Likelihood-function sensitivity and the non-identifiability of the interflow recession constant $KK_{SS}$ in GLUE uncertainty analysis of the Xin'anjiang rainfall-runoff model

A 2019 GLUE (Generalized Likelihood Uncertainty Estimation) study of the Xin'anjiang daily rainfall-runoff model for the Hongjiata basin (Zhejiang, China) left one explicitly open question and one unexplained reported result: (1) the question — how the choice of likelihood/objective functions and their behavioral thresholds affects the retained behavioral parameter sets, the posterior parameter distributions, and the prediction-uncertainty bounds — the study used only a joint threshold on the relative total-runoff error $L_1$ and the deterministic coefficient $L_2$; and (2) the result — that the posterior of the interflow recession constant $KK_{SS}$ remained approximately uniform over the whole prior range after filtering one million Monte Carlo samples down to 9,375 behavioral sets, so that no preferred value region could be identified; the study neither explained this behavior nor examined whether it can be remedied. No prior study quantifies these effects for the multi-objective $L_1$+$L_2$ scheme used in practice, and none explains the reported near-uniform $KK_{SS}$ posterior or the conditions under which it would concentrate.

Background

The Xin'anjiang (XAJ) model, developed by Zhao Renjun in the 1970s for humid and semi-humid regions of China, is a lumped conceptual rainfall-runoff model whose central assumption is saturation-excess runoff generation: soil-moisture storage capacity is distributed spatially over the basin by a storage-capacity curve, and runoff is produced only where and when that capacity is full (Zhao 1992). In its widely used "three-source" version, the runoff generated on saturated areas is split into surface runoff, interflow (subsurface stormflow), and groundwater, each routed through its own linear reservoir, while evapotranspiration is drawn from a three-layer soil tank. Model behavior is governed by empirical parameters that cannot be measured directly and must be calibrated against observed discharge; they include the evapotranspiration conversion coefficient $K$, the deep-layer evaporation coefficient $C$, the mean tension water capacity $W_M$ (basin-average soil-moisture storage capacity), the free water capacity of the surface soil layer $S_M$ (which controls the partitioning of generated runoff among the three flow paths), the outflow coefficient of free water to groundwater $K_G$, and the recession constant of the lower interflow storage $KK_{SS}$, which sets how quickly the interflow reservoir drains.

Different parameter combinations can reproduce observed discharge almost equally well — the equifinality phenomenon — so single "optimal" parameter sets are unreliable. The GLUE procedure (Generalized Likelihood Uncertainty Estimation; Beven and Binley 1992; Beven and Freer 2001) responds to this by sampling very many parameter vectors from prior ranges, scoring each with one or more informal "likelihood" (goodness-of-fit) measures, retaining the vectors above behavioral thresholds as "behavioral" parameter sets, and reading the distribution of parameter values among the behavioral sets as a posterior distribution quantifying parameter uncertainty. Chen, Ge, Liu and Wu (2019) applied this to the XAJ daily model for the Hongjiata basin (151 km², Zhejiang; 52 years of daily rainfall, evaporation and runoff data, 1966-2018). They drew $10^6$ uniform random samples over the prior ranges of six parameters ($K\in[0.7,1.2]$, $C\in[0.05,0.3]$, $W_M\in[80,200]$ mm, $S_M\in[2,40]$ mm, $K_G\in[0.1,0.9]$, $KK_{SS}\in[0,0.9]$) and retained 9,375 behavioral sets under two simultaneous objectives drawn from the Chinese hydrological standard GB/T 22482-2008: the relative error of total runoff volume,

$$L_1 = \left|\frac{\sum_{i=1}^{N} q_i - \sum_{i=1}^{N} q'_i}{\sum_{i=1}^{N} q_i}\right|\times 100\%,$$

required below 10%, and the deterministic coefficient (Nash-Sutcliffe efficiency, the fraction of observed discharge variance explained by the simulation),

$$L_2 = 1 - \frac{\sum_{i=1}^{N}(q_i - q'_i)^2}{\sum_{i=1}^{N}(q_i - \bar{q})^2},$$

required above 0.7. Five parameters ($K$, $C$, $W_M$, $S_M$, $K_G$) emerged with concentrated, approximately bell-shaped posteriors, but the posterior of $KK_{SS}$ remained approximately uniform over its whole prior range: no preferred value region could be identified, so the parameter could not be meaningfully calibrated for this basin. The authors state that "the effect of setting different objective functions on the results remains to be further explored."

The sensitivity of GLUE results to the likelihood measure has long been recognized in the broader literature: Montanari (2005) analyzed the large-sample statistical behavior of GLUE, Stedinger, Vogel, Lee and Batchelder (2008) critically appraised its statistical foundations, Zhang, Liu and Houseworth (2011) modified GLUE to handle the subjectivity of likelihood-measure selection, and Mirzaei, Huang, El-Shafie and Akib (2015) reviewed the approach. Specifically for the XAJ model, Alazzy, Lü and Zhu (2015) compared four likelihood functions (Nash-Sutcliffe efficiency, normalized absolute error, index of agreement, and the Chiew-McMahon measure) within GLUE in a monthly-streamflow application of that model, and found parameter uncertainty to be more sensitive to the likelihood choice than prediction uncertainty. GLUE-based parameter-sensitivity studies of the XAJ model (Zhang, Zhang, Guan, Chen and Chen 2012) report evident equifinality in both the water-balance and the runoff-routing parameter groups. No study provides a quantification of these effects for the multi-objective threshold scheme actually used in the Hongjiata study ($L_1$ and $L_2$ applied jointly), or an explanation of why the posterior of the interflow recession constant $KK_{SS}$ remained approximately uniform in such calibrations and under which conditions, if any, it would concentrate.

Problem Statement

For the three-source Xin'anjiang daily model (the lumped conceptual rainfall-runoff model, standard in China's humid basins, whose generated runoff is split into surface runoff, interflow, and groundwater components) calibrated by multi-objective GLUE (Generalized Likelihood Uncertainty Estimation: Monte Carlo sampling of parameter vectors, retention of those whose simulated discharge passes goodness-of-fit thresholds, and reading the retained parameter values as a posterior distribution), as instantiated for the Hongjiata basin by Chen et al. (2019) — prior ranges of the six calibrated parameters, namely the evapotranspiration conversion coefficient $K\in[0.7,1.2]$, the deep-layer evaporation coefficient $C\in[0.05,0.3]$, the mean tension water capacity (basin-average soil-moisture storage capacity) $W_M\in[80,200]$ mm, the free water capacity of the surface soil layer $S_M\in[2,40]$ mm, the outflow coefficient of free water to groundwater $K_G\in[0.1,0.9]$, and the recession constant of the lower interflow storage $KK_{SS}\in[0,0.9]$; behavioral thresholds $L_1<10\%$ on the relative error of total runoff volume and $L_2>0.7$ on the deterministic coefficient (Nash-Sutcliffe efficiency) — two linked questions arise, the first posed explicitly by the source and the second motivated by its reported, unexplained $KK_{SS}$ result:

(1) Likelihood-function sensitivity. Determine how the choice and combination of likelihood/objective functions, and their behavioral thresholds, affect the GLUE results: the number of retained behavioral sets, the posterior distributions of the six parameters, and the resulting prediction-uncertainty bounds. This must include the multi-objective scheme actually used ($L_1$ and $L_2$ applied jointly with fixed thresholds), in comparison with alternative single and composite likelihood formulations and threshold levels, so that the sensitivity of the published posterior statements (including the concentrated posteriors of $K$, $C$, $W_M$, $S_M$, $K_G$) to these choices is quantified rather than assumed.

(2) Non-identifiability of $KK_{SS}$. The source reported, without diagnosing it, that the posterior of the recession constant of the lower interflow storage, $KK_{SS}$, remained approximately uniform across its prior range — in contrast to the concentrated, approximately bell-shaped posteriors of $K$, $C$, $W_M$, $S_M$ and $K_G$ — so that no preferred value region could be identified and the parameter could not be meaningfully calibrated for this basin. Diagnose the causes of this reported behavior: to what extent it is attributable to the joint $L_1$+$L_2$ likelihood formulation (linking to part (1)), to the information content of daily aggregate discharge about interflow recession (data insufficiency), or to structural insensitivity (the simulated output being nearly invariant to $KK_{SS}$ over part or all of its range, which equifinality couples with $K_G$, $S_M$ and the channel-routing parameters); and determine under which data and likelihood conditions, if any, a concentrated $KK_{SS}$ posterior with a preferred value region — analogous to the other five parameters — arises.

The two parts are linked: part (1) quantifies how much of the published GLUE results depends on the likelihood choices, and part (2) examines the one parameter whose published posterior shows no preferred value region. Because the Hongjiata record is held by the authors' institute and the source does not report the full model configuration, the study may be conducted on the published Hongjiata setup or on an equivalent humid-basin dataset; in either case, likelihood variants and candidate conditions are to be compared under an otherwise identical model configuration, prior ranges and data, so that differences in retained behavioral sets, posterior distributions and prediction-uncertainty bounds are attributable to the likelihood formulation and thresholds rather than to setup differences.

References

1. Chen, Ce; Ge, Shilei; Liu, Lijun; Wu, Yi (2019). Research and Application of Xin'anjiang Model Parameters Uncertainty Based on GLUE Algorithm. IOP Conference Series: Earth and Environmental Science 304, 032028. doi:10.1088/1755-1315/304/3/032028. https://doi.org/10.1088/1755-1315/304/3/032028
2. Zhao, Ren-Jun (1992). The Xinanjiang model applied in China. Journal of Hydrology 135, 371-381. doi:10.1016/0022-1694(92)90096-E. https://doi.org/10.1016/0022-1694(92)90096-E
3. Beven, Keith; Binley, Andrew (1992). The future of distributed models: Model calibration and uncertainty prediction. Hydrological Processes 6(3), 279-298. doi:10.1002/hyp.3360060305. https://doi.org/10.1002/hyp.3360060305
4. Beven, Keith; Freer, Jim (2001). Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the GLUE methodology. Journal of Hydrology 249(1-4), 11-29. doi:10.1016/S0022-1694(01)00421-8. https://doi.org/10.1016/S0022-1694(01)00421-8
5. Montanari, Alberto (2005). Large sample behaviors of the generalized likelihood uncertainty estimation (GLUE) in assessing the uncertainty of rainfall-runoff simulations. Water Resources Research 41, W05006. doi:10.1029/2004WR003826. https://doi.org/10.1029/2004WR003826
6. Stedinger, Jery R.; Vogel, Richard M.; Lee, Seung Uk; Batchelder, Rebecca (2008). Appraisal of the generalized likelihood uncertainty estimation (GLUE) method. Water Resources Research 44, W00B06. doi:10.1029/2008WR006822. https://doi.org/10.1029/2008WR006822
7. Zhang, Yingqi; Liu, Hui-Hai; Houseworth, James (2011). Modified Generalized Likelihood Uncertainty Estimation (GLUE) Methodology for Considering the Subjectivity of Likelihood Measure Selection. Journal of Hydrologic Engineering 16(6), 558-561. doi:10.1061/(ASCE)HE.1943-5584.0000341. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000341
8. Zhang, Danrong; Zhang, Liru; Guan, Yiqing; Chen, Xi; Chen, Xinfang (2012). Sensitivity analysis of Xinanjiang rainfall-runoff model parameters: a case study in Lianghui, Zhejiang province, China. Hydrology Research 43(1-2), 123-134. doi:10.2166/nh.2011.131. https://doi.org/10.2166/nh.2011.131
9. Lü, Haishen; Hou, Ting; Horton, Robert; Zhu, Yonghua; Chen, Xi; Jia, Yangwen; Wang, Wen; Fu, Xiaolei (2013). The streamflow estimation using the Xinanjiang rainfall runoff model and dual state-parameter estimation method. Journal of Hydrology 480, 102-114. doi:10.1016/j.jhydrol.2012.12.011. https://doi.org/10.1016/j.jhydrol.2012.12.011
10. Li, Zhijia; Xin, Penglei; Tang, Jiahui (2013). Study of the Xinanjiang Model Parameter Calibration. Journal of Hydrologic Engineering 18(11), 1513-1521. doi:10.1061/(ASCE)HE.1943-5584.0000527. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000527
11. Alazzy, Alaa Alden; Lü, Haishen; Zhu, Yonghua (2015). Assessing the Uncertainty of the Xinanjiang Rainfall-Runoff Model: Effect of the Likelihood Function Choice on the GLUE Method. Journal of Hydrologic Engineering 20(10), 04015016. doi:10.1061/(ASCE)HE.1943-5584.0001174. https://doi.org/10.1061/(ASCE)HE.1943-5584.0001174
12. Mirzaei, Majid; Huang, Yuk Feng; El-Shafie, Ahmed; Shatirah, Akib (2015). Application of the generalized likelihood uncertainty estimation (GLUE) approach for assessing uncertainty in hydrological models: a review. Stochastic Environmental Research and Risk Assessment 29(5), 1265-1273. doi:10.1007/s00477-014-1000-6. https://doi.org/10.1007/s00477-014-1000-6
13. Cho, Huidae; Park, Jeongha; Kim, Dongkyun (2019). Evaluation of Four GLUE Likelihood Measures and Behavior of Large Parameter Samples in ISPSO-GLUE for TOPMODEL. Water 11(3), 447. doi:10.3390/w11030447. https://doi.org/10.3390/w11030447
14. Li, Zhenya; Yang, Tao; Zhang, Na; Zhang, Yandong; Wang, Jiahu; Xu, Chong-Yu; Shi, Pengfei; Qin, Youwei (2022). Understanding the impacts induced by cut-off thresholds and likelihood measures on confidence interval when applying GLUE approach. Stochastic Environmental Research and Risk Assessment 36(5). doi:10.1007/s00477-022-02182-6. https://doi.org/10.1007/s00477-022-02182-6
15. Zin, Thandar Tun; Lu, Minjiao (2022). Influence of Data Length on the Determination of Data Adjustment Parameters in Conceptual Hydrological Modeling: A Case Study Using the Xinanjiang Model. Water 14(19), 3012. doi:10.3390/w14193012. https://doi.org/10.3390/w14193012
16. Kan, Guangyuan; Li, Chenliang; Zuo, Depeng; Fu, Xiaodi; Liang, Ke (2023). Massively Parallel Monte Carlo Sampling for Xinanjiang Hydrological Model Parameter Optimization Using CPU-GPU Computer Cluster. Water 15(15), 2810. doi:10.3390/w15152810. https://doi.org/10.3390/w15152810
17. Zin, Thandar Tun; Lu, Minjiao; Ogura, Takahiro (2024). Assessing Recession Constant Sensitivity and Its Interaction with Data Adjustment Parameters in Continuous Hydrological Modeling in Data-Scarce Basins: A Case Study Using the Xinanjiang Model. Water 16(2), 286. doi:10.3390/w16020286. https://doi.org/10.3390/w16020286