#### 3.1.1. Model Structure

The Snowmelt Runoff Model + Glacier is a semi-distributed hydrological model for simulating runoff on a daily scale in watersheds where snow and glacier melt dominate the runoff generation process. It has been successfully applied in over 100 snow-affected watersheds around the world, with areas up to 900,000 km

^{2} and elevations up to 8840 m.a.s.l. [

28]. The standard version has two runoff components, i.e., snow and rain; therefore, for this study a third module was used to add the glacier component [

29]. The model has been used in other studies to simulate the hydrological response of watersheds in Chile. The effectiveness of the model has been notably verified in the Tinguiririca River at Bajo Briones, downstream of the studied sub-watershed, where a correlation coefficient of 0.88 was found between observed and simulated streamflow [

34].

The model estimates the total quantity of water produced by glacier melt and snow, along with liquid precipitation in the watershed, with a daily time step through Equation (1):

where

Q is the average daily discharge (m

^{3} s

^{−1});

M,

R and

G are average daily snowmelt, liquid precipitation, and glacier melt (cm day

^{−1}), respectively;

k is the recession coefficient (dimensionless);

n is the index of the current day;

i and

m are the indices and total number of elevation bands. The expression

$\left[\frac{10,000}{86,400}\right]$ represents a conversion factor to obtain the daily discharge units (m

^{3} s

^{−1}).

Daily runoff due to snowmelt (

M), glacier melt (

G) and precipitation (

R) in cm day

^{−1} is calculated using Equations (2)–(4):

where

S,

G and

R indices refer to the snow, glacier and rain components of runoff;

A represents the areas dominated by the different components in km

^{2};

c is the runoff coefficient in these areas;

ddf represents the degree-day factor for snow and

a_{G} for glacier ice (cm °C

^{−1} day

^{−1});

T is the number of positive degree days of air temperature (°C day) and

P is liquid precipitation (cm). The SRM+G calculates snow and glacier melt based on positive temperatures; therefore, no glacier or snowmelt occurs when air temperatures are negative.

In this study, discharge observations available at the San Andrés station for the hydrological years from April 2008 to March 2014 were used for model calibration and validation, using the even years for calibration and odd years for validation, to reduce climatic dependence in the calibration [

35]. The data from the on-glacier automatic weather stations, available from November 2012 to November 2014, were used to determine the precipitation and temperature gradients on the watershed. Daily precipitation and temperatures from the permanent Río Tinguirirca weather station were used to force the model.

#### 3.1.2. Model Parameters

In watersheds such as the one studied here, the runoff modelled by SRM+G depends on air temperature, the degree-day factors for snow and glacier melt, and the snow and glacier areas in the watershed [

29]. The degree-factor for snowmelt is calculated following the relationship established by Rango and Martinec (1996) [

36], as shown in Equation (5):

where

${\rho}_{s}$ is the snow density and

${\rho}_{w}$ is water density, both in kg m

^{−3}. The degree-day factors for snow varies during the ablation months (November to March), with a maximum value of 0.55 cm °C

^{−1} day

^{−1} derived from Equation (5) in March, with a snow density of 500 kg m

^{−3}, and a minimum value of 0.10 cm °C

^{−1} day

^{−1} in November. The minimum value and the intermediate values (for December to February) are calibrated through density in order to better adjust the representation of the model during the thaw period. The seasonality in the snow

ddf accounts for the unaccounted seasonal variation in incoming solar radiation and snow aging, the latter of which leads to increasing snow density, decreasing albedo and increased ablation [

37,

38,

39,

40]. The value of

ddf is set to zero on the coldest winter months (April to October) to account for reduced incoming solar radiation and limited or no melting in winter. Unlike snow, glacial ice does not have significant albedo variations during the year [

40]. Furthermore, this particular glacier has few debris on its surface, which are only located in the lower ablation zone); therefore, a constant degree-day factor for glacier melt (

${a}_{G}$) was used based on the study by Bravo et al. (2017) [

31] at Universidad Glacier, who used a value of 0.8 cm °C

^{−1} day

^{−1} based on values for glacier ice from Hock (2005) [

37]. Constant values have also been used in the Himalayas [

29,

41] and in other basins of central and southern Andes [

42,

43,

44].

Constant runoff coefficients were used for snow, glacier, and rain areas. The values used in this work for the snow and ice runoff coefficients were taken from Ismail and Bogacki (2018) [

29], for glacial watersheds in India, where

c_{S} = 0.8 and

c_{G} = 0.7. The runoff coefficient for rain area was adjusted as part of the model calibration. The recession coefficient

k was calculated according to Rango and Martinec (1996) [

36] using Equation (6):

where

k is the recession coefficient;

x and

y are constants that must be determined for the analyzed watershed and

Q is the observed discharge. The recession coefficient varies on a daily time step, while the values of

x and

y are established as constants. Here, one pair of values (

x,

y) was used for the ablation period (November to March) and another for the precipitation period (April to October). The

x and

y values were set as model calibration parameters and determined through an optimization method using genetic algorithms. The automatic calibration method using genetic algorithms [

45] seeks values for the calibration parameter that optimize the statistical indicators of model fit, which are presented in

Section 3.2.

Air temperature was distributed to the catchment using a mean altitudinal temperature gradient of −6.8 °C km

^{−1} calculated from the two weather stations located on the glacier (

Figure 2). Precipitation were distributed over the catchment on each elevation band with a precipitation gradient, expressed by Equation (7):

where

PP_{i} represents a percentage of precipitation measured at the Río Tinguiririca meteorological station in elevation band

i;

Z_{i} is the mean height in meters above sea level of the band

i. This precipitation gradient was obtained through the relationship between the rainfall measured in AWS1 in summer months (

Figure 2) (at 2790 m.a.s.l.) and the Río Tinguiririca meteorological station (at 1134 m.a.s.l.) for a one-year period of analysis. The obtained parameter for this study are consistent with those calculated by Ragettli et al. (2014) in the central Andes [

46]. Four elevation bands were used to distribute precipitation and to calculate snow and glacier melt over the catchment, based on the precipitation and temperature gradients. A threshold temperature of 0 °C was used to separate liquid and solid precipitation. The parameters used in the model are listed in

Table 1.

Glacierized areas were obtained from the public glacier inventory available from the Dirección General de Aguas (hereafter denoted DGA) in 2014 [

47]. According to DGA (2014), Universidad Glacier retreated by 0.03 km

^{2} year

^{−1} during de 1945–2011 period, reaching an area of 27.44 km

^{2} in April 2013. The snow cover area was determined on a daily scale using the Moderate Resolution Imaging Spectroradiometer (MODIS) Snow Cover Daily L3 Global 500-m grid satellite product (MOD10A1) [

48,

49]. This product provides the normalized difference snow index (NDSI), calculated using Equation (8):

where

MODIS_{B4,B6} are bands 4 and 6 of the sensor. Grid cells with a NDSI index larger than 0.4 were classified as snow [

50]. A filter was applied to discard the whole daily grid when cloud cover was greater than 20%, and the missing grid was interpolated using the previous and next day with available information through nearest neighbor interpolation. Finally, the area of liquid precipitation is determined with the temperature lapse rate; if precipitation occurs within an elevation band with temperature greater than zero, then the area of that band not covered by snow or ice is contributes liquid precipitation.