Uncertainty Analysis of Bivariate Drought Frequency Curves Based on Copula Functions

Abstract

Compared with other natural hazards, the impacts of drought are generally very difficult to quantify, and nonstructural and very complex phenomena both in terms of definition and causes. A precise and objective definition of drought is lacking and it sometimes contributes to indecision and inaction on the part of managers, policy makers, meteorologists, hydrologists, agricultural scientists, and others. Differences in hydrometeorological valuables, socioeconomic factor, and the stochastic nature of water demands in different regions have become an obstacle to having a precise definition of drought. However, the drought is usually defined as a relative term. The onset of meteorological drought is the first stage of drought disaster and is the driving force for the other drought categories, such as agricultural drought, hydrological drought, and social economic drought. The univariate frequency analysis of drought usually faces with the problem of discrepancies between drought duration and severity which make the use of univariate characterization not to be desirable. This study performed the bivariate drought frequency analysis for duration and severity of drought, using copula functions which allow considering the correlation structure of joint features of drought. And then, this study suggested the confidence intervals of duration-severity-frequency (DSF) curves for the given drought duration using stochastic scheme of monthly rainfall generation for the 57 sites in South Korea. The return periods of drought duration were compared with those of drought severity based on the univariate frequency analysis for historical maximum drought. The results showed the inconsistency of return periods between drought duration and severity, which indicates that bivariate drought frequency analysis is necessary for more consistent drought frequency analysis. The bivariate drought frequency analysis in this study allowed the quantification of various properties of extreme drought events for each site. In this study, the frequency curves were derived and used to determine return periods of severity given duration for historical maximum drought event. This study also investigated drought risk via the largest drought events on record over 50 and 100 consecutive years. It appears that drought risks are much higher in some parts of the Nakdong River basin, southern and east coastal areas. Generally, the main cause of uncertainties of hydrologic frequency analysis is associated with the lack of sample data. So far, the bivariate drought frequency analyses have been performed mostly based on historical drought events using observed monthly rainfalls. However, such analyses are not always reliable, especially when the frequency analysis is performed based on the data observed over relatively short period of time. For this reason, there is a need to develop of confidence interval within which the estimated drought severity for a given duration and a given recurrence interval can be considered reliable. This study applied two stochastic schemes in multivariate regression model; copula-based random generation and semi-nonparametric generation. The synthetic data were used to quantify the uncertainty of drought frequency curves. As a result, the droughts were filtered by different durations. Then, the 5%, 25%, 50%, 75%, and 95% quantiles of the drought severity for a given duration were estimated based on the simulated rainfall time series. Also, the 95% of the drought severity was encapsulated by the region delineated by the upper and lower confidence limits which represent the 2.5% and 97.5% quantiles of the drought severity. In this study, the bivariate return periods were estimated for a given duration (3months, 6months, 9months, and 11months) and the 95% confidence interval at each site. Five sites such as No.2 (100), No.7 (114), No.9 (127), No.13 (133), and No.31 (211) turned to have significantly large uncertainties in bivariate frequency curves. Also, we can find the drought hazard region there appears to be a short bivariate return period, and it means that drought risks are much higher in some parts of the Nakdong River and Han River basin, southern and east coastal areas. Finally, the reason for growing uncertainties was addressed in this study, in that the estimation of the joint probability using the two marginal distributions that the correlation coefficient of two variables is relatively low. In addition, incorporating climate change variation into precipitation sequences in a systematic manner remains non-feasible. In this regard, this study needs to quantify the impact of climate change on an extreme drought event to the drought forecasting. Therefore, further research is currently being carried out to quantify the impact of climate change on drought, after collecting the RCP 4.5 scenario precipitation provided by the Korean Meteorological Agency and the National Institute for Meteorological Research for the period of 1976–2005, 2011–2040, 2041–2070, and 2071–2100. In future study, this study will analyze the uncertainty of drought based on climate change scenario. It is presumed that there are different drought risks of 2011-2040 to 1976-2005, 2041-2070 to 1976-2005, and 2071-2100 to 1976-2005 for the largest drought events based on the RCP 4.5 scenario precipitation. The approaches proposed in this study made it possible to find cause and extent of the uncertainty of bivariate drought, that confirm a possible use of bivariate copula function in the frequency analysis considering uncertainty to project future drought risks. |일반적으로 가뭄은 다른 자연재해와 비교하여 복잡한 특성을 갖고 있으며, 실제 가뭄관리 담당자들은 자신들의 목적에 부합되는 다양한 방법을 이용하여 가뭄을 정의한다. 그 중 강수 및 하천유량 등의 수문기상학적 변량은 기상 및 수문학적 가뭄을 판단하기 위한 지표로 널리 사용되고 있다. 특히 강수의 부족은 가뭄의 주된 요인이라 할 수 있으며, 가뭄의 정량적 평가에 효과적으로 이용될 수 있다. 본 연구에서는 이변량 확률분포함수를 적용하여 가뭄빈도해석을 수행하였으며, 가뭄 특성(가뭄 지속기간과 심도)의 상호 관계를 고려하여 지역적 가뭄특성을 종합적으로 판단하였다. 또한 단변량 가뭄해석의 한계점을 극복하기 위한 방안으로 이변량 가뭄해석을 수행했으며, 이를 위해 코플라 함수를 적용하였다. 가뭄 발생의 확률 및 경향성을 종합적으로 나타내어 줄 수 있는 결합 확률밀도함수를 추정한 후, 지점별 가뭄빈도해석 및 과거 최대가뭄사상에 대한 단변량 및 이변량 재현기간을 산정하고 비교•분석하였다. 또한, 우리나라의 과거 최대가뭄사상에 대한 가뭄위험도분석을 위해, 연속되는 50년과 100년 동안 최소 한번 발생하는 확률(과거 최대가뭄사상 크기의 가뭄)을 강우관측지점별로 계산하여 가뭄위험지역을 예상하였다. 일반적으로 우리나라와 같이 강수자료의 기록연한이 짧은 경우에는 이변량 가뭄빈도해석의 수행에 큰 불확실성을 야기할 수 있다. 이러한 경우에는 추계학적인 모의발생기법을 이용하여 분석자료를 확충한 후 가뭄해석을 수행한다. 이와 같은 모의강수자료는 미래에 발생 가능한 시나리오를 제공함과 동시에 가뭄발생의 불확실성을 정량화하는 데 매우 중요한 역할을 하고 있다. 따라서 월강수량 모의기법의 정확성 및 적합성은 가뭄해석 결과에 대한 신뢰성 확보에 결정적인 역할을 한다. 본 연구에서는 가뭄분석 시 사용되는 월단위 강수자료를 모의발생하기 위해, 두 가지 강수모의기법을 적용하였으며, 이 중 반-비모수 방법을 활용하여 모의강수자료를 구축한 후, 이변량 가뭄빈도해석을 수행하였다. 이를 위해 가뭄 지속기간과 심도의 상호 의존구조를 충분히 반영하여 해석할 수 있는 코플라 함수를 적용하였다. 그 결과, 앞서 관측된 가뭄사상으로 추정된 이변량 가뭄빈도곡선에 대한 95%, 90%, 75%, 50%의 신뢰구간을 제시할 수 있었다. 또한 가뭄 지속기간(2개월∼8개월)과 심도에 해당하는 95% 신뢰구간의 이변량 가뭄재현기간의 경계값(상한값 및 하한값)을 추정하였다. 이를 바탕으로 5개의 관측지점(#100, #114, #127, #133, #211)이 상대적으로 큰 가뭄빈도곡선의 불확실성을 가지는 것으로 나타났다. 이에 대한 원인분석을 위해, 불확실성이 큰 5개 지점에 대한 가뭄 특성 및 빈도해석의 연구결과를 심층적으로 검토하였다. 그 결과 불확실성의 원인은 가뭄빈도해석 시 고려되었던 두 변량에 대한 낮은 상관성으로 인해 확률적인 방법으로 결합분포모형을 추정하는 데 있어 발생한 불확실성인 것으로 확인되었다. 본 연구에서는 미래 기후변화에 의한 가뭄의 불확실성에 대한 연구를 향후에 지속적으로 수행하고자, 기상청에서 제공되는 기후변화 시나리오(RCP4.5)를 이용한 가뭄특성분석을 수행하였다. 그 결과 미래 2011년∼2040년은 우리나라의 낙동강 유역의 일부와 강원도 산간 지역에 가뭄위험이 크게 나타나는 반면, 미래 2041년∼2070년은 한강 유역의 대부분과 영산강 유역에 가뭄위험이 클 것으로 예상되었다. 미래 2071년∼2100년에는 우리나라의 남부 지역에서 가뭄위험이 상대적으로 클 것으로 예상되었다. 이러한 결과는 지역적으로 장기적인 가뭄대비계획을 수립하는데 중요한 참고자료가 될 것이다.; copula-based random generation and semi-nonparametric generation. The synthetic data were used to quantify the uncertainty of drought frequency curves. As a result, the droughts were filtered by different durations. Then, the 5%, 25%, 50%, 75%, and 95% quantiles of the drought severity for a given duration were estimated based on the simulated rainfall time series. Also, the 95% of the drought severity was encapsulated by the region delineated by the upper and lower confidence limits which represent the 2.5% and 97.5% quantiles of the drought severity. In this study, the bivariate return periods were estimated for a given duration (3months, 6months, 9months, and 11months) and the 95% confidence interval at each site. Five sites such as No.2 (100), No.7 (114), No.9 (127), No.13 (133), and No.31 (211) turned to have significantly large uncertainties in bivariate frequency curves. Also, we can find the drought hazard region there appears to be a short bivariate return period, and it means that drought risks are much higher in some parts of the Nakdong River and Han River basin, southern and east coastal areas. Finally, the reason for growing uncertainties was addressed in this study, in that the estimation of the joint probability using the two marginal distributions that the correlation coefficient of two variables is relatively low. In addition, incorporating climate change variation into precipitation sequences in a systematic manner remains non-feasible. In this regard, this study needs to quantify the impact of climate change on an extreme drought event to the drought forecasting. Therefore, further research is currently being carried out to quantify the impact of climate change on drought, after collecting the RCP 4.5 scenario precipitation provided by the Korean Meteorological Agency and the National Institute for Meteorological Research for the period of 1976–2005, 2011–2040, 2041–2070, and 2071–2100. In future study, this study will analyze the uncertainty of drought based on climate change scenario. It is presumed that there are different drought risks of 2011-2040 to 1976-2005, 2041-2070 to 1976-2005, and 2071-2100 to 1976-2005 for the largest drought events based on the RCP 4.5 scenario precipitation. The approaches proposed in this study made it possible to find cause and extent of the uncertainty of bivariate drought, that confirm a possible use of bivariate copula function in the frequency analysis considering uncertainty to project future drought risks.