Preliminary Estimations of Precipitation
Based on the observed precipitation, the mean annual precipitation (Pann, mean) was 423.0 mm over the TRHR during 1961–2014, and the annual precipitation showed a significant increasing trend with the change rate of 7.11 mm/decade at the significance level of p < 0.1. The maximum annual precipitation (i.e., 514.8 mm) occurred in 1989, while the minimum one (i.e., 362.9 mm) occurred in 1969 (Fig. 3a). Moreover, the annual precipitation showed a decreasing trend with the change rate of −2.6 mm/decade during 1961–2002 but a significant increasing trend with the change rate of 19.3 mm/decade during 2003–2014 (Shi et al. 2016). Therefore, it was infeasible to calibrate and validate the statistical method for computing the preliminary estimations of precipitation; instead, to present the overall trend during 1961–2014 which would be regarded as the trend during 2015–2050, all the data during 1961–2014 were used to make the preliminary estimations of precipitation be close to the observed precipitation as much as possible. Following the method proposed in Subsection 2.2.1, the preliminary estimations of annual precipitation were computed. Based on the linear regression method, the tendency part of annual precipitation (i.e., Pann, trend) could be preliminarily estimated as follows:
$$ {P}_{\mathrm{ann},\mathrm{trend}}=0.711\cdot Y-990.13 $$
(11)
Then, for each year during 1961–2014, the difference between the observed annual precipitation and the tendency part was calculated, and the result (i.e., Sdiff) is shown in Fig. 3b. It is observed that the differences varied between −61.32 mm and 90.74 mm, with the mean of 0.0085 mm and the standard deviation of 36.69 mm. Moreover, the distribution of Sdiff was checked by Kolmogorov-Smirnov test at the significance level of p < 0.05, which confirmed that Sdiff obeyed the normal distribution. Thus, the variation part of annual precipitation (i.e., Pann, var) for each year during 1961–2014 could be preliminarily estimated, and the preliminary estimation of annual precipitation (Pann, est) for each year during 1961–2014 was computed as the sum of Pann, trend and Pann, var.
It is worth noting that random numbers were generated for Pann, var, which might bring uncertainty in estimation; thus, one set of Pann, var would not be enough to represent the performance of the statistical method for computing the preliminary estimations of precipitation. In this study, ten sets of Pann, var were generated, and thus ten sets of Pann, est were obtained. For each year during 1961–2014, there were ten Pann, est values, among which, the mean, maximum and minimum Pann, est values were selected, respectively. The left part of Fig. 4 shows the preliminary estimations of annual precipitation during 1961–2014, which would be used for calibration and validation of the proposed recursive approach in the next subsection. The observed annual precipitation during 1961–2014 (marked by “+”) could basically be covered by the range between the maximum and minimum Pann, est values (i.e., the solid lines), and the mean Pann, est values (marked by “o”) generally showed the increasing trend.
In this study, the mean Pann, est values were used to compute the preliminary estimation of monthly precipitation for each month of a year (Pmon, est, i, i = 1, 2, ..., 12). According to the method proposed in Subsection 2.2.2, the percentage of the mean monthly precipitation for each month of a year (PCTmon, mean, i, i = 1, 2, ..., 12) should be firstly computed. Based on the observed precipitation, the mean monthly precipitation for each month of a year (Pmon, mean, i, i = 1, 2, ..., 12) during 1961–2014 was obtained (Table 2). It is observed that the maximum value of the mean monthly precipitation (i.e., 92.7 mm) occurred in July, followed by the mean monthly precipitation in June (i.e., 81.3 mm) and August (i.e., 78.4 mm), and the minimum value (i.e., 1.9 mm) occurred in December. Because the mean annual precipitation (Pann, mean) was 423.0 mm during 1961–2014, the percentage of the mean monthly precipitation for each month of a year (PCTmon, mean, i, i = 1, 2, ..., 12) could be computed (Table 2). Then, for each year during 1961–2014, the preliminary estimation of monthly precipitation for each month (Pmon, est, i, i = 1, 2, ..., 12) was computed using Eq. (3).
Table 2 The mean monthly precipitation and the percentage accounted for by the mean monthly precipitation for each month of a year during 1961–2014 Figure 5 shows the comparison of the preliminary estimations of monthly precipitation (marked by “□”) against the observations for each month of a year during 1961–2014. It is observed that the preliminary estimations were basically close to the observations and uniformly distributed on both sides of the red dash line, and the NSCE value was relatively high (i.e., 0.9195). However, there were more underestimated preliminary estimations in the case of large observations. The result of residual analysis (marked by “□” in Fig. 6) showed that the standardized residuals of the preliminary estimations were generally within ±2 when the preliminary estimations were smaller than 60 mm, but a number of points were beyond the range of ±2 when the preliminary estimations were larger than 60 mm (Fig. 6). This could also be represented by the significant deviations of the points from the red dash line in Fig. 5. It is worth noting that the preliminary estimations larger than 60 mm mainly appeared in the wet season, especially from June to September. Therefore, the preliminary estimations of monthly precipitation should be further improved, especially for those in the wet season.
Improved Estimations of Precipitation
In this study, GP was considered as the optimization method for improving the preliminary estimations of monthly precipitation, and the main GP relevant parameters included size of population (100), number of generation (100), tournament size (6), and operator functions (times, minus, plus, sqrt, square). As mentioned above, the observations and preliminary estimations during 1961–2000 were used for calibration of the proposed method, while the remaining data during 2001–2014 were used for validation. Before optimization, the NSCE values derived from the observations and preliminary estimations (i.e., Case I) were 0.9190 and 0.9205 for the two periods of 1961–2000 and 2001–2014, respectively (Table 3). Following the procedures in Subsection 2.3.2, the optimization results were obtained and presented as follows.
Table 3 The NSCE values of the three cases Calibration
In this study, GP was applied to derive the functional relationship as given in Eq. (5), and the improved estimations of monthly precipitation were calculated using Eq. (6). It is worth noting that two cases of the improved estimations of monthly precipitation (i.e., Case II and Case III) were considered during the calibration period of 1961–2000. Case II denoted that the improved estimations, which were computed considering all the preliminary estimations as a whole, were compared against the observations. Case III denoted that the improved estimations, which were computed considering the preliminary estimations with the standardized residuals beyond ±2 separately, were compared against the observations. It is observed from Table 3 that the NSCE value derived from Case II (i.e., 0.9222) was only a little higher than that derived from Case I (i.e., 0.9190), while the NSCE value derived from Case III (i.e., 0.9520) was much higher than those derived from Case I and Case II. As a result, this study regarded the improved estimations derived from Case III as the better ones and the relevant functional relationships derived from GP were expressed as follows:
$$ {\displaystyle \begin{array}{c}{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}\le 60\ \mathrm{mm}:\\ {}{P}_{\mathrm{mon},\mathrm{imp},\mathrm{t}\_\mathrm{next}}=1.023\cdot {P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-1.023\cdot \mid {P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\mid \\ {}-0.00004854\cdot {P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}^2\cdot \left[{\varepsilon}_{\mathrm{t}}+\left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)\right]\\ {}+0.00141\cdot {\varepsilon}_{\mathrm{t}}\cdot {P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\cdot \mid {P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\mid +0.0001584\\ {}{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}>60\ \mathrm{mm}:\\ {}{P}_{\mathrm{mon},\mathrm{imp},\mathrm{t}\_\mathrm{next}}={P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}+\Big[12.6166\cdot {\varepsilon}_{\mathrm{t}}+71.9757\cdot \left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)\\ {}-0.3897\cdot {\varepsilon}_{\mathrm{t}}^2-10.4948\cdot {\left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)}^2\\ {}-3.9947\cdot {\varepsilon}_{\mathrm{t}}\cdot \left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)-51.3250\Big]\\ {}/\Big[1-0.3686\cdot {\varepsilon}_{\mathrm{t}}-1.9015\cdot \left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)\\ {}+0.04282\cdot {\varepsilon}_{\mathrm{t}}^2+1.1741\cdot {\left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)}^2\\ {}+0.4571\cdot {\varepsilon}_{\mathrm{t}}\cdot \left({P}_{\mathrm{mon},\mathrm{est},\mathrm{t}\_\mathrm{next}}-{P}_{\mathrm{mon},\mathrm{est},\mathrm{t}}\right)\Big]\end{array}} $$
(12)
Figure 5 shows the comparison of the improved estimations of monthly precipitation derived from Case III (marked by blue “×”) against the observations for each month of a year during the calibration period of 1961–2000. The improved estimations were closer to the observations than the preliminary estimations (marked by “□”), especially for those larger than 60 mm. The result of residual analysis on the improved estimations (marked by blue “×” in Fig. 6) showed that most of the standardized residuals were within ±2 even if the improved estimations were larger than 60 mm (Fig. 6). Moreover, it is worth noting that there were more improved estimations than preliminary estimations which were distributed between 90 mm and 120 mm, especially for those around 120 mm. This indicated that the larger observations could be better estimated by the improved estimations obtained from the proposed method.
Validation
Based on Eq. (12), the improved estimations of monthly precipitation for each month of a year during the period of 2001–2014 could be computed, and then, the validation of the proposed method was conducted. Figure 5 also shows the comparison of the improved estimations of monthly precipitation (marked by orange “·”) against the observations for each month of a year during the validation period of 2001–2014. The result indicated that the improved estimations were closer to the observations than the preliminary estimations (marked by “□”), and the distribution of the improved estimations during the validation period was similar to that during the calibration period. Moreover, Fig. 6 shows that most of the standardized residuals of the improved estimations during the validation period (marked by orange “·”) were within ±2, similar to those during the calibration period. Table 3 lists the NSCE values of the three cases (i.e., Case I, Case II and Case III), and the results derived from Case II (i.e., 0.9243) was only a little higher than that derived from Case I (i.e., 0.9205), while the NSCE value derived from Case III (i.e., 0.9517) was much higher than those derived from Case I and Case II.
To further evaluate the performance of the proposed recursive approach in precipitation prediction, additional validation was conducted as follows. For a designated year during the validation period of 2001–2014, the observations in the previous year were substituted by the improved estimations of monthly precipitation in the previous year, and then, the improved estimations of monthly precipitation in the designated year were computed for Case II and Case III, respectively. The relevant NSCE values were 0.9237 for Case II and 0.9524 for Case III, respectively. It indicated that, even when the observations were unavailable and the improved estimations were used as the substitutions of the observations for prediction, the performance of the proposed recursive approach could be as good as that using the observations directly. Therefore, the recursive approach proposed in this study could be adopted for the long-term prediction of monthly precipitation over the TRHR (see the next subsection for details). Nevertheless, the estimations of monthly precipitation could be further updated if the observations became available for subsequent predictions (Khu et al. 2001).
Long-Term Prediction of Monthly Precipitation
After calibration and validation, the proposed recursive approach has been proved to be applicable to long-term prediction of monthly precipitation over the TRHR. In this study, the predictions of monthly precipitation over the TRHR till 2050 were computed based on this recursive approach.
First, the preliminary estimations of annual precipitation were computed. For each year during 2015–2050, the tendency part (Pann, trend) and the variation part (Pann, var) of annual precipitation were estimated, and the preliminary estimation of annual precipitation (Pann, est) was computed as the sum of Pann, trend and Pann, var. Similar to that in Subsection 3.1, ten sets of Pann, var (as well as Pann, est) were generated, and the mean, maximum and minimum Pann, est values for each year during 2015–2050 were selected, respectively. The right part of Fig. 4 shows the relevant results.
Second, the mean Pann, est values were used to compute the preliminary estimations of monthly precipitation for each month of a year during 2015–2050 (Pmon, est, i, i = 1, 2, ..., 12), following the method proposed in Subsection 2.2.2.
Third, the improved estimations of monthly precipitation were calculated using Eq. (12). Since the observations during 2015–2050 were unavailable, the improved estimations of monthly precipitation in the previous year were regarded as the substitution of the observations in the previous year when applying GP.
Figure 7a shows the predictions (both the preliminary estimations and the improved estimations) of monthly precipitation for each month of a year during 2015–2050. It is observed that the series of the improved estimations had the more significant variation than the series of the preliminary estimations. In addition, some months in the wet season (e.g., July 2029 and July 2041) would have much higher values of precipitation than other months. Figure 7b shows the predictions of annual precipitation derived from the improved estimations for each year during 2015–2050. It is observed that the annual precipitation would present an overall increasing trend with significant inter-annual variation. This is reasonable because the same variation of the annual precipitation was found over the TRHR during 1961–2014 (Shi et al. 2016) (see Fig. 3a).
Discussion
It is worth noting that, in this study, ten sets of Pann, est were generated to address the problem that one set of Pann, est would not be enough to represent the performance of the statistical method; therefore, the impact of the number of sets should be discussed. In this section, comparative analysis is conducted between two results from ten sets and twenty sets of Pann, est, respectively. For the result from ten sets of Pann, est, the variation ranges of the mean, maximum and minimum values were 381~469 mm, 406~549 mm, and 320~426 mm, respectively; the averages of the mean, maximum and minimum values were 425 mm, 472 mm, and 375 mm, respectively. For the result from twenty sets of Pann, est, the variation ranges of the mean, maximum and minimum values were 383~468 mm, 445~567 mm, and 294~413 mm, respectively; the averages of the mean, maximum and minimum values were 424 mm, 490 mm, and 358 mm, respectively. It could be concluded that the maximum values would generally increase with larger number of sets, while the minimum values would generally decrease with larger number of sets. In contrast, the number of sets would not significantly influence the mean values. Since the mean values were used to compute the preliminary estimations of monthly precipitation in this study, the impact of the number of sets could be neglected.
Applying the proposed recursive approach for predicting monthly precipitation, we also need fully aware of the following four limitations. First, to some extent, the proposed recursive approach is region-dependent. The study area is located in a semi-arid region, and this approach is applicable. For different regions (e.g., humid regions), the variation characteristics of precipitation may be different, and thus the equations (e.g., Eqs. (11) and (12)) should be reestablished based on the observed data and the estimations. Second, this study adopted the mean Pann, est values for calculating the preliminary estimations of monthly precipitation, and thus, the results were in an average sense. If the maximum Pann, est, the minimum Pann, est, or any one of the ten sets of Pann, est values were used, the results might be different. Therefore, how to determine the most appropriate one is an important issue that is worth investigating in future works. Third, due to the quite different trends of annual precipitation between the two periods of 1961–2002 and 2003–2014, the statistical method for computing the preliminary estimations of precipitation was not calibrated and validated in this study, making the proposed recursive approach not a complete prediction scheme. However, for other regions with a consistent trend of annual precipitation, the proposed recursive approach could be further improved. Fourth, when predicting monthly precipitation over the TRHR during 2015–2050, the variation characteristics of precipitation were assumed to be the same as those during 1961–2014. Therefore, whether the variation characteristics of precipitation would change in the future is another issue that needs further study.
Nevertheless, with the awareness of the above limitations, the recursive approach proposed in this study can provide a new avenue of long-term prediction of monthly precipitation independent of other climatic indices and variables, which would be valuable for hydrology research around the world.