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Nowadays, one of the most important effects on water resources under climate change is increasing of free water surface evaporation which depends on the increasing of temperature. In basins, where there are no observed data, free water surface evaporation is taken into account depending on historical temperature and similar data and their long-term statistics. Predicting of real value of evaporation contains some uncertainties. The modeling of evaporation with a small number of predictors has crucial importance on the regions and basins where measurements are not sufficient and/or not exist. In this presented study, daily evaporation prediction models were prepared by using empirical Penman equation, Levenberg-Marquardt algorithm based on "Feed Forward Back Propagation Artificial Neural Networks (LMANN)", radial basis neural networks (RBNN), generalized regression neural networks (GRNN). When the models were compared, it was noticed that the results of neural network models are statistically more meaningful than the Penman equation.

Climate change becomes more and more important and effects hydrologic circle and water resources day by day. Evaporation is one of the most important parameters of hydrologic circle and its prediction is getting more important because of climate change effects. Both designing the water works and planning and management of water resources systems, modeling of evaporation have significant importance. The modeling of evaporation, being one of its most prominent predictor, is of crucial importance on the regions and basins where measurements are not sufficient or are not exist.

The purpose of this study is to apply different ANN techniques and compare them with Penman-Monteith method in order to estimate daily pan evaporation and reference evapotranspiration. FAO suggests PenmanMonteith Method to estimate daily evaporation as standard and this method is widely used in the world but it requires many parameters. We investigated that if modeling of daily pan evaporation with using much less parameters than Penman-Monteith, is possible and compared the results of employed ANN techniques including Levenberg-Marquardt algorithm based Feed Forward Back Propagation neural networks (LMANN), radial basis neural networks (RBNN), generalized regression neural networks (GRNN).

There are many studies about modeling of evaporation and reference evapotranspiration. Reference [

Reference [

Evaporation, which changes related with energy, mass and energy transfer mechanism, water depth and surface area, is basic component of hydrological circle and the function of radiation, temperature, wind speed, relative humidity and atmospheric pressure [

In this study, the availability of ANN models, which is developed as an alternative technique against to empirical methods such as Penman, were examined. Penman needs complex calculations and too many parameters belongs to study area, but ANN can obtain more significant results with less parameters than empirical equations such as Penman. ANN is an artificial technique which developed by using properties of neural system such as definition, prediction and reproduce of knowledge [

Daily mean temperature (T_{mean}) and relative humidity (RH_{mean}), wind speed at 2 m level (W_{mean}), maximum and minimum temperatures (T_{max}, and T_{min}) and total solar radiation (Rad) have been used as input-data for Penman-Monteith equation and ANN-models have been set by using these inputs. In order to examine the generalization-abilities of ANN-models, the models have been set by using 70% of data for training and tested by remaining data-set. At this stage, MATLAB codes have been used written by the authors. In order to increase the performance of the models and the speed of their convergence during the training period, all data have been scaled at the range of 0 - 1 and given to the networks. The scaled data have been converted to their original values by inverse operation after the completion of training process. By the training of the models, parameters related to algorithms and network structures have been determined so that the mean square errors (MSE) have to converge to a minimum while coefficient of determination (R^{2}) to a maximum.

By the training of Levenberg-Marquardt ANN (LMANN), Levenberg-Marquardt Standard Error (LMSE) value has been obtained by using cell number of hidden layer is 5, the first value of Marquardt parameter (μ_{0}) is 0.001 and decay rate (β) is 0.1. Furthermore, ten iterations have been found satisfactorily because of to prevent deterioration of test period performances as a result of extreme learning of LMANN model. Sigmoid function was selected as activation function in LMANN model (Equation (1)) by the training of,

The results of LMANN model were evaluated by using scatter diagrams and in terms of statistical parameters. Scatter diagrams of testing and training periods of LMANN model are presented in

In training period of RTANN, cell number in hidden layer is selected as 5 and σ is determined so as to find minimum mean square error. Accordingly, optimum σ value is found as 0.85. In this model, Gauss function is selected as activation function (Equation (2)).

In this equation, x refers to inputs, c_{k} to centre and σ to dispersion parameter, respectively. Scatter diagrams of testing and training period of RTANN model are presented in

In training period of GRNN, dispersion parameter σ was determined so as to find minimum mean square error. So, “σ” is found as 0.14. In GRNN model, Gauss function is also selected as activation function. Scatter diagrams of testing and training periods of GRNN model are presented in

For comparison of the ANN results with Penman’s predictions, Penman estimates have been also evaluated under two periods of 70% and 30% of the data (

In this stage of the study, the dimension of significant variable has been tried to minimize by using stepwise regression method that uses training data. In this method, T_{min} is automatically discarded from the models and focused on 5 alternative models relevant the results of stepwise regression (

According to the ^{2} and standard error are considerably less by the models with 4- and 5-inputs following the structure of the 3^{th} model containing T_{mean}, W_{mean}, R_{ad}, R_{mean}, T_{max}. For this purpose, ANN models have been set again by using 3-inputs approach. LMANN with 3-inputs converged in 6 iteration with one cell used on hidden layer and then minimum mean SE has been obtained. In LMANN model, Marquardt parameter (μ_{0}) and decay rate (β) have been selected as 0.1. Scatter diagrams of testing and training periods of LMANN model are presented in

In training period of RBANN with 3-number of inputs, cell number is selected as 9 and dispersion parameter σ is determined so as to find minimum mean square error. So optimum σ value was found then as 0.85. Scatter diagrams of model are presented in

In the training period of GRNN model with triple inputs, dispersion parameter (σ) has been determined such that the mean square error might be minimum. Thus, the optimum dispersion parameter has been found as 0.10. Scatter diagrams of the model are presented in

The applications and performances of different type artificial neural network-methods that model natural events of multivariable and nonlinear features present a great importance. In this study, the constructed models have been prepared through the inputs of Penman equation and 3 inputs obtained by the application of stepwise regression. Statistical comparisons evaluated in two phases, namely training and testing, are given in Tables 1 and 2; scattering diagrams in ^{2}-value during the testing-period has been found successfully on RTANN model.

The performance of LMANN model is more reliable and prevails over the other types of models if compared

of mean square errors and basic statistics. Because of higher correlation with each other of certain parameters used in Penman equation (i.e., T_{max} and T_{mean}), the number of inputs have been diminished with stepwise regression and, error performances of 3-inputs models during testing period has been found more effective. This situa

tion is worth considering the modeling studies in the regions and basins of no or insufficient daily evaporation data.

Models may be utilized mainly in winter seasons by the completion studies of missing pan evaporation data. The completed evaporation data by filling process may be also beneficial for the inputs of daily parametric models of water budget and also usable on stream basins as well as on reservoir basins of neighborhood dams.

On the farseeing directions under climate change, in the regions of either non-existed or insufficiently observed pan evaporation data depending on the increase of temperature and decrease of precipitation has also great importance. So, the computation of pan evaporation data will be then realized at the scenario-based critics.