Study area

The study was conducted in the Río Grande de Morelia basin, Michoacán, Mexico, where the urban zone of the city of Morelia is located, from upstream of the Cointzio Dam to where the river empties into the Cuitzeo Lake, an area of 1 748 km² (Figure 1).

Figure 1 Location of the Río Grande de Morelia Basin, Michoacán, México. 

Information used

The digital elevation model (DEM) used has a pixel resolution of 15 m and a scale of 1:50 000 (^{INEGI, 2013}).

The meteorological information used is from the National Weather Service’s Conventional Weather Stations (CWS) network (Estaciones Meteorológicas Convencionales del Servicio Meteorological Nacional [SMN, Spanish acronym]). This study included 34 CWS, from which daily precipitation registries from 1923 to 2015 and location were obtained. An average of 0.48% of the data in the series were missing (^{SMN, 2017}).

Methodology

The methodology applied in this study is presented in Figure 2.

Figure 2 Diagram for calculating the concentration indexes (CI) by sub-basin in the basin of the Río Grande de Morelia. 

Delimitation of the hydrological basin

The basins were delimited based on the Continuo de Elevación Mexicano (CEM 3.0) provided by ^{INEGI (2013)}. The basin of the Rio Grande de Morelia basin was delimited using the extension in the Soil and Water Assessment Tool (^{Winchell, Srinivasan, Di Luzio, & Arnold, 2013}) (Figure 3).

Figure 3 Location of the conventional weather stations in the Río Grande de Morelia basin, Michoacán. 

CWS selection, location and georeferencing

The geographic location of each conventional weather station was extracted from the database, using those that are located within and closest to the basin of the Rio Grande de Morelia, as described in Figure 3.

Collection of information on daily precipitation from the CWS registers

The 34 CWSs, their official codes and names, operational condition, georeferencing and length of their registers were identified up to 2015 (^{SMN, 2017}) (Table 1). These CWSs were located in a GIS (Figure 3).

Station 18145 (Zinapécuaro) contained the longest registry, with 91 years (1923-2014) and station 16257 (Santa Isabel de Ajuno) had the shortest registry, with 6 years (1982-1988).

Calculation of the Lorenz curve

Concentration indexes were determined by calculating and constructing the Lorenz curve as the basis. To calculate the Lorenz curve, Y is the accumulated percentage of precipitation, to which the accumulated percentage of days, X, contributes, where X represents the days that had that precipitation value ^{Martín-Vide, 2004}).

Based on this daily precipitation information from each CWS, the following variables are described:

Calculation of the Daily Precipitation Concentration Index, CI

^{Martín-Vide (2004)} proposed the concentration index, CI, as an approximation of the numerical representation of differences, shown by the Lorenz curves (Figure 4). In this case, it is used to estimate the importance of rainy days relative to the total accumulated rain in a time series and thereby determine the relative impact of daily precipitation to evaluate the weight of the registered daily maximums relative to the total.

Figure 4 Lorenz curve for the concentration of daily precipitation at the Iramuco Station (1929-1979). 

^{Martín-Vide (2004)} associates these curves with exponential type functions:

Where a and b are parameters of the corresponding Lorenz curve.

To determine the values of a and b in the equation, ^{Martín-Vide (2004)} obtained the following relationships:

The integral is the area, A, defined by the area under the Lorenz curve between 0 and 100 (Figure 4):

Area, S, is the area under the line of perfect equality (45º) where CI = 0, (100 * 100 / 2 = 5 000) and A is:

The concentration index, CI, is defined as the proportion between S and the area under the line of perfect equality (Figure 4):

The CI are calculated for the 34 conventional weather stations (Table 2 and Figure 5).

Figure 5 Daily precipitation concentration curves for 34 CWS in the Río Grande de Morelia basin, Michoacán. 

Equation 17 was used to calculate precipitation for the last quartile (25%) (Table 3) of rainiest days, Ppd 25% , as a percentage, where:

Calculation of the Lorenz Curve for each CWS

The process is illustrated with daily precipitation data registered at Station 11027 (Iramuco), which began recording on September 1, 1929; the last record was August 31, 1979, and a total of 15 709 data were recorded.

The daily precipitation data were arranged from lowest to highest; data of no precipitation (zero or not available) were excluded. What resulted was a reduced register of Daily Precipitation, Prdr _j , in which j = 1 is the minimum value and j = NJ is the maximum.

To estimate the frequency of Prdr _j in I _i , values were assigned to Ii, namely, I = 1 for its minimum value and I = NI for its maximum value. In this case, I ₁ = 0.1 and I _NI = 72.2, respectively (355 values). In the reduced register of the station, a value of 0.1 mm of precipitation is presented in 66 days, a value of 0.2 mm occurs 127 days, and so on until reaching a value of 77.2 mm, which occurs only once. The calculation of the Lorenz curve for CWS 11027 (Iramuco) is presented in Table 3 and the curve in Figure 4.

This calculation was performed for the 34 conventional weather stations; each station has its own Lorenz graph (Figure 5).

Figure 5 presents the Lorenz curves for the 34 weather stations. If we analyze the elements of areas A, S and CI, we find that they are the same elements as those presented by ^{Sarricolea and Martín-Vide (2012)}.

Calculation of the Daily Precipitation Concentration Index, CI, for each CWS

By applying the variables from Equations 16 and 17 for each conventional weather station, we obtain the following results (Table 2).

Parameters a and b varied from a maximum of 0.134 to a minimum of 0.033 and from a maximum of 0.034 to a minimum value of 0.020, respectively. This range of values is also presented by ^{Espinoza et al. (2013)}, ^{Martín-Vide (2004)}, and ^{Benhamrouche and Martín-Vide (2012)}. The values of A ranged from 2 622.2 to 1 964.3, those of S from 3 035.8 to 2 377.8 and of CI from 0.48 to 0.61. If we compare these values with those obtained by ^{Monjo and Martín-Vide (2016)}, we observe a value of CI = 0.6 for the latitude 19º 42’, presented in world maps and where the region of Morelia, Michoacán, is located. The values of precipitation of 25% of the rainiest days varied from 52.44% to 68.29%.

In the concentration curves, Station 16146 (Zirahuen), with a CI = 0.4756, is the curve that is closest to the line of perfect equality where CI = 0, while Station 16001 (Acuitzio del Canje), with CI = 0.6072, is the farthest from the line, where CI = 1.

The CI was compared with the precipitation of 25% of the rainiest days, and a linear relationship was found (Figure 6).

Figure 6 Relationship between concentration index and precipitation of 25% of the rainiest days. 

For low CI values (0.476), the percentage of precipitation of the last quartile of the rainiest days is low (52.4%). In contrast, for high CI values (0.607), values of 25% of the rainiest days are high 68.29%. This permits determining torrentiality in a physical sense, since 25% of the rainiest days contributes 68.29% of all the rain registered. Javier ^{Martín-Vide (2004)} did this analysis with the CI and 25% of the rainiest days; he found better contrast in regionalization and in data analysis.

Classification of the Sub-basins using CI

First, each sub-basin was delimited. The CI isopleths were then generated, and the CI was calculated by sub-basin.

Generation of CI Isopleths

Using the CI results for each conventional weather station, the data were imported in a GIS and a map was constructed by interpolating, in order to obtain the spatial distribution of CI, with a separation of CI = 0.01, with which a raster-type layer of the micro-basins is generated (Figure 7).

Figure 7 Isopleths of daily precipitation concentration index, CI, for the Río Grande de Morelia basin. 

Calculation of CI by Sub-basin

With ArcMap (^{ESRI, 2016}) software, the statistical zone tool was used. Input data included the sub-basins layer and the data raster of the interpolated CI grid, and an average value of CI was obtained for each sub-basin (Figure 8).

Figure 8 Average CI values by sub-basin for the Río Grande de Morelia Basin. 

Values in the sub-basins ranged from 0.519 to 0.585. Values in the same range of 0.6 were obtained by ^{Monjo and Martín-Vide (2016)}. The highest CI values were found in the basins upstream from the Cointzio dam and the lowest near the Río Chiquito micro-basin, which according to the nomenclature used, is basin SC10 (Table 4).

Table 4 CI values by sub-basin for the Río Grande de Morelia Basin. 

Num.	Code	Area (km2)	CI	Level
1	SC1	62.08	0.585	Highly torrential
2	SC2	122.14	0.582	Highly torrential
3	SC3	133.28	0.564	Highly torrential
4	SC4	16.61	0.569	Highly torrential
5	SC5	159.40	0.548	Torrential
6	SC6	28.88	0.562	Highly torrential
7	SC7	17.76	0.551	Torrential
8	SC8	81.77	0.562	Highly torrential
9	SC9	111.09	0.562	Highly torrential
10	SC10	85.52	0.520	Mediumly torrential
11	SC11	61.65	0.550	Torrential
12	SC12	18.09	0.554	Torrential
13	SC13	99.62	0.539	Torrential
14	SC14	43.74	0.570	Highly torrential
15	SC15	80.83	0.522	Mediumly torrential
16	SC16	37.86	0.543	Torrential
17	SC17	74.22	0.536	Mediumly torrential
18	SC18	124.48	0.561	Highly torrential
19	SC19	96.10	0.534	Mediumly torrential
20	SC20	19.01	0.534	Mediumly torrential
21	SC21	50.01	0.550	Torrential
22	SC22	73.82	0.551	Torrential
23	SC23	150.99	0.552	Torrential

Classification of sub-basin torrentiality by CI

To give the physical context of the CI values in the sub-basins and the basin itself, from the results of the CI values reported in the literature, such as ^{Monjo and Martín-Vide (2016)}, which range worldwide from 0.38 to 0.87, and those of our study, we propose a manner to compare torrentiality levels within a basin, that is, to determine which is more torrential. Using the concentration index obtained for each sub-basin, a histogram was constructed to identify the distribution of CI in the basin. A normal distribution was obtained, and a normality test (Shapiro Wilks) was applied. The test indicator showed a significant difference of nearly 1; thus, its normal distribution is acceptable and the p-value obtained is higher than the alpha of 0.005 tables. Considering the normality of the data, we propose four levels of torrentiality in the Río Grande de Morelia basin based on the CI value. To this end, we used the quantiles 0% (0.476), 25% (0.515), 50% (0.538), 75% (0.560) and 100% (0.607), which correspond to the boundaries of each class (Table 5).

The weighted average was calculated for the Río Grande de Morelia basin, obtaining CI = 0.5524; according to the proposed classification, it is of the torrential type.

Association between CI and Climate

To determine the behavior of the sub-basins, it is necessary to relate it to the climate that predominates in the study region. The average CI values in the 23 sub-basins range from 0.5195 to 0.5851. The highest CI values occur in the sub-basins in the upper part of the basin upstream from the Cointzio dam. These sub-basins have a C(E)(w₂)(w) climate, which is cool subhumid, the most humid of the subhumid, with summer rains. The months with the highest precipitation are May to October, with more than ten times more rain than the driest month of the year. The percentage of winter rain is < 5% of the annual total; precipitation in the driest month is < 40 mm and mean annual temperature ranges between 5 and 12 º C.

The CI values are between 0.60 and 0.55 in sub-basins downstream from the Cointzio dam, with C(w₂)(w) climate type, temperate subhumid, the most humid of the subhumid, with summer rains. The months with the highest rainfall are May to October, when precipitation is at least ten times more than in the driest month of the year. The percentage of winter rain is < 5%, precipitation during the driest month is < 40 mm, with a mean annual temperature that ranges between 12 and 18 ºC.

The lowest CI values, between 0.55 and 0.52, occur in the climate type C(w₁)(w), temperate subhumid with medium humidity and summer rains. The highest precipitation occurs from May to October when rainfall is at least ten times that of the driest month of the year. The percentage of winter rain is <5% and precipitation in the driest month is < 40 mm; mean annual temperature ranges between 12 and 18 ºC.

The lowest CI values, between 0.52 and 0.50, are found in the climate type C(w₀)(w), which is temperate and less humid than the subhumid climates with summer rains. Rainfall is highest from May to October when it rains at least ten times more than in the driest month of the year. The percentage of winter rain is < 5% and precipitation in the driest month is < 40 mm; mean annual temperature ranges between 12 and 18 º C (^{INEGI, 2000}).