GaS_GeoT: A computer program for an effective use of newly improved gas geothermometers in predicting reliable geothermal reservoir temperatures

A geochemometric study based on a multi-criteria decision analysis was applied, for the first time, for the optimal evaluation and selection of artificial neural networks, and the prediction of geothermal reservoir temperatures. Eight new gas geothermometers (GasG1 to GasG8) were derived from this study. For an effective and practical application of these geothermometers, a new computer program GaS_GeoT was developed. The prediction efficiency of the new geothermometers was compared with temperature estimates inferred from twenty-five existing geothermometers using gas-phase compositions of fluids from liquid- (LIQDR) and vapour-dominated (VAPDR) reservoirs. After applying evaluation statistical metrics (DIFF%, RMSE, MAE, MAPE, and the Theil's U test) to the temperature estimates obtained by using all the geothermometers, the following inferences were accomplished: (1) the new eight gas geothermometers (GasG1 to GasG8) provided reliable and systematic temperature estimates with performance wise occupying the first eight positions for LIQDR; (2) the GasG3 and GasG1 geothermometers exhibited consistency as the best predictor models by occupying the first two positions over all the geothermometers for VAPDR; (3) the GasG3 geothermometer exhibited a wider applicability, and a better prediction efficiency over all geothermometers in terms of a large number of samples used (up to 96% and 85% for LIQDR and VAPDR, respectively), and showed the smallest differences between predicted and measured temperatures in VAPDR and LIQDR; and lastly (4) for the VAPDR, the existing geothermometers ND84c, A98c, and ND98b sometimes showed a better prediction than some of the new gas geothermometers, except for GasG3 and GasG1. These results indicate that the new gas geothermometers may have the potential to become one of the most preferred tools for the estimation of the reservoir temperatures in geothermal systems.


Introduction
Geothermal energy has emerged as a clean alternative source of renewable energy for electric power generation and other direct uses (Wu and Li 2020). The enormous amount of energy stored in geothermal systems makes them an important renewable and sustainable energy source (Nieva et al. 2018;Gutiérrez-Negrín et al. 2020). Among the geothermal systems today exploited, hydrothermal systems stand out for their storage capability of hot fluids, which are used for the generation of electricity (Thien et al. 2015). The chemical composition of liquid and steam (gas) phases of geothermal fluids provides useful information on hydrogeological processes, thermal and recharge conditions of reservoirs, and underground flow patterns (Nicholson 1993). Within these applications, the reliable estimation of reservoir temperatures is a crucial task to evaluate the energy potential of geothermal resources (Gutiérrez-Negrín 2019). To carry out this task, several chemical geothermometers have been proposed for the prediction of deep equilibrium temperatures in geothermal systems (Guo et al. 2017). Chemical geothermometers are low-cost tools used for predicting reservoir temperatures in the early exploration and exploitation stages (Yan-guang et al. 2017). Solute geothermometers are mostly recommended for the prediction of reservoir temperatures in liquid-dominated reservoirs, LIQDR (Verma et al. 2008), whereas gas geothermometers are predominantly suggested for the calculation of reservoir temperatures in vapour-dominated (VAPDR) reservoirs (García-López et al. 2014).
The gas chemistry of geothermal fluids has been applied not only for the estimation of geothermal reservoir temperatures but also to elucidate water-rock interaction processes (Pang, 2001). The direct escape from deep magmatic sources, and the occurrence of gas-gas and gas-mineral reactions may explain the consumption and production of gases in these systems (Minissale et al. 1997).
The equilibria attainment among gas species is considered as a fundamental thermodynamic basis for gas geothermometry. This physicochemical process has been widely studied for the prediction of reservoir temperatures in LIQDR and VAPDR using the composition of vapour (or gas) samples collected from wells and fumaroles. To our knowledge, Ellis (1957) was the first geochemist to point out that gases in magmatic steam might be used to estimate reservoir temperatures. The first gas geothermometer is attributed to Tonani (1973) who found some relationships among geothermometric models, and the composition of geothermal gases.
From these studies, numerous gas geothermometers have been proposed based upon the analysis of gas-phase compositions, and the thermodynamic study of gasgas and gas-mineral equilibria reactions. A comprehensive review on gas geothermometers has not been published yet in indexed journals, although some efforts have been conducted in compiling some of the geothermometers most commonly used (e.g. Henley et al. 1985;Nicholson 1993;Powell 2000;Powell and Cumming 2010). In this work, an update compilation of the most commonly used gas geothermometers is reported in Table 1. Most of these geothermometers may be roughly grouped as follows: • Simple equations calibrated with databases of gas-phase compositions of fluids collected from geothermal wells; whereas Numerical method refers to a more complex function that correlate multiple variables (i.e., gas concentration, pressure, the fraction of water, temperature, among others); Grid geothermometric method refers to a gas geothermometer that provides a complex grid-numerical algorithm for the estimation of temperatures and other key parameters of the reservoir (i.e., steam fraction, distribution coefficients, and the steam/gas ratio) • Complex grid-numerical geothermometers that calculate reservoir temperatures and other key parameters (e.g. the steam excess, the distribution coefficients of gases between liquid and steam, etc.); and • Geothermometric equations derived from fluid-rock interaction experiments.
Although these geothermometers have been proposed for the prediction of geothermal temperatures, their generalized application has been limited by the following issues: 1. The significant statistical differences among the temperature estimates predicted by a group of gas geothermometers; 2. The scarcity of the gas-phase compositions at lower temperature levels (between 90 and 150 °C) which has hindered the proposal of new improved equations; 3. The limited intervals of the gas composition and temperature for the application of the geothermometric equations; 4. The scarcity of geochemometric studies for the evaluation of the temperature estimates and their uncertainties; and 5. The lack of practical computer programs to estimate reservoir temperatures by using a wide variety of concentration units, and the complicated calculations involved in some equations.
To address the referred issues (1-3), geochemical databases with representative compositions of gas and/or steam phases from LIQDR and VAPDR are required. Considering the complex nature of the gas-mineral equilibria processes, it is also necessary to explore new regression tools based on artificial intelligence techniques for calibrating multivariate gas geothermometers, and so, to predict reservoir temperatures with a better accuracy. Among these techniques, the artificial neural networks (ANN) have been used in solving multivariate problems in Earth sciences (Poulton, 2001). For example, in geothermal studies, the use of ANN has been applied for (i) the development of Na-K geothermometers (Díaz-González et al. 2008;and Serpen et al. 2009); (ii) the prediction of mass and heat transport in geothermal wells (Bassam et al. 2010;Álvarez del Castillo et al. 2012;Porkhial et al. 2015); and (iii) the optimization of geothermal power plants (Arslan and Yetik 2011), among others. One of the latest ANN applications reported for geothermal studies was conducted by Pérez-Zárate et al. (2019) who performed a preliminary study to evaluate ANNs for the prediction of geothermal temperatures using gas-phase compositions from which the present research work has been comprehensively completed.
To evaluate the uncertainties of gas geothermometers (the issue referred as 4), a limited number of geochemometric studies have been reported (e.g. D'Amore and Panichi 1980; Arnórsson et al. 2006;García-López et al. 2014). These studies stated that the prediction efficiency of gas geothermometers is affected by several error sources, such as gas sampling errors, analytical errors, coefficient errors, and the total propagated errors, associated with the calculation of temperatures (e.g. Kacandes and Grandstaff 1989;García-López et al. 2014).
The availability of new computer programs to calculate reservoir temperatures (the issue referred as 5) still constitutes a current necessity for the geothermometric studies. Computer programs to apply solute geothermometers in estimating temperatures are widely available for the study of geothermal systems (e.g. Verma et al. 2008;Spycher et al. 2016), whereas for the gas geothermometry, these programs are rarely shared in the literature. A short listing of open-source programs commonly used in gas geothermometry is included in Table 2. To fulfil the constraints (1-5), the development of new improved gas geothermometers and computer programs are still claimed by the geothermal industry.
To address a reliable prediction of geothermal reservoir temperatures, new improved gas geothermometers and a practical computer program called GaS_GeoT have been developed in this work. GaS_GeoT was calibrated for the reliable prediction of reservoir temperatures in LIQDR and VAPDR for a temperature interval between 170 and 374 °C.
To evaluate the prediction efficiency of the new gas geothermometers in geothermal wells, an updated worldwide geochemical database containing gas-phase compositions of fluids were compiled. The prediction efficiency of these geothermometers was compared against twenty-five existing gas geothermometers (listed in Table 3). Details of this geochemometric study are also outlined in this work.

Work methodology
A comprehensive computational methodology was developed to achieve the following research objectives (Fig. 1a, b): (i) to evaluate ANNs based on a Multi-Criteria Decision Analysis (MCDA) for selecting optimal prediction models that enable new improved gas   Blamey (2006) geothermometers to be developed; (ii) to create a new computer program for the effective use of the new gas geothermometers; (iii) to evaluate the prediction efficiency of the new gas geothermometers using the gas-phase compositions from well samples collected in LIQDR and VAPDR; and (iv) to compare the temperature estimates inferred from new gas geothermometers with those temperatures predicted by some existing geothermometers. In order to address these research goals, six computational modules were structured: GasGeo_ANN: To carry out the optimal evaluation and selection of ANN architectures by using a novel application of the MCDA method; GasGeo_Eq: To describe the new gas geothermometer equations developed and their applicability conditions; GasGeo_Geochem: To perform a comprehensive geochemometric analysis among the temperature estimates (predicted from all the geothermometers) and the bottom-hole temperature measurements (BHT m ).
A summary of these computational modules is described as follows:

GasGeo_ANN
A previous evaluation study of ANNs to predict geothermal reservoir temperatures was preliminarily conducted by Pérez-Zárate et al. (2019), which constitutes the foundational basis of the present research work. 455 ANN architectures were originally trained to predict reservoir temperatures using three Worldwide Geochemical Subdatabases (WG_SubDB 1 : q 1 = 527; WG_SubDB 2 : q 2 = 498; and WG_SubDB 3 : q 3 = 97) containing gas-phase (CO 2 , H 2 S, CH 4 , and H 2 ) compositions of geothermal fluids as input variables. According to Fig. 2, a multilayer perceptron model was used for the design of these ANNs using the conventional matrix notation (Eq. 1) for the determination of the ANN output (or target): where X is the input variable, and IW and b 1 represent the matrix and the vector of coefficients between the input and hidden layers. LW and b 2 are the layer weighting coefficients and the vector of coefficients between the hidden and output layers, and f corresponds to an activation function for the hidden layer.
A full description of the multilayer perceptron model is reported by Pérez-Zárate et al. (2019), and schematically summarized in Fig. 2 by listing the fundamental equations used for adjusting weighting and bias coefficients in the learning process of each ANN layer (referred as Eqs. 1.1-1.6). Such a perceptron model was solved by creating several Matlab numerical scripts that were previously reported (https ://githu b.com/ ANNGr oup/GasG-Scrip ts-ANNs.git).
(1) (1.5) Jacobian matrix, where p is the total number of neurons; Eq. (1.6) iterative update of weights using the LM algorithm, where w t+1 is the next weight, w t is the current weight, μ is the learning rate factor, and I is the identity matrix For the learning process of ANNs, the well-known Levenberg-Marquardt algorithm (LM), the hyperbolic tangent function, and the linear function were applied. All the ANNs were characterized by an input layer, one hidden layer, and an output layer. For the input and output layers, the data were normalized between -1 and 1, whereas for the hidden layer, the number of neurons varied from 1 to 35. Input data sets were randomly divided into training (80%), validation (10%), and testing (10%). From this study, thirty-nine ANNs were preliminarily reported as the most acceptable prediction models to correlate multivariate relationships among gas-phase compositions and BHT data (referred as ANN-1 to ANN-39 in Pérez-Zárate et al. 2019).
By using traditional evaluation metrics based on small differences and acceptable correlation coefficients between measured (BHT m ) and simulated (BHT ANN ) temperatures, six ANNs were proposed as the 'better prediction models' (cited as ANN-12, ANN-13, ANN-22, ANN-25, ANN-33, and ANN-38 in the same paper). Although acceptable results were obtained from these earlier prediction models, some applicability problems were later identified in two ANNs that used a small data set for the learning process (WG_SubDB 3 : q 3 = 97, which was used for the ANN-33 and ANN-38 models). The low representativeness of this sub-data set was lastly reflected on very limited intervals of applicability.
A generalization problem was therefore identified in the final development stage of new gas geothermometers, which may affect the future application of these tools in geothermometric studies for the geothermal prospection and exploitation. To correct these problems, a new evaluation methodology based on a novel MCDA method was applied for the optimal selection of the most reliable ANNs among the thirty-nine architectures previously proposed (Fig. 1b). The MCDA method was used, for the first time in the ANN literature, to comprehensively evaluate the efficiency of these ANN prediction models by utilizing the evaluation results derived from the following: I. The analysis Case 1 (AC 1 ) described by the global learning processes of three subdatabases (WG_SubDB 1 , WG_SubDB 2 , and WG_SubDB 3 ) applying the thirty-nine ANNs (pre-selected) and the following statistical metrics: the number of data used by each sub-database (q 1 = 527; q 2 = 498 and q 3 = 97), the correlation coefficients (r) obtained between measured (BHT m ) and simulated (BHT ANN ) temperatures for the training, validation, and testing stages, and the residuals calculated between measured and predicted temperatures (RMSE, MAE, and MAPE); II. The analysis Case 2 (AC 2 ) described by the application of the thirty-nine ANNs to the largest sub-database (q 1 = 527), as a generalization case, using the same statistical residuals between measured and predicted temperatures (RMSE, MAE, and MAPE); and III. The analysis Case 3 (AC 3 ) described by the application of the thirty-nine ANNs to the same sub-database (q 1 = 527) but strongly restricted by the applicability conditions of each ANN prediction model, using the same residuals between measured and predicted temperatures.   To apply the MCDA, it was necessary to normalize the resulting thirteen statistical metrics (Table 4). The normalization was performed either to minimize or maximize the evaluation metrics using the methodology proposed by Dincer and Acar (2015).
For example, in the particular case of the residual errors (RMSE, MAE, and MAPE), the best efficiency of the ANN prediction models would be given when these metrics achieve lower values, whereas for the linear correlation coefficient (r), higher values would be desirable. For the statistical residual (SR i ), or the minimization obtained during the global learning and application processes of the ANNs, normalized data were calculated by using the RMSE, MAE, and MAPE results and the following equation: whereas for the maximization, the normalized data was determined by using the following equation: where SP i are the statistical regression parameters: n (the data number used by each database) and r, the correlation coefficients, which were obtained from the training, validation, and testing of the ANNs.
The MCDA is suggested as an optimized method for the integral evaluation of various statistical metrics among predictor models using different scenarios to achieve a specific objective function (Adem and Geneletti 2018). To apply the MCDA, the Multi-Attribute Value Theory (MAVT) algorithm was used (Santoyo-Castelazo 2011;Santoyo-Castelazo et al. 2011;Santoyo-Castelazo and Azapagic 2014;Estévez et al. 2018). This algorithm required the determination of partial value functions, and the estimation of weighting factors for each evaluation metric. The global value function V(s) which represents the total score to be obtained by any ANN architecture in each scenario s is calculated as follows: where w i and u(s) are the weighting factors used by the evaluation metric, and the value function which provides the metric efficiency for each ANN architecture and scenario, respectively; and E is the total number of the evaluation metrics. Wang et al. (2009) suggested the use of variable weighting factors by considering the importance of the scenarios assumed on the predictor model efficiency. To apply the MCDA method, Santoyo-Castelazo and Azapagic (2014) suggested the use of a sensitivity analysis to determine the ranking score of prediction models based on different scenarios that enable indicators (or statistical metrics) to be varied under certain weighting criterion. In this work, the sensitivity analysis was applied for evaluating the prediction efficiency of each ANN model in the calculation of reservoir temperatures using four different scenarios (S), where thirteen statistical metrics were comprehensively analysed by assigning the following weighting criteria: -(S-1) Equal weighting factors assigned to the statistical metrics computed for the three analysis cases already postulated: AC 1 (global learning processes); AC 2 (the generalization case); and AC 3 (the applicability restrictions imposed by each ANN model), i.e. AC 1 = AC 2 = AC 3 , Fig. 1b; see Additional file 1: Table S1. -(S-2) Greater weighting factors assigned to the seven statistical metrics computed for the AC 1 (global learning processes) in comparison with equal weighting values assumed for the metrics of the AC 2 and AC 3 , i.e. AC 1 > (AC 2 = AC 3 ), Fig. 1b; see Additional file 1: Table S2. -(S-3) Greater weighting factors assigned to the three statistical metrics computed for the AC 2 (the generalization case) in comparison with equal weighting values assumed for the metrics of the AC 1 and AC 3 , i.e. AC 2 > (AC 1 = AC 3 ), Fig. 1b; see Additional file 1: Table S3. -(S-4) Greater weighting factors assigned to the three statistical metrics computed from the AC 3 (the restricted case of the applicability conditions) in comparison with equal weighting values assumed for the metrics of the AC 1 and AC 2 , i.e. AC 3 > (AC 1 = AC 2 ), Fig. 1b; see Additional file 1: Table S4.
After applying the MCDA (by means of the algorithm represented in Fig. 1b), the best ANN architectures were optimally selected and used for the development of the new gas geothermometers. A summarized description of the ranking score obtained by each ANN together with the optimal selection results are reported in Table 5. A full report containing all MCDA calculations obtained for the thirty-nine ANNs is reported in Additional file 1: Tables S1 to S4.

GasGeo_Eq
This module was created to describe the new gas geothermometer equations inferred from the best ANNs (from here referred as GasG 1 to GasG 8 ), and their applicability conditions. Table 6 summarizes the eight ANNs that were optimally selected for the development of the new gas geothermometer equations. The number of neurons used for the three layers of the ANNs (input, hidden, and output) are included. The input variables used by each ANN are also reported, including their relative contribution (in %) obtained from the sensitivity analysis (Garson 1991). Weighting and bias coefficients of the optimal ANNs are reported in Table 7. These coefficients were used for the development of each gas geothermometer (GasG i , i = 1 to 8) by using the following general equation (Eq. 5): where X is the input variable; IW and b 1 represent the associated coefficients to the input-hidden layers of the ANN; and LW and b 2 are the coefficients for the hidden-output layers. The subscripts k, m, and n refer to input, hidden, and output neurons, respectively, whereas α represents a normalization factor equal to 214 for the GasG 1 , GasG 2 , GasG 6 , and GasG 8 , and 193 for the remaining geothermometers (GasG 3 , GasG 4 , GasG 5 , and GasG 7 ).
Considering the complexity of the Eq. (5), it is pertinent to assume that a future application of the eight geothermometric equations could be discouraged by users when compared with simple equations already proposed from the existing gas geothermometers (Table 3). To overcome this limitation, a computer program (GaS_GeoT) codified

Table 6 New improved gas geothermometers proposed in this research work
According to the Garson's method (Garson 1991    The input variables of each new gas geothermometer are reported in Table 6. The subscripts k, m and n refer to input, hidden and output neurons, respectively in Java object-oriented programming was created. This program was developed for the effective use of the new gas geothermometer equations for the calculation of geothermal temperatures from gas-phase compositions (CO 2 , H 2 S, CH 4 , and H 2 ). GaS_GeoT and a quick user manual are available in the public server (https ://githu b.com/ANNGr oup/ GaS_GeoT.git). The applicability conditions of the new gas geothermometers (GasG 1 to GasG 8 ) are reported in Table 8. These limits are the min-max values of concentration and temperature used during the ANN learning processes. The gas concentrations must be given in mmol/mol units (dry-basis), whereas the temperature in °C.
The input variables for the GasG 1 -GasG 5 and GasG 8 are given by the natural logarithm values (dimensionless) because these equations use gas ratios, whereas for the GasG 6 and GasG 7 , the input variables are directly given in mmol/mol units.

GasGeo_Lit
This module was created to carry out a comparison of prediction efficiencies among the new (GasG 1 to GasG 8 ) and existing gas geothermometers using their temperature estimates. Twenty-five existing gas geothermometers have been included in Table 3. The fundamental criteria for selecting these geothermometers were as follows: (i) the determination of temperatures as a direct function of gas concentrations; and (ii) the use of gas concentrations (CO 2 , H 2 S, CH 4 , H 2 , and N 2 ) most commonly applied in geothermometric studies.
To perform the analysis of prediction efficiency in new and existing gas geothermometers, the input data compiled in NWGDB were classified in two major groups: Group-1, LIQDR (g 1 = 178), and Group-2, VAPDR (g 2 = 87). A complete version of the NWGDB is reported in Additional file 1: Table S5.

GasGeoT_Calc
This module was created to describe the new computer program GaS_GeoT and the effective use of the new gas geothermometers (GasG 1 to GasG 8 ). The user interface and a calculation routine of geothermal reservoir temperatures by using GaS_GeoT are also described. Before running GaS_GeoT and to apply the twenty-five geothermometer equations (Table 3), the concentration units of the gas-phase compositions and the applicability conditions of each gas geothermometer were verified (Table 8).
To proceed with the calculation of the reservoir temperatures, all gas geothermometers were executed (see Fig. 1a). As one of the goals of this work was focused on the effective use of the new gas geothermometers (GasG 1 to GasG 8 ), the calculation process used by GaS_GeoT is schematically referred in five major sections of Fig. 4 and summarized as follows.
To open or upload the input data through the File option (menu 4.1 in Fig. 4), an Excel spreadsheet (with a filename extension: xlsx) is required. This file must contain eight columns: Column 1, Sample ID (numerical data: integer); Columns 2-4: Geothermal Well, Geothermal Field, and Country information, respectively (in alphanumerical format); and the Columns 5-8, to enter the gas-phase compositions of the gases CO 2 , H 2 S, CH 4 , and H 2 in mmol/mol (dry-basis), respectively (numerical data in floating format). In the Help option (menu 4.2 in Fig. 4), a template file with some input data examples is available as a useful query. A User's Manual is also included in the Help option of the program, and the public server (https ://githu b.com/ANNGr oup/GaS_GeoT.git).
Two additional menu options appear in the main screen of GaS_GeoT: Validation and Geothermometers. The Validation menu option performs an input data validation for checking the existence of typographical mistakes in the CO 2 , H 2 S, CH 4 , and H 2 compositions (menu 4.3 in Fig. 4). The Geothermometers menu option invokes a computer subroutine which will make either the total or partial selection of the gas geothermometers (GasG 1 to GasG 8 ) by checking their corresponding boxes (menu 4.4 in Fig. 4). After selecting the geothermometers, the calculation of temperatures is performed, and the GaS_GeoT will prompt the user to provide an output file name to print a report with the gas-phase compositions, and the temperature estimates calculated by the new gas geothermometers. An alternative output option is also requested by GaS_GeoT for displaying the output results on screen (option 4.5 in Fig. 4).
After applying GaS_GeoT and twenty-five existing geothermometers to the gas-phase composition of NWGDB (LIQDR g 1 = 178 and VAPDR g 2 = 87), an output file with all the temperature estimates is generated both to carry out the geochemometric evaluation, and to estimate the prediction efficiencies of the thirty-three gas geothermometers (Fig. 1a). A complete version of the output file is reported in Additional file 1: Tables S6 and S7, which contain the results obtained from the new and existing gas geothermometers, respectively.

GasGeo_Geochem
This module was developed to perform the geochemometric analyses using the temperature estimates obtained from new and existing gas geothermometers. To carry out these analyses, the following statistical metrics were applied: (1) Percent Difference, DIFF%; (2) Root Mean Square Error, RMSE; (3) Mean Absolute Error, MAE; (4) Mean Absolute Percentage Error, MAPE; and (5) the Difference Coefficient Ratio or statistical Theil's U test. As these metrics require the knowledge of actual bottom-hole temperatures, most of these were used as accuracy measures. The calculation equations used by these metrics are summarized in Table 9. RMSE, MAE, and MAPE metrics are interpreted as statistical residuals obtained between the bottomhole temperatures (BHT m ) and the temperature estimates predicted by any gas geothermometer, whereas the metrics DIFF% and Theil's U require a previous analysis. For example, when the DIFF% value is positive, the calculated temperature is greater than the BHT m of the respective geothermal well (which means an overestimation) and vice versa. Differences (DIFF%) either positive or negative ≤ 20% are assumed as acceptable estimates by considering the total propagated errors that are commonly quantified in some solute geothermometers (Verma and Santoyo 1997;García-López et al. 2014).
On the other hand, the statistical Theil's U is recommended for evaluating the prediction efficiency among several predictor models (Theil 1961). This metric was therefore used to evaluate the efficiency of the new geothermometers (GasG 1 to GasG 8 ) among other existing gas geothermometers. If the calculated values for the Theil's coefficient ratios are less than 1, it means that the errors obtained by the new geothermometers are lower than that those obtained from the existing geothermometers and vice versa (Álvarez del Castillo et al. 2012).

Results and discussion
After applying thirteen evaluation metrics and the MCDA, the thirty-nine ANNs were ranked (Table 5). Based on these optimization results, eight new gas geothermometers were successfully developed (Table 6) By considering the gas-phase compositions and the BHT m measurements compiled in the NWGDB (n w = 265), thirteen geothermal fields of the world (Berlin, Zunil, Krafla, Kamojang, Sibayak, Amiata, Larderello, Olkaria, Cerro Prieto, Las Tres Virgenes, Los Azufres, Los Humeros, and Palinpinon) were used for the geochemometric evaluation (see Additional file 1: Table S5). The geothermal reservoir temperatures estimated by applying the new gas geothermometers (GaS_GeoT) along with twenty-five existing geothermometers are reported in Additional file 1: Tables S6 and S7, respectively.
The prediction efficiency for all the gas geothermometers to determine reservoir temperatures was compared with measured BHT m values using the evaluation metrics (DIFF%, RMSE, MAE, MAPE, and Theil's U). A summary of the prediction efficiency results is reported in Table 10. A full version of this statistical comparison is also reported in Additional file 1: Tables S8 to S13.

Results for wells located in LIQDR
The reservoir temperatures estimated for the gas phase compositions from LIQDR using the new gas geothermometers (GasG 1 to GasG 8 ) and twenty-five existing gas geothermometers were statistically compared with the corresponding BHTs, and the obtained results are presented as rounded-off values in the following sections:

Analysis of the DIFF% metric
The percent difference (DIFF%) calculated from the temperature estimates by all the thirty-three gas geothermometers (new and existing) and BHTs shows better efficiencies by the newly gas geothermometers (GasG 1 to GasG 8 ) with 91 to 96% of the predicted temperatures falling within the limits of acceptance (DIFF% ± 20%; Table 10). This behaviour is clearly observed when the temperature estimates are plotted against the measured BHTs ( Fig. 5a-d). However, only four existing geothermometers from the literature (ND84c, AG85b, AG85d, and AG85f ) predicted 85 to 88% of the temperatures in the same acceptable limits (Table 10 and Fig. 6a-d). About 55-63% of the estimated reservoir temperatures are overestimated (with DIFF% ≤ 20%), 30-34% of the temperatures are underestimated (with DIFF% ≤ 20%), and 5-12% are equal (with DIFF% ≤ 1%), by the eight new gas geothermometers (GasG 1 to GasG 8 ), when compared to the BHTs, respectively (Table 10).  (Table 8). Dotted and dashed lines represent a hypothetical variability of the ± 10% and ± 20% with respect to the BHT mean value, respectively Table 9 Statistical metrics used for evaluating the prediction efficiency of gas geothermometers (new and existing) BHT m(i) is the bottom-hole temperature measured in a geothermal well; BHT CALC(i) is the temperature calculated by any gas geothermometer; BHT GaS_GeoT(i) is the temperature calculated by the new gas geothermometers (GasG 1 to GasG 8 ); BHT GasGeo_ Lit(i) is the temperature calculated by the existing gas geothermometers (Table 3), and n is the total number of gas samples

Statistical metric Calculation equation Equation number
Reference  (2010) Difference coefficient (Theil's U)  Arnórsson and Gunnlaugsson (1985) reported reservoir temperatures in some geothermal fields where the gas geothermometers may yield both under-and overestimates for reservoir temperatures. Powell (2000) suggested that temperature underestimates may be due to the addition of un-equilibrated biogenic methane, whereas Barragán et al. (2000) stated that an overestimation of temperatures may derive from an excess of gases in total discharge of the wells.

Analysis of RMSE, MAE, and MAPE metrics
The RMSE values of the geothermometers vary between 32 and 136 (see Additional file 1: Table S9). The lowest values of RMSE ranging from 32 to 35 were obtained for the eight new gas geothermometers: GasG 1 to GasG 8 (Table 10). This systematic behaviour of RMSE replicates the better efficiency provided by the new gas geothermometers in comparison to those existing geothermometers, in predicting reservoir temperatures comparable to BHT m , which is clearly observed in Fig. 7a.
Moreover, all new gas geothermometers were characterized by the lowest values of MAE (ranging from 25 to 27), which also demonstrated that their temperature estimates have a better accuracy (Fig. 7b). With respect to the MAPE metric, small temperature differences between the temperature estimates and BHTs were also provided by the new gas geothermometers, which varied from 9 to 10 (Table 10). As MAPE represents an average value of absolute percentage errors, the lowest values calculated by the new gas geothermometers suggest a small better efficiency in predicting the reservoir temperatures, which is observed in Fig. 7c.

Analysis of the statistical Theil's U test (LIQDR)
After analysing the Theil's U results, the following inferences were accomplished: (1) the GasG 3 geothermometer may be considered with confidence as the best predictor model among all other geothermometers because the Theil´s U values were systematically lower than 1; (2) it was also observed that the new geothermometers (GasG 1 to GasG 8 ) systematically provided the lowest errors in comparison with twenty-five existing geothermometers (Table 10) gas geothermometers under evaluation using the statistical evaluation metrics (DIFF%, RMSE, MAE, MAPE, and Theil´s U) is reported in Additional file 1: Tables S8 to S10.

Results for wells located in VAPDR
The prediction efficiency of the reservoir temperatures calculated in VAPDR samples using the thirty-three gas geothermometers (new and existing) was also compared with the actual BHT measurements.

Analysis of the DIFF% metric
Based on the percent difference (DIFF%) values calculated by the new gas geothermometers, a good prediction efficiency was observed with 78 to 85% of the predicted temperatures falling within the limits of acceptance (DIFF% ± 20%), as it is observed in Fig. 5e-h. Out of the total twenty-five gas geothermometers, four geothermometers (ND84b, ND84c, AG85f, and A98c) from the literature have shown statistical differences comparable to these new geothermometers with 79 to 85% (Table 10, and Fig. 6e-h). About 44-59% of the estimated reservoir temperatures are overestimated (with DIFF% ≤ 20%), 34-49% of the temperatures are underestimated (with DIFF% ≤ 20%), and 5-9% are equal (with DIFF% ≤ 1%), by the eight new gas geothermometers (GasG 1 to GasG 8 ), when compared to the BHTs, respectively (Table 10). According to these results, the new ranking between 1st and 12th positions for the gas geothermometers under evaluation was given by A98c, GasG 3 , GasG 5 , GasG 1 , ND84c, GasG 2 , GasG 4 , GasG 8 , ND84b, GasG 6 , AG85f, and GasG 7 , respectively. In this ranking, the prediction efficiency was actually very close between the geothermometers A98c and GasG 3 , differing in only two decimals of the applicability percentage (i.e. 85.1% for A98c, and 84.9% for GasG 3 ; both percentages rounded-off as 85%). With these results, the gas geothermometer GasG 3 systematically shows a high prediction efficiency similar to those results obtained for LIQDR systems.

Analysis of the RMSE, MAE, and MAPE metrics
The RMSE values calculated in VAPDR samples for all geothermometers showed a wider variability between 36 and 181 in comparison with those values estimated for LIQDR samples (see Additional file 1: Table S12). For the new gas geothermometers (GasG 1 to GasG 1 ), seven out of the eight (except GasG 4 ) predict low values of RMSE ranging from 36 to 45 (Table 10 and Fig. 7d), whereas for the existing gas geothermometers, a wider interval of RMSE was calculated (from 39 to 181).
In relation to the MAE metric, seven out of the eight new geothermometers were characterized by the lowest values ranging from 28 to 34, which also demonstrated that the temperature estimates predicted by the seven new geothermometers (except GasG 4 ) are more accurate, as it is shown in Fig. 7e. Regarding the differences calculated in the MAPE values of the new gas geothermometers varied from 11 to 15 (Table 10 and Fig. 7f ); which also suggest that seven out of eight geothermometers (except GasG 4 ) provided a slight better efficiency.
Based on the integrated calculations obtained for all the metrics (RMSE, MAE, and MAPE), it was found that seven out of the eight new gas geothermometers show a better prediction efficiency to determine the reservoir temperatures in VAPDR gas samples.

Analysis of the statistical Theil's U test (VAPDR)
After applying the Theil's U results (Table 10), two interpretations are inferred: (1) it was systematically observed that the GasG 3 predictor model (Theil's U values lower than 1) is the most reliable and accurate gas geothermometer over the rest of the new geothermometers (GasG 1 -GasG 2 and GasG 4 -GasG 8 ); and (2) it was found that two (GasG 1 and GasG 3 ) out of the eight new geothermometers exhibited lower errors in estimating reservoir temperatures in comparison with those estimates predicted by the existing geothermometers.
A full description of the results obtained from the use of the thirty-three gas geothermometers in VAPDR systems is reported in Additional file 1: Tables S11 to S13.
As a final remark of this research work, it was demonstrated, for the first time, the effectiveness of the MCDA optimization method for ranking and selecting the most reliable ANN architectures to predict a dependent variable (BHT m ) as a function of multiple independent variables (gas-phase compositions: CO 2 , H 2 S, CH 4 , and H 2 ). We consider that this new evaluation proposal is actually innovating the evaluation methods commonly used in ANNs for evaluating their training, validation, and test stages, which typically rely on the simple use of the linear correlation coefficients (r) obtained between measured and simulated data.
With the optimal selection of the best predictor models used to correlate BHT m and gas-phase compositions, eight new improved gas geothermometers (GasG 1 to GasG 8 ) were successfully developed. Most of these new geothermometers provide reliable estimations of geothermal reservoir temperatures. We have also demonstrated that the prediction efficiency of these new geothermometric tools (mainly GasG 1 and GasG 3 geothermometers) exceeds the efficiency of those existing gas geothermometers available in the literature.

Conclusions
Eight new improved gas geothermometers (GasG 1 to GasG 8 ) based on an optimized selection of artificial neural networks by using the MCDA method were successfully developed for the reliable prediction of the geothermal reservoir temperatures. For an effective and practical use of these geothermometers, a new computer program GaS_GeoT was successfully developed. The evaluation of the efficiency of the new improved gas geothermometers in predicting the reservoir temperatures was successfully demonstrated for geothermal wells of LIQDR and VAPDR systems.
The new gas geothermometers (GasG 1 to GasG 8 ) provided the best prediction efficiencies for geothermal wells from LIQDR, whereas two out the eight (GasG 1 and GasG 3 ) demonstrated their best efficiency in predicting reservoir temperatures for VAPDR. Among the new gas geothermometers, the best geothermometric tool for predicting reservoir temperatures in LIQDR and VAPDR systems is the GasG 3