Figure 2 shows the spatial distribution of the frequency of heavy rainfall in Iran during the period from 1971 to 2011. As is evident from the map, the two realms with the maximum number of days of heavy rainfall events in Iran are the southern coast of the Caspian Sea and the western foothills of the Zagros Mountains. Some scientists believe that heavy rainfall over the Caspian Sea occurs because of a strong ridge over the Black Sea, eastern and central Europe, and the eastern Mediterranean, as well as a deep trough over the eastern Black Sea (Moradi, 2002).
However, in the territory of the southern shores of the Caspian Sea, days of rain have fallen from west to east, so the results of Nouri et al. (2012, 2013) and (Alijani et al., 2013) confirm the findings above. The first hotspot of the maximum number of days of heavy rains over the coastal shores of the Caspian Sea is located in Gilan and with a smaller number of days, is observed in Mazandaran (Sari). In this region towards the northeast, the frequency of heavy rainfall events has decreased (Fig. 2). The Caspian Sea coast located along a path that affect by humid westerly winds in the east valley, and then the low height area has penetrated between the Hezar Masjed and Binalud mountains to Mashhad. For this reason, there is a gradual trend of decreasing rainfall eastwards from the Caspian region. But in south and west sides of the Alborz mountain this decreasing is abrupt due to the Alborz range (Alijani, 1996). The second domain of the frequency of heavy rainfall, the western Zagros Mountains, is a dual-core maximum of rainfall occurring around the cities of Kermanshah and Shahrekord (Fig. 2). Such rainfall can occur in this region and is the result of a strengthened and intensified monsoon low pressure center and the Sudanese Red Sea converging into a dynamic and thermodynamic system (Azizi et al., 2009; Lashkari, 2001).
Of course, as the map clearly shows, the frequency of heavy precipitation events in west parts of the Zagros is smaller than those of the Caspian Sea. The Zagros mountains play a significant role in the abundance of rains in the territory. The largest bulk of the rainfall occurs in the entry of west winds to Iran and happens in the windward slopes of mountain barriers. However, mountains cannot be the only reason for increasing rainfall in the west of Iran. One should look for other influential factors. One of these factors can be the direction of arrival of Mediterranean cyclone and west winds (Alijani, 1995). There is also an area on the map that corresponds to the lowest frequency of heavy precipitation; this territory is located in the inner regions of Iran. Many researchers believe that the cause is being far from water sources (Alijani, 2002; Montazeri, 2009). Figure 3 indicates spatial distribution of the frequency of heavy precipitation in Iran during the study period. Compared to heavy precipitation, super heavy precipitation shows a lower frequency of occurrence during the period of study. In this map, in two realms, the maximum frequency of occurrence of heavy precipitation corresponds with heavy precipitation (Fig 3). The hotspot of heavy precipitation is observed in the territory of the southern coast of the Caspian Sea in Gilan, with its intensity being measurably reduced as one moves toward east and south (Fig 3). The map indicates a sharp decrease in the frequency of super heavy precipitation events in the North West compared to the frequency of heavy rainfall (specifically, along the eastern slopes of the Zagros Mountains and the southern slopes of the Alborz Mountains in the inner regions of Northwest Iran), to a rate of less than 200 days.
To gain a more accurate understanding, heavy and super-heavy rainfall is analyzed in different periods (1961-71, 1972-81, 1982-91, 1992-2001 and 2002-11) (see Fig. 4). The spatial distribution of heavy and super-heavy rainfall in the first period (1971-81) suggests that the frequency of these phenomena is mainly observed over the southern shores of the Caspian Sea, along the Zagros Mountains from the northwest to the southeast, and in Northwest, West and Northeast Iran. Figure 4 shows that the concentrations of heavy and super-heavy rainfall, or the hotspots of these rainfall types, are mainly located along the coastal shores of the Caspian Sea, especially in Gilan Province (heavy and super-heavy rainfall respectively have a frequency to 781 days and 416 days). The frequency of heavy and super-heavy rainfall reduces towards the east and south. This feature can be explained by the wind convection, which is one of the main factors causing autumn showers in Gilan (Kaviani and Alijani, 2001). However, in a vast area of the country——especially in the inner part, southern part, and southeastern part——the frequency of rainfall has dropped to its lowest level (12 days for heavy rainfall and 6 days for super-heavy rainfall) (Fig. 4). Some researchers (Masoudian, 2009) claim that the lack of rainfall in the inner regions of Iran, including the deserts of central and eastern Iran, can be attributed to the dominance of subtropical high pressure on hot days of the year and the location of this area (i.e. being far from the rainfall of the Zagros Mountains). The influence on this area of the rainfall over the Zagros Mountains is reduced because of the long distance between them. It can also be seen that, in the areas of the country where there is a high frequency of heavy rain, the maximum frequency for super-heavy rain occurs too. In fact, the frequency of occurrence of super-heavy rainfall areas follows that of heavy rainfall events. The interdependence of heavy and super-heavy rainfall is clearly apparent in the maps. The highest frequency of such rainfall is seen over the coastal shores of the Caspian Sea. Of particular importance in this regard is Gilan, which has experienced 416 days of super-heavy rain and 781 days of heavy rain (Fig. 4).
Estimations of heavy rainfall in the second period (1982-91) indicate a decreased frequency of rainfall occurrence in the north and northwest, west, east and southeast parts of the country, compared with 1971-81 decade. This phenomenon can be clearly observed in the anomaly map of the second decade (Fig. 4). In particular, the frequency of maximum rainfall in the provinces of Gilan and Mazandaran has reduced. Moreover, as the map shows, the number of rainy days has increased for southern coastal areas and some internal and northeastern areas of Iran. In southern and southeastern regions, the occurrence of heavy rainfall has increased from 12 days to 56 days. Super-heavy rainfall also has a similar pattern during this period; but, the bodies that covered by this two types of rainfall has reduced. Furthermore, during this period, for parts of Zahedan, Yazd, and the central desert of Iran, the number of days with heavy rain has reduced from 12 to 6 and the number of super-heavy rainfall days has decreased from 6 to 2 days. The hotspots for heavy (773 days) and super-heavy (411 days) rainfall in Gilan and Mazandaran are limited to Rasht, Anzali, and Sari. In general, the anomaly map illustrates that the proportion of days with heavy rain has reduced in 46.9% of the country's area and risen in 51.6%. The increase has reached a maximum of 153 days of heavy rainfall. Interestingly, this increase is seen in the southern part of the country and its hotspot is located around the province of Fars. At the same time, the core focus of the negative anomaly, with a maximum of -297 days, is in the northern part of the country. With respect to super-heavy rainfall, 37.5% of the country has experienced a negative anomaly of -174 days, with the hotspot being located in the northern regions. In contrast, 57.5% of the country has experienced a positive anomaly of 110 days, with the hotspot being located in southern areas, especially Fars (Fig. 5). This decline in northern parts and increase in southern parts may be attributable to the way precipitation systems enter the country.
As the anomaly map of the third decade (1992-2001) shows (Fig. 5), the frequency of heavy and super-heavy rainfall has considerably increased in comparison with the second period (1982-91). The maximum frequency of heavy rainfall (944 days) and super-heavy rainfall (599 days) is concentrated in Gilan (around Anzali). The important point is that, compared with the previous decade, most of the southern parts of the country has experienced a reduction in the frequency of rainy days. In this decade, about 56.2% of the total area of the country, including the southern slopes of the Alborz Mountains, western slopes of the Zagros Mountains, and vast areas of inner parts and southern parts of Iran, has experienced a frequency of less than 100 days of heavy rainfall. However, 92.8% of the area of the country has experienced less than 100 days of super-heavy rainfall. As is evident from the maps, the areas that have had over 100 days of heavy and super-heavy rainfall can be found mainly in northern coastal areas (next to the Caspian Sea), in the Zagros heights, and in the western and northwestern parts of the country.
(Qashqai, 1996) suggested that the frequency of heavy and super-heavy rainfall in coastal areas of the Caspian Sea, especially in Gilan, can be attributed to migratory anticyclones. He suggested that the Siberian high can cause heavy rainfall over the region, only if a core pressure of 1035 hPa is shaped over the Caspian Sea and, at the 500 hPa level, a deep trough is placed on the area (Qashqai, 1996). Moreover, according to previous research, the frequency of such rainfall over the Zagros heights can be explained based on the fact that westerly winds enter the country from this area and are blocked by the mountains——a phenomenon that causes rainfall in this region. Interestingly, with respect to the frequency of heavy rainfall, negative anomalies decrease in comparison with the previous decade. This kind of anomaly covers only 36.8% of the country's area (Table 1). However, with respect to the frequency of super-heavy rainfall, 45.2% of Iran has experienced a negative anomaly. Along the southern coasts of the country, both heavy and super-heavy rainfall show an inverse pattern in comparison to the previous period. Positive anomalies cover 61.5% of the area of the country for heavy rainfall, and 48.5% for super-heavy rainfall. This kind of anomaly mainly occurs in the northern, northwestern and western parts of the country.
The spatial distribution of the frequency of heavy and super-heavy rainfall in the period 2002-11 indicates that, similar to the previous decade, the frequency along the southern coast of the Caspian Sea, in the northwest, and over the Zagros Mountains, is higher than in other parts of the country. However, compared to the previous three decades, this decade experiences a smaller scope, with 55.3% and 90.7% of the area of the country having heavy and super-heavy rainfall, respectively, with a frequency of occurrence of less than 100 days (Table 2). This means that only 9.3% of the area of the country has experienced super-heavy rainfall for over 100 days and only 44.7% for more than 100 days. Additionally, the anomaly map indicates that the negative anomaly for heavy rainfall can be mainly found in the northern parts, with its hotspot (136 days) located in Gilan, and in the west, northwest, northeast, and southeast of the country. It is thus distributed across the country and covers an area of 50.1% (Fig. 5). Positive anomalies, on the other hand, cover 47.5% of the country's area and can be observed in southern coastal regions, inner parts of the country, in the east of the country, and in the heights of the southern Zagros Mountains. Positive anomalies can also be observed as sporadic cells in northwestern and western parts of Iran. With respect to super-heavy rainfall, negative and positive anomalies can be observed in 41.9% and 51.7% area of the country, respectively (Fig. 5). Like heavy rainfall, negative anomalies can be seen in coastal areas in the north, as well as parts of the northwest and southwest. Positive anomalies, on the other hand, can be detected over the Zagros Mountains, coastal areas of the south, and some parts of the northwest and west. Its hotspot (136 days of heavy rainfall and 159 days of super-heavy rainfall) is located near Fars.
To investigate the changes in heavy and super-heavy rainfall within the four decades, the spatial autocorrelation (hotspot index) is employed. The results are presented in Figs. 6 and 7. The Gi statistic, which is calculated for each region, is a type of Z-score. With respect to positive and statistically significant Z-scores, the larger the Z-score, the higher the clustering of values, and hence the formation of hotspots (in other words, having a positive spatial autocorrelation). With respect to negative and statistically significant Z-scores, the lower the Z-score, the greater the clustering of lower values (hence, negative autocorrelation), which indicates "coldspots". Figures 8 and 9 display the results of the spatial analysis of hotspots for heavy and super-heavy rainfall during the four periods.
Figure 5 indicates that, in all the periods, 99% of the coastal areas of the Caspian Sea and the Zagros Mountains have experienced heavy rainfall, hence forming high cluster patterns or positive spatial autocorrelation. A less strong hotspot can be observed in 95% of the areas around this focal point (Fig. 6). Therefore, based on the hotspot model, the probability of heavy rainfall in these areas is very high. Negative spatial autocorrelation at the level of 99% and 95% for all the periods can be mainly observed in southeastern and central parts of Iran. It is thus concluded that these areas experience less in terms of heavy rainfall.
Comparative analysis of the autocorrelation patterns of heavy rainfall in periods of heavy rainfall spatial displacement indicates that, although the oscillation is small, considerable changes can be seen in terms of the level of significance. For example, in the first period, 13.4% of the country's area has a pattern of negative spatial autocorrelation at the 95% level, while in the second period, 3.7% of the total area of the country has a low cluster model at the 95% level (Table 3). Overall, it can be said that, on average, 21% of the area of Iran has experienced a heavy rainfall pattern in low clusters at the 95% and 99% levels. Thus, 13% of the country's area has had a positive spatial autocorrelation (high clustering pattern). In general, in almost all the periods, around 62% of the country's area does not follow any specific pattern when it comes to heavy rainfall (Fig. 4). The annual pattern of heavy rainfall is similar to that on the decadal scale.
Figure 7 shows the spatial distribution pattern of super-heavy rainfall. Similar to the heavy rainfall pattern, the super-heavy rainfall pattern is a highly clustered one (positive spatial autocorrelation) along the Caspian coast and over the Zagros Mountains. The only difference is that, in comparison with heavy rainfall, a lower area is affected by super-heavy rainfall. For example, in all the periods, approximately 10.5% of the area of the country experiences super-heavy rainfall (positive spatial autocorrelation). This shows a 2% decline compared to heavy rainfall (Table 3).
In contrast to the patterns of heavy rainfall, negative spatial autocorrelation patterns of super-heavy rainfall are significant only at the 95% level. They mainly include central and southeastern parts of the country, particularly in Zabul (Fig. 8). Therefore, this part of the country has a very low probability of heavy rainfall occurrence. In the first and third periods, respectively, 7.7% and 1.6% of the area of the country has heavy rain with a low cluster pattern (negative spatial autocorrelation). Similar to heavy rainfall, super-heavy rainfall does not follow any specific pattern in almost 75% of the country's area (Fig. 9). In the first and third periods, negative spatial autocorrelation patterns show a significant decline. In the other two periods, however, they show insignificant fluctuations. As a result, heavy rainfall events form strong clusters only in parts of the west, northwest, and shores of the Caspian Sea, while they form weak clusters in central parts and the east of the country.
For investigating the relationship between two variables, the first logical step is to plot the data as points in a coordinate system, i.e. as a scatterplot. One of the aims of this model is exploring the relationship between different variables and the way they are influenced by each other. Generally, changes in one variable may lead to changes in the dependent variable. If the nature of the relationship between independent and dependent variables is clear, one can infer a model of the way the independent variables influence dependent ones. Furthermore, the dependent variables can be predicted based on the independent variable. To make accurate predictions, one can use the line of best fit of the regression analysis. The line of best fit indicates an equation for calculating one variable based on another. This line offers the means to calculate changes in a dependent variable based on one or more independent variables. In other words, the line is drawn along the greatest change observed in the plot. Figure 10 shows scatterplots and the lines of best fit for the relationship between heavy and super-heavy rainfall. Based on the Gi statistic, a significant direct relationship can be observed between heavy and super-heavy rainfall during all periods with a 95% confidence interval. This suggests that heavy and super-heavy rainfall events are interdependent, so an increase in heavy rainfall leads to a rise in super-heavy rainfall, and vice versa (Fig. 10).