كاربرد تئوري كوپلا جهت ارزيابي عملكرد مدل هيدرولوژيكي IHACRES (مطالعۀ موردي: حوضۀ آبريز طالقان)

Application of Copula theory for IHACRES hydrologic model evaluation (Case study: Taleghan watershed)

شرط اصلي استفاده از يك مدل هيدرولوژيك در يك حوضۀ آبخيز، ارزيابي عملكرد آن مدل است. در تمام معيارهاي معمول در هيدرولوژي از جمله ضريب ناش- ساتكليف (E)، ضريب همبستگي (R)، مجذور ميانگين مربعات خطا (RMSE) و شاخص تطابق (d) به مقايسۀ خروجي مدل با داده‌هاي مشاهده‌اي پرداخته مي‌شود و از توزيع احتمالاتي خروجي مدل و رابطۀ آن با توزيع مقادير ورودي غفلت مي‌شود. اين مسئله به‌ويژه زماني كه هدف، استفاده از خروجي مدل در تحليل فراواني هيدرولوژيك باشد، حائز اهميت است. در اين مطالعه كاربرد جديدي از تئوري كوپلا جهت ارزيابي عملكرد مدل در شبيه‌سازي صحيح رابطۀ احتمالاتي بارش-رواناب معرفي مي‌شود. بدين منظور مدل هيدرولوژيكي IHACRES انتخاب گرديد و جهت شبيه‌سازي دبي روزانۀ حوضۀ آبخيز طالقان به كار رفت. مدل مذكور در دورۀ پنج‌سالۀ 1995-2000 واسنجي شد و سپس جهت شبيه‌سازي دبي روزانۀ دورۀ پنج‌سالۀ 2000-2005 به كار رفت. نتايج ارزيابي عملكرد مدل با استفاده از معيارهاي ارزيابي عملكرد معمولي از جمله ضريب ناش- ساتكليف (0/75)، ضريب همبستگي (0/87)، RMSE (7/2) و شاخص تطابق (0/93) بر عملكرد مطلوب مدل دلالت دارد؛ با اين حال برازش توابع كوپلا به داده‌هاي شبيه‌سازي‌شده و مشاهده‌اي بارش-رواناب نشان داد توابعي با خصوصيات متفاوت به داده‌هاي شبيه‌سازي‌شده و مشاهده‌اي برازش يافتند. تابع كوپلا كلايتون و گوس با دارابودن حداقل مقادير AIC، BIC و LL به‌ترتيب به داده‌هاي مشاهده‌اي و شبيه‌سازي‌شده برازش داده شدند كه دليل اين امر ناتواني مدل در شبيه‌سازي دبي‌هاي حداكثر است كه در انتخاب نوع تابع كوپلا نمود پيدا كرده است.

Hydrologic modeling plays an important role in hydrologic response prediction for water resources managements, flood control and soil and water conservation. Performance evaluation is necessary before using a hydrologic model in a watershed. Various traditional evaluation criteria such as Nash-Sutcliff (E), correlation coefficient (R), Root Mean Square Error (RMSE) and index of agreement (d) are commonly used for the performance evaluation of hydrologic models. All these criteria compare the model output with observed data, however they cannot capture all the features reproduced by the hydrologic models such as information content of data and stochastic relationship between rainfall and run-off. This is doubly important when carrying out frequency analysis on the model output. This study aims at introducing a new application of copulas for the performance evaluation of hydrologic models in accurate simulation of stochastic relationship between rainfall and runoff. To this end, IHACRES hydrologic model was selected for daily flow simulation in the Taleghan watershed. The selected model was calibrated for the 5-year period of 1995-2000 and evaluated for 5-year period of 2000-2005. The non-linear module calibration optimized the parameters C, τw and F as 0.002 mm, 22 days and 4 degree Celsius respectively. The linear module parameters was also optimized as τq = 0.02 days, τs = 41.25 days and vs = 1. Performance evaluation of the model via Nash-Sutcliff (E=0.75), Pearson correlation coefficient (R= 0.87), root mean square error (RMSE=7.2) and index of agreement (d=0.93) indicated a good performance of the model for the evaluation period. Although the numerical performance evaluation criteria show satisfying results, however visual inspection of the scatter plot of observed and simulated flow showed that the model significantly underestimated the peak flows in spring. In contrast, the winter flow rate is noticeably overestimated. Regarding the Mediterranean climate of the study area, the greater portion of the winter precipitation falls as snow which piles up in mountains without draining into the river network. With the onset of spring and warming weather, the snow pack melting accompanied with spring precipitations lead to the peak flow generation in spring. As the snowmelt simulation module is not included in the model, the model considered all precipitation as rain in January, February and March. This lead to the overestimation of winter flow rate. On the other hand, the spring peak flows is underestimated because of neglecting the snowmelt runoff. In the next step various copula functions with different tail dependence structure in the upper and lower tail including Archimedean copulas (Clayton, rotated Clayton, Gumbel, rotated Gumbel, Frank) and elliptical copulas (Gaussian, t-copulas) were fitted on observed and simulated rainfall-runoff time series. Various goodness-of-fit test criteria AIC, BIC and Log likelihood (LL) were employed to choose the best fit copula functions for the observed and modeled time series. Results showed that the Clayton copula with the lowest AIC, BIC and LL values (AIC = –236.28, BIC = –236.28 and LL = –118.13) best fitted observed rainfall-runoff time series. Gaussian copula with lowest AIC, BIC and LL values (AIC = –217.08 BIC = –217.08 and LL = –108.54) was also selected as the best fitted copula for modeled rainfall-runoff time series. Nevertheless, the poor performance of the model in simulating the spring peak flows due to snowmelt runoff, the model has been approved by traditional performance evaluation criteria. However fitting the observed and simulated rainfall-runoff time series with Clayton and Gaussian copulas with different tail dependencies indicated inability of the model to properly simulate stochastic relationship between rainfall and runoff.