Evaporation estimation using artificial neural networks and adaptive neuro-fuzzy inference system techniques

Advances in Water Resources 32 (2009) 88–97

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Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres

Evaporation estimation using artificial neural networks and adaptive neuro-fuzzy inference system techniques A. Moghaddamnia a,*, M. Ghafari Gousheh b, J. Piri c, S. Amin d, D. Han e a

Department of Watershed and Range Management, Faculty of Natural Resources, University of Zabol, Iran Department of Range and Watershed Management, Faculty of Natural Resources, University of Zabol, Iran c Management of Agriculture, Zabol, Iran d Department of Water Engineering, Faculty of Agriculture, University of Shiraz, Iran e Department of Civil Engineering, Faculty of Engineering, University of Bristol, UK b

a r t i c l e

i n f o

Article history: Received 13 June 2008 Received in revised form 4 October 2008 Accepted 12 October 2008 Available online 22 October 2008 Keywords: Evaporation Artificial neural networks Adaptive neuro-fuzzy inference system Gamma test Input data selection

a b s t r a c t Evaporation, as a major component of the hydrologic cycle, plays a key role in water resources development and management in arid and semi-arid climatic regions. Although there are empirical formulas available, their performances are not all satisfactory due to the complicated nature of the evaporation process and the data availability. This paper explores evaporation estimation methods based on artificial neural networks (ANN) and adaptive neuro-fuzzy inference system (ANFIS) techniques. It has been found that ANN and ANFIS techniques have much better performances than the empirical formulas (for the test data set, ANN R2 = 0.97, ANFIS R2 = 0.92 and Marciano R2 = 0.54). Between ANN and ANFIS, ANN model is slightly better albeit the difference is small. Although ANN and ANFIS techniques seem to be powerful, their data input selection process is quite complicated. In this research, the Gamma test (GT) has been used to tackle the problem of the best input data combination and how many data points should be used in the model calibration. More studies are needed to gain wider experience about this data selection tool and how it could be used in assessing the validation data. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Evaporation takes place whenever there is a vapour pressure deficit between a water surface and the overlying atmosphere and sufficient energy is available. The most common and important factors affecting evaporation are solar radiation, temperature, relative humidity, vapour pressure deficit, atmospheric pressure, and wind. Evaporation losses should be considered in the design of various water resources and irrigation systems. In areas with little rainfall, evaporation losses can represent a significant part of the water budget for a lake or reservoir, and may contribute significantly to the lowering of the water surface elevation [1]. Therefore, accurate estimation of evaporation loss from the water body is of primary importance for monitoring and allocation of water resources, at farm scales as well as at regional scales. A large number of experimental formulae exist for evaporation estimation. There are direct and indirect methods available for estimating potential evaporation from free water surfaces. The direct method is an evaporation pan (e.g., the US Weather Bureau Class A pan measurement, which is 4 ft in diameter and 10 in. deep and is mounted on a timber grill about 6 in. above the soil * Corresponding author. Tel.: +98 915 1441381; fax: +98 542 2232600. E-mail address: [email protected] (A. Moghaddamnia). 0309-1708/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2008.10.005

surface [40,41]). The indirect methods, in increasing order of complexity and data requirements, include temperature-based formulas [3]; radiation-based approximations [4]; humiditybased formulas [5]; combination formulas, which include allowance for humidity and wind speed [6]; or even more intensive evaluations of an energy balance at the evaporation surface [7]. These and similar methods have been used and compared for evaporation estimation by many researchers [8–11,7,12]. Mosner and Aulenbach [17] compared four empirical methods of evaporation estimation including Priestly–Taylor, Penman, DeBruin– Keijman and Papadakis equations for Lake Seminole, south-western Georgia and north western Florida, for April 2000–September2001. It has been found that average monthly lake evaporation estimates derived from the empirical equations were as much as 16% in error. In recent years, other methods have been explored by many researchers, such as the mass transfer method [18–22], and eddy correlation techniques [23,24]. Despite the large amount of literature published, most of the reported methods are too demanding for observed meteorological data and prone to errors if local parameters are not available. However, most of the formulas are difficult to use in the real world due to the data availability. In this study, only three empirical methods suitable for the investigation site are used as benchmarks for checking the data based artificial intelligence models.

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Because evaporation is an incidental, nonlinear, complex, and unsteady process, it is difficult to derive an accurate formula to represent all the physical processes involved. As a result, there is a new trend in using data mining techniques such as artificial neural networks to estimate evaporation. Some typical studies reported so far are ANN in modelling daily soil evaporation [25], daily evapotranspiration [26], daily pan evaporation [2,27–31], and hourly pan evaporation [32]. Most hydrological processes are highly nonlinear, time varying and spatially distributed. ANN models have the ability to learn the underlying relationship between inputs and outputs of a process from historical data without the physical rules being explicitly attached [33]. In this paper, two data mining methods, adaptive neural-based fuzzy inference system (ANFIS) and artificial neural network (ANN), are proposed for estimating evaporation measured by an evaporation pan. In comparison with wider applications of data mining techniques in other fields (such as flood forecasting [34–39, 59, 61,62]), the modelling experience of these methods in evaporation estimation is still quite limited and there is an urgent need to study and report the trial of them in different climate regions so that some generalization of this method could be achieved. One major problem with all the data mining methods is the selection of input data for the model. For example, there are several potential inputs that could be used in the model in this study: wind speed, temperature, relative humidity and daily saturation vapour pressure deficit. It has been very tedious to trial different input combinations in the modelling process. Moreover, there is no guideline about how many data points are sufficient for model calibration (i.e., how many year records are needed to adequately calibrate an evaporation model). In this study, a new approach call the Gamma test is introduced to find the best input combination before a data mining model is calibrated, and it also informs the modeller how many training data points are required. This can greatly reduce the work involved the model development. The objectives of this study are: (1) to develop both ANN and ANFIS models with the aid of the Gamma test; (2) to evaluate ANN and ANFIS models for evaporation estimation in a hot and dry climate; and (3) to compare the results obtained from ANN and ANFIS models with the empirical methods. 2. Materials and methods 2.1. Artificial neural networks ANN was first introduced as a mathematical aid by McCulloch and Pitts [42]. They were inspired by the neural structure of the brain [43,44]. Fig. 1 is a general architecture of a Feed Forward ANN, with one hidden layer. Most ANNs have three layers or more: an input layer, which is used to present data to the network; an output layer, which is used to produce an appropriate response(s) to the given input; and one or more intermediate layers, which are used to act as a collection of feature detectors [27]. The ability of a neural network to process information is obtained through a learning process, which is the adaptation of link weights so that the network can produce an approximate output(s). In general, the learning process of an ANN will reward a correct response of the system to an input by increasing the strength of the current matrix of nodal weights [27]. There are several features in ANN that distinguish it from the empirical models. First, neural networks have flexible nonlinear function mapping capability which can approximate any continuous measurable function with arbitrarily desired accuracy [14,15], whereas most of the commonly used empirical models do not have this property. Second, being non-parametric and data-driven, neural networks impose few prior assumptions on the underlying process from which data are generated. Because of these properties, neural networks are less susceptible to model

89

Fig. 1. Architecture of a multi layer feed forward neural network (Wikipedia ‘Artificial neural network’, 2008).

misspecification than most parametric nonlinear methods [16]. Given the advantages of neural networks, it is not surprising that this methodology has attracted overwhelming attention in many application areas. There are a wide variety of algorithms available for training a network and adjusting its weights. In this study, an adaptive technique embedded in Matlab called ‘momentum Levenberg–Marquardt’ based on the ‘generalized delta rule’ was adopted [45]. Since there are large number of resources available on ANN (books, papers and web sites), no further introduction is provide here and readers are encouraged to explore the papers list above (also [64] and Wikipedia ‘Artificial neural network’). 2.2. Adaptive neural-based fuzzy inference system (ANFIS) Although ANN is quite powerful for modelling various real world problems, it also has its shortcomings. If the input data are less accurate or ambiguous, ANN would be struggling to handle them and a fuzzy system such as ANFIS might be a better option. Jang first proposed the ANFIS method and applied its principles successfully to many problems [13,48]. It identifies a set of parameters through a hybrid learning rule combining the back-propagation gradient descent error digestion and a least-squares method. It can be used as a basis for constructing a set of fuzzy ‘‘If–Then” rules with appropriate membership functions in order to generate the preliminary stipulated input–output pairs. Fundamentally, ANFIS is a network representation of Sugeno-type fuzzy systems, endowed by neural learning capabilities. The network is comprised of nodes and with specific functions, or duties, collected in layers with specific functions [47]. The Gaussian membership function is adopted in this study since it is the most popular form. An ANFIS toolbox from Matlab is used and its operation is explained in its user guide. For its theoretical background, interested readers are referred to Chang and Chang for further details [46]. 2.3. Selection of model inputs using Gamma test The Gamma test estimates the minimum mean square error (MSE) that can be achieved when modelling the unseen data using any continuous nonlinear models. The Gamma test was firstly reported by Koncˇar [49] and Agalbjörn et al. [50], and later enhanced and discussed in detail by many researchers [51–56]. Only a brief introduction on the Gamma test is given here and the interested readers should consult the aforementioned papers for further details. The basic idea is quite distinct from the earlier

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attempts with nonlinear analysis. Suppose we have a set of data observations of the form:

fðxi ; yi Þ; 1 6 i 6 Mg

ð1Þ m

where the input vectors xi 2 R are m dimensional vectors (with a record length of M) confined to some closed bounded set C 2 Rm and, without loss of generality, the corresponding outputs yi 2 R are scalars. The vectors x contain predicatively useful factors influencing the output y. The only assumption made is that the underlying relationship of the system is of the following form:

y ¼ f ðx1    xm Þ þ r

ð2Þ

where f is a smooth function and r is a random variable that represents noise. Without loss of generality it can be assumed that the mean of the r’s distribution is zero (since any constant bias can be subsumed into the unknown function f) and that the variance of the noise Var(r) is bounded. The domain of a possible model is now restricted to the class of smooth functions which have bounded first partial derivatives. The Gamma statistic C is an estimate of the model’s output variance that cannot be accounted for by a smooth data model. The Gamma test is based on N[i, k], which are the kth ð1 6 k 6 pÞ nearest neighbours xN½i;k ð1 6 k 6 pÞ for each vector xi ð1 6 i 6 MÞ. Specifically, the Gamma test is derived from the Delta function of the input vectors: M 1X dM ðkÞ ¼ jxNði;kÞ  xi j2 M i¼1

ð1 6 k 6 pÞ

ð3Þ

where |  | denotes Euclidean distance, and the corresponding Gamma function of the output values:

cM ðkÞ ¼

M 1 X jy  yi j2 2M i¼1 Nði;kÞ

ð1 6 k 6 pÞ

ð4Þ

where yN(i,k) is the corresponding y-value for the kth nearest neighbour of xi in Eq. (3). In order to compute C a least-squares regression line is constructed for the p points (dM(k), cM(k)) :

c ¼ Ad þ C

ð5Þ

The intercept on the vertical axis ðd ¼ 0Þ is the C value, as can be shown:

cM ðkÞ ! VarðrÞ in probability as dM ðkÞ ! 0

ð6Þ

Calculating the regression line gradient can also provide helpful information on the complexity of the system under investigation. A formal mathematical justification of the method can be found in Evans and Jones [57]. The graphical output of this regression line (Eq. (5)) provides very useful information (to be presented later on). First, it is remarkable that the vertical intercept C of the y (or Gamma) axis offers an estimate of the best MSE achievable utilising a modelling technique for unknown smooth functions of continuous variables [57]. Second, the gradient offers an indication of model’s complexity (a steeper gradient indicates a model of greater complexity). The Gamma test is a non-parametric method and the results apply regardless of the particular techniques used to subsequently build a model of f. We can standardise the result by considering another term Vratio, which returns a scale invariant noise estimate between zero and one. The Vratio is defined as

V ratio ¼

C

r2 ðyÞ

ð7Þ

where r2(y) is the variance of output y, which allows a judgment to be formed independent of the output range as to how well the output can be modelled by a smooth function. A Vratio close to zero indicates that there is a high degree of predictability of the given output

y. If the standard error (SE) value is close to zero we have more confidence in the value of the Gamma statistic as an estimate for the noise variance on the given output. Gradient is actually a rough measure of the complexity of the smooth function that we are seeking to construct. We can also determine the reliability of C statistic by running a series of the Gamma test for increasing M, to establish the size of data set required to produce a stable asymptote. This is known as M-test. M-test result would help us to avoid the wasteful attempts of fitting the model beyond the stage where the MSE on the training data is smaller than Var(r), which may lead to overfitting. The M-test also helps us to decide how many data points are sufficient for building a model with a mean squared error approximate to the estimated noise variance (when the M-test plot becomes flat). In practice, the Gamma test can be achieved through winGammaTM software implementation [54]. Corcoran et al. [58] applied the Gamma test as a method for crime incident forecasting by focusing upon geographical areas of concern that transcend traditional policing boundaries. The authors believed this technique was very effective and could be potentially used for water management including flood prediction and other hydrological nonlinear modelling. 2.4. Study area and data set Chahnimeh reservoirs are located in the Sistan region that is one of the arid regions generally characterized by water scarcity and low per capita water allocation. The Sistan region is located in the Southeast of Iran, one of the driest regions of Iran and famous for it’s ‘‘120 day wind” (ba¯d-e sad-o-bist-roz), a highly persistent dust storm in the summer which blows from north to south with velocities of nearly 10 m/s. Hirmand River, originated from Afghanistan, is bifurcated into two branches when it reaches the Iranian border, namely Parian and Sistan. Sistan is the only water supply known in the Sistan va Baluchistan province. It is the main stream of Hirmand River, which flows through Sistan plain and discharges into the natural swamp of Hamun-e-Hirmand (Fig. 3). As can be seen in the figure, Sistan plain is essentially an inland delta with its major watercourses leading to a series of lakes. The Sistan delta has a very hot and dry climate. In summer, the temperature exceeds 50 °C. Rainfall is about 60 mm/year and occurs only in autumn and winter. The open water evaporation is very high and is estimated at 3200 mm/year. Strong winds in the region are quite unique and are an important contributing factor for the high evaporation. The Chahnimeh reservoirs are a series of natural depressions used primarily to store water for irrigation. However, they also play an important part in attenuating floods. During periods of high flows, water is diverted to these reservoirs via an intake and canal which has a capacity of up to 1000 m3/s. For better control over the distribution of the water reaching the Sistan irrigated plain, the Chahnimeh reservoir was constructed on the Iranian side immediately downstream from the Hirmand fork, where the Helmand River separates into the Sistan and the Common Parian rivers. These reservoirs included three parts (Chahnimeh) and have been constructed for public water supply with a fourth reservoir under preparation. The present capacity of the Chahnimeh reservoirs is sufficient to guarantee a reliable supply for both the Sistan area as well as the agreed upon delivery for Zahedan, one of most important cities in the southeast of Iran. The use of water for irrigated agriculture in Sistan is mainly restricted by the variability of the supply and not by the total supply; the use of Chahnimeh reservoirs for irrigation will improve the performance of irrigated agriculture. The daily weather variables of automated weather station namely, Chahnime Station of Zabol (latitude 30°450 longitude 61°400 ) operated by the IR Sistan va Balochastan Regional Water

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(IR SBRW) are used in this study. The measured daily meteorological data for the Chahnime station were obtained from the IR SBRW (http://www.sbrw.ir). The data sample consisted of 11 years (1983–2005) of daily records of air temperature (T, °C), wind speed (W, km/day), saturation vapour pressure deficit (Ed, kPa), relative humidity (RH, %) and pan evaporation (E, mm). The basic statistics for the weather station is listed in Table 1 and the time series plots of all the weather variables are presented in Fig. 2 (with one year data for illustration). 3. Results and discussion 3.1. Results obtained from the Gamma test Traditionally, a modeller needs to use trial and error to build mathematical models (such as ANN) for different input combinations. This is very time consuming since the modeller needs to calibrate and test different models with all the likely input combinations. In addition, there is no guidance about how many data points should be used in the calibration. The Gamma test

can greatly reduce the model development workload and provide input data guidance before a model is developed (i.e., its result is independent of the models to be developed). Basically, the Gamma test is able to provide the best mean square error that can possibly be achieved using any nonlinear smooth models. In this study, different combinations of input data were explored to assess their influence on the evaporation estimation modelling. There were 2n  1 meaningful combinations of inputs; from which, the best one can be determined by observing the Gamma value, which indicates a measure of the best estimation attainable using any modelling methods for unseen smooth functions of continuous variables. Table 2 presented some of the combinations tested. In the table, the minimum value of C was observed when we used (W, RH, Ed) data set; i.e., daily wind speed (W), relative humidity (RH) and daily saturation vapour pressure deficit (Ed). This Gamma value is the only criterion for selecting the best input combination in this study and other three factors (Gradient, Standard error of C and V-ratio) are not quantitatively used. This is because the Gamma test is still a new tool in model input selection and there are still many knowledge gaps in this technique. We are aware that

Table 1 Basic statistics for Chahnime weather station. Data set

Unit

Xmean

Sx

Cv (Sx/Xmean)

Csx

Xmin

Xmax

T W Ed RH E

°C km/day kPa % mm

23.2 262.5 2.93 41.05 12.94

9.69 185.97 1.41 16.00 9.19

0.417 0.708 0.481 0.389 0.710

0.20 0.77 0.50 0.48 0.63

0 5 0.08 10.33 0

41 915 6.53 91.90 39.44

Fig. 2. Time series plots for the weather variables.

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Fig. 3. Location of the Sistan Region and the Chahnimeh reservoirs.

Table 2 The Gamma test results on the evaporation estimation data sets. Parameters

Different combinations

2

Gamma (C) (mm ) Gradient (A) Standard error of C (mm2) V-Ratio Near neighbours M Mask

W, T, RH, Ed

T, RH, Ed

W, RH, Ed

W, T, Ed

W, T, RH

0.0218 0.0595 0.00049 0.0874 10 4019 1111

0.0530 0.3065 0.00146 0.2118 10 4019 0111

0.0216 0.2738 0.00059 0.0864 10 4019 1011

0.0218 0.0950 0.00026 0.0870 10 4019 1101

0.0221 0.0950 0.00046 0.0883 10 4019 1110

the other three factors should be very important indicators since they represent how good the Gamma value is and how complex the model should be developed. However, there is a lack of study and experience about the quantitative utilisation of these factors conjunctively with the Gamma estimate. With more researchers to start using this tool in their model development, it is expected that the dos and dont’s with this technique will become clearer. Basically, the gradient (A) is considered as an indicator of model complexity (the higher of the gradient, the more complicated the model should be fitted). The standard error of C illustrates how reliable the Gamma value is (the smaller it is, the more reliable the Gamma value is). V-ratio is the measure of predictability of given outputs using the available inputs (it is dimensionless). If the best input data set selected by the Gamma value is with low values of the three factors (gradient, Standard error of C and V-ratio), it is possible to develop a mathematical model (e.g., ANN)) of high quality. From Table 2, the best input data combination based on the Gamma value (0.0216) has a set of reasonably low values of Gradient, Standard error of C and V-ratio. More studies are needed to further understand the significance of those three factors. It has been found that the relative importance of inputs is W > Ed > RH > T. The significance of the daily mean temperature data was relatively small when compared with other weather variables since the elimination of this input made small variation in the Gamma statistic value. It is interesting to note that the Gamma test result seems against our intuition. From Fig. 2, temperature is the most obvious

weather variable to influence evaporation since it correlates well with the output. The second variable is more likely to be Ed (saturation vapour pressure deficit). The link between the evaporation and wind is the least evident from these plots. However, the Gamma test indicates that it does not matter to drop any one of the three weather variables (T, Ed and RH), but wind is the most important variable and must be included in the model input. This is baffling at the first sight and to place the importance of wind over temperature is difficult to understand. If we explore this further, two important explanations could be elaborated. First, although the temperature seems an obvious input variable in Fig. 2, it may not be a significant variable when several other input variables are considered altogether. This is because all the information carried by temperature can be readily provided from the combination of the other two variables (i.e., temperature can be derived from Ed and RH, and vice versa). Among the three weather variables (T, Ed and RH), only two are independent and the third one is redundant in its information content. That is why the modelling performance has a minimum effect when one of the three variables is excluded as shown in Fig. 5. In addition, according to ‘Occam’s Razer’ principle, a better model could be built by excluding redundant variables. In linear systems, the redundant variables can cause the multicollinearity problem and make matrix inversion impossible to perform. On the other hand, wind is independent to other three variables and can not be derived by them. So if wind is excluded, significant amount information would be lost, as demonstrated in Fig. 5. Second, it is difficult to extend linear correlation to non-

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linear problems. Two variables with zero linear correlation might have a very strong nonlinear relationship. The relationship becomes more complicated for multivariable systems and pairwise linear correlations between output and individual input variables are not easily applicable in nonlinear data input selection. This is a very encouraging result from the Gamma test that is not only able to provide the useful information on input variable selection, but also to prompt us thinking about the implication and interrelations among these input variables. We can also determine the reliability of C statistic by running a series of the Gamma tests for increasing M (i.e., the number of input data points), to establish the size of data set required to produce a stable asymptote. This is known as M-test. The M-test results would help us to determine whether there were sufficient data to provide an asymptotic Gamma estimate and subsequently a reliable model. The M-test analysis results are shown in Fig. 4 for the best combination of 1011 model (i.e., with W, RH, Ed). The test produced an asymptotic convergence of the Gamma statistic to a value of 0.0216 at around 2413 data points (i.e., M = 2413), while the M-test was carried out up to 4019. The variation of the standard error (SE) corresponding to the data points also shows a similar pattern. We also performed the M-tests in different dimensions varying the number of inputs to the model (Fig. 5), which clearly

0.06

Gamma

0.05

0.08 0.07

0.04

0.06 0.05

0.03

0.04

0.02

0.03 0.02

0.01 0 10

SE

Standard Error (SE)

0.09 Gamma

0.01 1010

2010

3010

0 4010

Data Points (M-Test) Fig. 4. Relation between standard error and Gamma test for different training data length.

presented the response of the data models to different combinations of inputs data sets. It can be found that when wind is excluded in the model inputs, there would be very large modelling errors. If wind is included, the influence of any one of the other three variables is not significant. It should be noted although the Gamma value represents the error variance and should not be negative, the estimated Gamma values can be negative when there are insufficient data points used in the estimation (e.g., Fig. 5). This is because the Gamma values are derived from the regression of the nearest neighbour analysis and its result become unreliable for a small number of data points. 3.2. Analysis of evaporation modelling results The best input combination (with W, RH, Ed) derived from the Gamma test is used to develop ANN and ANFIS models. To assess their performance against the empirical approach, we selected three empirical models as benchmarks (Table 3). Although there are many empirical models available for open water evaporation estimation, most of them are not applicable due to the data availability problem (for example, there is no solar radiation data at the study site). A scale factor of 1/0.7 has been used to convert the larger water body formulas to the pan evaporation equivalent. The training data length has been set to 2413 and the rest is for testing. The ANN mode has been calibrated with the standard Matlab ANN toolbox and readers can refer to Matlab user manual for further details. For ANFIS, the number of membership functions for each input was set to 3 and the input data were scaled between [0, 1]. The membership function type was selected as the Gauss bell shaped. The Matlab ANFIS toolbox was used for its calibration and validation. The results in the training and testing of ANN and ANFIS are listed in Table 4, along with the empirical methods. It can be observed that ANN and ANFIS have much better performances than the empirical equations. The best result for the empirical formula (Marciano) is R2 = 0.54 for the test data set. As a contrast, the R2 for ANN is 0.97 and ANFIS 0.92, that are both significantly better than all the three empirical formulas. Between ANN and ANFIS, ANN has a slightly better performance, which indicates that fuzzy approach has not helped to improve the evaporation modelling results. Figs. 6 and 7 illustrate the scatter plots for all the models (since no training is needed for the empirical models, only the estimation results on the testing data are presented in Fig. 7).

Table 3 Empirical formulae used for evaporation estimation. Formula name

Equation

References

Hefner Lincare Marciano

E ¼ 0:028  U  ðes  ea Þ E = f(T, Tdew, Latitude) E ¼ 0:03  U  ðes  ea Þ

Alizadeh [63] Warnaka and Pochop [11] Alizadeh [63]

E: evaporation rate (mm/day), es: saturation vapour pressure (mm of Hg), ea: actual vapour pressure (mm of Hg), U: average wind velocity (km/h) at a height of 2 m above the lake or surrounding land areas, T and Tdew: air temperature and dew point.

Table 4 Comparison between ANN and ANFIS techniques for evaporation estimation. Type

Models

Training RMSE (%)

Fig. 5. Variation of Gamma statistic (C) for the data corresponding to different combinations of input data sets. (Note: three lines for ‘No RH, No Ed and All incl’ are not distinguished by line styles since they are very close to each other.)

AI

ANN ANFIS

Empirical

Hefner Lincare Marciano

Testing R2

RMSE (%)

R2

1.72 2.90

0.97 0.91

1.51 2.47

0.97 0.92

43.23 63.54 43.00

0.62 0.24 0.64

48.27 61.45 45.82

0.49 0.23 0.54

A. Moghaddamnia et al. / Advances in Water Resources 32 (2009) 88–97

Estimation (mm/day)

94

45 40 35 30 25

ANFIS Training R^2=0.91, RMSE=2.90%

20 15 10 5 0 0

5

10

15 20 25 Observation (mm/day)

30

35

40

40 ANFIS Testing R^2 =0.92, RMSE=2.47%

Estimation (mm/day)

35 30 25 20 15 10 5 0 -5 0

5

10

15 20 25 Observation (mm/day)

30

35

40

Estimation (mm/day)

40 ANN Training R^2=0.97, RMSE=1.72%

35 30 25 20 15 10 5 0 0

5

10

15

20 25 30 Observation (mm/day)

35

40

45

Estimation (mm/day)

35 ANN Testing R^2=0.97, RMSE=1.51%

30 25 20 15 10 5 0 0

5

10

15

20

25

30

35

40

Observation (mm/day) Fig. 6. Scatter plot of observed versus estimated values of evaporation for training and testing of ANFIS (a, b) and ANN (c, d) models. (Note: the bands in the observed data are due to discrete readings of the evaporation pan.)

4. Discussion This paper gives an overview of fairly recent AI techniques on the topic of evaporation estimation. Empirical formulae seem lim-

ited and therefore much research has been done to find methods that are more accurate for different localities. Since its application in evaporation estimation from 1997, artificial intelligence techniques have been studies by many researchers in different climate

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Estimation (mm/day)

90 80

Hefner on t esting dat a R^2 =0.49, RMSE=48.3%

70 60 50 40 30 20 10 0 0

5

10

15

20

25

30

35

40

Observation (mm/day)

Estimation (mm/day)

25 Linc are on t esting dat a R^2=0.23, RMSE=61.5%

20 15 10 5 0 0

5

10

15

20

25

30

35

40

Estmation (mm/day)

Observation (mm/day) 100 90 80 70 60 50 40 30 20 10 0

Marciano on t esting dat a R^2 =0.54, RMSE=45.8%

0

5

10

15

20

25

30

35

40

Observation (mm/day) Fig. 7. Scatter plot of observed versus estimated values of evaporation for the testing data set of Hefner (a), Lincare (b) and Marciano (c) models. (Note: the bands in the observed data are due to discrete readings of the evaporation pan.)

regions. The modelling experiences have been reported from the USA [25,26], India [27], Turkey [28,30,29] and Singapore [32]. This paper is the first attempt in applying ANN and ANFIS models in a Middle East region. It is confirmed that ANN and ANFIS worked well for a hot and dry place such as Iran. However, the fuzzy element introduced by ANFIS failed to outperform a simpler ANN in this study and it would be interesting to explore it in other climates to obtain a general conclusion on this model for evaporation estimation. Overall, artificial intelligence models such as ANN and ANFIS are quite robust, for a number of reasons. First of all, it is not necessary to make an analysis of a problem or of the internal structure of a system, based on the in-depth knowledge. Second, they can handle a great deal of data. Furthermore, they do not assume any probability distribution like normality or equal dispersion and covariance matrix requirements. However, they also suffer

from some major problems. First, they need the measured data for training and validation (that is why they perform much better than the empirical methods since ANN/ANFIS have been tuned to fit the local data). Second, they need a lot of time and effort to be applied for model development. This is because there are still many knowledge gaps in these models and the approach based ‘trial and error’ is currently the mainstay in this field. We hope more study on AI techniques by hydrologists in the future will help to establish some well-tested methodologies to reduce ‘trial and error’ and improve the efficiency for model development. Another interesting finding in this study is the unique combination of the contributing variables. It has been demonstrated that the important weather factors to be included in the model input are: wind, saturation vapour pressure deficit and relative humidity. The inclusion of saturation vapour pressure deficit coincides well

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with a suggestion by Anderson [60]. This result is different to all these reported in the literature. In the USA, Han and Felker [25], used three weather input variables to estimate the evaporation from the soil: relative air humidity, air temperature, wind speed. Kumar et al. [26], selected six input variables: minimum and maximum temperature, minimum and maximum relative humidity, wind speed, and solar radiation. In India with its hot and humid climate, Sudheer et al. [27], found that their model worked best with six inputs: minimum and maximum temperature, minimum and maximum relative humidity, sunshine hours and wind speed (albeit the improvement of using the minimum and maximum temperature over the mean temperature was quite small). In Turkey, there are mixed results from the published papers. Terzi and Keskin [28], investigated the evaporation at Lake Egirdir and found the important weather factors were (in order of their importance): air temperature, solar radiation and air pressure. One year later, the authors reported that only two weather input variables (air temperature and solar radiation) were good enough and the air pressure was dropped. Another paper by the authors later on [31], then reported three contributing weather factors should be considered in their model: solar radiation, air temperature, and relative humidity (wind and air pressure were ignored). The changing choice of the weather variables may indicate that their model input selection schemes were not very stable. In Singapore, Tan et al. [32], found the important variables were: solar radiation, relative humidity, air temperature and surface wind speed. There are three linked problems bothering all model developers: (1) what are the best input data combination; (2) how long record should be used for model calibration; (3) how long record should be used to validate a calibrated model. Despite decades of research in hydrological modelling, there is a lacking of guidelines on how to solve those problems. All those issues are also intertwined with noise and uncertainties in data [66,69,71,72] and model structures [67,68,70]. As a result, many modellers have been developing their models with a poor combination of input variables and with either too little or too large calibration data. Those problems are even more crucial for intensive hydrological experiments. For example, the HYREX experiment was funded by UK NERC with three weather radars, 49 tipping bucket rain gauges, an automatic weather station and an automatic soil station [65]. The experiment was very costly and ran from 1993 to 1997. A tough question facing hydrologists was how many years the experiment should run. If the period was too short, insufficient data would be collected and hydrologists would have problems to develop reliable mathematical models from the data. If the period was too long, it would be very costly and the extra data beyond the sufficiency would not be necessary. Although some further funding was provided to extend the experiment for another three years, there were no powerful tools available to guide hydrologists to choose an optimum experiment duration. The Gamma test may have a potential to help hydrologists in this aspect. It should be pointed out that the Gamma test introduced in this paper is very helpful for the aforementioned first two problems and we hope more experience from other researchers on using this tool would formalise some useful guidelines on the input data selection and the data sufficiency for model calibration. Unfortunately, the Gamma test is not designed for checking if the validation data are sufficient and clearly more research is needed for a tool to tackle this problem. 5. Conclusions Overall, the objectives of this study were to develop ANN and ANFIS models with the aid of the Gamma test, to evaluate ANN and ANFIS models for evaporation estimation in a hot and dry climate, and to compare the results obtained from ANN and ANFIS

models with the empirical methods. It is quite clear that all the planned objectives have been achieved with some interesting findings about the weather variable selections. Both ANN and ANFIS performed better than the empirical methods. It is useful to know that the more complicated approach from ANFIS did not achieve a better performance, hence the extra time and effort on developing ANFIS might not be worthwhile in the future if other researchers also find similar results from their case studies. The Gamma test has a huge potential to save a great amount of time and effort for a modeller to select the best input data combination. However, it is not a panacea (e.g., how to check the sufficiency of validation data) and more studies are needed to trial this technique in other catchments with different climates to gain wider valuable experiences. It should be pointed out that although the methodology described here is mainly used for repairing and extending the measured evaporation records, there is a potential for using the scheme to find out the important contributing variables in developing empirical formulae that could be used for ungauged catchments via a regionalisation approach. References [1] McCuen RH. Hydrologic analysis and design. Englewood Cliffs, New Jersey: Prentice Hall; 1998. [2] Kisi O. The potential of different ANN techniques in evapotranspiration modelling. Hydrol Process 2008;22:2449–60. doi:10.1002/hyp.6837. [3] Thornthwaite CW. An approach toward a rational classification of climate. Geograph Rev 1948;38:55–94. [4] Turc L. Estimation of irrigation water requirements, potential evapotranspiration: a simple climatic formula evolved up to date. Ann Agron 1961;12:13–49. [5] Romanenko VA. Computation of the autumn soil moisture using a universal relationship for a large area. In: Proceedings of the Ukrainian hydrometeorological research institute, Kiev, Ukraine; 1961. p. 3. [6] Penman HL. Natural evaporation from open water, bare soil, and grass. Proc Roy Soc London 1948;193:120–45. [7] McKenzie RS, Craig AR. Evaluation of river losses from the Orange River using hydraulic modelling. J Hydrol 2001;241(1–2):62–9. [8] Abtew W. Evaporation estimation for Lake Okeechobee in south Florida. J Irrig Drain Eng 2001;127(3):140–7. [9] Hanson CL. Prediction of Class A pan evaporation in southwest southwest Idaho. J Irrig Drain Eng 1989;115(2):166–71. [10] Knapp HV, Yu Y-S, Pogge EC. Monthly evaporation for Milford Lake in Kansas. J Irrig Drain Eng 1984;110(2):138–48. [11] Warnaka K, Pochop L. Analyses of equations for free water evaporation estimates. Water Resour Res 1988;24(7):979–84; Warnaka K, Pochop L. Neural networks for runoff prediction. J Hydrol Eng 1988;5(4):424–7. [12] Choudhury BJ. Evaluation of empirical equation for annual evaporation using field observations and results from a biophysica model. J Hydrol 1999;216 (1–2):99–110. [13] Jang J-SR. ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 1993;23(3):665–85. [14] Cybenko G. Approximations by superpositions of a sigmoidal function. Math Control Signals Syst 1989;2:303–14. [15] Hornik K, Stinnchcombe M, White H. Multi-layer feed forward networks are universal approximators. Neural Networks 1989;2:359–66. [16] Barron AR. Universal approximation bonds for superpositions of a sigmoidal function. Technical report No. 58, Department of Statistics, University of Illinois, Urbana Champaign; 1991. [17] Mosner MS, Aulenbach BT. Comparison of methods used to estimate lake evaporation for a water budget of lake Seminole, southwestern Georgia and northwestern Florida. In: Proceedings of the 2003 Georgia water resources conference, April 23–24, 2003, University of Georgia; 2003. [18] Yu YS, Knapp HV. Weekly, monthly, and annual evaporations for Elk City Lake. J Hydrol 1985;80:93–110. [19] Ikebuchi S, Seki M, Ohtoh A. Evaporation from lake Biwa. J Hydrol 1988;102:427–44; Laird NF, Kristovich DAR. Variations of sensible and latent heat fluxes from a great lakes buoy and associated synoptic weather patterns. J Hydrometeorol 2002;3(1):3–12. [20] Laird NF, Kristovich DAR. Variations of sensible and latent heat fluxes from a great lakes buoy and associated synoptic weather patterns. J Hydrometeorol 2002;3(1):3–12. [21] Hostetler SW, Bartlein PJ. Simulation of lake evaporation with application to modelling lake level variations of Harney–Malheur Lake, Oregon. Water Resour Res 1990;26(10):2603–12. [22] Blodgett TA, Lenters JD, Isacks BL. Constraints on the origin of paleo lake expansions in the Central Andes. Earth Interact 1997;1.

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