The height variability of Lake Patzcuaro, in central Mexico, remained constant for several decades, and during that period, the lake maintained its functionality and environmental services. However, in the last twenty years, there has been a significant decrease in the variability range. In order to estimate the water balance, in this work, an approach was developed to understand how water moves in time and space, and to determine possible inherent thresholds to lake variability. The historical Lake Patzcuaro height above 2035 masl was correlated with several climatic variables. The correlation with monthly rainfall shows that there is a delay of three months, indicating that the lake volume depends on three factors: precipitation, surface runoff and underground contribution, of which precipitation over the lake surface is the least significant. In the long term, using annual data, there is a remarkable memory of precipitation extending beyond five years, seen as a correlation with the accumulated precipitation anomaly ( r 2 = 0.76). This behavior should be explained by understanding relationship between the lake and its aquifer. Also important is the population growth, which affects the lake level in two ways: water extraction and water discharge. The models pointed out a nonlinear relationship between water level and the aforementioned variables, and suggest that the carrying capacity of the basin is around 90 thousand people, under current patterns of water consumption. It also indicates that, in order to allow lake recovery or to maintain system resilience, the anomaly of precipitation accumulated during several years should exceed 1000 mm. Although a correlation with land use was attempted, historical data do not reflect a trend that could be associated with water balance.
Lake Patzcuaro is a tropical, high-altitude, endorheic and fresh water body located on the Mexican plateau at 2036 meters above sea level (masl). Its origin is due to tectonic-volcanic processes associated with the Mexican Neovolcanic Axis [
In general, variations in water levels of lakes are due to several factors that can be classified into: (1) evolutionary processes, (2) climate natural variability, (3) extreme natural events and (4) events that are associated to human activities in the catchment area. Aside from rainfall patterns, volume variability in lakes shows that historical fluctuation trends are generally explained by variations in catchment land use and extractions [
Imagery and remote sensing techniques have been used to estimate the Patzcuaro Lake’s surface. According to aerial photographs taken in 1974, it was estimated to be 116.48 km2 [
The annual water balance has been calculated by integrating surface and underground water, over several years. Surface water includes estimates of the natural runoff, direct discharge to the lake by sub-basins. Underground water accounts for infiltrated water, springs and wells. For 2004, the inputs and outputs were estimated at 138.3 and 119.3 Mm3 respectively, leading to lake variations of approximately 18.7 Mm3. For this balance, the interaction with groundwater was identified as output rather than input [
The Equation (1) shows the most extensive monthly water balance of the lake (Mm3/month) [
∆V = (Vll + Re + B + Vman) − (ET + In + Inter + Ab + Uc + f) (1)
where,
∆V = Change in water storage
Vll = Rain volume in the basin = P A (P = precipitation in A = basin area)
Re = Water returns from agricultural fields
B = Extractions and water pumping
Vman = Creeks volume
ET = Evapotranspiration
In = Rain infiltration
Inter = Rain intercepted by vegetation
Ab = Volume downstream
Uc = water consumption (superficial and underground)
f = Losses in public networks
Based on data from the seven sub-basins, it has been estimated that only 10% of rainfall is infiltrated and just 6% reaches the lake. However, the lake response to fluctuations in precipitation (P) and evaporation (E) depends on the relative contribution of groundwater inflow and outflow [
Some closed basin lakes show an immediate response to an increase in rainfall, but it may exhibit a delayed response to a reduction in rainfall, as was observed in Lake Rukwa. In general lake levels fall when evaporation exceeds the total input from precipitation plus inflow, which has been the case in Lake Patzcuaro for several years. On the other hand, Lake Tanganyika shows a close relationship between rainfall variability and lake levels. Correlations were high (0.80) between lake-levels and mean rainfall for the previous five years, with lagged effects from 2 to 5 years [
After the Limnological research station was established in Lake Patzcuaro (1939), Yamashita measured the silica concentrations in water and suggested that an underground spring is discharging in the lake [
An analysis of the relationship between environmental and social transformations in the Patzcuaro Lake during the Pre-Columbian Tarascan (Purepecha) empire, results in a challenge to common conceptions regarding the impact of agriculture, urbanism and state collapse on ancient landscapes [
Indeed, the data gathered on the lake level over 500 years [
The main objectives of this paper are: 1) to use statistical models to find the physical and anthropogenic factors that best describe the annual cyclic and long-term behavior of Lake Patzcuaro’s level; 2) to identify variables that negatively affect the water balance and resilience of the system; and 3) to identify steps to mitigate the damage and to enhance conservation of the lake.
Lake Patzcuaro Basin is located between 19˚27' and 19˚44' North latitude and longitude 101˚26' and 101˚53' West (
Of the watershed area, 37% of the land has slopes less than two percent and 29% has slopes of less than six percent, this being the only land suitable for agriculture. More than 12% has slopes greater than 12%, with little resistance to erosion especially if deforested.
At the arrival of the Spaniards, the study area was occupied by a Tarascan estimated population of 60,000 inhabitants, who had been exploiting natural resources for several centuries. It became an important colonial settlement whose center was the town of Patzcuaro. Nowadays, the population of the study area is mainly composed of indigenous people originally dedicated to farming, fishing and timber harvesting. More recently handcrafts and tourism have become very important. Today Patzcuaro is a cultural tourism site that receives more than 200,000 visitors annually.
Three data sets were analyzed in this study: (1) Historical lake levels, (2) Local and regional climate data, 3) Population and land use data.
The historical lake level data were recorded by the Michoacán Government Fisheries Commission. Almost daily data are available since January 1950. In addition, from 2003 to 2010 a more detailed data set was generated in a specific study made by the Mexican Institute of Water Technology (IMTA for its acronym in Spanish). In this study, the 2035 masl was set as the inferior reference limit, and we have used this value here.
The main source of climatological data is the Maya database which contains daily values of rain, maximum and minimum temperature data, from January 1961 to December 2010, on a 0.2 × 0.2 degrees grid over the whole Mexican soil. Maya is an interpolated climate data base that uses all existing daily data, interpolated using inverse square distance criteria [
The monthly average maximum temperature was taken from the National Meteorological Service (SMN in Spanish) database for station 16087 located at 16˚32', 101˚37' and 2043 masl, in the Patzcuaro town.
In order to validate the historical data, short term data were collected by IMTA, during the 2003-2010 period, over a dense grid of 10 automatic stations across the basin. These stations were equipped with a rain gauge, temperature and evaporation sensors and a wind gauge.
Because evaporation and evapotranspiration data from the national database had many gaps, the monthly maximum average temperature was used instead as proxy value for evaporation. The coefficient of correlation found between these variables over several periods is above 0.8.
Population and land use were obtained from the INEGI (National Institute of Geography and Statistics, México) database. Population was estimated from a decadal INEGI census, which takes into account the number of inhabitants of the four municipalities included in the Patzcuaro watershed. The annual and monthly data were interpolated linearly. To determine urban and rural population it was assumed that all villages with more than two thousand people where urban. The information on land use is mainly concentrated in recent years, although some authors prior to 1980 give rough estimates. The information is not homogeneous but shows no significant changes in the distribution of forest area (30%), agriculture (7%), grasslands (20%), mixed or degraded forest (28%), and the area occupied by the lake, which is around 10%.
The methodology consisted of the following steps:
1) Obtaining data and apply quality control measures
2) Calculate annual and monthly data for precipitation, average maximum temperature, and lake level above 2035 masl, population and urban population and derived variables such as moving averages
3) Explore data qualitatively through plotting
4) Calculate auto correlations and cross correlations between data
5) Construct statistical models to predict and explain changes in the lake level
Cross correlations were calculated to identify relationships between variables, but mainly to explain lake level. The rainfall anomaly was calculated by removing the annual average over the period 1961 to 2010, which was 938.2 mm, Equation (2):
Because the lake acts as a reservoir, the 5 year rainfall moving average was calculated. Also a series of cumulative rainfall anomaly was created, adding the rainfall anomaly year by year, Equation (3):
Furthermore, because the rain regime of the area results in the majority of precipitation falling between June and October, the monthly rainfall anomaly was calculated by subtracting the average rainfall for each particular month, and the series of the monthly cumulative rainfall anomaly was calculated adding the monthly anomalies, Equation (4).
Two statistical models were developed using Generalized Additive Models, in the mgcv package [
The strong decrease observed in the lake levels during the decade of the 1980’s frightened both the population and the government, because it appeared that the lake was going to disappear. However the historical data (
During the 1960’s and 1970’s, the five-year moving average rainfall was well above the average, which is 950 mm (
Over the next 5 years the annual rainfall returned to 950 mm but this was not sufficient to raise the lake level. Between 1996 and 1997 rain was scarce and the lake lost more water. The lake level (
Because the lake is a dynamic storage system it has memory, therefore the cumulative rainfall anomalies has to take into account what happened in the years before. The time series (
Although the average precipitation during the 1961-2010 fifty years period was 938 mm, a high variability was observed. During the 1961-1985 interval it was 985 mm, 50 mm above average; while for the 1986-2010 period, it was only 891 mm, about 100 mm less than average, with a significant downward trend of −3.5 mm/year.
In order to investigate the lake seasonal behavior, the following variables were used: lake level, rainfall, the rainfall anomaly, rainfall five year moving average, cumulative rainfall anomaly and the average maximum temperature. Aside from the temperature, all available data cover the January 1961 to December 2010 period. There are a few missing values in the lake level data in 1992 and 1993. Seasonal cycles are observable in precipitation, lake level and maximum temperature (
After removing the annual seasonal cycles, the lake level fluctuations appear to follow the cumulative rainfall anomaly better than the five year moving average. It is interesting to point out the differences in the data cycles: the level, rainfall and temperature are out of phase. The lake level has its minimum from May to July, when the rain season begins, and it generally reaches a maximum between November and December.
In
Evaporation also follows an annual cycle that is out of phase with the annual rainfall cycle (
On the other hand, the population in the basin has been rising at a rate of almost 2% each year. From 1960 to 2010 the number of people in the area changed from 60 to almost 140 thousand. The urban population (i.e., towns with more than two thousand inhabitants) grew at even higher rates, from 15 to 106 thousand in 50 years, mainly in the Patzcuaro and the Quiroga cities and their surroundings. In
In spite of counting with some data related to changes in land use, these do not reflect important changes or a trend that can be correlated with patterns of rain or runoff in the basin. Therefore, data are included in order to let know the dynamic of registered change in the area but at 1:250000 scale. Updated land uses are related to the precision of tools to measure areas in images.
Because the climatic seasonal cycles are out of phase, investigating cross-correlations with different lags can be important.
The population and the lake level also appear to be highly inversely correlated: when the population increases, the lake level decreases. Similarly, the correlation with maximum temperature is negative and the maximum correlation occurs with a 2 - 3 months lag, which also indicates the expected seasonal cyclical behavior.
The regression model that yields the best fit (
Land use and vegetation | SERIES 1 (1980) | SERIES 2 (1990) | SERIES 3 (2005) | SERIES 4 (2010) | SERIES 5 (2013) | |||||
---|---|---|---|---|---|---|---|---|---|---|
Surf (ha) | % | Surf (ha) | % | Surf (ha) | % | Surf (ha) | % | Surf (ha) | % | |
Agriculture | 37,112.5 | 28.4 | 36,691.1 | 28.2 | 36,763.0 | 28.2 | 39,752.7 | 29.8 | 40,571.5 | 30.3 |
Forest | 75,646.4 | 57.9 | 75,522.9 | 58.0 | 75,595.6 | 58.0 | 77,458.7 | 58.1 | 78,210.0 | 58.3 |
Waterbody | 8653.6 | 6.6 | 9481.2 | 7.3 | 9480.6 | 7.3 | 9128.2 | 6.9 | 7869.5 | 5.9 |
Subtropical scrub | 1513.8 | 1.2 | 1355.8 | 1.0 | 1660.6 | 1.3 | 1300.2 | 1.0 | 1300.2 | 1.0 |
Pasture | 3912.1 | 3.0 | 3114.7 | 2.4 | 3016.7 | 2.3 | 1905.5 | 1.4 | 1774.9 | 1.3 |
Wetland | 3320.8 | 2.5 | 1956.3 | 1.5 | 1678.6 | 1.3 | 1218.4 | 0.9 | 1837.1 | 1.4 |
Urban Area | 446.9 | 0.3 | 2062.7 | 1.6 | 2061.6 | 1.6 | 2482.6 | 1.9 | 2501.9 | 1.9 |
Total | 130,606.1 | 100.0 | 130,184.7 | 100.0 | 130,256.6 | 100.0 | 133,246.3 | 100.0 | 134,065.1 | 100.0 |
Family: Gaussian | ||||
---|---|---|---|---|
Link function: identity | ||||
Formula: | ||||
level ~ s(rain) + s(acum) + s(pop) + s(tmax) | ||||
Parametric coefficients: | ||||
Estimate | Std. Error | t value | Pr(>|t|) | |
(Intercept) | 2.286712 | 0.007811 | 292.8 | <2e−16*** |
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||
Approximate significance of smooth terms: | ||||
edf | Ref.df | F | p-value | |
s1(rain) | 1.000 | 1.000 | 202.78 | <2e−16*** |
s2(acum) | 6.376 | 7.595 | 16.97 | <2e−16*** |
s3(pop) | 8.849 | 8.976 | 159.46 | <2e−16*** |
s4(tmax) | 3.781 | 4.701 | 26.26 | <2e−16*** |
R-sq.(adj) = 0.986 Deviance explained = 98.6% | ||||
GCV score = 0.025149 Scale est. = 0.023794 n = 390 |
term fluctuation explains more than 90% of the variability of the lake level. The following predictor variables were used: rainfall, rainfall anomaly, cumulative rainfall anomaly, population in the basin and average maximum temperature. Actually, the absolute rainfall and the rainfall anomaly are linearly correlated, indicating that only one of them is necessary.
The general additive model shows that the lake level is mainly determined by the value of the cumulative rainfall anomaly: if the cumulative anomaly exceeds 900 mm, the level of the lake will rise, but if the cumulative anomaly decreases below 400 mm the lake will lose water (
The climatological factors: monthly rainfall and maximum monthly temperature have less influence. The rainfall shows an almost inverse linear relationship with the lake level, which corresponds to the cyclic behavior observed in the cross correlation (
The final monthly model takes into account all the smoothed terms (
This model is able to reproduce the observed monthly variations in the lake level.
In order to explore the long term Lake level trend, the annual variation of the variable using the annual average values during the last half century were analyzed. The following variables were considered: rainfall, five-year moving average rainfall (
Our best annual model uses four predictor variables: population, cumulative rainfall anomaly, annual rainfall and the average annual maximum temperature (
The obtained model explains almost 99% of the variability (
As in the case of the monthly model, the annual model (
In contrast, a major predictor is the cumulative rainfall anomaly, which presents a positive slope: the greater
Family: Gaussian | ||||
---|---|---|---|---|
Link function: identity | ||||
Formula: | ||||
level ~ s(rain) + s(pop) + s(acum) + s(Tmax1) | ||||
Parametric coefficients: | ||||
Estimate | Std. Error | t value | Pr(>|t|) | |
(Intercept) | 2.41260 | 0.01365 | 176.7 | <2e−16*** |
Approximate significance of smooth terms: | ||||
edf | Ref.df | F | p-value | |
s1(rain) | 4.286 | 5.222 | 8.686 | 1.87e−05*** |
s2(pop) | 8.044 | 8.720 | 154.511 | <2e−16 |
s3(acum) | 7.264 | 8.209 | 13.829 | 1.11e−10 |
s4(tmax1) | 1.000 | 1.000 | 3.790 | 0.0614 |
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 | ||||
R-sq.(adj) = 0.993 Deviance explained = 99.6% | ||||
GCV score = 0.016408 Scale est. = 0.0093222 n = 50 |
the anomaly, the higher the lake level. The zero point on the trend line corresponds to a cumulative anomaly of about 1000 mm: below this point, the lake level declines. In 2010 the cumulative anomaly was around 290 mm, which means that a six or seven year-long wet period with at least a 1000 mm of rainfall would be needed to allow some recovery of the lake level.
The five-year moving average rain does not appear in the model; because it does not correlate as well as the cumulative rainfall anomaly.
This is the first work to explain the behavior of Lake Patzcuaro using data from an extended period of 50 years. Daily data from the Maya database have undergone certain quality controls; the lake height comes from the Michoacán Government Fisheries Commission with monitoring of the National Water Commission since the early 50’s. Population data were taken from the institute responsible from the national decadal censuses.
The correlation analysis and the statistical models obtained using monthly and annual data, point out that climate variables such as: rain, evaporation and temperature are responsible for a natural annual variation of the lake level of about 0.5 m, with certain time lags. The maximum level occurs between October and November, just after the rainy season, and the lowest level occurs in May-June.
The two models suggest that the main causes that explain the long term behavior of the lake are both natural and anthropogenic. Anthropogenic effects are measured by the size of the population, which in turn can be seen as a proxy for water consumption per person for both individual use and for different economic activities. An important question that needs to be addressed is why the lake degradation is so much related to population, given that the population is becoming more urban, while agriculture is decreasing in the region.
The next most important component is the variable derived from precipitation. It is implicit in the history of this phenomenon, which we find is best determined as the accumulation of the precipitation anomaly. This result coincides with the correlation coefficients obtained for different lags. The interpolation functions obtained from the models show that, in the case of the monthly data model, the lake is in a stable condition as long as the accumulated precipitation anomaly remains between 400 mm and 1000 mm. above this range, the lake increases its volume, while it decreases rapidly below 400 mm. In the model with annual data, a cumulative anomaly of 1000 mm allows the lake to maintain its level, but below this value, the level decays rapidly.
The memory of almost five years can be best explained by coupling the lake to its aquifers. This interaction occurs at different depths and by path of different lengths as proposed by [
It is needed to know the lake-aquifer interactions in order to understand how the rainfall, infiltration, runoff, lake and aquifer processes influence the lake behavior. The memory of at least five years cannot be explained by rain runoff only. Groundwater monitoring wells would be needed to determine the water depth and flow directions at different times throughout the year. Existing data are sporadic, and for only few wells, and do not cover extended periods.
The population has grown from 62,000 to almost 140,000 inhabitants during this period. In addition the floating population almost doubles these numbers: it is especially concentrated at certain dates with massive tourism. On the other hand, agriculture, which used to be the largest water user as well as the main economic generator, now covers only 45% of the surface, while the main activity is related to crafts made and tourist services. Because there is not enough data, it was not possible to employ land use as a predictor for lake level changes.
The Patzcuaro Lake has lost almost 4 m in depth between 1982 and 2010. This is translated into approximately 25% of the water volume, with a surface area estimated between 130 to 90 km2 at different times. The reasons for this variability have to do with both natural causes and human activities.
The average annual rainfall during last 50 years is about 938 mm, but with a considerable variation between 670 mm to 1300 mm, and decreasing at a trend of −3.5 mm/year, with dry periods lasting for four to five consecutive years after 1985 (
Lake level statistical models suggest that the main causes are natural and anthropogenic effects, which are mostly non-linear. In particular the results indicate that beyond a level of 90,000 inhabitants the lake level loses its resilience.
Because it is a closed basin, the most important contribution indicated by the model is that the lake acts as a reservoir with a long memory that takes into account the rainfall of the previous years. The cumulative rainfall anomaly shows best correlation, with the monthly level of lake.
IsabelQuintas,María AntonietaGómez-Balandra,WillemVervoort, (2016) Rain Variability and Population Growth to Explain Historical Levels of the Patzcuaro Lake in Mexico. Journal of Water Resource and Protection,08,168-182. doi: 10.4236/jwarp.2016.82014