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Working Document 22/01
Hydroeconomic analysis of droughts
in the Ebro basin using copulas for
streamflow simulation
Daniel Crespo, Taher Kahil, Jose Albiac, Franziska Gaupp
and Encarna Esteban
Department of Agricultural Economics
Agrifood Research and Technology Center (CITA-
Government of Aragon)
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This Working Document draws from the Ph.D. dissertation by Daniel Crespo in the
program of the Department of Economic Analysis, University of Zaragoza, Spain. The
research has been supported by projects RTA2017-00082-00-00 and RTA2014-00050-
00-00 from the Spanish Ministry of Science and Innovation. The ideas and opinions in
this document are responsibility of the authors and not those of the institutions supporting
the research.
To obtain reprints of the Working Document contact:
Jose Albiac
Departamento de Análisis Económico
Universidad de Zaragoza
Gran Via 2
50005 Zaragoza
Spain
email: maella@unizar.es
Phone: +34 639958988
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Abstract
Climate change intensifies water scarcity in arid and semi-arid regions where pressures
on water resources are significant, further compromising the sustainability of water
systems. Climate change triggers more frequent, longer and intense droughts that bring
about serious challenges for management. Hydroeconomic analysis provides a modeling
framework for policy design at basin scale, taking into consideration the spatial and
temporal relationships between water sectors. In this study, an integrated hydroeconomic
model of the Ebro basin is used to analyze the economic impacts of climate change under
several water management alternatives. An innovative approach, the Copula procedure,
is used to generate longer, and more intense and frequent drought events. Several policy
scenarios are simulated by combining two water allocation rules, proportional share or
water markets, with the possibility of investments in advanced irrigation systems. The
sustainability of the Ebro water system is evaluated by looking at its reliability, resilience
and vulnerability under each policy alternative. The risk assessment of the benefit losses
informs on the water system exposure to extreme drought events, and the contribution of
management options in reducing potential losses. The results highlight that climate
change exacerbates the likelihood of substantial economic losses from droughts, which
compromise the sustainability of the water system. Water markets and irrigation
efficiency enhancements reduce uncertainty and losses from droughts, although there is
a trade-off between irrigation benefits and damages to aquatic ecosystems. However, the
effectiveness of this policy combination decreases for longer and intense droughts.
Keywords: Hydro-economic model; climate change; drought impacts; water
management; copula; drought intensity, duration and frequency
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1. Introduction
Droughts are a natural hazard that impair the capacity of water systems to support
economic activities (Borgomeo et al., 2015). Agriculture, natural ecosystems,
hydropower, and urban and industrial supply are substantially exposed to water scarcity
and can sustain significant economic losses (Naumann et al., 2015). Droughts and water
scarcity are already a serious problem in arid and semiarid regions across the world, with
increasing pressures from the impending climate change. In Europe, the evidence during
recent years indicates that the drought anomaly in Europe is unprecedented in the past
2,000 years (Büntgen et al., 2021).
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Conflicts among users often arise from water scarcity, compounded by unsustainable
management policies and lack of cooperation (Quiroga et al., 2011; Iglesias et al., 2007).
The management challenge is serious because climate change widens the uncertainty of
water planning (Herman et al., 2015; Sandoval-Solis et al., 2011) and the drought
damages.
Costs of drought damages have been estimated at 9 billion € per year in the European
Union (Cammalleri et al., 2020), and $8 billion per year in the United States (NOAA,
2021). These costs represent between 0.05% and 0.1% of the gross domestic product,
although costs could be exceptionally higher some years. Kirby et al. (2014) estimate at
1% of GDP the costs of the 2009 drought in Australia, and Hernández et al. (2013)
estimate the cost of the 2005 drought in the Ebro basin (Spain) at 0.5% of GDP.
The countries in Europe with large drought damages in billion € per year are Spain
(1.5), Italy (1.4) and France (1.2), where drought planning efforts and climate adaptation
actions are being developed. Most damages affect the agriculture (50%) and energy (35%)
sectors, followed by the urban water supply sector (13%). Future damages would depend
on the increase in the global warming temperature, with damages increasing up to five
times for a +3°C scenario (Feyen et al., 2020).
Water management needs information to compare water system performance under a
wide range of climate conditions in order to identify suitable governance alternatives.
Sustainable water management faces the challenge of meeting human and environmental
1
The anomaly seems to be driven by anthropogenic warming, which is changing the position of the
summer jet stream.
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water requirements while reducing the adverse impacts of droughts (Sandoval-Solis et
al., 2011).
In Europe, drought frequency and intensity are highest in the Mediterranean area
(Spinoni et al., 2015), with considerable damages sustained by Spain, France, Italy and
Greece. Climate change will increase the frequency, intensity and duration of drought
spells (IPCC, 2014). The effects would include reductions of crop and pasture production,
higher risk of crop failures, livestock losses, land and ecosystems degradation, and
negative impacts on hydropower and urban supply (Fallon and Betts, 2010; Li et al.,
2009). Climate change will increase the vulnerability of water systems to droughts
leading to critical failures, and the acute water scarcity will force adjustments in
management to confront drought events (Vicente-Serrano et al., 2014; García-Ruiz et al.,
2011).
The impact of droughts is driven by the intensity, duration and frequency of drought
spells, and by the capacity of the system to endure these adverse events. River basin
management policies should enhance the performance of the water system to confront
disruptive events. Reliability, resilience and vulnerability are sustainability indicators that
inform of the adequacy of management and policy alternatives.
Extreme droughts are climate events with low frequency, and they are rarely
represented in climate projections (Rocheta et al., 2014). Water system vulnerabilities
and management performance could be identified by generating synthetic stream flows
that replicate historical and projected weather conditions.
Water management is challenging when drought spells entail large economic costs and
environmental damages, especially in arid and semiarid regions. The difficulties of water
management are compounded by economic growth, the increasing social concern for
environmental protection, and climate change. This implies that water management
becomes and adaptive process under constant revision based on updated information and
knowledge. Hydroeconomic analysis (HEA) integrates biophysical, economic and
ecosystem components in a framework that accounts for the temporal and spatial
dimensions of water scarcity problems. Therefore, HEA is an important tool for
evaluating water management adaptation to climate change (Ward 2021).
This paper analyzes the economic impacts of drought and water scarcity in the Ebro
basin under alternative water allocation policies, taking into account that climate change
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results in more frequent, intense and longer droughts. The reliability, resilience and
vulnerability of the water system to droughts are analyzed under different water
management policies using HEA. An important innovation in this hydroeconomic
modeling study is that the inflows in the model are generated using a statistical method
denominated copula. This procedure generates more accurate streamflow series, which
replicate historical stream flows and projected weather conditions with longer and more
intense droughts.
2. Modeling framework
There is a wide variety of procedures to calculate runoff and streamflow from climate
and environment variables, such as temperature, radiation, precipitation and vegetation
cover. Drought studies use this information together with climate change scenarios, for
enhancing the estimation of the intensity, duration and frequency of droughts. The
statistical models used to model droughts could be based on regression analysis,
variations of the autoregressive integrated moving average (ARIMA) model, Markov
chains, artificial neural networks (ANN) or probabilistic characterization using copulas,
among others (Mishra and Singh, 2011).
The approach taken in this study to drought modeling is to develop a model that can
inform water management at basin level, with the objective of enhancing long-term water
security. This requires an overall risk analysis as part of the selection of measures to be
taken in drought planning. This study does not focus on the biophysical drought
processes, but rather on the human-water interactions in the basin by looking at impact
linkages and finding accurate representations of the human-water interactions (Brunner
et al. 2020). This is in line with the essence of water systems analysis, which is the
prediction of the hydrologic, socioeconomic and environmental consequences of water
management (Brown et al., 2015).
The human-water interactions are represented using hydroeconomic modeling, which
is a spatially and temporally distributed mathematical model, where water demand and
supply nodes are characterized hydrologically and economically. This HEA approach
could address the challenges faced by stakeholders in the management of water systems,
because of the systematic integration of the hydrologic, engineering, economic,
environmental and institutional dimensions of basins in a unique framework. The HEA
framework has clear advantages in evaluating management and policy strategies for
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adaptation to climate change, by providing efficient water allocations across water uses,
spatial locations, and time periods (Ward 2021).
The hydrological component of the model is represented by a simplified reduced form
of the basin hydrology. Stream flows are stochastic, and therefore management decisions
should be taken in a risk-based framework. Different risk metrics are used in water studies
to compare the policy options for adaptation to climate change. Here we use the concepts
of reliability (probability of failure), resiliency (recovery duration) and vulnerability
(failure damages) for water system performance, which were proposed by Hashimoto et
al. (1982).
Modeling the hydrology requires the consideration of the joint distribution of random
variables, and we use the copula procedure to generate the headwaters entering the
hydrological network. The advantage of the copula approach is that the dependence
between the variables is independent from the choice of the marginal distributions of
individual variables.
Generating synthetic stream flows overcomes the constraints imposed by the lack of
long series of historical information. Streamflow generation is important in hydrology
studies, and estimation methods cover a broad range of techniques. Representing extreme
values using multivariate analysis is uncommon because of the limited number of
multivariate distributions that represent extreme values. Distributions like bivariate
Pareto and bivariate Gamma distributions could represent extreme values of two random
variables. The problems with those distributions are that: the same distribution is needed
for each marginal distribution; the estimation of parameter for these distributions could
be difficult; and the extension to more than two variables are problematic.
The copula approach resolves these problems because the marginal distributions are
fitted independently, parameters are estimated by maximum likelihood, and extensions to
more than two variables are straightforward by regular and canonical copulas. The
univariate marginal distributions and the multivariate dependence structure are separated,
with the marginal distributions fitted independently and the dependence structure
represented by a copula. Compared to other methods of streamflow simulation, copulas
are flexible in the selection of marginal distributions, the dependence structure of
variables, and the extension to multiple variables (Chen et al., 2015).
2.1 Streamflow simulation methodology
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Duration, frequency and intensity are temporal variables that characterize droughts.
Monthly stream flows are stochastic variables with temporal dependence, and the drought
persistence is driven by this temporal dependence. Here the objective is to generate
streamflow series matching the behavior of historic data but capable of including the
climate change effects on droughts’ intensity and duration. Several methods are used to
simulate stream flows, such as autoregressive moving average, block bootstrapping,
Markov chain processes, and copulas. The copula-based method is gaining traction to
characterize the joint probability distributions of stream flows, and droughts with longer
duration and larger intensity than previously observed can be generated by perturbing the
copula parameter (Borgomeo et al., 2015; Salvadori and De Michele, 2004).
Monthly streamflow in month is a stochastic variable with a cumulative
distribution function (cdf) . The probability integral transform states that
, where has the standard uniform distribution. Two
consecutive monthly stream flows are correlated and their dependence is represented by
the joint cdf , where the marginal
cumulative distributions of are and
. Standard multivariate modeling is a complex task that may not yield the best fit
for hydrological variables, and the copula-based method can overcome the difficulties in
modeling multivariate distributions.
The copula is a function that links univariate cumulative distributions to create a
multivariate distribution function. A copula is a joint distribution of two uniform random
variables, and the Sklar’s theorem states that the joint distribution function
can be expressed in terms of their marginal distributions
and , by defining a copula as:
where denotes the copula function, and
. The copula captures the dependence of two consecutive months. In
order to generate random values of stream flows, the conditional distribution method was
used in this study. The conditional probability of flow given the flow at ,
, can be obtained from the copula as:
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This relationship feeds the simulations of the correlated random variables. If the flow
at month is known, the value of is given by evaluating . Then,
the value of can be simulated from the inverse function and a uniform
random number . The flow at month is obtained from the inverse function or quantile
function
as
.
To simulate monthly stream flows, the distribution function of monthly stream flows
and the copula have to be fitted. There are several distribution functions to model monthly
streamflow, and the usual distribution functions are employed for characterizing
hydrological variables. The Clayton copula is selected in this study for modeling droughts
because it characterizes variables with low tail correlation, such as droughts.
The monthly streamflow is a random variable with unknown distribution function. The
distribution functions tested to fit the marginals of the copulas were Gamma, Lognormal,
Weibull, Pearson III and Generalized Extreme Values. The Lognormal, Pearson III and
GEV distributions have been selected to represent the marginals of the copula. The
parameter of the distributions is estimated maximizing the likelihood function of the
density function. The goodness of fit is computed by the Kolmogorov-Smirnov (KS),
Anderson-Darling (AD) and Cramer-Von Mises (CVM) tests, which identify the
distribution that better fits the observed data. Finally, the distribution is selected
comparing the Bayesian and Akaike information criteria (BIC and AIC).
There are many types of copula structures . The Archimedean copulas are a family
of copulas commonly used to describe different correlation structures between variables.
The copula considered in this study is the Clayton copula of the Archimedean copula
family. The Archimedean copulas are defined as follows:
where is the generator function that is a strictly decreasing function from onto
. The Clayton copula of two consecutive months is defined as:
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Figure 1. Streamflow simulation procedure based on Copulas
The conditional distribution of the Clayton copula is expressed as:
and the inverse function of the conditional distribution of the Clayton copula is:
Figure 1 shows the procedure followed for simulation. If the streamflow at
month is known, it is possible to simulate the streamflow at month using the
inverse function of the conditional distribution. The first step is to obtain the value of
using the cdf , and then a uniform random number between
zero and one is generated. The second step is to find using the inverse function of the
conditional distribution of the copula. The value of the simulated flow is obtained
with the inverse function of the cdf .
The simulation of the monthly stream flow using conditional copulas are summarized
as follows
1) Fit the marginal distribution for each month .
2) Fit the joint distribution using copulas for each pair of months and estimate their
parameters where .
Initial value
Random generator
…
…
…
…
…
…
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3) Given the streamflow in month , can be calculated with the
marginal . A uniform random variable between zero and
one is generated, and the value of is obtained from the inverse conditional
function.
4) The value of the streamflow at month is calculated with the inverse distribution
function
The copula simulation procedure can be used to simulate longer droughts by
multiplying the parameter of the copula by a factor , where the values one, two, six
and ten are selected for factor . For each perturbation of , 1.000 sequences were
generated using the conditional method described before. The streamflow generation
method has been used to generate 40 years of monthly stream flows.
2.2 Model components and scenarios
The hydroeconomic model of the Ebro basin integrates hydrological, economic,
environmental and institutional aspects. The model includes a reduced-form hydrological
model, a regional economy component, and an environmental benefit component.
2.2.1 Reduced-form hydrological component
The hydrological component represents flows between supply and demand nodes,
using the hydrological principles of water mass balance and flow continuity in the river.
The hydrological component shows the spatial distribution of water flows used by
economic sectors and environmental flows. The mathematical formulation is as follows:
(1)
(2)
(3)
Equation (1) is the mass balance equation, and it determines water outflow
in river reach , in year and month , which is equal to water inflow , plus
water release
from reservoir , minus water losses , water
abstraction for irrigation
, and abstraction for urban and industrial use
.
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Equation (2) guarantees river flow continuity, in which water inflow in the following river
reach is the sum of the water outflow from the previous reach , return
flows from previous irrigation districts
, urban return flows
, and flows entering this river reach from tributaries . Equation (3)
states that water outflow at the mouth of the Ebro must be above the
minimum environmental flow level.
The model dynamics is driven by the water storage in reservoirs. The formulation of
the reservoirs’ storage is as follows:
(4)
(5)
(6)
(7)
(8)
(9)
Equation (4) states that the reservoir stored water is equal to the stock in the
previous period, , minus both net release (outflow minus inflow),
, and
evaporation,
. Equation (5) is the initial reservoir water stock at 2 and
, . Equations (6) and (7) are upper and lower bounds on reservoir storage, given
by maximum capacity,
, and dead storage
. Equation (8) states that reservoir
evaporation,
, is proportional to the reservoir surface area, . The
parameter is the water evaporation per hectare of reservoir surface area (Mm3/ha).
Equation states the linear relationship between reservoir surface area and stored
water, where parameters and are the intercept and the linear coefficients of the
surface-storage equation. This equation gives a good approximation because storage
variations are limited between the upper and lower bounds.
The hydrological component has been calibrated introducing auxiliary variables for
river reaches, so that so that the predicted gauged flows are broadly consistent with
observed flows at each river gauge where measurement data are available. Calibration is
used to close the mass balance equation, since there are water inflows and outflows in the
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system that cannot be observed (for example, underground flows, evaporation, or some
return flows). Calibration includes non-observed flows, which are the difference between
flows estimated with the model and flows measured at gauging stations. The parameters
of the surface-storage equation are obtained using the database in Yigzaw et al. (2018).
2.2.2 Regional economic component
The regional economic component includes agricultural irrigation and urban water use.
There is an optimization model for agricultural activities in every irrigation district, which
maximizes farmers’ private benefits from crop production subject to technical and
resource constraints. Crop yield functions are assumed linear and decreasing in cropland
acreage, and output and input prices are constant. The optimization problem is formulated
as follows:
(10)
s.t.
(11)
(12)
(13)
(14)
(15)
where
is private benefit in irrigation district and year , and
is net income
of crop using irrigation technology . The decision variable of the optimization problem
is , which is acreage of crop under irrigation technology , in year . Equation
(11) represents the restriction of available land in irrigation district equipped
with irrigation system . Equation (12) states that water applied in an irrigation district ,
in year and month , is restricted to water availability , where is the
water requirement of crop with technology , in month . The water available
in irrigation district in year at month is the variable linking the optimization
model of irrigation districts and the hydrological component. The labor constraint (13)
represents labor availability in irrigation district , where is the labor
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requirement of crop with irrigation system . Equation (14) states that fruit trees for each
irrigation district, at year , cannot exceed the fruit trees irrigated the previous year,
. This constraint represent future loss of capital investment in fruit trees if farmers decide
not to irrigate in the current time period.
This optimization model includes the major crops in every irrigation district. Irrigation
systems for field crops are flood and sprinkler, and for fruit trees and vegetables the
irrigation systems are drip and flood. Net income per hectare is the difference
between crop revenue and direct and indirect costs (including capital amortization) and it
is expressed by where is price of crop , is yield of crop
under technology in the irrigation district , and are direct and indirect costs of crop
(including water costs).
The crop yield function is linear and represents decreasing crop yields when additional
land is assigned to crop production, based on the principle of Ricardian rent. The first
lands in production have the highest yields, and yields fall off as less-suitable lands enter
production. The crop function relates yields with acreage of crop under irrigation
technology , and is defined as:
(16)
The agricultural component is calibrated using the Positive mathematical
programming (PMP) to reproduce the observed land and water use under baseline
conditions, and to address the problem of crop overspecialization (Howitt 1995).
Calibration follows the PMP procedure by Dagnino and Ward (2012), where parameters
are estimated for a linear yield function [Equation (16)] based on the first-order conditions
of benefit maximization. Data on yields, prices, crop water requirements, production
costs, availability of water resources, land and labor, together with information on
biophysical parameters have been obtained from statistical databases, reports, previous
studies and expert consultation (MARM, 2010; MAGRAMA, 2015; INE, 2009; DGA,
2009; GC, 2009; GN, 2009).
The modeling of urban water maximizes economic surplus, the sum of consumer and
producer surpluses in the basin’s main cities. The optimization problem of the urban
sector is expressed by:
(17)
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s.t.
(18)
(19)
where
is the consumer and producer surplus in city . The variables and
are water supply and demand in city , respectively. The parameters and are the
constant term and the slope of the inverse demand function, and the parameters and
are the constant term and the slope of the water supply function. Equation (17) states
that the supply must be equal to or greater than the demand for water. The water supply
is the variable linking urban water with the hydrological component. The equation
parameters have been obtained from the studies by Arbués et al. (2004) and (all unit prices
are expressed in euros at 2009).
2.2.3 Model optimization, scenarios and sustainability outcomes
The net present value (NPV) of the benefits of economic sectors is maximized over
the planning horizon, where NPV is the sum of present benefits from agricultural
irrigation and urban water use. The model optimizes the objective function:
(20)
subject to the basin’s hydrological, land use, institutional, and environmental constrains
stated in equations (1) to (19).
The performance of the Ebro water system will be threatened by climate change and
the increasing frequency, duration and intensity of droughts. Several indicators such as
reliability, resilience or vulnerability are used to assess water system performance to
disruptive events like droughts. Reliability is the probability that water supply could meet
water demand during the simulation period, where reliability is simply one minus the risk
of system failure. Resiliency describes the capacity of the system to recover after a system
failure, and vulnerability can be measured by the economic losses of drought spells.
Sustainability and risk-based indicators contribute to the assessment of the likelihood
and impact of disruptive events on water systems. A comprehensive sustainability index
can be built by combining reliability, resilience and vulnerability indicators in a general
sustainability index. This sustainability index is used in the Ebro to compare the
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performance of the water system under different management and policy strategies for
climate change adaptation.
The analysis of the impact of water scarcity on the sustainability of the water system
is undertaken under several scenarios, which combine climate change conditions, water
allocation policies, and investments in advanced irrigation systems.
The assumptions for climate change in the Ebro for the next 40 years simulation
period are the following: the mean inflows in the basin decline progressively up to 12%,
and higher temperatures increase crop evapotranspiration and dam evaporation by 5.7%
and 6%, respectively. Climate change will also increase drought persistence, with longer
droughts spells. The copula procedure is used to account for the longer duration of
droughts. Ten inflow series of forty-year each have been simulated by the copula for a
given value of parameter , which regulates drought duration. The duration of historical
droughts are represented by parameter , and then the parameter is increased to ,
and to represent longer drought durations.
There are three climate scenarios: 1) current climate, which replicates historical
inflows, temperature, and drought duration; 2) future climate with decreasing inflows,
increasing temperature, and historical drought duration; and 3) future climate with
decreasing inflows, increasing temperature, and longer drought duration. Scenarios 2 and
3 compare future climate with historical or with longer drought duration, and the reason
is to discern the effects of drought duration and intensity.
The water allocation policies analyzed are institutional cooperation and water
markets. Institutional cooperation is the current policy applied by the water authority in
the Ebro basin. Under drought conditions, the basin authority reduces water allocations
for irrigation in relation to drought intensity, assigning the fall of inflows by proportional
share. Under the water markets policy, farmers receive the water allocations of
institutional cooperation, but then these water allocations can be exchanged among
irrigation districts, maximizing the private benefits of water use. There is no direct
exchange of water between selling and buying irrigation districts, but rather the selling
district reduces withdrawals and the buying district augments withdrawals in their
respective river reaches.
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Table 1. Climate change, water allocation and efficiency investment scenarios.
Climate change conditions
Water allocation
policy
Efficiency
investments
No climate change
Proportional share
No efficiency enhancements
Inflows
Parameter does not
change. Inflows and
drought spells replicate
historical behavior. Mean
inflows are the observed
levels
Crop
evapotranspiration
and dam evaporation
do not change
Under drought, water
allocations for
irrigation are reduced
in proportion to
drought intensity
Irrigation technology does
not change, with channel
efficiency between 70% and
90%
Climate change
Water markets
Efficiency enhancements
Inflows
Drought duration is
prolonged by increasing
the copula parameter .
Stream flows are
simulated for , and
10. Inflows fall steadily
up to 12% in 2040
Crop
evapotranspiration
and dam evaporation
increase up to 5.7
and 6% in 2040,
respectively.
Water is exchanged
among irrigation
districts Water is
exchanged among
irrigation districts
More efficient irrigation
technologies. Irrigation
systems change to sprinkle
for field crops (except rice),
and drip for fruit trees and
vegetables. Channel
efficiency is also increased
The investments in advanced irrigation systems is the preferred solution by decision
makers to confront water scarcity in most arid and semiarid basins around the world.
These investments improve the water conveyance systems and the irrigation equipment
in parcels, with gains in water efficiency at irrigation district level and higher crop yields.
However, Grafton et al. (2018) indicate that these investments tend to reduce stream flows
in basins, and call it “the paradox of irrigation efficiency”. Channel efficiency in the Ebro
range between 70% and 90% at present, and investments improve all channels up to 90%
efficiency. Current parcel irrigation technologies include flood, sprinkle and drip
irrigation, and investments will expand in the basin sprinkle irrigation to all field crops
(except rice), and drip irrigation to all vegetable and fruit crops.
2
Table 1 summarizes the
main aspects of the simulated scenarios.
3. Results
The results correspond to each combination of climate change and policy scenarios,
by performing eleven series of forty-year length simulations. These simulations are
replicated with the data on water inflows that are generated for the different values of the
factor in the copula procedure (=1, 2, 6 and 10), which regulate drought duration.
2
The investment costs of these advanced irrigation technologies are included in the benefits of crops.
18
Figure 2. Expected irrigation area under climate and policy scenario (103 ha)
Then, the outcomes from each combination of policies are calculated. The results show
the impacts of droughts on irrigated cropland, water extractions and irrigation benefits.
The analysis in irrigation districts shows the adaption strategies in cropland
distribution undertaken by districts. Simulations provide monthly information on water
withdrawals, environmental stream flows, water scarcity, and water system stress. The
information is used to estimate reliability, resilience, vulnerability and sustainability
indicators, which reveal the system performance.
3.1 Sustainability of the water systems for irrigation
Drought intensity and duration generate benefit losses, which indicate the system
sensibility to drought events. The performance of the different policies is compared, in
order to identify which are the tradeoffs between policies.
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Figure 3. Annual cropland distribution under climate and policy scenarios (103 ha)
In figure 2, the grey points display the irrigated area by year for climate change and
policy scenarios. The lines are smoothed trends, assuming a logarithmic relationship
between irrigated area and time. The top panels show the irrigated area with and without
climate change, for water markets (top-left) and proportional share (top-right) policies
under current irrigation technologies. The bottom panels show the irrigated area for
improved irrigation technologies. The irrigated area declines in the future because of the
recurrent drought events and the growing trend in water scarcity from climate change.
Annual cropland, water diversions and benefits under climate and policy scenarios, are
presented in figures 3, 4 and 5, respectively. The size of boxplots range between the first
and third quartiles of the distribution, providing information on dispersion and
uncertainty. The first quantile indicates the 25% of worst cases or the 75% of best cases.
Climate change exacerbates water scarcity problems, worsening the drought impacts.
Enhancements in the efficiency of parcel irrigation systems and conveyance channels
contributes to moderate the fall in irrigated cropland. The efficiency enhancements
contribute to meet the water consumed by crops, especially under climate change when
crop water requirements increase. The water market policy slows down cropland
reductions, compared with the proportional share policy. The combination of water
markets and efficiency enhancements maintains more cropland in production, but also
20
Figure 4. Annual water diversions distribution by climate and policy scenarios (Mm3).
shrinks environmental flows. Under climate change, the first quartile of irrigated area is
around 100.000 ha lower for all policy scenarios.
The vertical line dividing the boxplots is the median of the cropland distribution
ranging between 525,000 and 580,000 ha for all scenarios (figure 3), with the median
close to the baseline irrigated area (580,000 ha). Normal or wet weather conditions occur
in half of the time periods, during which the baseline crop production and water
extractions are maintained.
The impact of droughts is determinate by several factors like water stored in dams,
monthly inflows and policy. The relationship between irrigated area reductions and
annual water inflows is estimated by a Beta regression (see table A1 in the appendix for
details). Regression parameters are used to estimate the expected land reduction under
alternative policies. For example, investments in efficiency enhancement cut by half land
reductions, compared with maintaining current irrigation efficiency (Table A1).
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Table 2. Percentage cropland reduction over the baseline, under climate change and
policy.
First period of the simulation
Last period of the simulation
Moderate
drought (-15%)
Extreme
drought (-50%)
Moderate
drought (-15%)
Extreme
drought (-50%)
Without climate change
Without efficiency
enhancements
Proportional share
8.40%
33.40%
19.10%
56.20%
Water markets
6.90%
28.60%
15.90%
50.70%
With efficiency
enhancements
Proportional share
4.00%
18.60%
9.70%
36.80%
Water markets
3.30%
15.50%
7.90%
31.80%
With climate change
Without efficiency
enhancements
Proportional share
9.10%
35.20%
20.40%
58.10%
Water markets
7.40%
30.30%
17.00%
52.70%
With efficiency
enhancements
Proportional share
4.40%
19.80%
10.40%
38.70%
Water markets
3.50%
16.50%
8.50%
33.60%
The drought scenarios are moderate drought where water inflows fall by 15% and
extreme drought where water inflows fall by 50%. Table 2 presents the policy results
from the beta regression predictions. The percentages indicate the expected land
reductions under moderate and extreme drought conditions, for the first and last periods
of simulation. Results show that under the same policy and climate condition, the fall in
irrigated area doubles between the first and last periods. Climate change increases land
reduction between one and two percentage points. The mitigation capacity of the
combined policies declines with drought severity, climate change and time. The market
policy cuts the reduction in cropland by 20%, compared with the proportional policy,
while investments in irrigation efficiency cut cropland reductions by 55% compared with
no investments (Figure A4 and table 1 in the appendix).
The response of farmers to drought is reducing field crops and maintaining fruit trees
and vegetables. Corn represents around 18% of total crop mix and remains constant under
drought conditions, while the other field crops fall. Wheat and rice acreage diminish
progressively until they get out of production. Fruit trees and vegetables share of crop
mix grow, in particular vineyard and peach. Five crops account for 75% of the irrigated
area in the baseline, but under drought the cropping pattern is more diversified and
22
Figure 5. Annual benefits distribution under climate and policy scenarios (106 €).
minority crops gain importance (Figure A5 and figure A6 in appendix). Adaptation to
drought involves the retirement of crops with high water consumption and low
profitability.
Water scarcity takes place during droughts, and their impact depends on the policy mix
that combines proportional share or water markets, with current or enhanced water
efficiency. Stream flows lower than the median correspond to 50% of the worst cropland
reductions, and the ensuing likelihood and size of benefit losses. Water extractions are
driven by climate conditions and policies. The enhancement of irrigation systems reduces
water extractions from 4,500 Mm3 to 3,500 Mm3 under normal weather, where basin
inflows are around the historic mean (Figure 4).
Climate change increases water diversions because of the rising temperatures and
evaporation, even under normal and wet weather conditions. Also, the likelihood of water
scarcity and droughts increases under climate change, which shrinks mean inflows and
enlarges the duration of drought spells. Under climate change, the first quartile of water
diversions is around 3,000 Mm3 for all policy scenarios, which shows the fall of water in
the basin when droughts are severe.
23
Table 3. Min, 1st quartile, mean, median, 3rd quartile and max of annual benefits under
drought by policy-mix.
Min
1st
Quartile
Mean
Median
3rd
Quartile
Max
Without efficiency
enhancements
Proportional share
573
729
762
797
806
812
Water markets
617
741
769
800
808
812
With efficiency
enhancements
Proportional share
609
776
785
808
814
818
Water markets
660
786
791
809
815
818
The index of water stress is the proportion of water diversions over water inflows, so
the water deficit is the gap between water diversions and water inflows. Water scarcity
and water stress are especially intense in summer, when crop water requirements are large
and water inflows small, demonstrating the importance of dams to meet water demand.
Water stress is more likely from June to September when water extractions double basin
inflows, and the water system could be in stress (Figure A7 in appendix). However, water
stress and water scarcity can be underestimated because environmental flows are not
included in the supply-demand balance.
Larger irrigated cropland involves more water extractions and consumption
(evapotranspiration), reducing stream flows in the basin. Under drought, investments in
irrigation efficiency increase water extractions by 25% to 30% in comparison with
maintaining current irrigation efficiency (Figure A8). The consequence is lower stream
flows at the river mouth, with an average fall around 200 Mm3 (Figure A9).
The annual benefits for policy and climate scenarios are presented in Figure 5. The
median benefits are close to baseline benefits (820 M€), and range between 790 and 820
M€. To assess the effects of climate change on the distribution of benefits, we have pooled
the data of benefits from all policy scenarios, Climate change displaces the distribution
of benefits, since without climate change the benefits exceed 775 M€ in 75% of the cases,
but with climate change the benefits exceed only 725 M€ in 75% of the cases.
Weather conditions are stochastic, and therefore the impacts of drought are also
stochastic. These impacts are measured by the benefits obtained under each combination
of policies. Then, the benefit outcomes from each combination of policies are compared.
The first quartile of the distribution of benefit contains the worst benefit outcomes from
24
drought events. Also, climate change amplifies the dispersion of benefits, while
increasing both the uncertainty and likelihood of the fall in benefits.
Table 3 shows the minimum, first quartile, mean, median, third quartile and maximum
of the annual benefits by policy mix, which combines proportional share or water markets,
and current or enhanced water efficiency. Benefits are higher for water markets compared
to proportional share, and benefits are also higher for efficiency enhancements compared
to current efficiency. The mean benefits fall from 820 M€ of the baseline scenario to
between 760 and 790 M€, depending of the policy combination.
The likelihood and size of benefit losses from drought impacts reveal the degree of
exposure of the water system to adverse events. Risk is measured by the probability
(likelihood) of withstanding a certain level of benefit damages (size), and risk
management plays an important role in decision making. Value at risk (VaR) is a standard
risk measure that calculates the benefit level that is not exceeded for a given probability
or confidence interval. VaR can also be calculated in terms of benefit losses by the
exceedance of probability, which is the probability of exceeding a certain benefit loss.
Therefore, in terms of benefits losses, VaR is the benefit loss that is exceeded for a given
probability or confidence interval (see section 3 of the appendix for details).
The VaR for a 5% probability is widely used for risk assessment, and the combination
of water markets and efficiency enhancements reduce in 70 M€ the benefit loss level of
the VaR at 5%, compared with proportional share and current irrigation efficiency. This
reduction in benefit losses represent around 8% of baseline benefits (Figure A10). Figure
6 shows the results by irrigation district of the mean percentage reductions from the
baseline scenario of variables benefits, water diversions, labor, cropland, and cultivated
areas of field trees, field crops, and vegetables. Under the water markets policy, Canal de
Bardenas (CB), Canal Imperial (CI), Delta, and Zadorra sell water to other irrigation
districts under drought conditions. These water selling districts reduce field crops, while
buying districts have higher crop profitability and irrigation efficiency and could maintain
fruit trees, vegetables, and even field crops. Investments in efficiency enhancement retain
more cropland under production, and when combined with the market policy the water
exchanges go down because of lower water scarcity and a more uniform efficiency among
districts. The Jalon irrigation district is heavily damaged by drought because water is quite
scarce in the left bank of the Ebro, with considerably reductions of field crops for all
policy combinations.
25
Figure 6. Reductions from the baseline scenario of benefits, labor, water diverted, land (field crops, fruit trees, vegetables) in irrigation
districts by combinations of market, proportional and efficiency enhancement policies
26
Table 4. Reliability, resilience, vulnerability and sustainability index by climate and
policy scenarios.
Reliability
Resiliency
Vulnerability
Sustainability
Without climate change
Without efficiency enhancements
Proportional share
0.76
0.73
0.78
0.43
Water markets
0.78
0.76
0.81
0.48
With efficiency enhancements
Proportional share
0.85
0.83
0.83
0.58
Water markets
0.86
0.83
0.86
0.62
With climate change
Without efficiency enhancements
Proportional share
0.63
0.45
0.77
0.22
Water markets
0.64
0.46
0.80
0.24
With efficiency enhancements
Proportional share
0.79
0.57
0.82
0.37
Water markets
0.80
0.59
0.85
0.40
Labor reductions depend on crop patterns, with large declines in districts specializing
in vegetables and fruit trees that use labor intensively. Labor losses are up to 20% in Jalon
and Riegos del Alto Aragon districts. The combination of market and efficiency
enhancement policies maintains labor, although losses are important in water selling
districts.
Reliability is measured by the proportion of time periods in which baseline cropland
water demand is met by the water system, resilience is measured by the recovery duration
after the water system fails, and vulnerability is measured by the benefits losses from
water system failure (index decreases for more vulnerability). Then, the sustainability
index is defined as the product of reliability, resilience and vulnerability (see details in
section 2 of the appendix).
The indexes for reliability, resilience, vulnerability and sustainability by policy
combination are shown in Table 4. Climate change raises the likelihood of system failure
with longer recovery periods and lower benefits. These reduced reliability and resilience
and increased vulnerability, make the system less sustainable. The sustainability of the
27
Figure 7. Joint distribution of drought duration (months) and drought intensity (Mm3).
system improves by combining water markets and efficiency enhancements, with gains
in reliability, resilience and vulnerability.
Figure 7 shows the joint distribution of drought duration and drought intensity.
Drought events are identified by falling basin inflows below a threshold, defined at 75
percent of baseline monthly inflows. The drought period starts when inflows fall below
the threshold and finishes when inflows recuperate, with the duration being the number
of months under drought. The monthly deficit is the gap between the drought observed
inflows and the drought threshold, and drought intensity is the sum of monthly deficits
over the drought spell. Around 90 percent of drought spells are shorter than 12 months
with water deficit below 2,500 Mm3. Drought spells longer than two years with water
deficits above 5,000 Mm3 have a 5 percent probability. In extreme cases, the drought
duration reaches 60 months with deficits above 20,000 Mm3.
28
Figure 8. Conditional exceedance of probability of benefit losses for increasing drought
severity.
Under a severe drought spell (19 months duration and 4.128 Mm3 deficit), the
probability of accumulated benefit losses exceeding 250 M€, range between 12.5% and
37.5% for the different policy combinations. Combining water markets and efficiency
enhancements divides by three the probability that benefit losses exceed 250 M€ under
severe drought. The probability than benefit losses exceed 500 M€ is close to 5% in a
severe drought spell for proportional share without efficiency enhancements, but the
probability decreases below 1.5% for all other policy combinations.
Extreme droughts (47 months and a deficit of 11,146 Mm3) trigger benefit losses that
exceed 500 M€ in half of the cases for the proportional share policy, and 250 M€ with the
combination of water markets and efficiency enhancements (Figure A11). For the 5% of
worst cases, benefits losses exceed 1,000 M€ under proportional share without efficiency
enhancements, and 650 M€ under water markets and efficiency enhancements (Figure 8).
Figure 9 shows how conditional benefit losses depend on the duration and intensity of
drought, by policy combination. Benefit losses correspond to quantiles 0.5 (50%) and
0.99 (99%), and contour lines show additional benefit losses of 100 M€. The benefit
losses in the first and second columns are for proportional share and water market, without
29
Figure 9. Contour lines of benefit losses (106 €), by policy at quantiles 0.5 and 0.99
efficiency enhancements, while the third and fourth columns are for combinations with
efficiency enhancements. The rows show the conditional quantile benefit losses at 0.5
and 0.99 probabilities.
3
The combination of water markets and efficiency enhancements reduces benefit losses
by almost half (probability 0.5), and by one third in extreme cases (probability 0.99).
However, this policy combination also shrinks environmental flows, further degrading
water dependent ecosystems.
The more frequent droughts are shorter than 12 months with water deficits under
2,500 Mm3, and their benefit losses are below 100 M€ with a probability of 50% for all
policies. In extreme cases (probability 0.99), the droughts up to 12 months have benefit
losses around 300 M€ for proportional share, shrinking to 200 M€ when combining water
markets and efficiency enhancements. This indicates that droughts of short duration and
low intensity could involve substantial benefit losses, and that the gap in policy
performance is greater in adverse cases.
3
This information is extend to quantiles 0.5, 0.9, 0.95 and 0.99 in figure A12 of the appendix.
30
Table 5. Percentage of flow gaps between non-complying and minimum environmental
flows by policy combination*
Without climate change
With climate change
Without efficiency enhancements
Proportional share
8.9%
11.9%
Water markets
9.9%
13.7%
With efficiency enhancements
Proportional share
11.6%
14.6%
Water markets
12.3%
16.2%
* These are the average percentages during drought periods.
For longer and more intense droughts (between 12 and 24 months and between 2,500
and 5000 Mm3), there is a sharp increase in benefit losses. In extreme cases (probability
of 0.99), benefit losses rise to 500 M€ for proportional share, but only to 300 M€ for the
combination of water markets and efficiency enhancements. The join probability of
having a drought longer than 4 years and with deficits greater than 12,500 Mm3 is lower
than one percent. In these extreme drought events, the combination of water markets and
efficiency enhancement reduces benefits losses by 300 M€ compared to proportional
share and current irrigation efficiency. The sequence of annual droughts is usually a mix
of moderate, severe and extreme drought events, and extreme droughts lasting
consecutive years are very rare. For prolonged droughts, the system cannot longer relieve
water scarcity because water storage in dams is depleted. Once the dam storage is
exhausted, the only response to drought is sharing the remaining water with adjustments
in crop patterns.
3.2 Sustainability of the water systems for irrigation and the environment
Ecosystems in the Ebro basin are protected by minimum environmental flows in river
reaches across the basin, which are set-up by the Ebro Basin Authority in the basin plan.
These environmental flows are minimum levels of stream flows that maintain the status
of water-dependent ecosystem. The environmental flows are gauged in 15 river reaches
in the basin.
The percentage of the flow gap between non-complying and minimum environmental
flows provide information on expected environmental damages. Table 5 shows the
average percentage of the flow gap between non-complying and minimum environmental
flows during drought periods. The percentages indicate the size of the flow gap for each
policy combination, and the ensuing impairment of ecosystems. Efficiency enhancements
31
Table 6. Sustainability indexes for irrigation, environment and the whole system by
climate and policy scenario.
Irrigation
sustainability
Environmental
sustainability
Water system
sustainability
Without climate change
Without efficiency enhancements
Proportional share
0.43
0.91
0.39
Water markets
0.48
0.90
0.43
With efficiency enhancements
Proportional share
0.58
0.88
0.51
Water markets
0.62
0.87
0.54
With climate change
Without efficiency enhancements
Proportional share
0.22
0.88
0.19
Water markets
0.24
0.86
0.21
With efficiency enhancements
Proportional share
0.37
0.85
0.31
Water markets
0.40
0.83
0.33
and water market policies, together with climate change, aggravate non-compliance,
especially the policy of efficiency enhancements that maintains crop production at the
expense of ecosystems’ degradation.
The environmental sustainability is measured by the average proportion of minimum
environmental flows covered by each policy combination during droughts (Table 6). This
environmental sustainability index is highest for proportional share water allocation
without efficiency enhancements, and lowest for water markets with efficiency
enhancements. Water allocation by the current proportional share policy promotes
environmental sustainability, while water markets promote irrigation sustainability.
However, the differences in environmental sustainability by policy combination are small
compared with the differences in irrigation sustainability.
The entire water system sustainability is assessed by multiplying the irrigation
sustainability and the environmental sustainability indexes (Table 6). The combination of
water markets and efficiency enhancements provide the highest ranking for the water
system, and the combination of proportional share without efficiency enhancements the
lowest. Decision makers have to decide the policy mix to be chosen by considering the
tradeoff between irrigation benefits and environmental protection. The tradeoff indicates
32
that small increases in environmental protection require incurring in large losses in
irrigation benefits.
However, the environmental damages can be underestimated for irrigation efficiency
investments, as a consequence of the “paradox of irrigation efficiency” (Grafton et al.
2018). The paradox states that higher efficiency rarely reduces water consumption
(evapotranspiration) by crops for the same water withdrawals, and the consequence is a
fall in irrigation returns that reduce basin stream flows. To confront the paradox,
investments in efficiency gains must be coupled with virtuous collective action outcomes
capable of preventing the expansion of water consumption by crops.
The result of higher benefits from water markets compared with proportional share,
highlighted in this study, is consistent with the findings in the hydroeconomic analysis
literature (e.g. Crespo et al., 2019; Escriva-Bou et al., 2017 and Salman et al., 2017).
Many studies find that gains in irrigation efficiency reduce the impacts of drought, water
scarcity and climate change (Bekchanov et al., 2016; Sánchez-Chóliz and Sarasa, 2019).
However, the “paradox of irrigation efficiency” mentioned above will undermine this
finding, unless the expansion of water consumption by efficiency gains is prevented.
Connor and Kaczan (2013) describe this downside of water markets for in-stream flows
in the Murray-Darling basin in Australia, indicating that Australia has chosen trading on
water extractions instead of trading in water consumption, in order to reduce the
transactions costs of water markets.
Climate change modifies the reliability, resilience and vulnerability of water systems,
because of the shift in both water supply and demand (Gau et al. 2019). Despite
considerable water management efforts, the sustainability of water systems can be
threatened by the conflicting goals of maintaining irrigation and protecting the
environment. Folke et al. (2004) recommend a secure range of water allocations for
ecosystems, given the uncertainty in ecological responses involving irreversible regime
shifts and tipping points.
The present study could be further expanded for a better assessment of the Ebro water
system. Possible improvements include modeling the environmental benefits linked to
the response of ecosystems to stream flows, and adding the hydropower sector to the
economic activities in the basin. Other possible enhancement is to consider the spatial
33
variability of water inflows in stream flows simulations, given the local heterogeneity of
climate change impacts across sub-basins.
4. Conclusion
This study analyzes the economic impacts of drought and water scarcity in the Ebro
basin, under current and future climate conditions. Climate change projections point
toward more frequent, intense and longer drought spells. Reliability, resilience, and
vulnerability of the water system to droughts are examined under several water
management alternatives. The evaluated policies are the combinations of proportional
share or water markets, with current or enhanced irrigation water efficiencies. The
assessment of risk provides information on the water system exposure to extreme events,
which is important for water management.
Hydroeconomic analysis is conducted with a model that integrates hydrological,
economic, and environmental components into a framework that emphasizes the temporal
and spatial relationships between water usages. Water inflows are generated using the
copula approach, in order to represent both historical stream flows and projected stream
flows under weather conditions with longer and more intense droughts. This study is
innovative in the assessment of water scarcity impacts, by taking into account that the
duration, intensity, and frequency of droughts are changing in the coming decades. The
results are examined in terms of probability given the intrinsic uncertainty of drought
events and climate change.
Growing crop water requirements, reductions in water inflows, and longer and more
intense drought events are expected outcomes from climate change in the Ebro basin.
Droughts under historical and climate change conditions entail cutbacks in water
extractions, triggering substantial benefit losses in irrigation. Climate change increases
the likelihood of longer and more intense droughts and their negative impacts, and impairs
the resilience of the water system exposed to longer recovery periods. The frequency of
extreme weather conditions will be also higher, boosting the risk of severe benefit losses.
The climate stress will reduce the reliability of the water system to meet both human water
security and environmental flows.
The impacts of droughts and climate change can be reduced by combining water
markets with investments in irrigation efficiency. This policy mix expands cropland in
production, water extractions and irrigation benefits, while lowering their dispersion and
34
uncertainty. The impacts of extreme droughts are reduced somewhat, although the policy
effectiveness decreases when droughts are more intense. The combination of water
markets and efficiency enhancements improves the system sustainability because
vulnerability is reduced, and reliability and resilience are reinforced. However, the gains
in efficiency increase water consumption, reducing the basin in-stream flows.
Consequently, this policy hinders the compliance with environmental flows and
jeopardizes the status of aquatic ecosystem. Therefore, irrigation benefits are maintained
at the expense of the environment.
The duration of most droughts is less than one year, and the likelihood of longer
droughts shrinks because the probability of consecutive years of drought declines with
duration. However, ten percent of droughts are longer than one year, and the ensuing
benefit losses over several years can be important. The capacity of the water system to
avoid cutbacks during droughts depends on the water storage available in dams. Once the
capability of dams to offset water scarcity is exceeded, the reactions to droughts are
limited to adjustments in crop patters and water trading. Water markets and efficiency
enhancements are good interventions to maintain irrigation activities, at the cost of
degrading aquatic ecosystems, and the trade-off has to be settled by decision makers.
Acknowledgements
This study has been financed by the projects INIA RTA2014-00050-00-00 and INIA
RTA2017-00082-00-00 of the Ministry of Science and Innovation, partly financed by
European ERDF funds, and by support received from the ECONATURA research group
of the Government of Aragon. Daniel Crespo has conducted this study with the support
of a PhD grant from INIA. Part of the work by Daniel Crespo was developed at the Water
Program of the International Institute for Applied Systems Analysis (IIASA). Special
assistance has been provided by María Ángeles Lorenzo and Daniel Isidoro (CITA-
DGA), and by Rogelio Galván and Miguel Ángel García Vera (CHE-MITECO).
35
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39
Appendix
1. Beta regression for modelling proportions and bounded data
The performance of each policy alternative under different climate conditions is
analyzed, by regression, comparing the percentage of cropland reductions over the
baseline. OLS estimation is the simplest method, but is not appropriate since the irrigation
area and the percentage of cropland reductions are bounded variables.
Sinusoidal regression or logit regression analyze bounded data, like rates and
proportions, but they are difficult to interpret and do not treat heteroskedastic problems.
In addition, the distribution of proportions could be asymmetric and Gaussian
approximations are not appropriate. Beta regression overcome these difficulties assuming
that the response variable is beta distributed.
Following the model proposed by Ferrari and Cribari-Nato (2004), the response
variable is Beta distributed, , and its density function is defined by
two parameters and , and takes the expression:
where and are the precision and dispersion parameters, respectively, and satisfy that
and . The mean and variance of the beta distribution are expressed with
the precision and dispersion parameters as:
and
As in generalized linear models (GLM), the precision parameter is linked to the
covariates, , by a link function, and a linear predictor, ; and the dispersion
parameter is linked to another set of covariates, , by a second link function, and
a linear predictor, .
The coefficient sets and are estimated by Maximum Likelihood (ML). The logit,
probit or loglog are functions commonly used as link functions. A comprehensive
explanation of the model can be found at Cribari-Neto and Zeileis (2010).
40
The assumptions in the model are:
1) The percentage of cropland reductions over the baseline is beta distributed.
2) The precision and dispersion parameters are fitted by the percentage of the annual
inflows over the baseline inflows, year, climate change and policy combination.
3) The logit link function and log link function connect the precision parameter and
dispersion parameter, respectively, with the covariate sets.
The logit link function has been selected because the coefficients estimated indicate
odds ratios, and the log link function ensures that dispersion parameter is greater than
zero. Table A1 in the appendix shows the results of the estimation. The percentage crop
reduction under different drought intensity, climate change, year, and policy showed in
table 2 are computed with the results of the beta regression.
41
Table A1. Beta regression estimation results (standard deviation in brackets)
Dependent variable:
Percentage of cropland reductions over the baseline
M.1
M.2
Precision (µ) model with logit link
Year
0.023***
0.023***
(0.001)
(0.001)
Proportional share
0.226***
0.221***
(0.019)
(0.016)
Climate Change
0.081***
0.081**
(0.028)
(0.028)
Enhancement of
efficiency
-0.787***
-0.787***
(0.019)
(0.019)
Inflows
-4.232***
-4.230***
(0.035)
(0.035)
Intercept
-0.516***
-0.513***
(0.040)
(0.040)
Dispersion (ɸ) model with log link
Year
0.010***
0.008***
(0.001)
(0.001)
Proportional share
-0.018
(0.030)
Climate Change
0.470***
0.470***
(0.040)
(0.040)
Enhancement of
efficiency
0.277***
0.277***
(0.030)
(0.030)
Inflows
2.314***
2.310***
(0.053)
(0.053)
Intercept
0.817***
0.809***
(0.058)
(0.056)
Observations
9,020
9,020
R2
0.650
0.650
Log Likelihood
15,536
15,535
Note:
***p<0.001; **p<0.01
42
2. Reliability, resilience, vulnerability and sustainability index
Water management seeks to maintain the water system in a satisfactory state. The
threshold for determining system failure is settled as the 75 percent of the baseline
cropland water demand, and therefore the water system is in a satisfactory state if the 75
percent of the water demand at the baseline is meet. Reliability () measures the
capacity of the water system to maintain a satisfactory state, and it is the number of the
years over the total number of years () in which water system operates satisfactorily ().
In terms of probability, reliability is the probability of the water system to satisfy the 75
percent of the irrigation water demand.
System resilience () measures the recovery capacity of the system after a system
failure. Then, resilience is the frequency with which the system recovers from failure
Vulnerability () of the water system is measured by the benefit losses in the water
system. Vulnerability is defined by the mean value of irrigation benefits () over
irrigation benefits in the baseline (), when the system is in an unsatisfactory state:
Sustainability is measured as the product of the reliability, resilience and
vulnerability
3. Probably, exceedance of probability and conditional probability
Probability is an important concept in this study that is used in several indicators used
in the analysis of the results. In order to clarify concepts as exceedance of probability or
Value at Risk, the concept of probability is briefly explained. Probability is a measure of
the likelihood of an event happening. For a continuous random variable , the probability
43
that takes a value lower than is expressed as ; and the probability that the
variable exceeds a certain value is named exceedance of probability, that is
.
The cumulative distribution function (c.d.f.) is defined as
and the complementary cumulative distribution function (c.c.d.f.) (tail
distribution or exceedance of probability function) is a function that account the
probability that is equal or greater than , and it is expressed as:
The inverse function of c.d.f. is the quantile function , and is the minimum value
of the amongst x that the c.d.f. excess the value :
Value at Risk (VaR) is a risk metric that indicates the maximum benefit losses given
a probability level, and is obtain with the quantile function. Benefits losses can be
accounted with a positive value or with a negative value. In case that benefit losses are
expressed as a negative value, , the VaR is the lower - quantile of the random
variable and is defined as
and when benefit losses are greater than zero, the VaR is obtained by the
exceedance of probability:
VaR provides information about uncertainty and risk, and it is appropriate to compare
water management alternatives.
The economic impact of a drought spell is the accumulated annual benefit losses in
relation to the baseline throughout the episode, and depends on its duration and intensity.
The conditional probability of benefit losses given certain duration and intensity of a
drought spell indicates the exposure of the system to that drought event, since it measures
the probability of an event given the occurrence of another events. The conditional
probability of the random variable Y given and is expressed as
44
and the conditional quantile is given by the expression
Its definition is straightforward from quantile definition: the minimum value of from
amongst all those values of whose c.d.f. value excess the value given the events
and . A different way to express the conditional quantile function is
The maximum benefit losses at a certain level of probability given a drought duration
and a drought intensity is obtain from the quantile that satisfices
where is benefit losses in M€, D is drought duration in months and I is drought intensity
in Mm3. The VaR of irrigation benefits given a drought spell with certain duration and
intensity is obtain from the conditional probability and the conditional quantile.
Non-parametric estimators and copulas approach estimate the conditional probability
and the quantile required to compute the VaR. Non parametric estimators are
computationally demanding and copula approach could be preferred when the number of
estimations is large. The non-parametric method proposed by Li et al. (2013) estimates
the conditional quantile function by numerically inverting the estimated
conditional distribution function
and solving:
The conditional distribution function and the conditional quantile function are
estimated in R with the package np developed by Hayfield and Racine (2008). The
exceedance of probability at figure 8 and figure A11 are estimated by the non-parametric
method.
The conditional quantile based on copulas is an alternative method to non-parametric
(Kraus and Czado, 2016). The joint distribution of a multivariate random
vector is expressed in terms of a copula . Then, the density
function can be expressed as
45
Table A2. Copula and parameters of the join function of benefits losses (L), drought
duration (D) and drought intensity (I).
Scenario
Copula
Type
Without efficiency enhancements
Proportional share
Gumbel
3.62
-
0.72
Survival Clayton
2.37
-
0.54
BB8
1.42
0.99
0.18
Water markets
Gumbel
3.62
-
0.72
Survival Clayton
2.37
-
0.54
BB8
1.40
0.99
0.18
With efficiency enhancements
Proportional share
Gumbel
3.62
-
0.72
Survival Clayton
2.09
-
0.72
BB8
1.43
0.99
0.19
Water markets
Gumbel
3.62
-
0.72
Survival Clayton
2.04
-
0.51
BB8
1.41
1.00
0.18
where is the copula density (Aas et al., 2009) and the estimation of the conditional
quantile function is obtained following Kraus and Czado (2016).
The conditional quantile in figure 9 and figure A12 is estimated by the copula method,
since the non-parametric approach is impracticable due to the number of estimations
needed for the grid. The join function distribution of benefit losses, drought duration and
drought intensity is estimated with a C-Vine copula that combine bivariate copulas. Tabla
A2 shows the structure of the C-Vine and parameter estimations.
4. The join probability of drought duration and drought intensity
Gumbel copula describes asymmetry dependence and it is used to capture strong upper
tail dependence and weak lower tail dependence. The joint distribution of drought
duration and drought intensity is obtained from a Gumbel Copula and it takes the form:
46
The marginal distributions of the copula are the empirical distributions and the
parameter of the copula is estimated by maximum likelihood.
47
5. Figures
Figure A1. Boxplot of the mean (left) and standard deviation (right) of 1.000 realizations
of monthly streamflow generated with copula parameter
Figure A2. Boxplot of the mean (left) and standard deviation (right) of 1.000 realizations
of monthly streamflow generated with copula parameter
Figure A3. Boxplot of the autorrelation function of 1.000 realizations of monthly
streamflow generated with copula parameter (left) and
48
Figure A4. Estimated marginal means of cropland reduction over the baseline scenario
by policy and climate change scenario. Confidence interval at 95%.
EE means Enhance of efficiency and CC is climate change
Figure A5. Cropland distribution under irrigation land reduction
49
Figure A6. Concentration index under policy scenario
The figure A6 shows the Herfindahl-Hirschman concentration index under policy
scenario and it is defined as:
where is the share of the crop over the total land and N is the number of crops. The
index measures the production concentration, specialization, and sharing. Higher values
of H means higher concentration.
50
Figure A7. Monthly water deficit (Mm3) and water scarcity index
Figure A8. Comparison of water use by policy with and without efficiency
enhancements under the same climate conditions
51
Figure A9. Distribution of the fall in streamflow at the river mouth from investments in
efficiency enhancements (Mm3)
52
Figure A10. Probability of annual benefits and exceedance of probability of annual
benefit losses (106 €)
Figure A10 shows the cumulative probability of the annual irrigation benefits and the
exceedance of probability of losses under market and proportional policies with and
without efficiency enhancement.
53
Figure A11. Conditional exceedance of probability of benefit losses under different
drought spells.
54
Figure A12. Conditional loss of benefits (106 €) by policy at quantiles 0.5, 0.9, 0.95 and 0.99.
55
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