In this work, we evaluated the relationship between evapotranspiration and precipitation, based on the data recently made available by the Brazilian Meteorological Institute. ETP tend to be lower in rainy periods and vice-versa. This relationship was assessed both in physical and statistical ways, identifying the contribution of each explaining variable of ETP. We derived regression equations between monthly rainfall and ETP, which can be useful in studies where ETP time series are not available, such as reservoir design, irrigation management and flow forecast.

Evaporation and evapotranspiration (

Moreover,

The Brazilian hydropower network, which accounts for more than 70 % of
power supply in the country, is designed to minimize the risk of power
shortages in the case of repetition of a so-called critical hydrological
period, which was a very dry sequence of years in the country as a whole,
from 1949 to 1956. But hydrological data during that period was very scarce,
leading to the need of hydrological modelling, mostly using

Again, given the low variability of

However small, interannual variability in

Therefore, a way of overcoming

Recently, the Brazilian National Institute of Meteorology (INMET) has put its daily hydrometeorological time series available for research purposes. Data since 1961 over 290 stations around the country can be obtained through the internet after registering, attending a long-time demand of water managers and research community in Brazil.

In the present work, some statistics were calculated using this database, in
order to identify a more physical explanation for the inverse relationship
between rainfall and

Moreover, regression equations between monthly

The Penman-Monteith equation, which is considered to be the most suited
method for quantifying

The inverse relationship between rainfall and

Rainy days also generally show higher relative humidity. This leads to a
decrease of

The effects of the remaining variables are not so evident. Mean temperature,
for example, is positively correlated with

In the case of the other variables (atmospheric pressure and wind velocity), it is harder to establish a correlation intuitively, as made above. The magnitude of each meteorological variable in dry and wet years were statistically tested as follows.

For each station, daily time series were separated between dry and rainy
days, the latter defined as days with precipitation lower or equal than 5 mm.
For each of the five variables, we tested the hypothesis that the mean of
the variable in dry days is equal to the mean in rainy days (null
hypothesis), using Student's ^{®} for a 5 % significance level. If the
null hypothesis was rejected, it was verified that the mean in rainy days
was higher or lower than in dry days.

Seasonality was also accounted for, by performing the tests for each month of the year. Months where less than 5 % of the days were rainy were discarded (typically the period between July and September in the Southern Hemisphere), in order to avoid non-rejection of the null hypothesis only because of the small size of samples.

INMET stations were grouped according to the five Brazilian climate zones (IBGE, 2013). For each zone, we calculated the frequency of months/stations where magnitude was higher in dry and rainy days, as well as the percentage of months were the null hypothesis could not be rejected (in this case, we considered that no significant difference exists).

Table 1 shows the condensed results of the hypothesis test for wind velocity. For each climate zone, we calculated the frequency of months/stations where wind was stronger during rainy and dry days, as well as no significant difference.

In most regions, there is no difference between wind velocity during rainy and dry days. In the semi-arid and coastal northeast, stronger wind occurs slightly more frequently during dry days. In the more temperate south, it is the other way round. In general, it was observed that coastal stations showed stronger wind during dry weather, while stations in hillier regions have stronger winds more frequently during rainy days.

Wind velocity influences

Statistical analyses were performed considering minimum, maximum and mean daily temperature. Table 2 shows the analysis for the maximum temperature. In almost 90 % of the cases, maximum temperature was significantly higher during dry days than during rainy days, more so in the temperate south and less so in the equatorial amazon. It confirms the intuitive notion, also stated by Allen (1998), that dry days have maximum temperatures higher than rainy days.

For minimum temperatures, aggregate results are shown in Table 3. Its results show that the assumption by Allen (1998), that rainy/cloudy days have higher minimum temperatures, is valid for temperate climates, but not necessarily for tropical/equatorial regions. Here, very often minimum temperatures were higher in dry days.

In the average, results shown in Table 4 allow the conclusion that dry days tend to be warmer in all regions of the country. As seen, this is in part due to higher maximum temperatures, as expected, but in some parts also due to higher minimum temperatures even in dry days, contrary to some literature, notably Allen et al. (1998).

This also helps to explain the negative relationship between rainfall and

Frequency of stronger wind during dry and rainy days.

Frequency of higher maximum temperature during dry and rainy days.

Frequency of higher minimum temperature during dry and rainy days.

Frequency of higher mean temperature during dry and rainy days.

Frequency of higher mean sunshine duration during dry and rainy days.

Frequency of higher relative humidity during dry and rainy days.

Frequency of higher atmospheric pressure during dry and rainy days.

In the case of sunshine duration (used here to estimate incoming radiation),
the correlation is quite intuitive, and is confirmed by the statistical
analysis: in almost 100 % of the months/stations, mean sunshine duration
was higher in dry days, as seen in Table 5. This is
the main factor explaining the inverse relationship between rainfall and

The same happens for relative humidity. Table 6
shows that in 100 % of the cases, relative humidity is higher during rainy
days. Since relative humidity is negatively correlated with

Finally, results for atmospheric pressure are shown in Table 7. For higher latitudes, there is a positive correlation between dry weather and atmospheric pressure, as expected. In regions closer do the Equator; however, this correlation is no longer valid, since there is no statistical difference between pressure during dry and rainy days. The variability of atmospheric pressure also plays a minor role in the variability of ETP; for one side, higher pressure increases the specific weight of air, allowing more moisture to be retained in the surrounding air; on the other hand, it increases the psychrometric constant, which comes into the denominator of the Penman-Monteith equation, thus decreasing ETP.

Example of regression for Brasilia, month of January.

Daily time series for ETP were calculated through the Penman-Monteith, based
on Shuttleworth (2012) and synthetized by Collischonn et al. (2007), using the
meteorological data recently made available by INMET in 290 meteorological
stations in Brazil. Incident radiation data was estimated based on sunshine
duration. Reference evapotranspiration was calculated adopting an albedo of
0.23 m, surface resistance of 70 s m

Monthly averages of

Spatialized values of

During time intervals where one or more of the variables were missing, ETP was not calculated, the only exception being missing atmospheric pressure. In this case, pressure was estimated based on a seasonal correlation between monthly mean pressure and station elevation. This is considered to be acceptable, given the low influence of this variable over the variability of ETP and the large number of missing data for atmospheric pressure.

ETP data where then accumulated into monthly totals. If missing data accounted for less than 5 days, missing data were filled using the mean ETP of the remaining days. Otherwise, the month was not included in the regression. The same was done for monthly rainfall.

Regression equations were then derived from the paired ETP/rainfall monthly data. Regressions were done for each month of the year, in order to account for seasonality. Given the expected inverse relationship between these variables, the slope of the regression should be negative.

Figure 1 shows an example of the resulting
regressions, for the meteorological station of Brasilia (tropical Brasil
central zone), for the month of January. It can be observed that, as
expected, rainfall variability is in general much higher than that of ETP.
In this case, while rainfall varied between 29 and 577 mm month

Although data dispersion is relatively high (also expressed in the low
coefficient of determination

This kind of regression is particularly useful during transition months between the dry and rain seasons in tropical regions (typically April/May and September/October in the Southern Hemisphere), which can be either very rainy or very dry, depending on the year. It is also useful in regions where the rain season presents periodical breaks, or periods of 10 to 15 days without rainfall. These periods, usually between December and March, can have very high ETP given the higher extraterrestrial radiation during summer months.

Only during very dry months, like July and August, ETP can have a higher variability, since rainfall is hardly larger than zero. In this case, the regressions are less useful, and the use of ETP averages seem to be more practical.

Figure 2 shows the spatialized values of the

As can be seen, the higher values for the coefficient were obtained in the tropical zone. In the temperate and equatorial zones, the inverse relationship between rainfall and ETP is not so clear, perhaps because of a higher frequency of cloudy, but not necessarily rainy days.

Finally, Table 8 shows the average of

As mentioned, the winter months in the Southern Hemisphere (June to August) showed the lowest average regression performance, because of the low variability of rainfall during these months in most regions of Brazil.

This work is a first attempt to further explore the meteorological time series recently made available by INMET, based on a hydrological point of view. In the following sections, the main specific conclusions are presented.

The statistical analysis performed over the meteorological variables explaining the evapotranspiration process supported the comprehension of the mechanisms behind the inverse relationship between rainfall and ETP. Incoming radiation, which is the main driver of ETP, is clearly higher during dry days than in rainy days. Relative humidity, which also strongly influences ETP, is clearly higher during rainy days. Both results are intuitive and are supported by the statistical analysis.

Mean air temperature, which is also positively correlated with ETP, is generally higher during dry days, in comparison to rainy days. The results also showed that minimum temperatures very often were higher during dry days, mostly in the tropics. This result shows that one very popular premise of hydrometeorology is not necessarily true for tropical regions, namely the belief that minimum temperatures should be lower during clear weather due to less heat retention in the lower atmosphere.

For wind velocity and atmospheric pressure, results were less clear. In most cases, null hypothesis of equal means could not be rejected, thus not allowing further conclusions. One singular fact that was observed is a tendency for stronger wind during dry days in coastal places, while in hilly locations wind was stronger in rainy days.

Thus, the inverse relationship between rainfall and ETP is explained mostly by the higher incoming radiation (or sunshine duration) and lower relative humidity in dry days, and by a lesser part, by higher mean air temperature in these conditions.

In general, the results of the regressions confirmed the inverse relationship between rainfall and ETP, since regression slopes were negative in most cases.

Although the numerical values of the coefficient of determination

The obtained relations are particularly useful in tropical regions, during months of transition between rain and dry seasons, which can be either very wet or very dry, thus showing varying ETP.

These indirect estimates of ETP can be useful, for example, for extending ETP series into the past, in regions or periods where rainfall data is more abundant than meteorological data. As mentioned, the Brazilian hydropower network is planned and operated to supply energy demand, at a low risk, even in the case of a repetition of a critical hydrological period occurred in the 1950s, when hydrometeorological data was scarce nationwide (and even more so in low-populated areas like the amazon, where hydropower was expanded in recent years). More accurate ETP estimates would be helpful both for a better assessment of inflows during that period (given the need of proper rainfall-runoff modelling) and for a better estimation of evaporation losses.

Other potential areas of usage are reservoir design and reservoir yield estimation, where long-term ETP averages are still widely used, leading to an overestimation of reservoir yields. Irrigation management would also be an area for application, through a better estimation of crop water needs, based only on rainfall measurements and the regressions obtained. Finally, assessment of future hydrological conditions would also benefit, since ETP conditions could be easily estimated based only on rainfall predictions.

Meteorological data used in this paper belongs to the Brazilian Meteorological Institute and is available (after registration) at

The authors thank the Brazilian National Institute of Meteorology (INMET) for making the hydrometerological time series available for academic purposes.