transfer function of Eq. 5. This reduces Eq. 6 to S
RDM
≈ |a|S
Al
60
–
80
= 38 cm
M. van den Berg et al. Agriculture, Ecosystems and Environment 81 2000 43–55 51
The model was run 100 times for each harvest for configurations 1–7, which adds up to 7×81×100 runs
±7 h run time PC, 266 MHz. The average yield po- tential per harvest, Y
sim
and the pooled standard devia- tion S
Y , sim
root mean of squared standard deviations were calculated from the results of these test runs.
3. Results
