Agricultural Water Management 109 (2012) 81–93

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Agricultural Water Management
journal homepage: www.elsevier.com/locate/agwat

Evaluation of subsurface drip irrigation design and management parameters foralfalfa
Maziar M. Kandelous a,∗, Tamir Kamai a, Jasper A. Vrugt b, Jiří Šimůnek c, Blaine Hanson a, Jan W. Hopmans aaDepartment of Land, Air and Water Resources, University of California, Davis, CA 95616, USA
Department of Civil and Environmental Engineering, University of California, Irvine, CA 92697, USAc
Department of Environmental Science, University of California Riverside, CA 92521, USAba r t i c l e

i n f oArticle history:
Received 19 September 2011
Accepted 26 February 2012
Available online 17 March 2012
Keywords:
Alfalfa
Subsurface drip irrigation
Optimization
HYDRUS-2D
AMALGAMa b s t r a c t
Alfalfa is one of the most cultivated crops in the US, and is being used as livestock feed for the dairy, beef,
and horse industries. About nine percent of that is grown in California, yet there is an increasing concern
about the large amounts of irrigation water required to attain maximum yield. We introduce a conceptual
framework to assist in the design and management of subsurface drip irrigation systems for alfalfa thatmaximize yield, while minimizing deep percolation water losses to groundwater. Our approach combines
the strengths of numerical modeling using HYDRUS-2D with nonlinear optimization using AMALGAM
and Pareto front analysis. The HYDRUS-2D model is used to simulate spatial and temporal distributions ofsoil moisture content, root water uptake, and deep drainage in response to drip-line installation depth anddistance, emitter discharge, irrigation duration and frequency. This model is coupled with the AMALGAM
optimization algorithm to explore tradeoffs between water application, irrigation system parameters, andcrop transpiration (Ta ), to evaluate best management practices for subsurface drip irrigation systems inalfalfa. Through analysis of various examples, we provide a framework that seeks optimal design andmanagement practices for different root distribution and soil textures.
Published by Elsevier B.V.

1. Introduction
Alfalfa is one of the most widely cultivated crops in the world,
and is used as livestock feed for the dairy, beef, and horse industries (Breazeale et al., 2000). The United States is the world’s largest
producer of alfalfa, with an annual value of about one billion dollars. About nine percent of all alfalfa produced in the United States isgrown in California, with an average acreage of 360,000 ha (900,000
acres, Alfalfa’s commodity fact sheet, 2011).
Because of California’s semi-arid climate, irrigation is necessary
to attain high yields. In California, about 1.2 billion m3 of water isused to irrigate alfalfa each year, which is about 20% of California’s developed water supply (Natural Resources Defense Council,
2001; Putnam et al., 2001). These large amounts of required irrigation water have encouraged the development and applicationof efﬁcient irrigation systems. Hutmacher et al. (2001) reported a
20% increase in water use efﬁciency for alfalfa by using subsurface
drip irrigation (SDI) rather than furrow irrigation. Another comparison between SDI and ﬂood irrigation by Godoy et al. (2003)∗ Corresponding author.
E-mail address: mkandelous@ucdavis.edu (M.M. Kandelous).
0378-3774/$ – see front matter. Published by Elsevier B.V.
doi:10.1016/j.agwat.2012.02.009

demonstrated that subsurface irrigation signiﬁcantly improved theyield by about 25%, while using about 40% less water than ﬂood irrigation. The study by Alam et al. (2002) showed that a well-designed
SDI system can potentially decrease the volume of applied waterby about 22%, while increasing the yield by 7%, compared to using
a center pivot sprinkler system.
Alfalfa, a perennial crop, is harvested between 3 and 11 times
throughout the year, depending on soil, irrigation practice, and climatic conditions. One of the biggest challenges in alfalfa productionis the selection of irrigation design and management practices thatmaximize yield (and thus income), while simultaneously minimizing water losses. In addition, at harvesting times the soil surface
should be sufﬁciently dry so that machinery can drive over theﬁeld without creating stressed root zone soil moisture conditions,allowing for quick re-growth of the cut alfalfa. In addition, thenonlinearity of soil–water–plant relationship makes it particularly
difﬁcult to ﬁnd irrigation systems and strategies that maximize
yield while simultaneously minimizing water losses. For all thesereasons the optimal irrigation system and design practices are notimmediately obvious for most climatic and soil conditions. A subsurface drip system, however, is ideally suited as it directly supplies
water to the rooting zone at high frequency, allowing control ofsurface soil moisture required, for dry soil surface conditions prior

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M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

Nomenclature
AWmax maximum possible applied water [L3 L−2 ]D
drip-line depth [L]DPdeep percolation [L3 L−2 ]DPmaxmaximum possible deep percolation [L3 L−2 ]actual evapotranspiration [LT−1 ]ETa
EToreference evapotranspiration [LT−1 ]ETppotential evapotranspiration [LT−1 ]irrigation frequency [T−1 ]fFEﬁnite elementshsoil water pressure head [L]h1 , h2 , h3 , and h4 parameters of Feddes uptake reduction
function [L]irrigation duration [T]ID
I1
optimization
scenariosuchthatωD = ωL = ωAW = ωDP = 1
optimization scenario such that ωD = 0 andI2
ωL = ωAW = ωDP = 1
optimization scenario such that ωD = ωDP = 0 andI3
ωL = ωAW = 1
ISA
irrigation scheduling Alfalfa modelK(h)unsaturated hydraulic conductivity [LT−1 ]crop coefﬁcientKc
Ks
saturated hydraulic conductivity [L T−1 ]Ldrip-line distance [L]lshape parameter in the van Genuchten soil
hydraulic functions
Lx
m
N
n
OFi
Q
q
R1
R2
S(x,z)
SDI
Se
Sp
t
T1
T2
T3
Ta
Tp
x
z
˛(h)
˛VG
ˇ(x,z)rs
˝width of the soil surface associated with transpiration [L]shape parameter in the van Genuchten soil
hydraulic functions, m = 1 − 1/nnumber of irrigation events
shape parameter in the van Genuchten soil
hydraulic functions
objective function (i)drip-line discharge [L3 L−1 T−1 ]drip-line discharge [L3 L−2 T−1 ]optimization scenario using uniform root distribution
optimization scenario using linear root distribution
sink term [L3 L−3 T−1 ]subsurface drip irrigation
effective saturation
potential root water uptake [L3 L−3 T−1 ]time [T]
optimization scenario using clay-loam soil
optimization scenario using loam soil
optimization scenario using sandy-loam soil
actual plant transpiration [LT−1 ]potential plant transpiration [LT−1 ]horizontal spatial coordinate [L]vertical spatial coordinate [L]Feddes’ uptake reduction function
shape parameter in the van Genuchten soil
hydraulic functions
normalized root density for any coordinate in thetwo-dimensional soil domain [L2 ]volumetric water content [L3 L−3 ]residual water content [L3 L−3 ]saturated water content [L3 L−3 ]root zone area [L2 ]ωD , ωL , ωAW , ωDP weighting factors for D, L, AW, and DP in
objective functions

to alfalfa cutting. We use detailed numerical soil water ﬂow modeling with HYDRUS-2D (Šimůnek et al., 2008), combined with amulti-criteria optimization framework to determine optimal irrigation water management strategies for subsurface drip irrigationof alfalfa.
Past sensitivity analysis (Gärdenäs et al., 2005) has shown thatthe root distribution exerts strong inﬂuence on subsurface drip irrigation design and management practices, as water uptake by plantroots determines spatial and temporal patterns in soil water availability. Whereas various past studies investigated root distributionsof alfalfa (Abdul-Jabbar et al., 1982; Meinzer, 1927), we know of nostudies that evaluate the effects of SDI on alfalfa root distribution.
The ever increasing pace of computational power along withsigniﬁcant advances in numerical modeling of soil–plant–water
relationships enables the application of numerical vadose zone
simulation models for analyses of micro-irrigation systems involving a wide range of crops. The HYDRUS-2D (Šimůnek et al., 2006,2008) model has been widely used for this purpose, including forSDI (Gärdenäs et al., 2005; Hanson et al., 2006; Skaggs et al., 2006;
Hanson et al., 2008). The effect of different irrigation design variables such as drip-line distance and drip-line installation depth
can be readily simulated by such a modeling system for a range ofsoil types. Whereas these design parameters are relevant to water
availability for crops and soil types, drip-line depth and distancealso have economic consequences as they essentially determine
irrigation system costs. Other factors to consider are leachinglosses, rodent damage, and requirement of soil dryness before
harvesting.
In this study, we present a general purpose multi-criteria optimization framework to help in the design of optimal subsurface
drip irrigation systems for alfalfa. Instead of providing speciﬁc irrigation system and water application recommendations, we present
a ﬂexible optimization tool that allows deﬁnition of a range ofobjective functions to be minimized, using multiple criteria andweights. Our approach combines the strengths of numerical vadosezone modeling using HYDRUS-2D with the AMALGAM evolutionary search (Vrugt and Robinson, 2007) and Pareto front (Wöhling
et al., 2008) algorithms, to provide water application strategies
that maximize yield and minimize water loss for a range of irrigation system designs. In particular, we seek to optimize drip-line
installation depth and distance, irrigation duration, and irrigation
frequency, while maximizing root water uptake and minimizing
irrigation water losses by leaching. In addition, our analysis is especially designed to ensure sufﬁciently dry soil surfaces at harvestingtimes. These optimal parameters are determined for different root
distribution (uniform and linear), and soil types (sandy loam, loam,
and clay loam). We realize that economics of irrigation system andwater costs should also be considered, however, we do not consider these factors in the present analysis, as dollar values can beeasily included in the formulation of the objective functions whenavailable.
2. Materials and methods
A schematic of the followed computational procedures to arriveat the ﬁnal set of optimal design and decision parameters is shownin the ﬂow chart of Fig. 1. The main computational loop is deﬁned by
the AMALGAM evolutionary search algorithm (Vrugt and Robinson,2007), selecting speciﬁc values of drip-line installation depth (D)and distance (L), irrigation duration (ID) and frequency (f) from

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

83

Start
AMALGAM
Generatesystem design andmanagement
parameters

Irrigation
frequency, f

Drip-line
installation
depth, D

Irrigation
duration, ID

HYDRUS-2D

Drip-line
distance, L

Calculate OF1

Calculate Ta and DP
Calculate average soil water
potential in top 30 cm

Calculate OF2 and OF3

Penalize the objective functions

Less thanthreshold?

No

Yes

Convergence?

No

Yes

Return Pareto optimal solution

End
Fig. 1. Flow diagram of the presented multi-objective optimization approach. In this diagram, Ta denotes actual transpiration, DP is deep percolation, and OFi represents theobjective function i.

the prior allowed ranges (Table 1). For each of the selected combinations, the HYDRUS-2D unsaturated water ﬂow model (Šimůnek
et al., 2008) simulates soil water ﬂow for the speciﬁc irrigation
system selected and computes the spatial and temporal distributions of soil water potential and water content, with corresponding
root water uptake, and drainage and actual transpiration rates
for the growing season. In addition, cumulative values of actual
transpiration (Ta ), deep percolation losses (DP), and average soil
water pressure head values for the top 30-cm at predeﬁned harvesting times are computed for each growing season simulation.
Upon completion of each HYDRUS simulation, AMALGAM evaluates the three objective functions (OF’s) of Eqs. (8) or (9) with

Table 1
Ranges of system design and water application parameters for HYDRUS model simulations, as deﬁned for AMALGAM search algorithm.

Drip-line depth (cm), D
Drip-line distance (cm), L
Duration of irrigation (h), ID
Irrigation frequency (days−1 ), f

Lower value

Upper value

Interval

20
40
8
1/7

70
300
24
1

5
10
1
1/7

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M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

corresponding weighting factors (Eq. (8) only), and uses the OFvalues to seek new system design parameters from the priorallowed ranges for a next iteration with HYDRUS-2D calculations.
It is here where the AMALGAM search algorithm is novel and efﬁcient, in seeking all possible optimal solutions with a minimum
number of HYDRUS-2D simulation iterations required. Upon completion of the optimization, as determined by various pre-deﬁned
convergence criteria, AMALGAM derives the Pareto solution set
that consists of a sample of optimal solutions representing different
tradeoffs to the multi-criteria optimization problem. For this studyof alfalfa subsurface drip irrigation, the main tradeoffs are between
water application and transpiration. As an example of these complex trade-offs, for large distance between drip-lines more watermust be applied in order to wet the root zone and satisfy non-stressconditions for high transpiration. However, adding more water
to the root zone also increases deep percolation, whereas otherparameters such as drip-line installation depth also inﬂuence transpiration and deep percolation. Thus, we seek a balance between
the different processes and parameters. This trade-off balance isachieved by the Pareto front analysis, and provides for development of optimal subsurface drip irrigation system parameters fora range of soil types (Table 2) and typical root distributions. In thefollowing we provide a detailed description of the various elementsof our optimization framework. Although we limit our attention toalfalfa, the methodology presented herein can readily be applied
to develop efﬁcient drip irrigation designs for a wide range ofcrops.
2.1. Numerical HYDRUS simulations
The two-dimensional module of the HYDRUS-2D/3D package(Šimůnek et al., 2008) was used to simulate soil water movement
and plant root water uptake for the speciﬁc irrigation system design
and water application parameters deﬁned in Table 1.
2.1.1. Soil water movement
The HYDRUS-2D model uses the two-dimensional form ofRichards’ equation to describe transient water ﬂow in isotropic
unsaturated soils:



∂h
∂
∂
K(h)
=
∂t
∂x
∂x



∂
+
∂z





∂h
K(h)
+ K(h) − S(h)∂z(1)

where  is the soil’s volumetric water content [L3 L−3 ], h denotes
the soil water pressure head [L], S(h) is a sink term [L3 L−3 T−1 ]representing plant root water uptake, t signiﬁes time [T], K(h) is
the unsaturated hydraulic conductivity function [LT−1 ], and x andz are the horizontal and vertical spatial coordinates [L], respectively, of the simulated soil domain. Solution of Eq. (1) requires
characterization of the soil hydraulic properties, as deﬁned by
the soil water retention, (h), and unsaturated hydraulic conductivity function, K(h). We used the constitutive relationships ofvan Genuchten-Mualem (van Genuchten, 1980) and represent theeffective saturation, Se by:Se = − r1=m
s − r(1 + |˛VG h|n )

and



1/m m

K(h) = Ks Sel 1 − (1 − Se)(2)(3), respectively. Eq. (1) is solved using a Galerkin type linear
ﬁnite element method applied to a network of triangular elements. Time integration is achieved using an implicit (backwards)
ﬁnite difference scheme, with the approximate equations solved
iteratively and time steps adjusted depending on convergence
rates.
2.1.2. Root distribution and crop water uptake
The spatial distribution of the plant roots of alfalfa exerts astrong inﬂuence on soil water ﬂow, root water uptake, and deep
drainage, and therefore essentially determines DP and Ta for a given
irrigation strategy. In the absence of detailed information about the
spatial distribution of alfalfa roots in the literature, we assumed two
different root distributions (linear and uniform), and conductedsimulations for each of these two cases. Speciﬁcally, we used thesink term, S(h), in Eq. (1) to quantify root water uptake, using thecommonly used approach of Feddes et al. (1976):
S(h) = ˛(h) × Sp ,where ˛(h) is a dimensionless root water uptake reduction function
with values between zero and one, to account for soil water stress. If
the soil maintains favorable conditions for root water uptake, S(h) is
equal to the potential root water uptake rate, Sp [L3 L−3 T−1 ]. However, if the soil is too dry or too wet at any given location (x,z), then˛ < 1, and the uptake at position (x,z) is linearly reduced with themagnitude determined by the reduction function parameters foralfalfa as selected from a data-base (Taylor and Ashcroft, 1972). The
potential root water uptake rate, Sp , is calculated from (Šimůnek
and Hopmans, 2009):
Sp (x, z) = ˇ(x, z)Lx Tp ,,(3)

where  s and  r represent the saturated and residual water content [L3 L−3 ], respectively, Ks is the saturated hydraulic conductivity[LT−1 ], ˛VG [L−1 ], n, and l are shape parameters, and m = 1 − 1/n.
Though these four parameters are directly related to pore size
distribution, pore connectivity and tortuosity, they are generally
obtained from ﬁtting of (h) and K(h) data to functions (2) and(5)

where ˇ(x,z) [L−2 ] represents the normalized root density for any
coordinate in the two-dimensional soil domain, Lx [L] denotes thewidth of the soil surface associated with the potential plant transpiration, Tp [LT−1 ]. We note that the actual plant transpiration, Ta[LT−1 ], is computed in HYDRUS-2D by numerical integration of Eq.
(4) across the entire root zone, ˝ [L2 ], or:
1
Ta =
Lx



1
S(h, x, z)d˝ =
L
x
˝



˛(h)Sp (x, z)d˝,

(6a)

˝

Substituting Eq. (5) into (6a), yields:Ta = Tpˇ (h, x, z)d˝,

(6b)

˝

where ˇ’ (h, x, z) = ˛(h) × ˇ(x, z).
Potential evapotranspiration of alfalfa was calculated using theirrigation scheduling alfalfa (ISA) model. This management model
was especially developed to help determine a suitable irrigation
strategy for alfalfa and to predict the effect of water stress on yield(Snyder and Bali, 2008). This model implements a standardized
method for estimating reference evapotranspiration, ETo (Allenet al., 1998, ASCE, 2005) and uses the crop coefﬁcient, Kc , to compute potential daily evapotranspiration, ETp , (Doorenbos and Pruitt,1977), or
ETp = ETo × Kc .

2

(4)

(7)

The Kc value varies during the growing season and depends on
the length of initial, rapid, midseason and late growth stages. To
illustrate this approach, we consider Fig. 2, presenting daily ETp as
calculated by the ISA model for a typical growing season of alfalfa,
from March 1 (day 0 in Fig. 2) to October 26 (day 240 in Fig. 2)
in the Sacramento Valley, USA (B.R. Hanson, unpublished data). As
illustrated in Fig. 2, ETp varies widely during the growing season,with Kc values ranging between near zero (after each cutting) andmaximum values near 1.0 in the mid season. Because of full soil

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

85

Table 2
Soil hydraulic function parameters for soils of scenarios T1 (clay loam), T2, (loam) and T3 (sandy loam), as taken from Carsel and Parrish (1988).

Clay loam
Loam
Sandy loam r (cm3 cm−3 ) s (cm3 cm−3 )˛VG (cm−1 )nKs (cm day−1 )

l

0.095
0.078
0.065

0.41
0.43
0.41

0.019
0.036
0.075

1.31
1.56
1.89

6.24
24.96
106.1

0.5
0.5
0.5

coverage by the alfalfa, soil evaporation can be considered negligible throughout the simulated growing season, so that Tp in Eqs.
(5) and (6) is equal to ETp . Typically, the alfalfa is cut about once amonth, starting in late April, through late October, for a total of 6
cuttings. In HYDRUS-2D, this time series of ETp deﬁnes the potential root water uptake rate, Sp . We assumed a growing season witha total of six growing cycles, corresponding with the six cuttings.
Despite the expected variations in alfalfa yield for the wide rangeof irrigation system parameters analyzed, the alfalfa cutting dates
were held ﬁxed to these same dates (Fig. 2) for all simulations.
2.1.3. Domain properties and boundary conditions
In all HYDRUS-2D simulations we assumed that the soil ishomogeneous and that a drip-line behaves as an inﬁnite line
source with a constant water discharge rate along the drip line.
Water ﬂow in a homogenous soil is symmetrical in the horizontal direction between drip lines, so that only one-half of thedomain space between drip-lines needs to be simulated. Therefore, the two-dimensional transport domain (Fig. 3) is rectangular,and L/2 cm wide (a half-length between drip-lines). We simulated
a soil domain with a depth of 200 cm, and placed the subsurface
drip line at depth D (right side of the domain). In each numerical simulation we assumed that the maximum rooting depth ofalfalfa was similar to 200 cm, the bottom boundary of our hypothetical soils (Table 2). The spatial domain was discretized using
triangular ﬁnite elements (FE). Depending on drip-line distance, L,
the number of elements varied between 1954 and 4897, with element size gradually increasing with distance from the drip-line.
The smallest FE size of 0.1 cm was selected around the drip-line,
whereas the large FE’s were about 5 cm, furthest away from thedrip line. A high nodal density is required in the immediate vicinityof the drip-lines to be able to accurately model the large spatial
gradients in soil water pressure head caused by the inﬁltratingwater.
A time-variable ﬂux boundary condition was used along theboundary elements representing the emitter, so that the total drip
emitter discharge when integrated around the emitter was equal
to the deﬁned emitter discharge, q (volume per unit emitter area
and time) (Gärdenäs et al., 2005). A free drainage (unit gradient) boundary condition was applied along the bottom boundary,allowing for downward drainage and leaching. All other remaining

Fig. 2. Assumed alfalfa ETp using ISA model during simulated growing season. The
arrows indicate the different cutting times.

boundaries were assigned a zero water ﬂux condition. Irrigation
system design parameters were optimized using HYDRUS-2D simulations for a sandy loam, loam and clay loam soil type, with van
Genuchten–Mualem soil hydraulic parameters of Eqs. (2) and (3)listed in Table 2.
2.2. Parameter optimization
Optimal irrigation system design and water application must
maximize root water uptake and Ta , while minimizing total irrigation system installation costs, applied irrigation water and drainage
water losses. For that purpose we optimized drip line depth (D) anddistance (L), irrigation frequency (f), and irrigation duration (ID),with the latter two parameters enabling minimization of applied
irrigation water (AW) using a ﬁxed emitter discharge rate. The
HYDRUS-2D model computes Ta , DP losses, and the near surface soil
water content at the alfalfa cutting times for each of the selectedparameter sets for each of the AMALGAM iterations, until it converges. In the ﬁnal step, the search algorithm derives the Pareto
solution set, consisting of a number of optimal solutions that each
represents a different trade-off between various optimal irrigation
system design parameters. The trade-offs are the result of minimizing three objective functions in Eqs. (8) or (9), and allows forselecting the optimal combination of parameters depending on relevance or cost differences between them. Hence, there is no single
optimum parameter set, but the Pareto analysis provides for many
combinations of parameter values that in combination result insimilar residual values for the three combined objective functions.
Typically, parameter optimization methods such as AMALGAM
are used to determine optimal parameter values by minimizing
residuals between measured and model-simulated variables. However, in the present study, the objective functions are deﬁned byminimizing residuals consisting of differences between optimum
and model parameter/variable values, as deﬁned in Eqs. (8) and(9). In some way, one can view our study as a sensitivity analysis, evaluating the impact of variable irrigation system design and

Fig. 3. Schematic overview of the two-dimensional numerical grid used by
HYDRUS-2D for the presented simulations.

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M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

water application parameters on water use efﬁciency, deﬁned by
the ratio of AW to Ta .
2.2.1. Irrigation system parameters
The optimized irrigation system design and water application
parameters with their upper and lower bound values are listed inTable 1. These parameters deﬁne the width of the prior uniform
distribution functions from which selected parameter values aretaken in the AMALGAM optimization procedure. They include thedrip-line installation depth, D (L), drip-line distance, L (L), irrigation
duration, ID (T), and irrigation frequency, f (T−1 ). As each of thesefour parameters are expected to largely affect soil water ﬂow, root
water uptake, and drainage, we assumed physically realistic ranges,
and used AMALGAM to ﬁnd Pareto optimal solutions that simultaneously minimize the three different objectives considered in
Section 2.2.2. In essence, we seek optimum combinations of D, L, ID
and f, that maximize irrigation efﬁciency (as deﬁned by minimizing
applied water (AW) and deep percolation (DP) losses) and Ta . Without a preference to any of the individual objective functions (OF’s)in the applied multi-criteria optimization analysis, the minimum
uncertainty that can be achieved for the optimized parameters isessentially determined by the trade-off in minimizing the threedifferent objectives. Speciﬁcally, if two objectives are conﬂicting,it is not possible to ﬁnd a single combination of parameter values that optimizes both of these two objectives simultaneously.
Instead, a trade-off will be determined in the ﬁtting of both multiple
objectives, resulting in parameter variations. The Pareto solution
set deﬁnes the trade-offs with minimum OF values.
With regard to ﬁnding the optimum drip line depth (D), thenear-surface soil must become sufﬁciently dry for each cutting
operation, without reducing crop yield. On the other hand, a toolarge installation depth may cause deep percolation losses andincrease installation costs. Whereas increasing drip-line distanceis cost-effective, a too large spacing may require higher irrigation
frequency to maintain optimum root zone soil moisture conditions
between the drip lines, and increase drainage losses by deep percolation as a result. Similarly, large irrigation duration values willaffect deep drainage and may cause plant water stress by introducing anaerobic soil conditions, thereby reducing Ta and yield. On theother hand, applying irrigation water at values less than Tp may
reduce plant transpiration and yield even more. Therefore, irrigation frequency and/or water application duration must be higher
in summer months compared to the early and late growing season.
However, rather by varying irrigation frequency during the growing season, we chose to vary irrigation water application time, using
average Tp rates for each of the six cutting cycles. In all our calculations reported herein, we assume a constant drip-line discharge
per unit length, Q = 2.3 × 10−3 m3 m−1 h−1 (Alam et al., 2002).
Initial AMALGAM optimizations demonstrated the need for theﬁnite element (FE) size of HYDRUS-2D domain to be ﬂexible, toaccommodate different irrigation designs and to minimize numerical model errors. For example, each combination of drip-line
distance (L) and drip-line installation depth (D) required a different two-dimensional nodal discretization in HYDRUS-2D. For that
purpose, we deﬁned discrete drip installation depth (D) and drip
line distance (L) increments of 5 and 10 cm, respectively (Table 1).
We followed a similar approach for irrigation duration (ID) and frequency (f) parameters, and used 1-hour and 1-day time incrementsinterval, respectively. While this approach signiﬁcantly increased
the efﬁciency of the multi-criteria analysis, it did not affect the ﬁnal
outcome.
2.2.2. Objective functions
For general application, we deﬁne the following three objectivefunctions to be minimized, OFi (i = 1, 2, 3), which in combination optimize irrigation system cost and beneﬁt by minimizing

drip-line depth (D), applied irrigation water (AW), and deep percolation (DP), while maximizing root water uptake and croptranspiration (Ta ) and distance between irrigation drip-lines (L):



OF1 =

(D − Dmin )
1
× ωD
+ ωL
ωL + ωD(Dmax − Dmin )





1−

(L − Lmin )(Lmax − Lmin )

1AW − AWmin
DP − DPmin
× ωAW
+ ωDP
ωAW + ωDP
DPmax − DPmin
AWmax − AWminTa
OF3 = 1 −
Tp
OF2 =



,

(8)

The weighting factors ωD , ωL , ωAW , and ωDP allow to differentiate between cost of the decision variables within each objectivefunction. In particular, ωD denotes the cost of installation per unit
depth, ωL signiﬁes the cost of unit length of drip-line, ωAW is thecost of unit volume of water, and ωDP represents either the cost ofunit volume of water or the cost needed for disposing the unit volume of drained water. Realizing that minimum values of both DP
and AW are zero and that DPmax = AWmax , each of the three objective functions are scaled functions with values ranging between
zero and one.
The ﬁrst objective function to be minimized, OF1 , minimizes theinstallation cost and equipment (i.e., drip-line) expense by minimizing the drip-line depth (D), while simultaneously maximizing
the distance between drip-lines (L). This performance measure issolely dependent on the actual values of L and D and does not
require a HYDRUS-2D run to calculate soil moisture dynamics. The
second objective function, OF2 , minimizes the cost of water byminimizing the applied irrigation water (AW) and drainage water
losses by deep percolation (DP). The third objective function, OF3 ,maximizes the yield proﬁt by minimizing soil water stress to maximize plant transpiration. These latter two criteria can be computed
only using HYDRUS-2D. Differential weighting allows ﬂexibility
in applying the objective functions to different costing scenarios, if applicable. For example, in OF1 one can assign the cost of
installation and equipment for the known speciﬁed region or evenselect either D or L as the main decision parameter by assigning azero weight to the speciﬁc parameter, if appropriate. We note thatassigning a zero weight to a parameter excludes it as an objectivefunction parameter, however, it will remain a variable parameter with an identiﬁed range (Table 1), thereby contributing to thePareto distribution of optimized parameters.
For most of the analysis in this study, we simpliﬁed the threeobjective functions of Eq. (8) by eliminating the cost factor, andinstead deﬁned the combined objectives such as to optimize irrigation system parameters by minimizing applied irrigation water(AW), while maximizing root water uptake and crop transpiration(Ta ) and distance between irrigation drip-lines (L):OF1 = 1 −(L − Lmin )(Lmax − Lmin )AW − AWmin ,AWmax − AWminTa
OF3 = 1 −
Tp

OF2 =

(9)

where AW = q × ID × N with N representing the number of irrigations of duration ID and drip-line discharge q (L T−1 ). Conveniently,
each of the three objective functions are scaled functions with values ranging between zero and one. The ﬁrst objective function to beminimized, OF1 , now maximizes the distance between drip-lines(L). The second objective function, OF2 , minimizes applied irrigation water (AW) and the third objective function, OF3 , seeks tomaximize plant transpiration by minimizing soil water stress. The
ﬁrst two performance measures can be calculated directly from theparameters considered herein and hence do not require HYDRUS2D simulations. Only for the third criteria, simulated soil moisturedynamics are required.

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

Minimization of objective functions in (Eq. (9)) might lead tophysically unrealistic results. We therefore introduce two main
constraints. First, only those solutions were accepted for which Ta
was equal or higher than 85% of Tp . Available data indicated that
alfalfa yields were only slightly affected for such a reduction, butthat signiﬁcant yield losses may occur for lower Ta values. Becauseof uncertainties in crop water production function (Grismer, 2001),
which is depended on irrigation management and climatic region,
we set the minimum acceptable value for Ta at 85% of Tp . Second, cutting of the alfalfa crop requires that the average soil waterpressure potential in the top 30 cm is lower (more negative) thanthe prescribed threshold value of −500 cm, (B.R. Hanson, unpublished data) to allow cutting and baling machinery to drive on
the ﬁeld without destroying soil structure and crop. If one of theconstraints was violated we penalized the corresponding objectivefunctions so that these solutions were not further considered during the Pareto search. In general, values for both constraints canbe changed for speciﬁc crops and irrigation water management
applications.
2.2.3. AMALGAM method
In the presence of multiple conﬂicting objectives, it is highly
unlikely that a single optimal parameter combination exists thatsatisﬁes all objectives. Instead, it is expected that a multi-criteria
optimization will result in considerable trade-offs, as with multiple optimal parameter combinations, as deﬁned by Pareto fronts(in case of two objectives) or Pareto surfaces (three or more objectives). The AMALGAM evolutionary search method of Vrugt andRobinson (2007) was used to explore the multiple parameter space
and to provide the Pareto solutions using a very computationalefﬁcient algorithm. A detailed explanation of this approach canbe found in Vrugt and Robinson (2007) and Vrugt et al. (2009),
and is beyond the scope of the current work. In brief, AMALGAM
runs multiple different optimization methods simultaneously thateach learns from each other using a common population of parameter values. The combination of optimization methods included in
AMALGAM are Genetic Algorithm, Particle Swarm Optimizer, Differential Evolution, and Random Walk Metropolis with adaptive
updating of the parameter covariance matrix. Improved parameter sets are generated in an iterative way, with each algorithm
contributing in proportion to its convergence rate towards a ﬁnal
optimal parameter set. This approach has proven to be a credible and computationally efﬁcient optimization model, especially
if many parameters are optimized simultaneously. It has shown tosigniﬁcantly improve the efﬁciency of Pareto optimization, and is
ﬁnding increasing use in many different study ﬁelds (e.g., Huisman
et al., 2010).
2.3. Example scenarios
To illustrate the generality of the irrigation design approach, we
present three different sensitivity analyses with identical climaticconditions and design constraints (Section 2.2.2.) to evaluate theeffect of (1) selected irrigation parameters (I1, I2, and I3), (2) root
distribution (R1 and R2), and (3) soil texture (T1, T2, and T3) on
system design and management (Table 3).
The effect of irrigation system parameter type on the optimization results was investigated using three different scenarios
considering simulations for the loam soil with uniform root distribution. For scenario I1, we evaluated the optimization results
for the case that minimizes drip-line depth (D), AW and DP, while
maximizing drip-line distance (L) and plant transpiration (Ta ). In
other words, scenario I1 minimizes the cost of installation, equipment, and water, while maximizing proﬁt. We note that the main
objective functions used in this study are those presented in Eq.
(8), but here we show the generality of the presented method by87

assigning the arbitrary cost of unity for all decision parameters. Scenario I2 is the same as I1, except that we excluded the parameter Dfrom the analysis (ωD = 0), as drip installation costs for the founded
range in this study are almost independent of installation depth(Toro, Inc., personal communication). Scenario I3 is the same as I2,
except we excluded DP as an optimization parameter (ωDP = 0), in
addition, thus optimizing solely on maximum drip-line distance, L(OF1 ) (minimizing installation costs), and minimizing plant water
stress (OF3 ) (maximizing plant transpiration, Ta , and crop yield),while applying a minimum amount of applied irrigation water, AW(OF2 ).
It has been shown before that depth distribution of roots andassociated root water uptake has a major inﬂuence of the soil water
regime in micro irrigation (Gärdenäs et al., 2005; Hanson et al.,2008). However, since information on alfalfa root distribution in
subsurface drip irrigation is absent, for the purpose of this optimization study we assumed two likely distribution functions that
describe a (1) uniform and constant root distribution across the rootzone (R1), and (2) linear decreasing rooting pattern with soil depth(R2). For the linear model, the maximum root density is at the soil
surface and decreases linearly to a zero value at the bottom of theroot zone. To minimize the number of optimizations for the root
distribution analysis, we limited the optimizations to those for aloam soil only, setting ωD in Eq. (8) equal to zero. Moreover, we set
ωDP in Eq. (8) to zero, to focus the analysis on minimizing applied
water (AW) and maximizing drip-line separation distance (L) andplant transpiration (Ta ).
The soil textural and hydraulic properties affect irrigation
design and management due to their effect on horizontal, downward and upward vertical water movement. Therefore, three soil
textures—clay loam (T1), loam (T2), and sandy loam (T3)–wereconsidered in this study, evaluating the effect of soil hydraulicproperties (Table 2) on irrigation design and management, for soil
domains with a uniform root distribution only.

3. Results and discussion
To better understand the results of a multi-criteria parameter optimization with AMALGAM, we present the Pareto contour
plot of Eq. (9) for a loam soil and uniform root distribution in
Fig. 4a, with corresponding two-dimensional bi-criterion plots forthe three-objective optimization in Figs. 4b–d. As described in Section 2.2.2, the multi-objective optimization approach used hereinmaximizes drip-line distance (L, OF1 ) and crop transpiration (Ta ,OF3 ), minimizing applied irrigation water (AW, OF2 ). In this speciﬁc
application, the drip line installation depth, D, was excluded fromOF1 , but their values are represented among the Pareto data points.
Both the contour plot and 2-dimensional Pareto graphs present
values for optimum parameters, with both AW and Ta normalizedwith respect to Tp . This ﬁnal optimum parameter space is relatively
small, because only limited water stress is allowed (Ta /Tp > 0.85),
while simultaneously requiring dry soil conditions at all six cutting
times, as achieved by penalizing the objective functions if either
constraint is not satisﬁed.
We clarify that all data points of Fig. 4b–d combined, makeup
the Pareto contour plot of Fig. 4a, with each data point representing a different Pareto optimal solution for any of the four design
parameters combinations used in HYDRUS for which the two design
constraints were satisﬁed. Speciﬁcally, the contour plot of Fig. 4a
projects the Pareto surface (representing the optimal solutions forthe three OF’s) on the two-dimensional surface, with the isolines ofdrip-line distance (L) representing the optimal solutions for eachspeciﬁc L. Hence, Fig. 4a and b presents identical data, with the isoline of L = 40 cm (smallest drip-line spacing) of Fig. 4a corresponding
with the Pareto front of Fig. 4b. The solid lines of Fig. 4b–d deﬁne

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M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

Table 3
Example scenarios of sensitivity analysis.
AnalysisType
Irrigation system
Root distribution
Soil texture

Root distribution

Soil texture

Uniform
Uniform
Uniform
Uniform
LinearUniform
Uniform
Uniform

Loam
Loam
Loam
Loam
Loam
Sandy loam
Loam
Clay loam

Scenario
I1
I2
I3
R1
R2
T1
T2
T3

the so-called Pareto fronts that represent the trade-offs between
respective objective functions.
In Fig. 4a the Pareto contour plot shows the trade-off between
Ta and AW, with the contour lines representing isolines of dripline distance, L. For example, Fig. 4a shows that for a given L value,larger Ta values will require supplying more AW to minimize soil
water stress. Moreover, to maintain identical Ta values, values ofAW must increase as L is larger. Speciﬁcally, the trade-off in Fig. 4bshows that increasing the AW/Tp ratio from 95% to 110% increases
the value of Ta /Tp from 85% to 94%. However, a further increase of
AW is much less sensitive to Ta , as optimum soil moisture conditions
will remain. The results of Fig. 4b are consistent with Fig. 4c, presenting the tradeoffs between root water uptake, as represented
by Ta /Tp and drip-line distance, indicating that Ta is increasingly
reduced as drip-line distance is increased. By changing drip-line
distance from 40 to 140 cm, the Ta value reduces very little, but
an additional increase of drip-line distance to 240 cm reduces therelative Ta by about 10%.
When considering Fig. 4d, comparing tradeoffs between L and
AW/Tp , we note that increased irrigation water applications areneeded with increasing distance between drip-lines. This is to
be expected as more irrigation water is required to reﬁll soil
water storage between irrigation events with increasing distance
between irrigation drip-lines. The Pareto front signiﬁes the tradeoff between L and AW. For example, large values for L result in low
values for OF1 and high values of OF2 (Eq. (9)), as more AW must be

Parameter included in the objective functions
OF1

OF2

OF3

L, D
L
L
L
L
L
L
L

AW, DPAW, DP
AW
AW
AW
AW
AW
AW

Ta
Ta
Ta
Ta
Ta
Ta
Ta
Ta

supplied to minimize soil water stress and to maximize Ta . In contrast, by the same reasoning, low L-values result in larger values for
OF1 and low values of OF2 . Consequently, the two extreme choicesof L produce about identical values for the total OF value, thereby
deﬁning the Pareto front.
3.1. Example scenarios
As before, the Pareto points that exceeded the wetness and transpiration constraints were excluded from further analysis. Thus,
for each different scenario the trade-off of applied water (AW) anddrip-line distance (L) are presented, whereby each point of the presented bi-criterion plots corresponds to different Pareto solutions.
To better illustrate the effect of drip-line depth (D), we used different symbols for the various drip-line installation depth classes
for all bi-criterion plots of Fig. 5 (I-scenarios), 6 (R scenarios), and
9 (T scenarios). We also presented the Pareto contour plot of eachscenario showing the trade-offs between drip-line installation distance, L (OF1 ), and applied irrigation water, AW (OF2 ) with thecontour lines representing isolines of alfalfa transpiration, Ta (OF3 ).
3.1.1. Irrigation systems – I1–I3
A comparison of scenarios I1 with I2 and I3 is presented in Fig. 5,
for the case of a uniform alfalfa root distribution in a loamy soil.
The plots 5a–c summarize the effect of different drip-line installation depths, D, on the Pareto distributions for AW and drip-line

Fig. 4. (a) Pareto contour plot of objective functions in Eq. (9), showing the trade-off between relative transpiration and AW, with the contour lines representing isolines ofdrip-line distance. The other three graphs are corresponding two-dimensional bi-criterion plots for (b) relative transpiration (Ta /Tp ) and (AW/Tp ), (c) relative transpiration(Ta /Tp ) and drip-line distance (L), and (d) drip-line distance (L) and (AW/Tp ).

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

89

Fig. 5. Bi-criterion (top) and Pareto contour (bottom) plots for I-scenarios, optimizing AW and drip-line distance, L, for (a and d) I1, including D and DP as optimization
parameters, (b and e) I2, excluding D and including DP as optimization parameters, and (c and f) I3, excluding both D and DP as optimization parameters. The drip-line depths
in plots a–c are indicated with different symbols. The contour lines in plots d–f represent isolines of relative transpiration (Ta /Tp ).

distance, L. We note that for the I2 scenario, ωD is set to zero, andfor I3 both ωD and ωDP values in Eq. (8) are set to zero. Similarly,
as in Fig. 4, each symbol represents a high-ranked optimum solution from the Pareto set of optimal solutions. Across all 3 scenarios,
we ﬁnd that AW must increase as drip-line distance (L) increases.
The optimization also showed that the shallowest possible installation depth in this study to satisfy the soil wetness constraint was
60 cm for the case of a uniform root distribution and a loamy soil.
That is, even for the I1 scenario (with drip-line depth, D, among the
optimized design parameters), no optimal installation depth solution was found for values of D close to the soil surface, hence, we
presented only results for drip-line installation depths larger than
60 cm.
A comparison of Fig. 5a–c indicates that the Pareto solutions
exhibit a tendency towards deeper drip-line installation depths as
one compares I1 with I2 and I3, with the number of AMALGAM
solutions for the shallow drip-line depth (D = 60 cm) decreasing. As
expected, the minimum number of optimal solutions for D = 70 cm
is the largest for the I3 scenario, as deep percolation was excluded
as a decision parameter for this scenario, thereby allowing for larger
drip installation depths.
The corresponding Pareto contour plots are presented inFig. 5d–f; with the contour lines representing isolines of relativetranspiration, Ta /Tp . When comparing Fig. 5a–c with their corresponding contour plots of Fig. 5d–f, we note that the Pareto frontof each scenario is represented by the isoline of Ta /Tp = 0.85 in thecorresponding Pareto contour plot, as this ratio was the minimum
allowable transpiration ratio. All Pareto contour plots show clearly
that higher transpiration rates are achieved by either increasing AW
or decreasing L, or both.
Overall though, when comparing Fig. 5a–f, relative small differences between the three different scenarios for D and Ta were
found. This indicates that the selected design parameters L and

AW are sufﬁcient when optimizing SDI for alfalfa for optimum
Ta , but with the assumption that DP losses are relatively unimportant. Indeed, the selected soil is a homogenous non-layered
loam, for which DP is relatively small and the installation cost
is not signiﬁcantly affected by varying the installation depth. For
coarse-textured or layered soils with dense layers at deeper depths,
we expect the results to be different for the I2 and I3 scenarios.
For these types of soils it is rather difﬁcult to maintain favorable
soil moisture conditions required to maximize alfalfa yield andsimultaneously minimize deep water percolation. Therefore, DP
is likely to vary considerably between parameter combinations,and for other than ﬁne textured soils, this variable will becomemuch more important relative to rooting and drip-line installation depths. Alternatively, it is possible to assign an appropriate
cost value for deep percolated water and installation depth in Eq.
(8), thereby signifying their importance in managing drip irrigation
systems.
3.1.2. Root distribution – R1 and R2
The depth distribution of alfalfa roots, in relation to drip-line
depth and drip-line distance will largely affect the soil water distribution in the root zone, as it will determine spatial distributionof root water uptake, soil water stress, and DP out of the root zone,
and thus affecting irrigation system design and management. We
therefore analyze the effect of two root distribution types (R1, uniform; and R2, linearly decreasing) on irrigation system design andmanagement parameters (Fig. 6). In addition, we address the effectof root distribution on soil water and root water uptake dynamics
for a given irrigation system (Fig. 7).
3.1.2.1. Analysis of irrigation system parameters. The effect of thetwo root distribution models (uniform and linear) on the design andmanagement of SDI system is presented in Figs. 6a and b for (R1)

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M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

Fig. 6. Bi-criterion (top) and Pareto contour (bottom) plots for R1 and R2, optimizing AW and drip-line distance, L, for (a and c) R1, uniform root distribution, and (b and d)
R2, linearly decreasing root distribution. The drip-line depths in plots a and b are indicated with different symbols. The contour lines in plots c and d represent isolines ofrelative transpiration (Ta /Tp ).

uniform and (R2) linearly decreasing root system. The bi-criterion
graphs illustrate the properties of the Pareto front for drip-line distance (L) and applied water (AW) for drip-line installation depth(D) values ranging between 50 and 70 cm. Moreover, the contour
plot of alfalfa relative transpiration, Ta /Tp , representing the tradeoffs between drip-line distance, L, and irrigation applied water,AW, for both R1 and R2 scenario are presented in Fig. 6c and d.

These plots convincingly show that larger values of L (cheaper irrigation system) can be compensated for by larger values of AW,
thereby minimizing soil water stress. For the uniform root distribution results (Fig. 6a, R1), none of the Pareto solutions derived withAMALGAM with a drip line installation depth (D) of less than 60 cm
satisfy the soil dryness constraint at harvesting times. This also
explains why the optimal AW values generally increase as drip-line

Fig. 7. Spatial distribution of (a) actual root water uptake distribution (ˇ , [L−2 ]) and corresponding volumetric water content (b) for R1 scenario (uniform root distribution),with columns representing the times in the middle of each of the six different cutting cycles (cutting cycle is indicated in the top-right corner). The drip-line is installed at
the 60 cm depth.

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91

Fig. 8. Spatial distribution of (a) actual root water uptake distribution (ˇ , [L−2 ]) and corresponding volumetric water content (b) for R2 scenario (linear decreasing root
distribution), with columns representing the times in the middle of each of the six cutting cycles (cutting cycle is indicated in the top-right corner). The drip-line is installedat the 60 cm depth.

installation depth increases, simply to minimize soil water stress.
Similarly, as in Fig. 5, increasing the drip-line distance will lead toan increase in AW. This highlights a general ﬁnding of our paper, anddemonstrates that by increasing the distance between the drip lines(less costly irrigation system) additional applied irrigation water isrequired to maintain adequate soil moisture conditions in the rooting zone to maximize yield. This trade-off is intuitively clear, andadvocates the use of the joint AMALGAM and HYDRUS-2D framework considered herein to ﬁnd optimal irrigation practices andsimultaneously satisfy speciﬁc constraints imposed by the farmer.
In contrast to the optimization results of R1, the scenario witha linearly decreasing rooting system (R2, Fig. 6b), allows for a shallower drip-line installation depth (D) of 50 cm, as the maximum
root activity at the near soil surface provides for intermittent periods of soil surface dryness. Moreover, the increasing soil water
availability with the shallower installation depths minimizes soil
water stress as most roots are located near the soil surface. Alternatively, increasing the drip-line installation depth for scenario R2
enhances irrigation water losses by DP. Therefore, the optimization
results in Fig. 6b show that there are no optimal irrigation systems
for D-values larger than 60 cm. Comparing the alfalfa transpiration(Ta ) ranges for R1 (Fig. 6c) and R2 (Fig. 6d), the results show that afraction of AW is likely available for the higher density of roots inthe upper part of the soil proﬁle by capillary rise for D < 65 cm. However, this upwards-directed water supply was insufﬁcient for the
loamy soil to obtain the Ta as high as that achieved for the uniform
root distribution scenario, R1. The AMALGAM optimization results
show that this upwards-directed water supply was not sufﬁcient
to satisfy the prior deﬁned constraints on Ta and thus no optimal
solution was found for D > 65 cm.
3.1.2.2. Analysis of soil water distribution and root water uptake. As
an example, we evaluate the effect of irrigation system parameters for ﬁxed values of drip-line distance (L = 170 cm) and drip-line
installation depth (D = 60 cm). This is the most economic irrigation

system design, with maximum spacing of the drip-lines, and nearest drip installation to the soil surface. The optimized irrigation
scheduling for this scenario resulted in a single Pareto irrigationinterval of two days only, with irrigation duration (ID) of 10 and
20 h for root systems R1 and R2, respectively. To maintain favorable soil moisture conditions and avoid water stress conditions,scenario R2 with a linearly decreasing root distribution requires
a doubling of irrigation water amounts, AW, to maintain similar Ta
values. Figs. 7 and 8 present the corresponding spatial distributionsof root water uptake (top), as represented by ˇ (h,x,z) of Eq. (6b) andthe sink term S computed using daily values (day−1 ), and volumetric soil water content (bottom) for R1 (Fig. 7) and R2 (Fig. 8). The
six different panels represent snapshots at the middle of each of
the six different cutting cycles. The spatial patterns or root water
uptake are largely determined by the spatial distribution of roots,
as deﬁned by ˇ(x,z) in Eq. (5), with reductions in uptake caused by
the soil water stress function, ˛(h), in Eq. (4).
For the uniform root distribution (Fig. 7), the roots initially takeup water uniformly from the entire soil proﬁle. In later cutting
cycles, the uptake rate varies spatially depending on available soil
water. For the ﬁrst three cutting cycles, the soil wetting patterns
remain in the center of the soil proﬁle (Fig. 7b), and is symmetric around the drip line (D = 60 cm), causing reduced root water
uptake at the soil surface and bottom of the soil proﬁle, and possibly
some cumulative soil water stress. However, for the latter 3 cutting
cycles, soil water redistributes downward by gravity, increasing thewetted soil root zone towards 200 cm, causing water losses by DP.
The ﬁnal optimized results for the R2 root system in Fig. 8 showthat root water uptake is concentrated in the top of the soil proﬁle(Fig. 8a), as expected, with signiﬁcant water losses by DP (Fig. 8b) forthe cutting cycles in mid season (cycles 3, 4, and 5). However, mostof the AW moves upward by capillary rise as a result of large negative gradients of soil water pressure head close to the surface causedby root water uptake. Maximum root water uptake occurs near thesoil surface, since there are relatively fewer roots at and below the92

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

Fig. 9. Bi-criterion (top) and Pareto contour (bottom) plots for T1 (clay loam), T2 (loam), and T3 (sandy loam), optimizing AW/Tp and drip-line distance, L, for drip-line
installation depths of 55–70 cm. The drip-line depths in plots a–c are indicated with different symbols. The contour lines in plots d–f represent isolines of relative transpiration(Ta /Tp ).

drip-line. Since the near-soil surface between the two drip-lines
was insufﬁciently wetted, root water uptake there is rather low,causing soil water stress.
In summary, for the linear root distribution case (R2) water isrequired at shallower depths and in higher amounts as comparedto the uniform root distribution (R1), thereby leading to signiﬁcant water losses by DP. In contrast, root water uptake is constantand independent of soil depth for R1, allowing AW to be much
lower to maintain adequate root water uptake with no soil water
stress. We note that these results may be different if the root watermodel allows for compensated uptake, as presented in Šimůnek andHopmans (2009). These results indicate that the root distribution
has a major effect on system design and irrigation management.
3.1.3. Analysis of soil texture – T1–T3
The bi-criterion Pareto plots for the clay loam (T1), loam (T2),and sandy loam (T3) soil with uniform root distribution are presented in Fig. 9a–c, respectively. Additionally, the Pareto contour
plots of Ta /Tp for each soil type representing trade-off between L
and AW are shown in Fig. 9d–f. No optimal Pareto parameter values were determined for drip-line installation depths of 55 and60 cm for T1 (clay loam) and 55 cm depth for T2 (loam), as theﬁner-textured soils allow for deeper optimum D-values becauseof sufﬁcient water holding capacity and low DP losses, while theshallower installations cause too wet soil conditions near the surface at harvesting. Because of its higher water holding capacity,much lower AW is required for the clay loam than the loam soil
to achieve high transpiration rates (Fig. 9d–e). However, for the
sandy loam soil, all of the Pareto solutions were determined fromthe shallow drip-line installation at D = 55 cm, as it is a well-drained
soil that satisﬁes the soil dryness constraint, while simultaneously
minimizing DP. Although, drip-line installation at D = 55 cm was
sufﬁcient for AMALGAM to ﬁnd Pareto solutions for the sandy loam
soil, it is shown in Fig. 9f that the soil water holding capacity was

not enough to avoid soil water stress and thus lower transpiration
obtained compared to those achieved for the loam and clay loam
soils, especially at the larger drip-line distances. As Fig. 9 shows,the ﬁner-textured soils of T1 and T2 allow for drip-line installation
depths of 70 cm, as DP losses are relatively small, while maintaining dry soil surface conditions. When comparing the Pareto
contour plot of scenarios T1–T3 it is shown that larger amountsof AW are required to maintain a speciﬁed transpiration rate as thesoil texture is coarser, as corresponding soil water holding capacity
decrease and DP losses increase.
When comparing optimal drip-line distance (L) between thevarious soil types, the results show that larger L-values are possible for ﬁner-textured soils (compare T3 with T1) for any given
installation depth, D. This is so because the lateral soil water movement in clay loam (T1) is larger as compared to the sandy loam soil
(T3). However, the increasing capillary rise for the clay loam soil
maintains a wet soil near the soil surface, thereby eliminating thepossibility of shallow drip line installation depths for soil types T1
and T2.
4. Conclusions
An irrigation design-management modeling tool was developed to simultaneously optimize subsurface drip irrigation systemdesign and management, using the optimization model AMALGAM.
With this multi-criteria optimization approach, in combination
with HYDRUS-2D modeling to simulate unsaturated soil waterﬂow and root water uptake, the presented sensitivity analysis
provides for optimal irrigation system and management parameters for a wide range of conditions, including drip-line installation
depth and distance, applied irrigation water, degree of crop water
stress, and drainage losses. The so-called optimal Pareto solutions
suggest multiple trade-offs, depending on the speciﬁc parameterof most interest. Though not explicitly presented, the proposed

M.M. Kandelous et al. / Agricultural Water Management 109 (2012) 81–93

optimization scheme with multiple objective functions allows forinclusion of costs for the different irrigation systems and management practices, thereby allowing for an economic analysis as
well.
The importance of root distribution in irrigation system design
and water management has been acknowledged for several
decades, and has been evaluated by consideration of two different depth distribution functions of root architecture. We concluded
that root distribution has a large effect on optimal applied irrigation water, irrigation water scheduling, and deep percolation losses.
Also, we recognize the general lack of information on the effects ofirrigation type on alfalfa root distribution, necessitating dedicatedexperiments to determine the inﬂuence of subsurface drip on root
growth and spatial distribution.
Analysis of the soil texture effects showed major differences
because of variations in water holding capacity, capillary forcesand drainage rates between soil types. Consequently, the numberof combinations of optimal Pareto parameter values was limited
for the coarse-textured soils because of increasing deep drainage
losses, whereas optimal solutions were limited to deeper drip-line
installation depths for the ﬁner-textured soils because of upward
capillary gradients thereby preventing dry soil surface conditions
required for alfalfa harvesting.
The study presents examples of the capabilities of ﬂexible irrigation design-management tool for alfalfa, but can be equally applied
to other crops and climates. Moreover, the presented framework
can be utilized to include cost estimates to the multi-criteria
objective functions, thereby allowing for an extensive economic
cost-beneﬁt analysis for a range of irrigation system designs andirrigation water management practices.
Acknowledgements
This project was partly possible through a graduate student fellowship by the Hydrologic Sciences Graduate Group at UC Davis.
The AMALGAM source code can be obtained from the third author
upon request.
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