WATER RESOURCES RESEARCH, VOL. 48, W08530, doi:10.1029/2012WR012040, 2012
Coupling hydrodynamic models and value of information for designing stage monitoring networks Leonardo Alfonso1 and Roland Price1 Received 24 February 2012; revised 27 June 2012; accepted 28 June 2012; published 31 August 2012.
[1] Because the collection of data in water systems is important for making informed decisions, monitoring networks are designed and installed in such systems. Traditionally, the design of hydrometric monitoring networks has been concentrated on measuring streamflow/precipitation at particular key (gauged) sites so that streamflow/precipitation can be estimated accurately at ungauged sites. Although many methods take into account a set of final users of the information, there appears to be no method that explicitly considers them in the mathematical formulation of the decision-making process. This paper presents a novel approach for designing monitoring networks in a water system using the concept of value of information (VOI). This concept takes into account three main factors: (1) the belief that the decision maker has about the state of the water system before having any information; (2) the consequences associated with the decision of having to choose among several possible management actions given the state of the water system; and (3) the evaluation and update of new information when it becomes available. The methodology uses a water level time series generated by a hydrodynamic model at every computational point, each one being a potential monitor site. The method is tested in a polder system in the Netherlands, where monitoring is required to make informed decisions about the operation of a set of hydraulic structures to reduce flood impacts. Citation: Alfonso, L., and R. Price (2012), Coupling hydrodynamic models and value of information for designing stage monitoring networks, Water Resour. Res., 48, W08530, doi:10.1029/2012WR012040.
1. Introduction 1.1. Design of Monitoring Networks [2] The collection of data in water systems is important in order to make informed decisions about a range of issues such as planning (e.g., land use policies) and operation (e.g., turning pumping stations), and for this purpose monitoring networks are designed and installed in such systems. Monitoring networks consist of sets of measuring devices that are strategically located within a water system in order to provide indications about its state and make water management decisions accordingly. There exists an extensive literature regarding methods for monitoring network design to measure different hydrological processes such as rainfall [see, e.g., Bogárdi et al., 1985; Bras and Rodríguez-Iturbe, 1976; Krstanovic and Singh, 1992; Moore et al., 2000; Pardo-Igúzquiza, 1998; Rodríguez-Iturbe and Mejia, 1974], surface water [e.g., Husain, 1989; Moss, 1976; Moss and Karlinger, 1974], groundwater [e.g., Ammar et al., 2011; Mogheir et al., 2006; Nunes et al., 2004; Yeh et al., 2006], and for general purposes [e.g., Karasev, 1968; Ruddell and Kumar, 2009]. For a comprehensive review of methods for 1 Hydroinformatics, Department of Integrated Water Systems and Governance, UNESCO‐IHE, Delft, Netherlands.
Corresponding author: L. Alfonso, Hydroinformatics, Department of Integrated Water Systems and Governance, UNESCO‐IHE, Westvest 7, NL-2611AX Delft, Netherlands. (
[email protected]) ©2012. American Geophysical Union. All Rights Reserved. 0043-1397/12/2012WR012040
designing monitoring networks the reader is referred to Mishra and Coulibaly [2009]. [3] In general, the existing approaches are based on interpolation methods that reduce the uncertainty in the estimation of a particular water variable at locations where direct monitoring is not taking place. In this paper, a method that explicitly considers the consequences of the decisions that are taken in view of the potential information coming from the monitors is presented within a mathematical procedure that follows the concepts of Value of Information, described below. The method is applied in a highly controlled, low-lying water system that requires monitoring to make informed decisions about the operation of a set of hydraulic structures to reduce flood impacts. 1.2. Value of Information (VOI) [4] The concept of Value of Information (VOI) was introduced first in the area of economics as a way to judge whether it would be rational to invest in additional information when making decisions under uncertainty [Grayson, 1960; Howard, 1966; 1968; Schlaifer, 1959]. This paper is based on the findings of Hirshleifer and Riley [1979], who present in a straightforward manner VOI concepts for monitoring network design. Their development takes into consideration decision making under uncertainty and how this process depends on the availability of information. If a decision maker has to make a decision without information, he will have to rely on his perception of the state of the system. This perception can be quantified as P(s), the prior probability of having a particular state, s, in the system (noted from now on as Ps). On the other hand, when new
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information is available, the decision maker can judge whether or not to believe the new information, thus updating his beliefs, or rejecting the new information. In the VOI concept, the new information has value if the individual uses it to make a decision, and has no value otherwise. The decision maker may or may not accept the new information. If he accepts it, he will change the probability vector Ps; if he does not accept it, he continues with his prior belief. This can be represented by Bayes theorem for updating belief: PðsjmÞ ¼
PðsjmÞPs Pm
ð1Þ
where, for a given state s, P(s|m) is the posterior or updated belief following the receipt of the message m and P(m|s) is the conditional probability (or likelihood) of receiving the message m given the state s; Pm or P(m) is the probability of receiving a message m, which is independent of the available set of states of the system, and it is related to P(m|s) by Pm ¼
X Ps PðmjsÞ
ð2Þ
S
where S is the total number of possible states of the system. One of the difficulties in using the VOI lies in how to assess the probabilities before and after receiving the new information. Some authors have tried to estimate them using knowledge elicitation methods by interviewing decision makers directly [see e.g., Bouma et al., 2009; Schimmelpfennig and Norton, 2003], while other authors have used empirical methods such as data from model outputs [see Dakins et al., 1996; Lin et al., 1999]. Yokota and Thompson [2004] offer a comprehensive review of various VOI applications in the field of management decisions concerning environmental health risk and the different approaches used to estimate these probabilities. In this paper we use model results to assess P(s) and P(m|s) as explained further. [5] For the following discussion, we define Information Service (IS) as a monitoring device that provides a message related to the measurement of a particular water variable and that is potentially useful for decision making. The value of information provided by one message received, Dm, can be estimated as the difference between the utility, u, of the action, am, that is chosen given a particular message, m, and the utility of the action, a0, that would have been chosen without additional information: Dm ¼ uðam ; PðsjmÞÞ uða0 ; PðsjmÞÞ
ð3Þ
Ps is included implicitly in the estimation of a0, which is evaluated on the basis of the IS. Therefore, the utility of a0 is in respect to the posterior probabilities of the messages received. Both utilities are calculated following the expected utility rule of Von Neumann and Morgenstern [1947] uða; Ps Þ ¼
X
Ps uðcas Þ
ð4Þ
S
where Ps, is the perceived probability of state, s, cas is the consequence associated with the decision of performing the action, a, when the system has a state, s, u(cas) is the utility of such a decision. Therefore, the expected utility u(a, Ps) is the probability-weighted average of the utilities of the associated consequences cas.
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[6] Since it is not possible to know in advance which message the information service will provide, the VOI is the expected utility of the values Dm for the separate messages, which is given by VOI ¼ EðDm Þ ¼
X
Pm D m
ð5Þ
m
Supposing that a rational decision maker will always choose the action with the maximum utility, and putting equation (3) into equation (5), the final expression for VOI becomes VOI ¼
X m
" Pm max a
S X s¼1
! ca;s PðsjmÞ max a
S X
!# ca;s Ps
s¼1
ð6Þ
Equation (6) shows that, ultimately, VOI is a function of cas, Ps and P(m|s) (see equation (1)). Therefore three sets of data are needed for its estimation: the set of actions a, the set of possible states s and the set of messages m that the monitors provide. First, the set a = (a1, a2, …, aA) contains A actions that are available for the decision maker to deal with the state of the system, for example, by turning a pump on, by releasing a warning or simply doing nothing. Second, the set s = (s1, s2, …, sS) contains S possible states of the system, namely, flooding or normal water levels, that should be defined for each point of the system. Third, the set m = (m1, m2, …, mM) contains M messages that the IS will provide to the decision maker as an indication of the corresponding state of the system; that is, m1 describing the state s1, m2 describing the state s2, etc., so S = M; examples of such messages can be “Danger” or “Relax.” From now on, x will denote the location of the monitor (where the messages m are produced) and y will denote the location of any other point in the system where the states s are to be inferred by the monitor. [7] Although other factors such as the cost of using the information to make decisions (when analyzing data is expensive) and the price of the next best substitute for the information may affect VOI [Macauley, 2005], the analysis in this paper is limited to the variables in equation (6).
2. VOI and the Role of Models in Its Estimation [8] The ultimate objective of the proposed method is to locate monitoring devices conveniently in a water system, by selecting, among predefined potential locations, where the information gives the highest value for the decision maker with respect to the state of the water system. [9] The first consideration regarding the methods presented in this paper for monitoring network design is the use of a model as a data generator and the use of the generated data to estimate the probabilities required for the VOI calculation. The model is required because the available measurements are generally limited to a few points, making them insufficient to draw conclusions from their analysis. In contrast, the model generates a dense set of points, such that every calculation point within a model is considered as a potential location for a monitoring point within a water system, in a similar fashion to the concept presented by Alfonso et al. [2010a]. Additionally, models can generate long time series, so that the frequency-based approaches to define the VOI variables (presented below) give reliable
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Table 1. Definition of the Cas Matrix Cas
a1: Action 1
a2: Action 2
s1: State 1 s2: State 2
Cost of doing a1 when s1 Cost of doing a1 when s2
Cost of doing a2 when s1 Cost of doing a2 when s2
estimations of the probabilities. The models used in this paper are hydrodynamic models, which generate data series of water levels and discharges at every computational point by numerically solving the shallow water equations as explained for instance by Vreugdenhil [1994]. [10] For a particular water variable, the states s observed at location y (and the messages m produced at location x that describe them) are defined by means of thresholds applied to the corresponding time series at x and y. This means that each record of the time series at x will be categorized, for example, as either “Danger” or “Relax” and each record of the time series at y will be categorized as either “Flood” or “Normal.” Therefore, the time series are transformed into a binary time series from which the probabilities are estimated using a frequency-based approach, as described below. 2.1. Estimation of Prior Beliefs Ps and Conditional Probabilities P(m|s) [11] The assessment of the prior beliefs, Ps, and the conditional probabilities, P(m|s), is difficult, because the probabilities before and after receiving the new information are not known. In order to overcome this problem, we estimate these variables using a frequency-based approach, with the assumption that the hydrodynamic model generates time series that are long enough to give acceptable estimates of the probabilities. [12] First, the prior beliefs: Ps is a vector with S elements that contains the perceived probabilities of the occurrence of the possible states s at a point y. Given a time series of n records generated by the hydrodynamic model, the estimation of Ps at the prescribed point of the water system is 2
3 2 3 P ðs1 Þ ns1 =n 6 Pðs2 Þ 7 6 ns2 =n 7 7 6 7 Ps ¼ 6 4 ⋮ 5¼4 ⋮ 5 P ðsS Þ nsS =n
ð7Þ
where nsj is the number of occurrences of the state j ( j = 1, 2, …, S), and therefore ns1 + ns2 + … + nsS = n. As a result, each element of Ps contains the relative frequency of each state, which can be regarded as the probability of a particular state occurring at the point, and in this respect it may be treated as the knowledge the decision maker has about what is happening there. [13] Second, the conditional probabilities: P(mx|sy), or the likelihood of receiving the message m (produced at a point x, where a monitor is placed), given the state s (at any other point y in the water system), is a matrix of size S (also equal to the size of M) that can be estimated using a frequencybased approach, as follows. Recalling that the point x produces the messages m1x, m2x, …, mMx to describe the corresponding states s1y, s2y, …, sSy occurring at y, let nm1x, s1y be the number of times m1 is produced by x when the state at y is s1, nm1x,s2y the number of times m1 is produced at x when the state at y is s2, and so on. Therefore, the total
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number of occurrences of m1 in the history of the point x is nm1x = nm1x,s1y + nm1x,s2y + … + nm1x,sSy. Similarly, nm2x = nm2x,s1y + nm2x,s2y + … + nm2x,sSy, nm3x = nm3x,s1y + nm3x,s2y + … + nm3x,sSy, etc., and n = nm1x + nm2x + … + nmMx. Consequently, the probability of the joint occurrence of m1 and s1 can be estimated as p(n1x, s1y) = nm1x,s1y/n, and therefore the likelihood of receiving the message m1x given the state s1y is P m1x ;s1y P m1x js1y ¼ P s1y
ð8Þ
where P(s1y) = ns1y/n. Here ns1y is the number of occurrences of the state s1 in the history of the point y. Therefore, the likelihood of receiving the message m at x, given the state s at y, can be estimated as 2 Pm1x js1y 6 P m1x js2y P mx jsy ¼ 6 4 ⋮ P m1x jsSy
Pm2x js1y P m2x js2y ⋮ P m2 jsSy
⋯ ⋯ ⋯ ⋯
3 PmMx js1y P mMx js2y 7 7 5 ⋮ P mMx jsSy
ð9Þ
It can be observed that the elements in each row sum up to one. 2.2. Definition of the Consequences cas [14] The consequences, cas, of taking an action, a, given a particular state s, form a matrix that contains the costs associated with having chosen to perform an action according to the state the decision maker thinks is happening. Table 1 presents an example of this matrix for the case of two actions and two states. [15] Similar contingency tables have been used to estimate the value of flood forecasts [Verkade and Werner, 2011]. This matrix depends on the type of water system; its definition is not straightforward because of the hypothetical character of the damages under the different scenarios, especially for extreme states. However, it can be built according to the judgment of decision makers, in a similar way as to that adopted by van Andel [2009, pp. 116–118]. Although direct, tangible costs are generally used, the estimation of indirect, intangible costs of hydrological disasters is an issue that is currently under research [Balbi et al., 2011]. [16] An important consideration in the generation of the matrices P(m|s), Ps, and Cas is that if a particular state does not occur in the history of the time series, the corresponding rows in these matrices are removed. Also, if a monitor at x cannot capture any of the states at y, then the VOI of x is zero for that particular point y. This means that a monitor at x makes sense only if it is able to say something about the state at y. 2.3. Assessing the Value of Monitor Locations [17] Before presenting the proposed approach for monitoring network design, a proper methodology to assess the value of monitor locations is needed. This methodology is developed below, first by considering the location value of one monitor and then extending it further for the case of the location of two monitors. 2.3.1. Value of the Location for One Monitor [18] For the subsequent discussion, x is the location in a water system where a monitoring device is placed to provide messages about the possible states of any point, y, at which a decision maker bases his water management related
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[20] Due to the fact that placing a monitor at each point in the water system is not feasible, it is necessary to identify the monitor location x that is able to describe the status of the maximum number of points in the system. One convenient way consists of looking at the total VOI that x gives about the state of the entire system, VOIx, by adding up the individual values of Vx(y) obtained for all the points in the water system. For the theoretical case of having infinite number of points in the water system, this is equivalent to finding the area below the curve Vx(y) (see Figure 1a). VOIx can be then estimated as Z VOIx ¼
Vx ðyÞdl
ð11Þ
y
Figure 1. Definition of Vx(y), Vx, and VOIx for a monitor located at x to give the state of the system at y for (a) infinite and (b) finite number of calculation points. decisions. For simplicity, it is considered, in principle, that the water system is a linear canal. In the remaining part of this paper, we denote Vx(y) as the value a decision maker is willing to pay for a monitor located at x to know the state at any other point y. From equation (6), Vx(y) can be estimated as
Vx ðyÞ ¼
X m
" Pm max a
S X s¼1
ca;s P sy jmx
! max a
S X
!# ca;s Ps
s¼1
ð10Þ
The method considers that m and s are dependent on the time series generated by the hydrodynamic model of the water system at x and y, respectively, and therefore Vx(y) is dependent on the geographical position of both points. As mentioned above, it is assumed that every computational point of the hydrodynamic model is either a potential location x to place a monitor or a location y where the states are to be observed. [19] If the geographical position of the monitor is fixed at the point x and we allow y to take all possible locations in the water system (y1, y2, …, yN, where N is the number of computational points used by the model), we obtain the vector Vx = [Vx(y1), Vx(y2), …, Vx(yN)], represented by the curve in Figure 1a. Provided that a monitor gives measurements within acceptable confidence bounds, it will always generate accurate messages concerning the state of the system at its own location. Consequently, in order to know the state of the world at point x, the most convenient point to place a monitor to know the state at x is precisely the point x itself. For this reason, Vx has a maximum at x and decreases progressively as y moves away from x (some of the messages produced at x will not provide the right states of a distant point y). In other words, a decision maker should be willing to pay a maximum of Vx(x) in order to know the state of the system at x.
where l is the spatial dimension on which the monitors are placed. However, from the practical viewpoint, the modeled water system will always have a finite set of N points such that the resulting curve Vx is a discrete one (Figure 1b); therefore, the total VOI that the location x gives about the state of the entire discrete water system can be defined as VOIx ¼
N X
Vx ðyÞ
ð12Þ
y¼1
Alternatively, the average VOI can be calculated by dividing the total VOIx by N, to obtain an idea of the average value of information per point that x gives about the system state. 2.3.2. Value of the Locations for Two Monitors [21] Suppose a monitor at point x1 is located to know the state at a point y and that an individual needs an additional monitor located at x2 to have better information about the state at the same point y. As the existing monitor at x1 is already giving some information about y with value Vx1(y), then the maximum value the individual should be willing to pay for the additional monitor located at x2 is Vx2(y) Vx1(y). Figure 2 where Vx1, Vx2 and Vy are obtained as in Figure 1. This approach is in agreement with the fact that the value an individual is willing to consider for placing additional monitors to know the state at the same point y is progressively lower as new monitors are added. [22] The difference Vx2(y) Vx1(y) is bounded in the interval [0, Vy(y)]. The lower bound implies that if Vx2(y)
0 fVxN ðyÞ … Vx3 ðyÞ Vx2 ðyÞ Vx1 ðyÞg > 0
where Ax1x2…xi is the positive area between the curve Vx1 + Vx2 +, …, + Vx(i1) and Vi, which represents the value that a monitor at xi can add to the set of monitors previously located at x1, x2, … xi1 about the state of the entire system. The solution of equation (15) gives the best set of monitors that are valuable in describing the state of the entire system with minimum redundancy among them. As above, this optimization problem is solved by looking at all possible combinations of the N locations in the water system, which implies that the computational cost is significant for large N, characteristic of NP-complete problems [Garey and Johnson, 1979]. This difficulty can be addressed either by dividing the problem into subsets of potential monitors in the water system, by drastically reducing the search space eliminating redundant monitors (e.g., analyzing correlation structure to reduce complexity), or by using heuristic optimization methods. [27] The application of the procedure for monitor location is described in the case study given below.
4. Case Study: Pijnacker Region, Netherlands ð13Þ
3.2. Locating Two New Monitors [25] Following the reasoning presented above, it is proposed to pose the problem of simultaneously locating two monitors x1 and x2, as an optimization problem: maxfVOI1x þ Ax1x2 g where Z Z VOI1x ¼ Vx1 dl and Ax1x2 ¼ ðVx2 Vx1 Þdl
ð15Þ
Z
ð14Þ
y
Subject to : Vx2 ðyÞ Vx1 ðyÞ > 0
where Ax1x2 is the positive area between the curves Vx1 and Vx2 and represents the value that x2 can add to x1 about the state of the entire system (see Figure 3). The solution of equation (14) allows the selection of nonredundant monitors that have value and this is obtained by looking at all possible combinations of the locations x1 and x2 in the water system.
[28] In order to test the VOI approach for siting monitors, the method is applied to the flat, highly controlled canal network of the Pijnacker polders in the Netherlands. In this system the water level is controlled through the operation of hydraulic structures, which in turn define the boundaries of independent hydrological units of a flat region. The system consists of 15 pump stations, 21 fixed weirs and flows mainly from the east (lower areas) to the west (higher areas), draining excess water into a canal with a bigger capacity that acts as storage basin (see Figure 4). [29] The need of a proper monitoring network of water stage in this area is very relevant since convenient procedures to operate the available pumping stations are needed in order to properly drain the area. Given the fact that there is a complex network of canals with flow discontinuities at weirs, the decision to operate the pumping stations is currently based on measurements within their locality (i.e., hydrological or drainage units), which lead to frequent flood situations at sites that are not measured. [30] The hydrodynamic model used in this analysis was provided by the Delfland Waterboard, which uses it for
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Figure 4.
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Canal network of the Pijnacker water system.
planning purposes and it uses a rainfall-runoff component with a fixed rainfall event of 5 year return period that is the input discharge of the canal system. The hydrodynamic model solves the one-dimensional shallow water equations, obtaining as output stages and discharges at a time step of 15 min. Water level time series are calculated at 1520 computational points that shape the canal network with a spatial separation of 15 m on average. Each time series has a length of 1682 records resulting from a simulation of 10 days. Further information about the model is given by Alfonso et al. [2010b]. [31] The actions, states and messages required for the VOI analysis are defined for both case studies as follows. First, the set of actions are a1: Release a flood warning; a2: Do nothing. Second, the set of possible states are s1: Flooding, s2: Normal. Finally, the messages the monitors can provide are m1 = “Danger,” m2 = “Relax.” The consequence matrix, cas, created arbitrarily for this example, is shown in Table 2, where the negative values indicate a cost. So far as costs are concerned, it is assumed that the cost of releasing a warning when there is flooding is $5 (the costs associated with communication and mobilization) and it is $30 if there is no flooding (the political and real costs of having mobilized people under a false alarm). Correspondingly, the cost of doing nothing when there is flooding is $100 (a high cost
due to the disaster associated with the flooding that was not warned), and it is $0 if there is no flooding. [32] For reasons related to computational effort, this section includes two different experiments: one with simplified inputs for the complete Pijnacker water system, and one with more complex inputs in a smaller system. 4.1. Entire Pijnacker Polder System [33] This section presents a numerical example of the calculation of the VOI of the point O in Figure 4 to know the state of the system at the point R. The states s1: “Flooding” and s2: “Normal” are defined at every point in the water system using the level at which flooding occurs, which is assumed equal to the local elevation of the lowest canal embankment. The water level time series at each of the 1520 points are then transformed into binary time series of 0 s (Normal) and 1’s (Flooding). For the point R, this results in Table 2. Arbitrary Consequence Matrix of Doing Action a Given State s (Costs Units) Cas
a1
a2
s1 s2
5 30
100 0
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Table 3. Number of Occurrences for Each Situation at R and O State at R
Message at O
s1 = “Flooding” s2 = “Normal”
m1 = “Danger”
m2 = “Relax”
nm1O,s1R = 140 nm1O,s2R = 340
nm2O,s1R = 0 nm2O,s2R = 1202
118 records of “Flooding” and 1564 records of “Normal.” Therefore, the associated prior beliefs Ps, interpreted as the knowledge from past experiences about both states at R, can be estimated using equation (7): Ps ¼
118=1682 0:07 ¼ 1564=1682 0:93
[34] From the Bayesian viewpoint, the data used to estimate the prior Ps should be independent to the data used to evaluate the likelihood P(m|s), so it is debatable to use the same data set to estimate the likelihoods (this issue is addressed in section 5). For the time being, let us assume that Ps was obtained from some knowledge elicitation method [see e.g., Bouma et al., 2009; Schimmelpfennig and Norton, 2003]. Then, the probability of the joint occurrence of the messages produced at O and the states observed at R can be estimated using equation (8). The number of occurrences of the transformed time series at O and R for each possible situation is summarized in Table 3. [35] From Table 3, P(m1O, s1R) = nm1O,s1R/n = 140/140 = 1.00 and similarly, P(m2O, s1R) = 0.22, P(m1O, s2R) = 0.00 and P(m2O, s2R) = 0.78. Then, the likelihood of receiving the message m1O given the state s1R is, from equation (9), Pðm1O js1R Þ ¼
Total State Occurrences
1:00 0:22 Pðm1O ; s1R Þ Pðm2O ; s1R Þ ¼ 0:00 0:78 Pðm1O ; s2R Þ Pðm2O ; s2R Þ
ns1R = 140 + 0 = 140 ns2R = 340 + 1202 = 1542
From equation (2), Pm = [0.27; 0.73] and from equation (1), P(s|m) = [0.26, 0.00; 0.74, 1.00]; from equation (4), the utility of the decision about performing action a1 without receiving new information is u(a1, Ps) = 5.00 * 0.07 + 30 * 0.93 = 28.25 and similarly u(a2, Ps) = 7.00 for action a2. Therefore, the action that maximizes the utility of the decision without receiving new information is max (28.25; 7.00) = 7.00. [36] After receiving the new information, the utility of message m1 associated to the decision of performing action a1 is u(P(s|m1), ca1, s) = 5.00 * 0.26 + 30.00 * 0.74 = 23.50. Similarly, u(P(s|m2), ca1, s) = 30.00, u(P(s|m1), ca2, s) = 26.00 and u(P(s|m2), ca2, s) = 0.00. Therefore, the maximized utility of message m1 is max (23.50, 26.00) = 23.50 and of message m2 is max (30.00, 0.00) = 0.00. From equation (10), V0 ðRÞ ¼ 0:27 ð23:50 þ 7:00Þ þ 0:73 ð0:00 þ 7:00Þ ¼ 0:655
Following the same procedure, but now allowing R to take all the positions in the system, the vector V0 with size N = 1520 is obtained, and it can be mapped as in Figure 5. [37] From Figure 5 it can be observed that the value of information that the monitor located at O is a maximum for the point O is Vo(O) = 21.45. This is high for some of the points surrounding it. This means that O is a good point at which to infer the states “flood” or “normal” at the surrounding points,
Figure 5. Value of Information of point O with respect to the water system (cost units). 7 of 13
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Figure 6. Average VOI map and solutions for the location of 1 and 2 monitors.
aspect that can be used as a possible criterion to reduce the problem complexity in further research. It can also be observed that it is capable of providing the states at distant points or points located at different hydrological units. [38] From equation (12), the total value of information that the point O gives about the state of the entire system is: 1520 X VO(y) = 5806.4 (monetary units), and its average VOIO = y¼1
Value of Information is VOIO = 5760.8/1520 = 3.79 cost units per point. If the same exercise is performed, but now allowing all points in the water system to take the place of point O, then the average value of information for each point is obtained, and these values can be represented in a VOI map of the system such as in Figure 6. [39] It can be observed, importantly, that a percentage of the points do not provide value in terms of describing the state of the entire system. This is because the prior beliefs at these points are at extremes, that is, additional information has no value since their states are known by the decision maker’s experience. Also, the spatial distribution of the VOI is dependent on the location of pumps and weirs, since they create independent regimes at different hydrological units. From Figure 6 it can also be noted that there is a region in which VOI is particularly high. This is because, on the one hand, the vector Ps at those points is close to [0.5; 0.5], which implies that there is high uncertainty about the states of the system. On the other hand, the quality of the messages produced at those points regarding the rest of the system (measured by the likelihood P(m|s)) decreases mainly because at some points the state “flooding” never occurs in the history of their time series. These points, therefore, should not be taken as monitoring locations. 4.1.1. Location of Monitors [40] The location of one monitor x0 is determined by solving the optimization problem posed in equation (13), which is equivalent to selecting the darkest point in Figure 6. In this
case, the maximization of VOIx yields a single point with VOI = 3.81, which is represented by a square in Figure 6. For the case of locating two monitors x1 and x2, the solution of the optimization problem presented in equation (14), yields the locations represented by circles in Figure 6. [41] It is interesting that the location obtained for one monitor is not selected for the case of two monitors. This is a consequence of solving the problem simultaneously for more than two monitors, which is the ideal condition for designing completely new monitoring networks. Certainly, if the selection of new monitors is performed by steps, in which the monitor with the highest value is selected first and the subsequent monitors are selected on the basis of the “next best option,” the monitors selected at the end of the procedure will have less and less value as monitors are added, so the final set of monitors will provide messages with diverse quality. [42] Figure 7a presents the maximum value of VOIy for each point in the system, and is provided to see how worthwhile it is to add new monitors to know the state of the system. [43] With this aim, the value of information that remains in the system once x0 is placed (calculated as the difference between max (VOIy) and the vector Vx0(y)) is presented in Figure 7b. Similarly, the value of information that remains in the system once x1 and x2 are placed, is presented in Figure 7c. [44] As the procedure to solve the optimization problem in equation (15) is based on an exhaustive search, it is guaranteed that the VOI at the locations shown in Figure 7 are maximum values, so no cross-validation procedure is needed. However, as the computational effort required for the calculations to locate three and more monitors is significant, the use of heuristic optimization methods to solve equation (15) would be an option. In this case, a crossvalidation procedure to check the soundness of the result would be needed.
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Figure 7. Change in VOI when new monitors are added. (a) With no monitors; (b) locating one monitor x0; (c) locating two monitors x1 and x2.
[45] In this regard, it is important to clarify that the method does not depend on the complexity of the models or their running time, because the method uses the resulting output of the modeling process. The computational cost or complexity of the solution is O(Nm), where N is the number of computational points considered in the model and m is the number of monitors to be selected. This difficulty can be overcome using at least two approaches. The first approach consists of reducing the number of computational points by conveniently selecting a representative point out of a set of neighboring points (e.g., using information theory criteria to cluster monitors in a water system as suggested by Alfonso et al. [2010a]). The second approach considers the use of heuristic methods (e.g, genetic algorithms) to solve the
Figure 8.
optimization problem in a similar fashion as Alfonso et al. [2010b]. The third approach consists of dividing the original system into sets of subsystems and solving each one independently. The first two approaches are subject of future research and the last one is illustrated below, where the reduction in the number of points allows more detailed inputs to be explored. 4.2. Selected Subsystem Within Pijnacker Polder System [46] The selected subsystem is located in the northeastern part of the Pijnacker water system. The water level in this area is controlled by four weirs and three pumping stations that define independent hydrological (drainage) units. The
Selected subsystem of the Pijnacker polder system. 9 of 13
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Table 4. Consequences Cas for Different Land Uses for the Pijnacker Region Urban
S1: S2: S3: S4:
Severe Flood Flood Normal Drought
Glasshouse
Pasture
Pump On
Pump Off
Pump On
Pump Off
Pump On
Pump Off
500 100 0 20
1000 200 0 10
250 50 0 10
500 100 0 5
50 10 0 2
100 20 0 1
land is mainly used for pasture, but glasshouses and urban developments also exist. This subsystem was modeled with 65 calculation points (see Figure 8) using rainfall data for 32 months and a reporting time step of 1 h. This gave data series with more than 20,000 records, long enough to estimate the matrices Ps and P(m|s) in equations (7) and (9), respectively. [47] The availability of the model data allows the introduction of three new features in the procedure for locating the monitors. First, more than two possible states of the system are allowed. Then different land uses (urban, glasshouse and pasture) are defined, each one with its own damage function (consequence matrix). Finally, experiencebased consequence matrices, considering the available operational status (ON/OFF) of the relevant pump, are included. These three features, discussed with staff members of the Delfland Waterboard in November 2009, are summarized in Table 4 and Figure 9, where four levels, namely severe flood (L1), flood (L2), normal (L3) and drought (L4) were defined as follows: L1 is the level that overtops the lowest field level of the canal embankment by a value of d = 10 cm, 20 cm and 30 cm for glasshouse, urban and pastureland uses, L2 is the lowest field level of the canal embankment, L3 is the minimum control level of the hydraulic structure downstream the point under consideration (either the off level of a pump or the lowest crest level of a weir) and L4 is the bed level of that point, respectively. A particular water level occurring between these levels defines the possible states S1, S2, S3, S4 at a given location as shown in Figure 9. It must be noted that the consequences shown in Table 4 do not represent monetary values but are relative costs (the norm is 1000 units, being the reference for the worse-case damage scenario of not pumping downstream when a severe flood is present in an urban area). The damages associated with droughts are related to subsidence in urban areas, to the lack of water supplied to glasshouses and to the low soil moisture for pasture. 4.2.1. Estimation of Ps and P(m|s) [48] As mentioned above, the data used to estimate the prior Ps must be independent to the data used to evaluate the likelihood P(m|s). Taking advantage of the long time series produced for the selected subsystem, the first half of the time series has been used in equation (7), while the second half has been used in equation (9). After testing several time thresholds to split the series it was found that this division has little impact on the results, due to the fact that the state S3 (normal) is dominant in the series. 4.2.2. VOIx for Different Decision Makers [49] An interesting experiment consists of a comparison of the generated results with the estimates of an ignorant (inexperienced) decision maker in the process, which can be simulated by having uniformly distributed priors Ps. With this objective, the mean value of information VOIx estimated
using the time series division explained above is mapped in Figure 10a next to the VOIx obtained by an ignorant decision maker (Figure 10b). It can be observed that the VOI distributions are very different in both cases, with a higher scale for the latter. For the case of an experienced decision maker, the points around urban areas and at a canal at the northeast provide better insights about the state of the subsystem, while for the inexperienced decision maker a monitor on a canal at the northwest part would be more valuable. This implies that the nature of the decision maker plays a significant role in how the monitors are distributed. [50] In both cases, however, VOIx is relatively low, considering that the maximum damage possible is 1000 units, when severe flood occurs in urban areas (see Table 4). In general, this is due to the fragmented nature of the system, in which the water level state occurring at a point upstream of a particular hydraulic structure does not match the corresponding water level state at points with a different hydraulic structure downstream. In spite of this, interesting values for local state descriptions are obtained for different drainage units. For instance, a monitor located at point a (Figure 10a) gives a value of 25.8 units when describing the state at point b (Va(b) = 25.8), while a monitor located at point f (Figure 10b) gives a value of 82.2 units when describing the state at point g (Vf (g) = 82.2). In both cases the points are located in different drainage units limited by a weir and a pumping station. 4.2.3. Monitor Location [51] The optimization problem posed in equation (15) is solved in order to locate the monitors in the case of an inexperienced decision maker, thus using the mean VOI shown in Figure 10b. Defining V*(*) as the vector of the maximum VOI at each point, the color scale used in
Figure 9. Definition of the possible states, land uses, and damage function (consequences), where d = 10 cm, 20 cm, or 30 cm for glasshouse, urban, or pasture locations, respectively.
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Figure 10. VOIx maps considering priors considering (a) experienced and (b) ignorant decision makers. Figure 11 for the map for one monitor x is V*(*) Vx1, while for the map of two monitors x1 and x2 is V*(*) max (Vx1, Vx2), for three monitors x1 x2 and x3 V*(*) max (Vx1, Vx2, Vx3) and so on. These scales are used to observe how the inclusion of one monitor reduces the value a decision maker
Figure 11.
is willing to pay for an additional monitor given the existence of the previous one. The results in Figure 11 show that the value that the decision maker has to pay for additional information approaches zero as the number of monitors increases. In addition, although 3 monitors reduce the VOI to
Results for the selected subsystem of the Pijnacker water system. 11 of 13
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be as low as 1 unit, 4 monitors do not reduce the value further, giving a situation that can be used as a new criterion to select the number of monitors worth placing.
5. Conclusions [52] In this paper a new methodology has been presented to design monitoring networks optimally based on the concept of Value of Information, defined as the value a decision maker is willing to pay for additional information prior to making a decision. The concept considers the prior beliefs of a decision maker within the mathematical formulation for monitoring network design, for whom new information has value if it would be useful to make a decision that maximizes his expected utility and has no value otherwise. [53] The VOI method for locating monitors optimizes the VOI such that the messages produced by them can provide insights about the state of the entire water system with a minimum redundancy between the monitors. [54] The proposed approach is flexible, because it can be applied to any type of water system, for any water variable and any set of statuses that may occur in the water system, provided there is a model that generates time series for the variables in question. [55] Depending on the type of system under study, additional considerations may be needed to use this approach. For example, in the case of natural rivers, it is necessary to include the lag time effect of flood waves, because the message produced by a monitor at x might not have value about the state of a remote point y downstream if the corresponding records at both points are used. [56] The resulting hydrometric monitoring network configuration, obtained with the proposed method will always depend on the past experience that the decision maker has about the state of his system and not only on the geospatial accuracy of the estimates at ungauged places. This implies that a different decision maker may find more value in different configurations. [57] In addition, the decision maker may eventually change his prior beliefs about the states described by the resulting network configuration, due to model inaccuracies or new hydraulic or hydrologic situations the model is not able to describe. This implies that the presented approach is part of a looped process in which the new data collected may be used to improve the model, which in turn will be used to adjust the configuration of the monitoring network. The implications of this loop is a topic for future research. [58] There are three main difficulties when applying the method. First, there is the need for a good definition of the consequences or damage functions. Second, a convenient definition of the states of the system that need to be monitored is required. Third, the huge computational effort is needed to solve the monitor location problem for more than three monitors. The first difficulty concerns the amount of data required to define the damage functions, while the second involves a clear definition of the objectives of the monitoring network. Nevertheless, it has been indicated that the objectives can be achieved by analyzing different scenarios. A workaround for the third difficulty, which is used in this paper, consists of dividing the water system into several subsystems and solving for each subsystem separately. Alternatively, the number of computational points can be reduced by selecting a representative point out of a set of neighboring points.
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[59] The presented approach can be modified easily in order to consider the existing monitors (e.g., at pumping stations), in which case the monitor selection would be done including the monitor which provides the highest VOI about the remaining, ungauged areas of the system at a given time. [60] Further research should focus on extra factors that may influence the final configuration of the monitoring network, such as the actual needs of multiple water users, available resources for installing and maintaining monitors, access to monitoring sites and political interests in particular areas.
Notation am action that is chosen given a particular message m. a0 action that would have been chosen without additional information. n number of records of the time series. N number of calculation points used to model the water system. m message or information generated by a monitor, sensor or measuring device. Ps perception, in probabilistic terms, of having a particular state, s, in the system. P(s|m) posterior or updated perception about the state s given the message m. P(m|s) likelihood of receiving the message m, given the state s. Pm unconditional probability of receiving a message m. s state of the water system at a particular point. VOIx total value of information that location x gives about the state of the entire system. Vx(y) value a decision maker is willing to pay for a monitor located at x to know the state at any other point y.
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