Mathematics Supporting the Theory of Global Warming
In California, summers are hot, so wintertime is used to store water supplies for the rest of the year but with temperatures rising and we as humans ignoring this effect, we underestimate the risk of extreme events. The increase in global temperatures significantly increases the chances of weather extremes like droughts and heatwaves occurring. For example, in 2014, California experienced an extensive heatwave and low precipitation. Although climate change does not cause droughts, it intensifies the severity of a drought. Since we are ignoring these extremes, studies using multivariate copulas have been used for assessing the relationship between climate variables and extremes. Multivariate copulas are important because they can be used for deriving probability occurrence and return period of dependent variables. In California during 2014, the state experienced not only drought that the state has yet to recover fully from, but it also experienced high temperatures, wildfires, and even snow drought. Researchers formed data by calculating monthly precipitation and temperature values from 1896-2014 (AghaKouchak 1).
Using Weibull’s approach, the researchers sorted from the most extreme to the least extreme with a univariate return period of an m-ranked extreme event in an N year. So, it is estimated as T=(N+1)/m but the concurrent extreme return period analysis is based on the concept of copulas designed to model the dependence of multiple variables. The researchers assumed two variables to be X (precipitation) and Y (temperature) with the cumulative distribution functions F_X (x)=Pr(X≤x) and F_Y (y)=Pr(Y≤y) and using the copula (C) to obtain their joint distribution function: F(x,y)=C(F_X (x), F_Y (y)). In other words, the joint distribution of X and Y is F(x,y)=Pr(X≤x, Y≤y). Now the researcher continues to solve and from the joint distribution function, obtains the joint survival distribution using the concept of survival copula giving the function F ̅(x,y)=C ̂(F ̅_X (x), F ̅_Y (y)) where F ̅_X and F ̅_Y are the marginal survival functions of X and Y, and (C ) ̂ is the survival copula. Next, the researcher acknowledges that there is a set of realizations of X and Y that share the same probability. The survival return period of X and Y is defined as follows K ̅_XY=µ/(1-K ̅(t)) where this is defined as Kendall’s return period assuming µ >0. K ̅ is the Kendall’s survival function defined as K ̅(t)=Pr(F ̅(X, Y)≥t)and referencing the survival copula, Pr(C ̂((F_x ) ̅(x),(F_Y ) ̅(y))≥t). Finally, by inverting Kendall’s survival function associated with K ̅(t) at the probability p=1-µ/T, the corresponding survival critical layer can be estimated q ̅=q ̅(p)=K ̅^(-1) (p) where q ̅ is the survival Kendall’s quantile of order p. Now that we have Kendall’s quantile, the corresponding survival critical layer presents the set of realizations sharing a joint return period T (AghaKouchak 2). In addition, various copula groups are sufficient in finding the return period analysis. In this analysis, the author uses the t copula that led to a significant p-value that provided 95% confidence (AghaKouchak 2-3).
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On the other hand, California has experienced extreme droughts that were worse than the one in 2014 including the worst one recorded in 1977. This argument supports the theory that climate change does not cause droughts but makes them more severe because, in 2014, the mean temperature of California was over 10°C which only happened in 1940 and 2013. Furthermore, the year 2014 was the highest recorded mean temperature in California. In January 2014, California experienced a heat wave that gave the state summer weather during one of the coldest months of winter usually experienced by California (AghaKouchak 3). The problem is when an extreme condition occurs combined with another nonextreme condition, it is possible that the compound extreme event could have significant impacts on the ecosystem and environment (AghaKouchak 3-4).
What happens when the phenomenon occurs? The more severe the extreme condition combined with a nonextreme condition, the longer effect it has on the ecosystem. Using the survival copula, talked about before, the conditions in 2014 appear to be a 200-year extreme event which means that this event has the probability of a .5% chance of ever happening again. However, the 1977 extreme event was originally classified as a 120-year event but turned out to be only a 50-year event (AghaKouchak 4). Therefore, the original definition of return period is used to determine the risk of an extreme event, but the purpose of this article is that due to global warming, the chances of concurrent droughts and heat waves increase! Therefore, the true probability of the 2014 event happening again could be much sooner than we realize (AghaKouchak 5). Considering the 2014 California drought and showing the univariate return period analysis based on the precipitation (which is used in hydrology ) drastically underestimates the occurrence probability of the 2014 California drought because they do not consider global temperature. This conclusion is extremely significant because, for regions like California where a drying trend continues to occur, it could have impacts on the ecosystem, water availability, energy production, and agriculture industry. The largest effect of looking for a solution is that farmers are demanding water supply so that they can feed the nation clash (AghaKouchak 5). In California, droughts are increasing and having an enormous effect on the ecosystem but simultaneously the occurrence of wildfires continues to grow as well.
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