# Correction of Precipitation Records through Inverse Modeling in Watersheds of South-Central Chile

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}), most of which are located in low-altitude areas. The spatial distribution of precipitation is, therefore, not properly recorded. In this study an inverse modeling approach is used to estimate the extent to which precipitation amounts must be corrected. Using a lumped water balance model, a factor for correcting precipitation data is calculated for 41 watersheds located in south-central Chile. Then, based on a geo-statistical interpolation, a map for correcting the precipitation amounts is proposed and a validation of these corrections is achieved. The results show that in gently sloping areas, the precipitation records are more representative than in steep mountain areas. In addition, the higher the mountains, the less representative the precipitation records become.

## 1. Introduction

^{2}, and most of these stations are concentrated in the Central Valley and along the coast. This density is very low in comparison to the recommendations of the World Meteorological Organization (WMO); according to the WMO [5], to have a good precipitation gauge network in terms of density, there must be at least one precipitation gauge every 250 km

^{2}in mountainous areas and one every 575 km

^{2}in interior plains [5,6]. In addition, the presence of mountainous reliefs complicates access and impedes the installation of precipitation gauge stations; therefore, there is a much lower density in areas such as the western slope of the Andes and the highlands of the Coastal Range (e.g., Nahuelbuta Range, lat. ~37°45′ S). Thus, in a watershed with areas that are mostly mountainous, precipitation may be underestimated since the increase in precipitation due to the orography may not correctly represented.

## 2. Study Area

^{2}and includes the Coastal Range (with an average elevation of 1200 m.a.s.l.), the Andes (with altitudes greater than 4000 m.a.s.l.) and the Central Valley. This last area is a flat valley located between the two mountain ranges (see topography in Figure 1). The area has an average precipitation gauge density of 1 precipitation gauge every 675 km

^{2}, which can be subdivided into 1 precipitation gauge every 540 km

^{2}in low zones (below 400 m.a.s.l.) and 1 precipitation gauge every 960 km

^{2}in mountainous areas (above 400 m.a.s.l.). In mountainous areas, the station density is a quarter of that recommended by the World Meteorological Organization [5].

## 3. Data

## 4. Methods

#### 4.1. Conceptual Model Description

#### 4.2. Calibration and Calculation of the Precipitation Correction Factor

_{50}) of the best 10% of the models according to the identifiability analysis and three objective functions. The root mean square error (RMSE), transformed root mean square error (TRMSE) and the runoff coefficient error (ROCE) functions, designed to calibrate a model as a function of high streamflows, low streamflows and long-term water balance closure, respectively, were used [21].

#### 4.3. Objective Function Description

- (i)
- Root mean square error (RMSE) (Equation (1)): this indicator operates as a function of simple differentials of the simulated streamflows (Q
_{s,t}) with respect to the observed streamflows (Q_{o,t}) and is focused on the high flows.$$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{m}}{\sum}_{\mathrm{j}=1}^{\mathrm{m}}{\left({\mathrm{Q}}_{\mathrm{s},\mathrm{t}}-{\mathrm{Q}}_{\mathrm{o},\mathrm{t}}\right)}^{2}}$$ - (ii)
- Transformed root mean square error (TRMSE) (Equations (2) and (3)): this function is equivalent to RMSE, with the difference that the simulated and observed values are first transformed through Box–Cox transformation, which results in low flows taking on greater importance than medium or high flows.$$\mathrm{TRMSE}=\sqrt{\frac{1}{\mathrm{m}}{\sum}_{\mathrm{j}=1}^{\mathrm{m}}{\left({\mathrm{Z}}_{\mathrm{s},\mathrm{t}}-{\mathrm{Z}}_{\mathrm{o},\mathrm{t}}\right)}^{2}}$$$$\mathrm{Z}=\frac{{\left(1+\mathrm{Q}\right)}^{0.3}-1}{0.3}$$
- (iii)
- Runoff coefficient error (ROCE) (Equation (4)): this indicator captures the overall water balance, as it combines the flows into just one descriptor of hydrological characteristics—the average annual runoff coefficient—defined as ($\overline{\mathrm{Q}}/\overline{\mathrm{P}}$). The differences in absolute value between the simulated $\frac{{\overline{\mathrm{Q}}}_{\mathrm{s}}}{\overline{\mathrm{P}}}$ and observed $\frac{{\overline{\mathrm{Q}}}_{\mathrm{o}}}{\overline{\mathrm{P}}}$ descriptors are compared. This descriptor calculates the volumetric differences between simulated and observed streamflows over a modeling period, and therefore is the most suitable function for the objective of this study.$$\mathrm{ROCE}=\mathrm{abs}\left(\frac{{\overline{\mathrm{Q}}}_{\mathrm{s}}}{\overline{\mathrm{P}}}-\frac{{\overline{\mathrm{Q}}}_{\mathrm{o}}}{\overline{\mathrm{P}}}\right)$$

## 5. Results

_{50}of the best 10% of the models, of a total of 10,000 simulations) of factor A. As a complement, Table 2 shows: (i) average annual precipitation calculated for each watershed from DGA records using the inverse distance weighting method to estimate the representative precipitation of each watershed (AAP

_{P}); (ii) average annual precipitation calculated from average annual precipitation isohyets from the national water balance made by the DGA [22] (AAP

_{I}); and (iii) average annual precipitation (AAP) calculated from inverse modeling using three objective functions (AAP

_{RMSE}, AAP

_{TRMSE}and AAP

_{ROCE}).

_{P}values as a reference point for the precipitation of each watershed, the percent difference in the AAP was calculated from isohyets and inverse modeling. Thus, the greater the difference in absolute value, the less representative the recorded precipitation is of the precipitation in the watershed, while positive differences suggest an underestimation of AAP

_{P}values and negative differences an overestimation. Consolidated results are shown in box plots grouped by watershed location (Andes, Central Valley and Coastal Range) (Figure 3).

_{ROCE}(Figure 4a) and the precipitation correction factor estimated using the ROCE function (A

_{ROCE}, Figure 4b) was carried out. AAP

_{ROCE}and A

_{ROCE}values were assumed to be at the geometric center of each modeled watershed. Figure 4 leads to the same conclusions as those obtained from Figure 3. On the map it is observed that the area with the closest calculated and measured precipitation values is the Central Valley, where factor A varies around 1.

^{2}) [23]. In addition, in the Andes and Coastal Range there is a strong interaction between the entry of frontal systems and the topography, producing an increase in precipitation as land elevation increases [26,29,30]. This variability, along with the low station density (1 station every 960 km

^{2}) explains the underestimation in the calculated precipitation amounts.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Study area, with locations of precipitation gauge stations (circles) and streamflow gauging stations (rhombuses). Made by the authors.

**Figure 3.**Percent difference in the AAP calculated by isohyets and inverse modeling for each watershed. Made by the authors.

**Figure 4.**(

**a**) Estimated average annual precipitation based on the results of inverse modeling and a cokriging-z interpolation; (

**b**) Spatially distributed factor for precipitation correction using an approximation of the inverse model and a cokriging-z interpolation. Made by the authors.

**Table 1.**Description of the model parameters, adjustment factors, and conceptual-physical range for the precipitation-runoff model.

Parameter | Description | Influence over | Range |
---|---|---|---|

C_{max} | Maximum runoff coefficient when the sub-surface layer is saturated | EI | 0.05–0.70 |

P_{Lim} (mm) | Precipitation limit over which PPD exists | PPD | 0–200 |

D | Percentage of precipitation over P_{Lim} transformed into PPD | PPD | 0–100 |

H_{max} (mm) | Maximum capacity of the soil layer to retain water | C_{max} and ER | 200–500 |

PORC | Fraction of H_{max} that defines the soil water content, restricting the evaporation processes | H_{max} and ER | 0–100 |

C_{k} | Subterranean runoff coefficient | ES | 0.05–0.80 |

A | Precipitation data adjustment factor | PM | 0.85–2.00 |

B | Potential evapotranspiration data adjustment factor | PET and ER | 0.85–2.00 |

**Table 2.**Precipitation correction factors estimated by inverse modeling for each watershed, using three objective functions. In addition, the average annual precipitation amounts from National Water Directorate (DGA) precipitation gauge records (AAP

_{P}), isohyets (AAP

_{I}) and inverse modeling (AAP

_{c}) using different objective functions are shown. Made by the authors.

Area | Watershed | A [-] | AAP [mm] | ||||||
---|---|---|---|---|---|---|---|---|---|

RMSE | TRMSE | ROCE | AAP_{P} | AAP_{I} | AAP_{RMSE} | AAP_{TRMSE} | AAP_{ROCE} | ||

Andes | Chillánen Esperanza | 1.38 | 1.38 | 1.43 | 2284 | 2250 | 3142 | 3151 | 3260 |

Diguillínen San Lorenzo | 1.46 | 1.48 | 1.5 | 2123 | 2500 | 3097 | 3145 | 3194 | |

Lircay en Puente las Rastras | 1.03 | 1.08 | 1.12 | 1673 | 1783 | 1731 | 1798 | 1879 | |

Longavíen la Quiriquina | 1.6 | 1.58 | 1.61 | 1727 | 2741 | 2762 | 2725 | 2775 | |

Perquilauquénen San Manuel | 1.44 | 1.37 | 1.43 | 1918 | 2733 | 2754 | 2632 | 2742 | |

RenegadoenInvernada | 0.72 | 0.7 | 0.67 | 2047 | 2483 | 1464 | 1428 | 1367 | |

Rucúe en Camino a Antuco | 1.45 | 1.39 | 1.46 | 1833 | 4695 | 2666 | 2554 | 2679 | |

Estero UpeoenUpeo | 1.43 | 1.39 | 1.46 | 1063 | 2000 | 1516 | 1476 | 1550 | |

Claro enlosQueñes | 1.03 | 1.05 | 1.23 | 1293 | 1996 | 1333 | 1352 | 1586 | |

Colorado en Junta con Palos | 1.59 | 1.66 | 1.87 | 1407 | 2109 | 2232 | 2329 | 2626 | |

Palos en Junta con Colorado | 1.69 | 1.81 | 2.24 | 1479 | 2222 | 2494 | 2678 | 3314 | |

Ancoa Antes Túnel Canal Melado | 1.9 | 1.82 | 2.03 | 2248 | 2500 | 4269 | 4093 | 4566 | |

Ñubleen la Punilla | 1.2 | 1.21 | 1.22 | 2252 | 2753 | 2698 | 2727 | 2754 | |

Sauces Antes Junta Con Ñuble | 1.24 | 1.22 | 1.21 | 2060 | 2300 | 2562 | 2509 | 2501 | |

AllipénenMelipeuco | 1.56 | 1.59 | 1.62 | 2209 | 3818 | 3452 | 3521 | 3569 | |

Central Valley | Claro enCamarico | 0.92 | 0.93 | 1.02 | 1374 | 1544 | 1264 | 1278 | 1401 |

Claro enRauquen | 1.37 | 1.45 | 1.49 | 1089 | 1109 | 1491 | 1578 | 1622 | |

PerquilauquenenGniquen | 1.11 | 1.09 | 1.15 | 1397 | 1907 | 1551 | 1523 | 1607 | |

PerquilauquenenQuella | 1.1 | 1.08 | 1.19 | 1292 | 1550 | 1421 | 1395 | 1537 | |

CauquenesenDesembocadura | 1.04 | 1.06 | 1.1 | 763 | 743 | 794 | 809 | 839 | |

PurapelenSauzal | 0.98 | 1 | 1.04 | 703 | 800 | 689 | 703 | 731 | |

Loncomillaen Bodega | 1.46 | 1.4 | 1.59 | 859 | 1059 | 1254 | 1203 | 1366 | |

Loncomillaen las Brisas | 1.26 | 1.31 | 1.42 | 986 | 1335 | 1242 | 1292 | 1400 | |

ItataenTrilaleo | 1.07 | 1.02 | 1.02 | 1731 | 1801 | 1852 | 1766 | 1766 | |

Itataen General Cruz | 1.08 | 1.06 | 1.13 | 1522 | 1666 | 1644 | 1613 | 1720 | |

LumacoenLumaco | 1.09 | 1.07 | 1.14 | 1020 | 1211 | 1112 | 1091 | 1163 | |

QuillenenGalvarino | 1.13 | 1.22 | 1.29 | 1250 | 1216 | 1413 | 1525 | 1613 | |

CholCholenCholChol | 1.32 | 1.26 | 1.39 | 1079 | 1259 | 1424 | 1360 | 1500 | |

QuepeenQuepe | 1.5 | 1.43 | 1.5 | 1633 | 2752 | 2450 | 2335 | 2450 | |

PuyehueenQuitratue | 1 | 0.96 | 0.99 | 2034 | 2000 | 2034 | 1953 | 2012 | |

DonguilenGorbea | 1.11 | 1.07 | 1.07 | 1942 | 2109 | 2156 | 2078 | 2078 | |

Coastal Range | CauquenesenArrayan | 1.32 | 1.12 | 1.15 | 804 | 759 | 1062 | 897 | 875 |

LoancoenDesembocadura | 1.56 | 1.39 | 1.47 | 739 | 800 | 1157 | 1024 | 1176 | |

PurapelenNirivilo | 1.5 | 1.47 | 1.59 | 699 | 800 | 1051 | 1027 | 1274 | |

Andalién Camino a Penco | 1.74 | 1.7 | 1.6 | 1011 | 1397 | 1761 | 1718 | 2235 | |

ButamalalenButamalal | 1.76 | 1.77 | 2.04 | 1139 | 2310 | 2006 | 2020 | 4721 | |

CarampangueenCarampangue | 2.08 | 1.98 | 2.05 | 1169 | 2048 | 2432 | 2315 | 4199 | |

CayucupilenCayucupil | 1.93 | 1.89 | 2.2 | 1149 | 2327 | 2222 | 2166 | 5121 | |

ReputoenReputo | 1.09 | 1.08 | 1.23 | 1190 | 1500 | 1291 | 1277 | 1838 | |

Tucapelen Cañete | 1.49 | 1.49 | 1.65 | 1228 | 2375 | 1823 | 1832 | 3912 | |

Mahuidancheen Santa Ana | 1.22 | 1.17 | 1.19 | 1876 | 1825 | 2287 | 2200 | 2175 |

_{P}, AAP

_{I}: Average annual precipitation calculated from isohyets; AAP

_{RMSE}: Average annual precipitation calculated inverse modeling and RMSE objective function; AAP

_{TRMSE}: Average annual precipitation calculated inverse modeling and TRMSE objective function; AAP

_{ROCE}: Average annual precipitation calculated inverse modeling and ROCE objective function.

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## Share and Cite

**MDPI and ACS Style**

Muñoz, E.; Acuña, M.; Lucero, J.; Rojas, I.
Correction of Precipitation Records through Inverse Modeling in Watersheds of South-Central Chile. *Water* **2018**, *10*, 1092.
https://doi.org/10.3390/w10081092

**AMA Style**

Muñoz E, Acuña M, Lucero J, Rojas I.
Correction of Precipitation Records through Inverse Modeling in Watersheds of South-Central Chile. *Water*. 2018; 10(8):1092.
https://doi.org/10.3390/w10081092

**Chicago/Turabian Style**

Muñoz, Enrique, Mauricio Acuña, Juan Lucero, and Ignacio Rojas.
2018. "Correction of Precipitation Records through Inverse Modeling in Watersheds of South-Central Chile" *Water* 10, no. 8: 1092.
https://doi.org/10.3390/w10081092