Journal of the Indian Society of Soil Science, Vol. 41, No. 4, pp 730-737 (1993)

A Reassessment of the Crop Tolerance Response Function
M. Th. VAN GENUCHTEN AND S.K.GUPTA*
USDA-ARS, U.S. Salinity Laboratory 4500 Glenwood Drive, Riverside, California, USA, 92501

Abstract: Crop salt tolerance data have traditionally been analyzed with a three parameter
threshold-slope model that assumes maximum yield until salinity threshold value, and a linear
decrease in yield beyond the threshold. This study shows that an alternative, S-shaped responsemodel provides an equally good or better fit to many experimental data sets, and with less bias in
the parameter estimation process. Analysis of 204 data sets from the original database compiled by
Mass and Hoffman (1977) revealed that a single dimensionless curve could be used to represent thesalt tolerance of most crops. The curve is given by Y, = 1/[1+ (C/CSO)~ ],where Yr is the relative
yield, c the average root zone salt concentration, and cso a parameter which &scribes the degree ofsalt tolerance of the crop (the average root zone salinity at which the yield has declined by 50%).
The presence of a unique dimensionless curve to describe the salt tolerance of many or most crops
may point to some common mechanism that could govern the yield response of crops to salinity,
andperhaps to other yield limiting factors as well. (Key words: Salt tolerance, response function,soil salinity, yield-limiting factors)

Salinity-induced crop yield losses may often be resulted in various salt tolerance lists (Harris &prevented by adopting soil and water management Pittman 1919; van den Berg 1950; U.S. Salinity
and agronomic practices that are appropriate for the Laboratory Staff 1954; Bernstein-1964; de Forges
local soil, crop and environmental conditions. 1970; Maas & Hoffman 1977; among others).
Unfortunately, high salinities are sometimes
Probably the most comprehensive analysis todifficult to prevent because of a lack of non-saline
date of available salt tolerance data was publishedirrigation water. When and where salinity is aby Maas and Hoffman (1977). and recently updated
problem, an effective use of available soil and waterby Maas (1991). Based on an extensive literature
resources requires the production of agricultural
search, Maas and Hoffman (1977) concluded thatcrops that are relatively tolerant to salinity. To docrop yield as a function of the average root zone
so, reliable data are needed to predict crop yields insalinity could be described reasonably well with a
response to various levels of salinity in the root
piecewise linear response function characterized byzone. Numerous field and laboratory experiments
a salinity “threshold” value below which the yield ishave been conducted over the past 80 years or so to
unaffected by soil salinity, and above which yieldobtain the necessary salt tolerance data. This has
decreases linearly with salinity. Results compiled by
Maas and Hoffman (1977). and since then by manylCentral Soil Salinity Research Institute, Kamal. 132001

1993]REASSESSMENT OF SALT TOLERANCE FUNCTION OF CROPS

731

others, also show that the crop salt tolerance model applicable to most data sets considered in the
response function is variety-specific, and may analysis.
depend, among other things, on the unique soil,
environmental and water management conditions ofan experiment.
Threshold-slope salt tolerance response
function: The threshold-slope model of Maas and
The threshold-slope model of Maas andHoffman(1977) is characterized mathematically by
Hoffman (1977) has proved to be extremely usefulapiecewise
linear function (Fig.l) containing three
for a variety of applications in research andindependent
parameters: the maximum yield undermanagement. Notwithstanding the popularity of thisnonsaline
conditions(Y,), the salinity thresholdmodel, other salinity response functions have been
(ct),
which
is
definedasthe maximum soil salinity
found equally successful in describing observed
without
yield
reduction,and the slope(s) of thecrop salt tolerance data (Feinerman et al. 1982; vanfunctiondetermining
thefractional
yield decline per
Genuchten & Hoffman 1984). The study b y van
unit
increaseinsalinitybeyondthethreshold. In
Genuchten and Hoffman (1984). in particular,
equationform,
thethreshold-slopemodel
is givenpointed to potential problems with thethreshold-slope model in describing experimental by
O<c<ctYllldata, namely the relatively poor definitidn of theY=
Ym-Yms(c-Ct)
ct<c<co. . . (1)salinity threshold value for data sets which are0c>copoorly defined, erratic or have limited observations, where, Y is crop yield, c is the average root zone
and the inability to accurately reproduce many salt salinity, and co the concentration beyond which thetolerance data sets at relatively high soil salinities.
yield is zero. Soil salinities in Eq. (1) can be
In this paper we will use one of the smooth expressed in terms of concentration, osmotic
S-shaped response functions of van Genuchten and potential, or electrical conductivity of either the soil
Hoffman (1984) to re-analyze the salt tolerance water extract (EL,) or the soil saturation extract
database of Massand Hoffman (1977). In (EC=). We note that co is not an independent
particular, we will illustrate the improved accuracy parameter, but can be expressed in terms of ct and sof this function in describing several data sets, and as follows:
also show how the function was used to derive a
co= c+ l/s. . . (2)single dimensionless (scaled) salt tolerance response
Maas and Hoffman (1977). among others,
found that Eq.(l) gives a fairly good description ofmany salt tolerance data sets. Unfortunately, thePiecewise Linear
function has at least two shortcomings, one of which
Response Modelis illustrated in fig. 2, using barely (forage) dataobtained by Saini (1972). Notice that the yield data
are expressed as a fraction of the control yield, Yc,of the first data point at the lowest salinity. ,The solid
curve in fig. 2b was obtained with a nonlinear
least-squares parameter optimization programme,
SALT (van Genuchten 1983). which simultaneously
fits the three unknown coefficients (Ym. ct and s) of
the threshold-slope model (Eq.1) to the observed
data. The SALT program minimizes the objectivefunction
Average rootzone concentration, C

Fig.1. Graphical representation of the piece-wise linear crop andsalt tolerance response function (Eq.1)

0(b) 1”: EYi-4 WI2. . . (3)where, Yi and ?i are measured and calculated crop

732

[vol. 41

JOURNAL OF THE INDIAN SOCIETY OF SOIL SCIENCE
I

+a

1.0-o

:e
0
.o
A
.g
g

0.8 -

I

0.6 0*40.20

I

I

I

’

I

0
0
0

I

(a)

14

00 0 0Y, =I.0004
p ~3.62

5
IO
15
Soil Solinity, EC,(dS/m)

20

20 25

_

30

Soi&Salinity, EC,,(dS/m)

Fig.2. Observed salt tolerance data for barley (a), and fitted Fig.3. Observed salt tolerance response data for meadow foxtail(a) and fitted response functions using the Mold-slope
response functions using the threshold slope model (b)model (b) and the S-shaped model (c)and the S-shaped model (c)yields at the n measured soil salinity ci and b is the at the higher yields. Straightforward application ofvector of unknowns coefficient: b = (Yrn, ct, s). the three-parameter inversion method to these data
Figure 2 shows that the salinity threshold-slope resulted in a threshold salinity value, ct. that wasmodel matches the data reasonably well at the lower considerably less than the salinity of the firstsalinities. However, the data at the higher salinities, measured data point (fitted curve not shown in fig.
notably when c > co, are not described well. The 3b). This situation, found to be typical of many salt
tailing phenomenon at the higher soil salinities is a tolerance data sets (van Genuchten & Hoffman
characteristic feature of many salt tolerance data sets 1984). leads to a uniquely defined response curveand cannot be described with the piecewise linear only for salinities exceeding the threshold value.
However, the fitted values of Ym and ct are
threshold-slope model.
meaningless in these types of examples since no
A second weakness of the threshold-slope data points at the lower salinities are available to fix
model is illustrated in fig. 3 using data for Meadow the two parameters. In fact, different initial
Foxtail grass (Brown & Bernstein 1953). Note that e s t i m a t e s for Ym and Q in the least- squares analysis
the experimental curve in this case is poorly defined1993]REASSESSMENT OF SALT TOLERANCE FUNCTION OF CROPS

733

will usually lead to different fitted values for these the curve (i.e., the higher the p, the steeper is thetwo coefficients.
curve). Notice that salinities and yields in eq(4) can
There are several ways to resolve this problem, be expressed in reduced form. Hence, eq.(4) leads toeach one having its disadvantages. One possibility a dimensionless plot in terms of two scaled variables
would be to assume that either ct is known before (relative yield, Y/Ym and relative concentration,
hand and equal to the salinity of the first data point, r&so), and one coefficient, p, reflecting theor that Ym is known and coincides with the yield of steepness of the curve.
that same first point. Both assumptions will fix the
Figure 2c shows that application of thethe end point of the fitted curve in the upper left nonlinear least square inversion program to thepoint of fig. 3b. Unfortunately, the approach results barley data resulted in an excellent fit of eq. (4) toeither in a Ym value that is still less than the yield of the experimental data. In particular, the tailing partthe first data point or in a threshold salinity that still at the higher concentrations is now described muchlies to the left of the point. Both situations appear better. An excellent fit was also obtained for the
unrealistic and are not further considered here.
Meadow Foxtail data (Fig. 3c). The inverse
A somewhat more realistic approach would be procedure resulted now in a uniquely defined curve,
to fix both ct and Ym at the values of the first data independent of the initial parameter estimates.
point in fig.3. This alternative leads to a one These, and many other examples not further
parameter fit for s and results in the solid curve of reported here, revealed the accuracy and flexibility
fig.3b. Yet another approach is possible by of eq. (4) in describing a large number of data sets.
reasoning that one should be interested mainly in the Encouraged by this success we applied the s higher yields, and hence that the outliers at the shaped model to all data sets originally used in thehigher salinities (the last four data points) should be analysis of Maas and Hoffman (1977).
neglected in the analysis. This reasoning leads to the Results and Discussion
dashed curve. Notice that the threshold value is now
The data base of Maas and-Hoffman (1977) was
three to four times larger than the threshold value divided into four groups: fields crops, forage crops,
for the solid line. This example demonstrates that vegetables and fruit trees. AU fruit tree data sets
the threshold- slope model in conjunction with the were discarded because of generally too few orpoorly defined data set can quickly lead t o unreliable experimental data. Of the remaining 256
ambiguities that are not easily resolved.
individual data sets, an additional 52 were also
Alternative S-shaped salinity response model : judged to be unsuitable for our analysis because of

In order to avoid some of uniqueness problems withthe threshold-slope model, we examined several
alternative response functions which would give amore accurate description of the above data, andalso would lead to a more stable and unbiased
inverse problem. A preliminary analysis of several
candidate functions, including exponential functionsof the type by Feinerman et al. (1982), caused us toselect the following smooth, sigmoidal function of
van Genuchten (1983) for further analysis:
Y=

ym _

. . . (4)

1 + (c/cso)P
where, cso and p are empirical constants. Thisfunction also contains three unknown parameters:
the maximum yield Ym, the soil salinity c50 at which
the yield is declined to 50% of its maximum value,
and a parameter p that determines the steepness of

insufficient resolution in the data. Typically, thesedata sets exhibited severe scattering in the data
points, or heavy clustering in only one part of the
response curve. The remaining salt tolerance database consisted of 56 *experiments involving field
crops, 80 forage crops, and 68 vegetable crops,
giving a total of 204 data sets.
Each data set was analysed for the unknown
parameters CSO, p and Ym. Table 1 summarizes
results for crops for which at least three separate
data sets were available. The table lists the means of
the estimated c50 and p-values, as well as the range
in fitted p-values. No clear correlation between pand ~50 is observed. A statistical analysis of the data
showed that p was independent of ~50. and hence ofthe degree of salt tolerance of a particular crop.
Also, no statistically significant differences were734

[Vol. 41

JOURNAL OF THE INDIAN SOCIETY OF SOIL SCIENCE

Table 1.

Average values for the parameters cx) andp in eq. (4) forselected experiment taken from theMaas-Hoffman salt tolerance
database

Crop

Number of data sets

BarleyCottonFlax
MilletRice
Sugarcane
Wheat

7
8
3
5
12
9
5

Alfalfa
BarleyBermuda

9
3
11
11
3
7
5
12

CloverCorn
Grasses
Lovegrass
Wheat grass
Bean
Red beetCabbage
Carrots
Corn
CucumisLettuce
Onion
Peppers
RadishSpinachsweet potato
Tomato

11
3
3
6
4
4
4
4
5
3
3
3
6

-

%I
p variable
p=3
FIELD CROPS
19.8
20.9
28.5
28.7
13.4
128
16.6
16.9
6.9
6.9
15.6
16.0
24.3
23.9
FORAGE CROPS
18.2
18.4
14.1
14.5
27.1
27.6
12.7
13.4
14.8
14.9
24.4
23.9
16.9
16.8
37.0
37.3
VEGETABLES CROPS
9 . 1
9.6
16.4
16.8
11.3
11.6
8.0
8.0
10.9
10.8
14.4
13.9
8.9
9.3
8.4
8.6
10.7
10.3
10.3
10.3
14.0
14.3
8.8
8.7
12.6
12.1

P
Range

Average

1.82-5.37
1.55-4.80
1.28-3.90
1.84-4.30
1.95-8.09
1.66-5.07
2.70-3.97

3.80
3.00
2.45
3.22
4.54
2.57
3.25

1.23-7.14
1.92-4.15
1.06-6.43
1.51-5.37
1.61-4.03
1.94-5.57
2.69-5.59
1.51-7.48

2.51
2.90
3.16
2.62
2.58
3.44
3.85
3.74

2.06-4.04
2.18-3.20
2.20-3.34
2.61-5.06
2.42-6.74
1.71-5.44
2.22-3.71
1.63-8.11
2.54-4.17
2.38-4.04
1.55-2.37
2.21-9.54
2.31-8.51

2.95
2.75
2.58
3.14
4.20
3.49
2.86
3.75
3.34
3.41
1.99
5.07
3.88

found between the average p-values for the 3 sets ofcrops. Figure 4 shows a frequency distribution of all
204 fitted p-values, along with the fitted normal and
log-normal distributions. Notice that the log-normal
frequency distribution fits the data quite well,yielding a mode of 2.55, a median of 3.05 and amean of 3.34. The variance of the distribution was
2.20. The “average” value of p was judged to be
sufficiently close to 3 to allow p to be fixed at thatvery convenient value. Table 1 shows that the
average p-values of many individual crops were alsoquite close to 3.
P

Fig.4. Frequency distribution of 204 fitted p- values

Assuming p=3.0; we decided to analyze all 256
experiments again for the two unknown, Ym and~50. We were especially interested in reanalyzing

19931REASSESSMENT OF SALT TOLERANCE FUNCTION OF CROPS

735

those experiments which yielded the outliers i.e. data as the variable case (Fig. 6a). The same is notdata sets characterized by very low or very high true for the gourd example (data from Paliwal andvalues for p. The results in table 1 indicate p(column 3) and fixed p (column 4). Differences in
c50 generally were in the range of 0-5% , and neverexceeded 6%.

Maliwal, 1972), which yielded the highest pvalueof all 204 experiments used in our analysis. While
the accuracy of the fit definitely decreased (SSQ
increased by a factor of about 3). we judge thedescription to be still quite acceptable ‘for many
applications.
The analyses leading to figures 2.3 and 5 show
the superiority of the S-shaped model (Eq.4). ascompared to the threshold-slope model, in
describing most or all salt tolerance data sets. In
particular, the optimization process with the
S-shaped model was found to be stable and provided
statistically unbiased fits of the data. Fixing thevalue of p at 3.0 only marginally affected theaccuracy of the least-square fit for most data sets,

Soil Salinity, EC,(dS/m)” 0.6

Fig. 5. Observed and fitted salt tolerance curves assuming p = 3 ineq. 4; (a) barley forage - o, meadow foxtail - x. and (b)cotton-o,oat-Aandryegrass-x

Figure 5 shows typical results obtained when p
was fixed at 3. Notice the excellent description in
0.6
each case, including the experiments for barleyt(Fig5a) and Meadow foxtail (Fig5b) which were0
0.6
previously analyzed assuming variable p (Figs 2c*
and 3c, respectively). Figure 6 gives two examples
z o-4which initially yielded extremely low and highestimate for p. When p was fixed at 3. the shape of
0.2 the curve for flax (data from Hayward and Spurr.
(b)1944) changed quite significantly. However the
residual sum of squares (SSQ) of the optimizationn[equal to the minimized value of the objectivefunction, Q(b), in eq.(3)] was only little affected by
fixing p at 3. Visibly, the more restrictive case of FIg.6. Observed and fitted salt tolerance curves for flax-o andfixed p (Fig. 6b) compares equally well with the
gourd-x (a) with variable p in cq.4. and (b) with fixed p =3.0 ill eq.4

736

JOURNAL OF THE INDIAN SOCIETY OF SOIL SCIENCE,[Vol. 41

although a few exceptions were apparent (notably the curve are also evident. Because of fewer
the gourd example in fig. 6). However, most unknowns in the response function, the informationcomparisons with p=3 were of the type shown in could be used to more accurately analyze poorly
fig.5. Keeping p constant at 3 results in one unique defined or incomplete salt tolerance data sets. This

Relative conc*ntration,C/cSo

Fig.7. Proposed general dimensionless salt tolerance response function

dimensionless salt tolerance response curve if point is illustrated in fig. 8 using data for sugar beets
expressed in terms of the reduced variables c/c50 (Eaton 1942). This experiment was initiallyand Y/Y,,, (Fig.7). The presence of such an unique discarded from our analysis because of the severe
curve raises the interesting question whether or not scattering among the data points. Notice that thecertain similarities exist in the response of different data in fig. 8 are also concentrated in only a limited
crops to salinity, and perhaps to other yield-limiting part of the curve, with none of the data points
stresses as well (notably to water stress). having relative yield of less than 0.5. Clearly, any
Considerably more work is necessary before these attempt to analyze this data set should be viewed
questions can be answered. This additional work with caution. However, since a considerable amountmust not only involve carefully executed and of time and labour is generally invested in thewell-controlled salt tolerance experiments, but experimental determination of these type of salt
probably also requires a statistically more tolerance data sets, it is also fully understandable
sophisticated analysis of the available data. Our that researchers would like to analyze the data, evenanalysis was restricted to the data collected from the if the data appear too incomplete for a reliable
literature by Maas and Hoffman in 1977. Numerous analysis. The fitted solid line in fig. 8a was obtained
experiments have been carried out since that time. when all 3 parameters (Y,, GO and p) were assumed
We believe that these and other data should be to be unknowns, resulting in a very high p-value ofcompiled into one computerized database which 15.9, and a maximum, yield Ym of 0.897 of thewould be internationally available to both control yield, Yc. When the maximum yield was
researchers and field practitioners. Such an updated fixed at the control yield (i.e., the yield of the data
database would be especially useful for point having the lowest average soil salinity), thephenomenological analyses of the type carried out two-parameter analysis produced the solid curve inin this study.
fig. 8b. Equating Ym with YC may sometimes be an
While the presence of a single and unique acceptable procedure, especially when the control
dimensionless curve may point to some common value is the average of several replicates. Notice thatmechanism governing crop yield response to the assumption Ym= Yc leads to a radically different
salinity, other and more immediate applications of p-value (1.45 as compared to 15.9 in fig. 8a); theREASSESSMENT OF SALTTOLERANCE FUNCTION OF CROPS

1993]
I

,

I

1

0.2 (b)

c/c50

1

737

early work by Maas and Hoffman (1977). crop salt
tolerance is generally described by a piecewise
linear model (Eq.1) which assumes no yield decline
until a “salinity threshold” value, and a linear
decrease in yield beyond the threshold . This study
shows that an alternative S-shaped response model
given by eq.(4) leads to an equally good or better
description of experimental data, and with a morestable and unbiased statisdcal fit of the data.
Because of fewer unknowns, the dimensionless
two-parameter response function may be used tomore accurately analyze poorly defined or
incomplete salt tolerance data sets. The presence ofa unique dimensionless salt tolerance responsemodel may point to some yet undefined common
mechanism governing the yield response of crops tosalinity, and perhaps to other yield-limiting factors
as well (especially to water stress).
References
Ashgar, A.G.. Ahmed. N. & Asghar. M. (1962) Pakistan J.
scient. Res. 14,162.
Bernstein. L (1964) U.S. Dep. of Agric. Info. Bull. 283. 23p.
Brown, J.W.. & Bernstein. L (1953) US. Salinity Laboratory
Report IO Collaborators, Riverside, California, pp 44-46.
De Forges. J.M. (1970) Natural Resources, 6.2.
Eaton, EM. (1942) J. ogric. Res. 64,357.
Feinerman. E.. Yaron D. & Bielorai. H. (1982) Wat. Resour. Res.
18.101.
Harris. F.S. & Pittman. D.W. (1919) Utah ogric. Exp. Sta. Bull.
168,23p.
Hayward, H.E. & Spurr, W.B.(1944) J. Am. Soc. Agron. 36.287.
Mass, E.V. (1991) In Agricultural Assessment and Management,

Fig.8 Observed and fitted salt tolerance curves for sugarbeet
obtained by fitting q.4 assuming (a) variable Ym. CYI and
ASCE ManuaIs and Report on Engineering Practicep; (b) fixed Y,,, = Y. and variable cst~ and p and (c) p=3
No.71. (K.K. Tanji, Ed.) Am. Soc.Civil Eng.,pp 262-304.
and variable Y, and cm
Maas. E.V. & Hoffman, G.J. (1977) Jlrrig. Drain. Div. ASCE.
103.115.
value of cso in this case also changed significantly. Paliwal. K.V. & Maliwal. G.L (1972)J. hort. Sci. 47;517.
Fixing p at 3, while keeping Ym and c50 variable, Saini. G.R. (1972) Ecology, 53.524.
leads to a fit (fig. 8c) which is only slightly lessaccurate than the complete 3-parameters fit of fig. Troughton, A. (1967) PI. Soil, 27.148.
U.S. Salinity Laboratory Staff (1954) Hondb. U.S. Dep. ofAgric.

8a, but somewhat better than the results of fig. 8b.
60.
Visually, the results of fig. 8c appear equally
van den Berg, C. (1950) Trans 4th int. CongrSoil Sci. 1.411.
acceptable as those of fig. 8a. Many other examples
M.Th. (1983) Research Report No. 120, U.S.
with similar behaviour could be given. However, van Genuchten,
SaIinity Laboratory, USDA-ARS, Riverside, California, 50
like the sugarbeet example, most of those additionalP.
data sets were considered to be too incomplete or van Genuchten, M.Th.& GJ. Hoffman (1984) In Soil Salinity
ill-defined for inclusion in our original analysis
under Irrigation, Processes and Management. Ecologicalinvolving the 204 salt tolerance data sets.
Studies No. 51. I. Shainberg and J. Shalhevet, (Eds),
Springer Verlag, New York. pp 258-271.
In conclusion, It can be said that following the(Received October 1991, Accepted August 1993)