# A Multi-Model Approach Using Statistical Index and Information Criteria to Evaluate the Adequacy of the Model Geometry in a Fissured Carbonate Aquifer (Italy)

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

- -
- Calcare massiccio formation (Hettangian-Pliensbachian p.p.): stratified limestone in tough blocks without noticeable vertical heterogeneities, approximately 700 m thick.
- -
- Bugarone group (Pliensbachian p.p.-Titonian p.p.): comprised of limestones and marly limestones with frankly greenish intercalations of marls in the marly calcareous facies; it outcrops only locally in substitution of the three previous formations, with an overall thickness of 30–40 meters.
- -
- Corniola formation (Sinemurian p.p.-Toarcian p.p.): finely stratified micritic limestones, with brown or light grey chert and with grey-green pelitic intercalations that are rather abundant both in the upper part and at the base; the thickness varies from 150 to 400 meters.
- -
- Bosso formation (Toarcian p.p.-Bajocian p.p.): comprised of a marly and marly calcareous unit with medium-thick strata (Marne del Serrone and Rosso Ammonitico); a third unit is made of limestones and marly limestones (Calcari a Posidonia). The overall thickness varies from 40 to 120 meters.
- -
- Calcari Diasprini formation (Bajocian p.p.-Titonian p.p.): limestones and cherty limestones with thick stratification; thickness can reach 100 meters.
- -
- Maiolica formation (Titonian p.p.-Aptian p.p.): whitish micritic limestones, in medium strata, with thin pelitic intercalations that increase towards the top of the formation. The thickness ranges from 150 and 400 meters.
- -
- Marne a Fucoidi formation (Aptian p.p.-Albian p.p.): the lower portion is made up of multicolored, closely layered marls and, subordinately, limestones that towards the top become cherty with frequent marly levels; 80–100 meters thick.
- -
- Scaglia rosata formation (Albian p.p.-Bartonian p.p.): limestones and marly limestones with polychrome chert, in medium and thin layers that vary in color from pink to red, (200–400 m);
- -
- Scaglia cinerea formation (Bartonian p.p.-Aquitanian p.p.): marly limestones, calcareous marls and clayey marls in thin layers (150–200 m).
- -
- Bisciaro formation (Aquitanian p.p.-Burdigalian p.p.): limestones, cherty limestones, marly limestones and calcareous and clayey marls, in medium and, sometimes, thick layers; the thickness ranges from 20 and 60 meters.

- -
- the Tennacola high group of springs; and
- -
- the Tennacola low group of springs.

^{3}/s. In this case, the Scaglia aquifer likely forms a unique hydrogeological complex with the Maiolica aquifer, which is stratigraphically reversed here (Figure 1b). The Marne a Fucoidi aquiclude, which is usually interposed between the two aquifers, is partially elided or absent due to the compressive tectonics and the presence of the thrust plane. Moreover, preliminary water budgets led us to additionally hypothesize a minor contribution (approximately 10%) from the Basal aquifer on the western side of the area (Figure 1b).

#### 2.2. Water Budget

^{3}/s that, however, were suddenly absorbed by the large thicknesses of the coarse materials (pebbles and blocks) that form the riverbed. Slightly more significant discharges (approximately 0.006 m

^{3}/s) were recorded at the lowest station, which was located at an altitude of 600 m a.s.l. These conditions certainly persisted until the first half of October when, during the periodical surveys, no significant changes were observed.

^{3}/s and then decreased significantly (0.287 m

^{3}/s) at the river station at 675 m a.s.l. (due to the large amounts of coarse materials in the river bed). After that, the flow increased again at the last station (altitude 600 m a.s.l.), where the fluvial discharge exceeded 0.450 m

^{3}/s. A similar trend was observed during the January 2014 measurement campaign.

^{3}, with a recharge of approximately 55,515,801 m

^{3}and an annual average flow of approximately 5,046,891 m

^{3}/year (Table 3).

#### 2.3. Conceptual Model

- -
- a western side, where the Maiolica and Calcare Massiccio aquifers are separated by an aquiclude comprised of the Bugarone, Calcari a Posidonia and Calcari Diasprigni formations: based on the water budget discussed in the previous section, a partial connection between the two aquifers (with a very limited contribution from the basal aquifer) was considered;
- -
- an eastern side, where the thrust plane and the underlying Scaglia cinerea low permeability formation, constitute an impermeable barrier;
- -
- a northern side, where the Tennacola low group of springs is located, and with no contribution from the Tennacola stream; and
- -
- a southern side, where inflow was excluded because of the presence of a structural underground water divide.

^{2}to be considered (Figure 1a); therefore, in relation to the water budget and annual discharge at the springs, the effective infiltration was determined to range from 70% to 80% of the mean yearly recharge.

#### 2.4. Model Design

- -
- Sector 1, which is characterized by two overlapping aquifer systems: the Maiolica aquifer (sMai) on the top and the Basal aquifer (sBas) below, which are separated by low permeability formations;
- -
- Sector 2, which corresponds to the Maiolica aquifer (sMai);
- -
- Sector 3, which is characterized by the Marne a Fucoidi aquiclude (sFuc);
- -
- Sector 4, which is characterized by the Scaglia aquifer (sScr); and
- -
- Sector 5, which corresponds to the valley floor of the Tennacola stream thick fluvial deposits (sTen).

- -
- two zone model, M2Z (sBas, sMai-sFuc-sScr-sTen);
- -
- three zone model, M3Z (sBas, sMai-sFuc-sScr, sTen);
- -
- four zone model, M4Z (sBas, sMai-sScr, sFuc, sTen); and
- -
- five zone model, M5Z (sBas, sMai, sFuc, sScr, sTen).

^{2}and was discretized into a mesh of 2085 elements and 2290 nodes. Because stratigraphic and water table data were missing in the study area, the initial piezometric surface was reconstructed using interpretative hydrogeological cross-sections, bibliographic data and the results of fluvial discharge monitoring carried out in the mountainous portion of the Tennacola stream.

#### 2.5. Hydrogeological Properties

#### 2.6. Boundary Conditions

#### 2.7. Model Calibration

_{i}is the weight of the ith-observation; ${\mathrm{c}}_{\mathrm{i}}$ is the ith-observation; ${\mathrm{c}}_{\mathrm{i}}^{\text{'}}$ is the simulated value corresponding to ${\mathrm{c}}_{\mathrm{i}}$; and ${\mathrm{r}}_{\mathrm{i}}$ is the residual value.

_{r}

^{2}[64]:

^{t}QJ is the normal matrix.

_{j}of the AIC, which is also generally used to indicate differences in the other parameters, BIC and KIC [37]. This value is then used together with the Akaike weight:

_{j}is the difference of the AIC of a specific model to the smallest AIC from all of the considered models.

_{j}indicates the best model among all of the evaluated models [37]. The difference value is inversely proportional to the weight w

_{j}, and for low values of the weight, the probability that the considered model is the best is null.

_{i}is the weight associated with the ith-observation.

_{j}(related to observation j) is instead calculated as follows [63]:

^{t}J is the Hessian matrix; j is the number of observations; and n is the number of adjustable parameters.

## 3. Results

#### 3.1. Uncertainty Analysis

#### 3.2. Sensitivity Analysis

## 4. Discussion

^{3}/s, which were too high with respect to the observed discharges at the springs. The error is comparable among the models but slightly increases with the number of zones (and consequently with the number of adjustable parameters); this is due to the problem of equifinality, in which increasing the parameter space can increase the number of optimum solutions available and reducing the performance of all “optimums” [67]. As a consequence, the model with two zones and a low number of adjustable parameters remained the most suitable among the models investigated in relation to the available data. The analysis of the AIC, BIC and KIC parameters confirmed these results.

_{j}, which was 100%, confirmed the goodness of the model with two zones among the models tested; on the contrary, the other models had values close to zero (Table 7). Engelhardt et al. [34] carried out an analysis of the significance of the BIC, KIC and AIC parameters. The first two parameters indicated that the best model had the lowest number of adjustable parameters, whereas AIC indicated that the best model had a higher number of parameters; the Akaike weight definitively confirmed the result expressed by the AIC parameter. However, because AIC ignores the potential error related to the geometry of the model, the authors stated that the choosing the KIC parameter could be the best solution, which confirmed the result achieved by [40]. Foglia et al. [30] conducted a multi-model analysis using AIC and BIC supported by a cross-validation, and they concluded that both information criteria selected the same optimal model with only three adjusted parameters.

^{3}/year on average (Table 6).

^{3}over the entire period of investigation, but the mean annual discharge calculated by means of the spring hydrograph analysis and water budget, on the other hand, indicated a value of 55,515,801 m

^{3}. The results confirmed that infiltration in the study area plays a very important role because it feeds approximately 80% of the total discharges to the springs, and, as a consequence, approximately 20% should come from a neighboring aquifer.

## 5. Conclusions

^{3}/s; therefore, it seems probably high compared to the discharges at the springs. In relation to the four models evaluated, the information criteria in parallel with the Akaike weight identified the model with two zones, i.e., the model with the smallest number of adjustable parameters, to be the most suitable. Using that model, a total discharge for the whole period 2004–2014 of approximately 44,021,000 m

^{3}was determined, compared with the average total observed at the springs of approximately 54,688,330 m

^{3}, which implied a contribution from infiltration of approximately 80% and a minor contribution of approximately 20% that was probably derived from the nearby basal aquifer on the western side of the area.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AIC | Akaike information criterion. |

AICc | Akaike information criterion. |

AIC w_{j} | Akaike weight. |

BC | Boundary condition. |

BIC | Bayesian information criterion. |

C.I.I.T. | Tennacola Water Consortium. |

C.I.P. | Potential Infiltration Coefficient. |

FAO | Food and Agriculture Organization of the United Nations. |

FEFLOW | Finite element groundwater flow and transport modeling tool. |

KIC | Kashyap’s information criterion. |

R | Correlation coefficient. |

PEST | Model-Independent Parameter Estimation and Uncertainty Analysis. |

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**Figure 1.**(

**a**) Hydrogeological sketch of the study area; and (

**b**) hydrogeological conceptual model for the two springs on study.

**Figure 3.**The pluviometric and termometric regime for the studied period at the Pintura di Bolognola station.

**Figure 5.**Tennacola low group of springs hydrograph compared to the rainfall measured to Pintura di Bolognola station.

**Figure 9.**Simulated versus observed spring hydrograph: (

**a**) model with two zones; (

**b**) model with three zones; (

**c**) model with four zones; and (

**d**) model with five zones.

**Figure 10.**Simulated versus observed spring hydrograph for the years 2006, 2008, 2010 and 2012: (

**a**) model with two zones; (

**b**) model with three zones; (

**c**) model with four zones; and (

**d**) model with five zones.

**Figure 11.**Regression analysis: (

**a**) model with two zones; (

**b**) model with three zones; (

**c**) model with four zones; and (

**d**) model with five zones.

**Figure 12.**Information criterion and sum of squared weighted residuals assessment of the calibrated models.

**Table 1.**Evapotranspiration assessment using different methods and next effective precipitation evaluation. P, rainfall; T, temperature; E

_{tr}, real evapotranspiration; E

_{tp}, potential evapotranspiration; E

_{t0}, reference evapotranspiration; P

_{eff}, effective rainfall.

Hydrologic Year | P (mm) | T (°C) | Turc | Thornthwaite | Hargreaves-Samani | Hargreaves-Samani Corrected | Penman-Monteith | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

E_{tr} (mm) | P_{eff} (mm) | E_{tp} (mm) | P_{eff} (mm) | ET_{0} (mm) | P_{eff} (mm) | ET_{0} (mm) | P_{eff} (mm) | ET_{0,PM} (mm) | P_{eff} (mm) | |||

2004 | 2027.80 | 7.91 | 507.55 | 1520.25 | 552.55 | 1475.25 | 411.53 | 1616.27 | 379.66 | 1648.14 | 556.48 | 1471.32 |

2005 | 1839.00 | 7.82 | 501.71 | 1337.29 | 519.13 | 1319.87 | 379.69 | 1459.31 | 348.50 | 1490.50 | 522.46 | 1316.54 |

2006 | 1401.40 | 7.85 | 490.86 | 910.54 | 521.31 | 880.09 | 400.34 | 1001.06 | 368.90 | 1032.50 | 588.04 | 813.36 |

2007 | 823.80 | 9.80 | 489.18 | 334.62 | 610.29 | 213.51 | 516.76 | 307.04 | 476.75 | 347.05 | 769.34 | 54.46 |

2008 | 1411.20 | 8.30 | 504.34 | 906.86 | 527.35 | 883.85 | 424.12 | 987.08 | 392.98 | 1018.22 | 633.10 | 778.1 |

2009 | 1814.40 | 8.26 | 514.93 | 1299.47 | 540.55 | 1273.45 | 369.99 | 1444.41 | 341.48 | 1472.92 | 567.63 | 1246.77 |

2010 | 1314.00 | 7.77 | 484.96 | 829.44 | 496.12 | 818.28 | 407.39 | 907.01 | 375.90 | 938.50 | 595.47 | 718.93 |

2011 | 1266.60 | 8.18 | 494.10 | 772.50 | 522.15 | 743.65 | 464.16 | 802.44 | 428.42 | 838.18 | 718.42 | 548.18 |

2012 | 1342.40 | 8.62 | 510.63 | 831.77 | 573.85 | 770.75 | 523.60 | 818.80 | 483.35 | 859.05 | 752.39 | 590.01 |

2013 | 1331.60 | 8.13 | 495.93 | 835.67 | 519.31 | 812.09 | 353.48 | 978.12 | 326.27 | 1005.33 | 567.77 | 763.83 |

2014 | 2058.00 | 8.74 | 534.84 | 1523.16 | 533.80 | 1523.40 | 368.47 | 1689.53 | 339.84 | 1718.16 | 588.36 | 1469.64 |

Mean value | 1511.87 | 8.31 | 502.64 | 1009.23 | 537.86 | 974.02 | 419.96 | 1091.92 | 387.46 | 1124.41 | 623.59 | 888.29 |

**Table 2.**Effective infiltration assessment using different methods for the period between 2004 and 2014.

Hydrologic Year | Parameter | Turc | Thornthwaite | Hargreaves-Samani | Hargreaves-Samani Corrected | Penman-Monteith |
---|---|---|---|---|---|---|

2004 | P (mm) | 2027.8 | 2027.8 | 2027.8 | 2027.8 | 2027.8 |

E_{t} (mm) | 507.6 | 552.6 | 411.5 | 379.7 | 556.5 | |

P_{eff} (mm) | 1520.3 | 1475.3 | 1616.3 | 1648.1 | 1471.3 | |

R (mm) | 76.0 | 73.8 | 80.8 | 82.4 | 73.6 | |

I_{eff} (mm) | 1444.2 | 1401.5 | 1535.5 | 1565.7 | 1397.8 | |

2005 | P (mm) | 1839.0 | 1839.0 | 1839.0 | 1839.0 | 1839.0 |

E_{t} (mm) | 501.7 | 519.1 | 379.7 | 348.5 | 522.5 | |

P_{eff} (mm) | 1337.3 | 1319.9 | 1459.3 | 1490.5 | 1316.5 | |

R (mm) | 66.9 | 66.0 | 73.0 | 74.5 | 65.8 | |

I_{eff} (mm) | 1270.4 | 1253.9 | 1386.4 | 1416.0 | 1250.7 | |

2006 | P (mm) | 1401.4 | 1401.4 | 1401.4 | 1401.4 | 1401.4 |

E_{t} (mm) | 490.9 | 521.3 | 400.3 | 368.9 | 588.0 | |

P_{eff} (mm) | 910.5 | 880.1 | 1001.1 | 1032.5 | 813.4 | |

R (mm) | 45.5 | 44.0 | 50.0 | 51.6 | 40.7 | |

I_{eff} (mm) | 865.0 | 836.1 | 951.0 | 980.9 | 772.7 | |

2007 | P (mm) | 823.8 | 823.8 | 823.8 | 823.8 | 823.8 |

E_{t} (mm) | 489.2 | 610.3 | 516.8 | 476.8 | 769.3 | |

P_{eff} (mm) | 334.6 | 213.5 | 307.0 | 347.1 | 54.5 | |

R (mm) | 16.7 | 10.7 | 15.4 | 17.4 | 2.7 | |

I_{eff} (mm) | 317.9 | 202.8 | 291.7 | 329.7 | 51.7 | |

2008 | P (mm) | 1411.2 | 1411.2 | 1411.2 | 1411.2 | 1411.2 |

E_{t} (mm) | 504.3 | 527.4 | 424.1 | 393.0 | 633.1 | |

P_{eff} (mm) | 906.9 | 883.9 | 987.1 | 1018.2 | 778.1 | |

R (mm) | 45.3 | 44.2 | 49.4 | 50.9 | 38.9 | |

I_{eff} (mm) | 861.5 | 839.7 | 937.7 | 967.3 | 739.2 | |

2009 | P (mm) | 1814.4 | 1814.4 | 1814.4 | 1814.4 | 1814.4 |

E_{t} (mm) | 514.9 | 540.6 | 370.0 | 341.5 | 567.6 | |

P_{eff} (mm) | 1299.5 | 1273.9 | 1444.4 | 1472.9 | 1246.8 | |

R (mm) | 65.0 | 63.7 | 72.2 | 73.7 | 62.3 | |

I_{eff} (mm) | 1234.5 | 1210.2 | 1372.2 | 1399.3 | 1184.4 | |

2010 | P (mm) | 1314.4 | 1314.4 | 1314.4 | 1314.4 | 1314.4 |

E_{t} (mm) | 485.0 | 496.1 | 407.4 | 375.9 | 595.5 | |

P_{eff} (mm) | 829.4 | 818.3 | 907.0 | 938.5 | 718.9 | |

R (mm) | 41.5 | 40.9 | 45.4 | 47.0 | 36.0 | |

I_{eff} (mm) | 788.0 | 777.4 | 861.7 | 892.0 | 683.0 | |

2011 | P (mm) | 1266.6 | 1266.6 | 1266.6 | 1266.6 | 1266.6 |

E_{t} (mm) | 494.1 | 522.2 | 464.2 | 428.4 | 718.4 | |

P_{eff} (mm) | 772.5 | 744.5 | 802.4 | 838.2 | 548.2 | |

R (mm) | 38.6 | 37.2 | 40.1 | 41.9 | 27.4 | |

I_{eff} (mm) | 733.9 | 707.2 | 762.3 | 796.3 | 520.8 | |

2012 | P (mm) | 1342.4 | 1342.4 | 1342.4 | 1342.4 | 1342.4 |

E_{t} (mm) | 510.6 | 573.9 | 523.6 | 483.4 | 752.4 | |

P_{eff} (mm) | 831.8 | 768.6 | 818.8 | 859.1 | 590.0 | |

R (mm) | 41.6 | 38.4 | 40.9 | 43.0 | 29.5 | |

I_{eff} (mm) | 790.2 | 730.1 | 777.9 | 816.1 | 560.5 | |

2013 | P (mm) | 1331.6 | 1331.6 | 1331.6 | 1331.6 | 1331.6 |

E_{t} (mm) | 495.9 | 519.3 | 353.5 | 326.3 | 567.8 | |

P_{eff} (mm) | 835.7 | 812.3 | 978.1 | 1005.3 | 763.8 | |

R (mm) | 41.8 | 40.6 | 48.9 | 50.3 | 38.2 | |

I_{eff} (mm) | 793.9 | 771.7 | 929.2 | 955.1 | 725.6 | |

2014 | P (mm) | 2058.0 | 2058.0 | 2058.0 | 2058.0 | 2058.0 |

E_{t} (mm) | 534.9 | 533.8 | 368.5 | 339.9 | 588.4 | |

P_{eff} (mm) | 1523.2 | 1524.2 | 1689.5 | 1718.2 | 1469.6 | |

R (mm) | 76.2 | 76.2 | 84.5 | 85.9 | 73.5 | |

I_{eff} (mm) | 1447.0 | 1448.0 | 1605.1 | 1632.3 | 1396.2 | |

Average | P (mm) | 1511.9 | 1511.9 | 1511.9 | 1511.9 | 1511.9 |

E_{t} (mm) | 502.6 | 537.9 | 420.0 | 387.5 | 623.6 | |

P_{eff} (mm) | 1009.2 | 974.0 | 1091.9 | 1124.4 | 888.3 | |

R (mm) | 50.5 | 48.7 | 54.6 | 56.2 | 44.4 | |

I_{eff} (mm) | 958.8 | 925.3 | 1037.3 | 1068.2 | 843.9 |

**Table 3.**Tennacola low group of springs budget analysis. Q

_{0}, discharge at the beginning of the period measurement; Q

_{t}, discharge at the time t; V

_{0}, dynamic reserve at the beginning of the period; V

_{t}, dynamic reserve at the end of the period; ΔV, dynamic reserve change during the water year; Q

_{ya}, groundwater discharge volume during the water year; R, groundwater recharge.

Hydrologic Year | Initial Discharges | Dynamic Reserve | Dynamic Reserve Change | Discharge | Recharge | ||
---|---|---|---|---|---|---|---|

Q_{0} (m^{3}/s) | Q_{t} (m^{3}/s) | V_{0} (m^{3}) | V_{t} (m^{3}) | ΔV (m^{3}) | Q_{ya} (m^{3}) | R (m^{3}) | |

2004 | 0.081405 | 0.167231 | 1,004,770.53 | 2,064,108.34 | 1,059,337.8 | 5,224,351 | 6,283,689 |

2005 | 0.163678 | 0.099915 | 2,020,254.57 | 1,233,234.84 | −787,019.72 | 5,832,717 | 5,045,698 |

2006 | 0.094687 | 0.095703 | 1,168,703.87 | 1,181,246.24 | 12,542.367 | 5,484,057 | 5,496,599 |

2007 | 0.078592 | 0.063809 | 970,045.879 | 787,591 | −182,454.88 | 2,407,626 | 2,225,171 |

2008 | 0.073567 | 0.087314 | 908,021.541 | 1,077,701.27 | 169,679.73 | 4,746,155 | 4,915,835 |

2009 | 0.073831 | 0.115963 | 911,288.695 | 1,431,319.19 | 520,030.49 | 5,599,006 | 6,119,037 |

2010 | 0.117108 | 0.097739 | 1,445,448.3 | 1,206,378.76 | −239,069.54 | 5,530,212 | 5,291,143 |

2011 | 0.097566 | 0.096214 | 1,204,238.86 | 1,187,551.71 | −16,687.148 | 5,501,548 | 5,484,861 |

2012 | 0.079799 | 0.144573 | 984,948.743 | 1,784,445.76 | 799,497.02 | 3,794,060 | 4,593,557 |

2013 | 0.153852 | 0.095275 | 1,898,976.22 | 1,175,969.27 | −723,006.95 | 5,600,413 | 4,877,406 |

2014 | 0.086282 | 0.10367 | 1,064,962.68 | 1,279,584.59 | 214,621.92 | 4,968,184 | 5,182,806 |

Total value | 1.100366 | 1.167406 | 13,581,659.9 | 14,409,131 | 827,471.09 | 54,688,330 | 55,515,801 |

Average value | 0.092189 | 0.096703 | 1,137,882 | 1,193,595.12 | 55,713.56 | 4,971,666 | 5,046,891 |

**Table 4.**The range of hydrogeological parameters for the five zones used in the analysis. K

_{xx,yy,zz}= hydraulic conductivity for the different space orientations; S

_{y}= specific yield; TR IN/OUT = transfer rate input/output; minimum (min), reference and maximum (max) values of hydrogeological parameters.

Model | Zone | Sector ID | K_{xx} (m/s) | K_{yy} (m/s) | K_{zz} (m/s) | S_{y} (-) | TR IN/OUT (1/day) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Min | Ref | Max | Min | Ref | Max | Min | Ref | Max | Min | Ref | Max | Min | Ref | Max | |||

M2Z | 1 | sBas | 1e^{−08} | 1e^{−06} | 1e^{−05} | 1e^{−08} | 1e^{−06} | 1e^{−05} | 1e^{−09} | 1e^{−07} | 1e^{−06} | 0.008 | 0.01 | 0.05 | 1e^{−01} | 8.64e^{−04} | 1e^{+10} |

2 | sSca,sMai,sFuc,sTen | 1e^{−04} | 5e^{−04} | 1e^{−03} | 1e^{−04} | 5e^{−04} | 1e^{−03} | 1e^{−05} | 5e^{−05} | 1e^{−04} | 0.002 | 0.005 | 0.01 | ||||

M3Z | 1 | sBas | 1e^{−08} | 1e^{−05} | 1e^{−03} | 1e^{−08} | 1e^{−05} | 1e^{−03} | 1e^{−08} | 1e^{−05} | 1e^{−03} | 0.001 | 0.005 | 0.1 | |||

2 | sSca, sMai,sFuc | 1e^{−09} | 1e^{−07} | 1e^{−05} | 1e^{−09} | 1e^{−07} | 1e^{−05} | 1e^{−09} | 1e^{−07} | 1e^{−05} | 0.005 | 0.05 | 0.5 | ||||

3 | sTen | 1e^{−03} | 1e^{−02} | 1e^{−01} | 1e^{−03} | 1e^{−02} | 1e^{−01} | 1e^{−03} | 1e^{−02} | 1e^{−01} | 0.05 | 0.1 | 0.5 | ||||

M4Z | 1 | sBas | 1e^{−09} | 3.5e^{−07} | 1e^{−05} | 1e^{−09} | 5.8e^{−08} | 1e^{−05} | 1e^{−09} | 1.6e^{−07} | 1e^{−05} | 0.005 | 0.009 | 0.5 | |||

2 | sSca-SMai | 1e^{−08} | 4e^{−05} | 1e^{−03} | 1e^{−08} | 1.6e^{−05} | 1e^{−03} | 1e^{−08} | 2.5e^{−05} | 1e^{−03} | 0.001 | 0.005 | 0.5 | ||||

3 | sTen | 1e^{−03} | 4.5e^{−03} | 1e^{−01} | 1e^{−03} | 1.7e^{−02} | 1e^{−01} | 1e^{−03} | 1.2e^{−02} | 1e^{−01} | 0.05 | 0.3 | 0.5 | ||||

4 | sFuc | 1e^{−08} | 1e^{−07} | 1e^{−05} | 1e^{−08} | 1e^{−07} | 1e^{−05} | 1e^{−08} | 1e^{−07} | 1e^{−05} | 0.001 | 0.002 | 0.1 | ||||

M5Z | 1 | sBas | 1e^{−09} | 3.5e^{−07} | 1e^{−05} | 1e^{−09} | 5.8e^{−08} | 1e^{−05} | 1e^{−09} | 1.6e^{−07} | 1e^{−05} | 0.005 | 0.009 | 0.5 | |||

2 | sSca | 1e^{−06} | 1e^{−05} | 1e^{−03} | 1e^{−06} | 1e^{−05} | 1e^{−03} | 1e^{−06} | 1e^{−05} | 1e^{−03} | 0.01 | 0.05 | 0.5 | ||||

3 | sTen | 1e^{−03} | 4.5e^{−03} | 1e^{−01} | 1e^{−03} | 1.7e^{−02} | 1e^{−01} | 1e^{−03} | 1.2e^{−02} | 1e^{−01} | 0.05 | 0.3 | 0.5 | ||||

4 | sFuc | 1e^{−08} | 1e^{−07} | 1e^{−05} | 1e^{−08} | 1e^{−07} | 1e^{−05} | 1e^{−08} | 1e^{−07} | 1e^{−05} | 0.001 | 0.005 | 0.1 | ||||

5 | sMai | 1e^{−06} | 1e^{−04} | 1e^{−03} | 1e^{−06} | 1e^{−04} | 1e^{−03} | 1e^{−06} | 1e^{−04} | 1e^{−03} | 0.01 | 0.05 | 0.5 |

**Table 5.**Statistical index derived from PEST analysis. ME

_{w}, Average value of the weighted residuals; MAE

_{w}, Absolute average value of the weighted residuals; σ

_{r}

^{2}, variance of weight residuals; σ

_{w}, standard deviation; Φ (SSWR), sum of squared of the weighted residuals; R, correlation index.

Model | ME_{w} | MAE_{w} | σ_{r}^{2} | σ | Φ | R |
---|---|---|---|---|---|---|

M2Z | −0.300 | 0.390 | 0.14 | 0.37 | 133 | 0.73 |

M3Z | −0.371 | 0.509 | 0.40 | 0.60 | 225 | 0.72 |

M4Z | −0.556 | 0.764 | 0.59 | 0.80 | 517 | 0.73 |

M5Z | −0.550 | 0.740 | 0.55 | 0.74 | 491 | 0.72 |

**Table 6.**Difference between simulated and observed total discharge for the single year considered (ΔQ).

Model | 2006 | 2008 | 2010 | 2012 |
---|---|---|---|---|

ΔQ (m^{3}) | ΔQ (m^{3}) | ΔQ (m^{3}) | ΔQ (m^{3}) | |

M2Z | 1,119,329 | 1,428,274 | 1,489,013 | 1,343,655 |

M3Z | 1,035,046 | 1,455,953 | 1,534,658 | 1,403,962 |

M4Z | 1,054,714 | 1,284,587 | 1,538,168 | 1,258,872 |

M5Z | 978,568 | 1,353,970 | 1,313,571 | 1,239,845 |

Average | 1,046,914 | 1,380,696 | 1,468,853 | 1,311,584 |

**Table 7.**Difference of the Akaike information criterion (AIC), Bayesian information criterion (BIC), and Kashyap’s information criterion (KIC) values and likelihood of the flow models from the Akaike Weights (AIC w

_{i}).

Model | AIC | KIC | BIC | AIC w_{i} |
---|---|---|---|---|

M2Z | 0 | 0 | 0 | 1.00 |

M3Z | 310.85 | 328.26 | 357.71 | 0.00 |

M4Z | 797.66 | 710.75 | 817.64 | 0.00 |

M5Z | 775.75 | 827.97 | 706.83 | 0.00 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Giacopetti, M.; Crestaz, E.; Materazzi, M.; Pambianchi, G.; Posavec, K.
A Multi-Model Approach Using Statistical Index and Information Criteria to Evaluate the Adequacy of the Model Geometry in a Fissured Carbonate Aquifer (Italy). *Water* **2016**, *8*, 271.
https://doi.org/10.3390/w8070271

**AMA Style**

Giacopetti M, Crestaz E, Materazzi M, Pambianchi G, Posavec K.
A Multi-Model Approach Using Statistical Index and Information Criteria to Evaluate the Adequacy of the Model Geometry in a Fissured Carbonate Aquifer (Italy). *Water*. 2016; 8(7):271.
https://doi.org/10.3390/w8070271

**Chicago/Turabian Style**

Giacopetti, Marco, Ezio Crestaz, Marco Materazzi, Gilberto Pambianchi, and Kristijan Posavec.
2016. "A Multi-Model Approach Using Statistical Index and Information Criteria to Evaluate the Adequacy of the Model Geometry in a Fissured Carbonate Aquifer (Italy)" *Water* 8, no. 7: 271.
https://doi.org/10.3390/w8070271