SMM is a widely used baseflow separation method (Aksoy et al., 2008; Piggott et al., 2005; Xie et al., 2020; Tan et al., 2020). It assumes that total streamflow is partitioned into baseflow and event-flow components and baseflow constitutes 100 % of streamflow during low-flow periods (Gustard et al., 1992). The SMM procedure involves partitioning daily streamflow into non-overlapping N-day intervals and identifying the minimum value within each segment. These minimum points (Q1, Q2, Qi, …) are then screened using a filtering coefficient (M): a point is discarded if M⋅Qi exceeds the value of either adjacent minimum. Finally, the baseflow series is constructed by linearly interpolating the remaining minima. The method involves two key parameters: the segment length parameter (N) and the filtering coefficient parameter (M). The segment length parameter N is a proxy of the flow event duration (Stoelzle et al., 2020). Generally, a smaller N results in a higher proportion of baseflow in streamflow, implying shorter surface flow duration. In the literature, N often defaults to 5 d or is predicted using a power-law relationship with catchment area, namely N=1.6×A0.2, where A is the catchment area in km² and N is expressed in days (Zhang et al., 2017; Aksoy et al., 2008). These two formulations of N are included in the comparison and are denoted as FD and FPL, respectively. To further examine the explanatory capacity of the areal-based power-law relationship, we considered a calibrated form of N=a⋅Ab (denoted as FPL′), where the coefficients a and b are estimated from the data by minimizing the squared error between the reference and predicted Ns.

The filtering coefficient parameter M is used to determine if a streamflow minimum qualifies as a strict baseflow point. Higher values of M (typically not exceeding 1) correspond to more stringent criteria for identifying pure baseflow conditions. Unlike N, the parameter M is less sensitive to the baseflow separation results and is commonly assigned a constant value of 0.9 (Stoelzle et al., 2020; Aksoy et al., 2008).

Symbolic regression (SR) is employed to derive expressions for the parameter N. SR represents an expression using a tree structure, where each node corresponds to a mathematical operator, and each leaf represents an input variable or constant. Structure of the tree evolves to identify expressions that best fit the inputted data through genetic programming (Koza, 1994). Five sample SR trees were demonstrated within the dotted box in Fig. 2a. In this study, the SR method is implemented using the PySR library in Python (Cranmer, 2023), which enforces syntactically valid mathematical structures through predefined operator sets and expression tree representations. The nine representative catchment attributes in original units (Table 1) were used as predictors without normalization to maintain interpretability for the SR expressions. The function space consists of the catchment attributes, free constants, and a set of mathematical operators: addition (+), subtraction (-), multiplication (×), division (/), power-law (power), and logarithm (log).

To control the search space and ensure physically interpretable expressions, several structural constraints were imposed in SR model training. Multiplication, division, power-law, and logarithmic operators were not allowed to be nested within operators of the same type. The internal complexity of expressions inside power-law and logarithmic operators was restricted to a maximum value of 3. The maximum allowable total complexity was set to 20. Expression complexity is defined as the sum of the complexity index assigned to each component in the equation. Take N=1.6×A0.2 as an example, if multiplication and power-law operators are each assigned a complexity of 2 and constants and input variables are assigned a complexity of 1, the total complexity of the expression is calculated as 2+2+1+1+1=7. In this study, all operators were assigned a uniform complexity index of 1 to avoid bias toward specific functional forms. Recursive formulations (i.e., expressions where the output variable appears as an input to itself) were not permitted to ensure model interpretability and avoid trivial or ill-posed solutions.

The SR search process was configured with the following hyperparameters: a population size of 33, populations of 15, the crossover rate of 0.066, and evolved over 40 generations. The goodness of fit between the reference and the predicted Ns is evaluated using the mean squared error (MSE): 1MSE=1C∑i=1CNi-N^i2, where C is the total number of catchments, and Ni and N^i represent the reference and predicted values of N for catchment i, respectively. To evaluate the robustness of the SR models, a ten-fold cross-validation strategy is employed (Fig. 2a). The 855 catchments are randomly partitioned into ten subsets of approximately equal size. In each iteration, the model is trained on nine subsets and tested on the remaining one to estimate the generalization error. This process is repeated ten times so that each subset serves once as the testing set. In each iteration, 7 to 10 expressions with varying levels of complexity are generated, resulting in a total of 91 expressions. Among these expressions, we identified recurrent equation forms across all ten iterations.

Three strategies are employed to evaluate the performance of different Ns calculated by the SR formulas (Fig. 2b). The first strategy compares the predicted and reference Ns using the mean bias (B) and coefficient of determination (R2): 2B=1C∑i=1CNi-N^i,3R2=∑i=1CNi-N‾r2-∑i=1CNi-N^i2∑i=1CNi-N‾r2, where N‾r=1C∑i=1CNi is the mean of the reference N over all catchments. Positive (negative) Bs indicate overestimation (underestimation), with lower magnitudes suggesting better performance. R2 measures the proportion of variance explained by the prediction; it ranges from 0 to 1, with higher values indicating better agreement.

The second strategy evaluates the SR-derived N values in the context of baseflow separation using the SMM method. Specifically, the SMM baseflow time series are calculated using different N values with the other parameter M fixed at 0.9 to eliminate its influence. Baseflow time series calculated with the reference N for each catchment (Ni) is used as the reference. The similarity between the calculated and reference baseflow is assessed using the Kling-Gupta Efficiency (KGE) metric (Gupta et al., 2009): 4KGE=1-ρ-12+σpσr-12+μpμr-12, where ρ is the correlation coefficient between the predicted and reference baseflow, and μ and σ denote the mean and standard deviation of baseflow. KGE ranges from negative infinity to 1, with higher values indicating better agreement.

The third strategy compares SEC calculated using the SMM baseflow with the observed SEC. Specifically, baseflow time series generated by SMM using Ns derived by the constant, the power-law, and the three SR formulas are used to estimate SEC based on the chemical and water balance relationship: 5SECt^=SECs,tQt-Bt+SECb,tBtQt, where Qt and Bt are the observed streamflow and the predicted baseflow at time t, respectively; SEC_s,t and SEC_b,t are the surface flow and baseflow SEC concentrations, respectively. The values of SEC_s,t and SEC_b,t are derived using the extreme value interpolation method, which connects the monthly maxima and minima of the observed SEC with spline interpolation to represent the stable variation of SEC in each of the flow components (Mei et al., 2024b). The derivation of Eq. (5) is documented in Supplement Sect. S1 in the Supplement. The agreement between observed and predicted streamflow SEC is assessed using the R2 metric (Eq. 1).

To assess whether the differences in SR-based SEC prediction performance are statistical significant, we apply the Diebold–Mariano (DM) test (Diebold and Mariano, 1995). Details on procedure and test statistics are provided in Section S2 in the Supplement. For each catchment, pairwise DM tests are performed among the three SR formulas to determine whether their SEC R2 differences are statistical significance at the 0.01 significance level. If the largest R2 is significantly different from the second-largest R2, the SR formula associated with the largest R2 is the single best formula of the catchment. If the difference between the largest and the second-largest R2s is insignificant but that between the second and the third ones is significant, the first two SR formulas are tied. Otherwise, the three SR are tied. Based on these catchment-scale rankings, the best-performing formula for the HUC2 and HUC4 regions is determined as the one most frequently ranked as the best among all catchments within the regions.

Three SR formulas with identical functional structures and increasing complexity levels of 5, 13, and 17 were consistently identified across the ten cross-validation iterations. As summarized in Table 2, the simplest formulation is F1:N=Aa1+b1, followed by the intermediate formulation F2:N=d2(A+c2)a2+(Ksat)e2+b2, and the most complex formulation F3:N=d3(A+c3)a3+(Ksat)e3+f3⋅fSWE+b3. The optimized coefficients of these replicated formulas vary only within narrow ranges (Table 2), indicating a high degree of consistency in parameterization across folds. From F1 to F3, the SR results progressively incorporate A, Ksat, and fSWE, suggesting that catchment size, subsurface permeability, and snow processes jointly govern the variability of N. The repeated identification of these three formulations across all iterations further demonstrates their structural robustness for predicting N. Accordingly, subsequent analyses focus on these three formulas.

Figure 3 further illustrates the behavior of these formulas. For F1, the nearly identical exponents (∼ 0.23) and intercepts (3.66–3.86 d) result in almost overlapping curves (Fig. 3a), indicating a stable power-law relationship between N and A, with a diminishing rate of increase. For F2, the similarly constrained exponents (0.28–0.32) and intercepts (1.37–2.78 d) produce tightly clustered response curves (Fig. 3b–c), showing that both A and Ksat contribute positively to N. F3 extends F2 by introducing fSWE as a linear term. The replicated formulas still exhibit closely grouped slopes (3.28–3.71 d) and intercepts (1.21–2.32 d), which explains the clustering of curves in Fig. 3d–f. The marginal relationships of N with A and Ksat in F3 remain consistent with those in F2, whereas increasing fSWE leads to an approximately linear increase in N, at a rate of about 0.3–0.4 d per 0.1 increment in snow fraction. Overall, SR identifies A as the most influential factor in predicting N, as evidenced by its presence in all SR-derived formulas. The narrower ranges of predicted Ns in Fig. 3b and d also suggest that A exerts greater influence than Ksat and fSWE.

Figure 4 presents the performance of N predicted using the two literature-suggested formulas (FD and FPL) and the three SR formulas (F1, F2, and F3) for the ten training and testing sets. It should be noted that the literature-suggested formulas do not require training; instead, they are directly applied to the training and testing sets separately. Overall, the performance of formula FD is poor, with negative R2 values less than -0.6. FPL shows higher performance but still yields modest R2 medians of 0.23 and 0.22 for the training and testing sets, respectively. The calibration with respect to SEC data improves the performance for FPL′ to median R2 values of 0.43 and 0.42 for the training and testing sets, respectively, and R2 of 0.42 for the combined testing set. For predictions from the three SR-derived formulas, we observed higher performance than the conventional formulas. The R2 medians for F1, F2, and F3 are 0.49, 0.53, and 0.56 for the training sets, and 0.45, 0.49, and 0.55 for the testing sets, respectively. The overall R2 values for the ten testing sets together for F1, F2, and F3 are 0.48, 0.52, and 0.55, respectively. In terms of the bias, the SR-derived formulas significantly reduce the underestimation of N by the conventional formulas (Fig. 4b). The medians B for F1, F2, and F3 are 0.00/0.06, 0.00/0.08, and 0.00/-0.03 d for the training/testing sets, respectively, comparing to FD and FPL at -4.44/-4.38 and -2.34/-2.28 d. These values for the combined testing set are -4.44, -2.33, 0.00, 0.00 and 0.00 d for FD, FPL, F1, F2, and F3, respectively. It is noteworthy that the testing sets exhibit wider ranges of R2 and B values compared to the training sets. This variability is primarily attributed to the differences in catchment attributes of the testing samples rather than the discrepancies among the 10 replicated formulas. This is supported by the nearly identical coefficients (i.e., a, b, and c) in the SR formulas (Table 2).

Figure 5a reveals the baseflow separation performance of different N predictions measured by KGE for the testing sets. The baseflow separation performance roughly follows the ranking of FD<FPL<FPL′<F1<F2<F3, in line with the N predictions (Fig. 4). The median KGEs for FD, FPL, FPL′, F1, F2, and F3 are 0.63, 0.73, 0.83, 0.84, 0.84, and 0.85, respectively. For FD and FPL, 12 % and 5 % of catchments exhibit KGE <0, while 64 % and 78 % report KGE >0.5, respectively. In contrast, the percentage of catchments with KGE <0 drops to 1 % for FPL′, F1, F2, and F3, and those with KGE >0.5 rise to more than 85 %. These observations indicate the benefits of performing regional calibration for the N prediction.

Figure 5b shows the performance of the three SR formulas across the 18 HUC2 regions. Overall, all formulas perform well for the HUC2 regions, with median KGEs ranging from 0.57 to 0.93. The best performing regions for F1, F2, and F3 are HUC 02, 01, and 07, respectively, exhibiting median KGE values above 0.89 and over 95 % of catchments achieving KGE values greater than 0.5. In contrast, the lowest performance for all three formulas occurs in HUC 12, where median KGE values fall below 0.65 and more than 25 % of catchments show KGE values below 0.5. This may be related to the relative arid climate and flashy hydrological response of HUC 12 (Kratzert et al., 2019; Feng et al., 2020), which is difficult for SMM to capture. Note that SMM is more skillful for smooth baseflow dynamics (Stewart, 2015). All three formulas exhibit larger performance variability in HUC 10–12 and 14 compared to other regions. This can be attributed to the fact that these regions are mountainous and plain areas, characterized by relatively high heterogeneity in catchment characteristics.

Figure 5c compares the relative performance of F1, F2, and F3 across regions. F2 outperforms F1 for HUC 04-09 located in the northern and northeastern CONUS, while it performs worse than F1 in HUC 12–15 on the south. F3 generally outperforms F2 in the mountainous region spanning HUC 10–11 and 13–18, and shows comparable performance to F2 in HUC 01, 02, 04, 05, and 07 with mild topography. In HUC 02, 04, 07, 08, 10, 11, 13, 14, 16, and 18, the most complex F3 outperforms the other SR formulas. However, F3 performs worse than F1 and F2 in HUC 12. This may be because the SR formulas were calibrated using all catchments to optimize performance at the CONUS scale; thus, their region-specific performance inevitably reflects trade-offs. In relatively warm regions such as HUC 12, where snow influence is weak, the SWE term included in F3 may add unnecessary complexity rather than useful information, resulting in slightly poorer performance than F1 and F2.

Figure 6 presents the average KGE of baseflow separation as functions of the three influential catchment attributes. Overall, the SR formulas consistently outperform the literature-suggested formulas for most of the predictor value ranges, with relatively minor performance differences among F1, F2, and F3, highlighting their robustness across diverse catchment conditions. According to Fig. 6a, when A exceeds 300 km², corresponding to N=5 d by FPL, the performance of FD deteriorates markedly. This suggests that using N=5 d cannot adequately represent its variation for larger As. Conversely, for A<300 km², the performance of FPL declines, indicating that the power function coefficients are not suitable for smaller catchments over CONUS. The calibrated power-function (FPL′) reveals improved performance to a similar level with the SR formulas except for the A<100 km² bin, indicating the necessity regional calibration. Across the full range of A, the SR formulas consistently achieve higher KGE values, emphasizing the benefits of regional calibration. For small catchment (A<100 km²), accounting for fSWE and Ksat leads to performance gains. In contrast, for larger basins (A>100 km²), the additional contribution of these factors becomes negligible.

Figure 6b examines the influence of Ksat. When Ksat<5 mm h⁻¹, all predictions show similar level of baseflow prediction performance, suggesting that Ksat is not a major factor for catchments with low soil hydraulic conductivity. In the intermediate ranges (5–25 mm h⁻¹), the two regionally fitted power-law formulas (F1 and FPL′) outperform the conventional power-law formula (FPL). Although F1 and FPL′ do not explicitly include Ksat, their performance remains comparable to F2 and F3, which incorporate Ksat. This indicates that regional fitting may compensate the effects of Ksat for small and medium values. When Ksat>25 mm h⁻¹, incorporating Ksat substantially improves the predictions, highlighting the increasing importance of subsurface processes in highly permeable catchments. Figure 6c reveals the influence of fSWE. F3 outperforms all formulas when fSWE>0.4, indicating the importance of fSWE for catchments with stronger snow persistency. In contrast, for fSWE<0.4, FPL′, F1, and F2 perform similarly to F3, demonstrating that regionally calibrated coefficients can partially offset the lack of explicit consideration of snow-related variables.

The SEC estimation performance by different N predictions for the testing set is evaluated using R2. The median R2 values range from 0.70 to 0.74 for the FD, FPL, FPL′, F1, F2, and F3 formulas, with similar interquartile ranges between 0.29 and 0.33, indicating relatively small overall differences among formulas. Despite these modest differences, both the medians and interquartile ranges of R2 follow a clear order of FD<FPL<FPL′<F1<F2<F3. This ranking is consistent with both the N predictions (Fig. 4) and the baseflow separation results (Fig. 5). The mean R2 differences between most formulas are statistically significant, except for the comparison between F1 and F2 (p≥0.01).

Figure 7 illustrates the spatial distribution of the best-performing formulas across the HUC4 regions. F3 demonstrates superior performance across most of the mountainous regions, including the Rocky Mountain foothills, Cascade Range, Sierra Nevada, and some regions within the Great Plains. The improvement for the mountainous regions is likely due to the explicit incorporation of fSWE in the SR formula, capturing the influence of snow processes; for the relatively flat Great Plains regions, the improvement is primarily driven by the consideration of Ksat, which better represents subsurface processes such as infiltration capacity, percolation, and lateral subsurface flow that regulate groundwater recharge and baseflow contributions. In contrast, F1 performs best in the eastern and southeastern CONUS, including the Appalachian Mountains, coastal plain, and Florida peninsular, where the effects of Ksat and fSWE are minimal. These regions are generally characterized by low BFI values (Fig. 1c), indicating surface runoff-dominated streamflow with limited relevance of delayed-flow processes (Wu et al., 2021; Mcmillan, 2020). F1 achieves the highest performance in fewer HUC4 regions, indicating that the effects of Ksat may be compensated by the regional calibration of F1. Interestingly, several regions located in the Great Plains exhibit mixed optimal performance for multiple formulas, suggesting a more complex interplay of hydrological drivers in these areas.

In this study, we used SR to derive mathematical expressions for the predictions of N using 9 catchment attributes. Across ten cross-validation iterations, the identified expressions exhibited consistent structures, predictors, and nearly identical regression coefficients, indicating that SR can yield stable functional relationships between catchment attributes and N. Compared to the RF-based predictions reported by Lin et al. (2026), the SR-based approach showed lower predictive skill (R2= 0.54 vs. 0.80), reflecting the trade-off between predictive accuracy and interpretability. While RF achieves superior predictive performance, it functions as a “black-box” ensemble, offering no explicit functional form to clarify whether environmental controls operate additively, multiplicatively, or through nonlinear transformations. In contrast, SR provides structural transparency by yielding a closed-form equation, facilitating direct analytical insights (Karpatne et al., 2024; Häfner et al., 2023). This explicit representation enables rigorous sensitivity analysis via differentiation; for instance, the marginal effects derived from equation F3 quantify how geomorphic and climatic factors jointly govern N. By trading a degree of predictive skill for parsimony, SR transforms the problem from simple estimation into a hypothesis-generating exercise, providing compact transfer functions that are easily integrated into regionalization frameworks (Feigl et al., 2022; Samaniego et al., 2010). Therefore, RF and SR should be viewed as complementary rather than competing approaches: RF provides a benchmark for predictive performance, while SR offers structural transparency that facilitates theoretical interpretation and model integration.

One important aspect to consider when training SR models is the diversity of mathematical operators. A wide variety of operators for model training enables the discovery of complex relationships among variables, but it also enlarges the search space, increasing training time and risk of overfitting (Li et al., 2025; Elsken et al., 2019). To address these challenges, we adopted two strategies. First, we incorporated domain knowledge to guide the choice of operators. Specifically, the well-established power-law relationship between streamflow response time and catchment area motivated the inclusion of the power-law operator in the SR operator set. The fact that all SR-derived formulas captured this power-law relationship indicates the effectiveness of using this domain knowledge. Second, cross-validation was applied to assess model generalizability and reduce overfitting, ensuring representativeness of the SR expressions across the catchments. Although SR can generate more complex formulas than F3 with better training fit, these structures are inconsistent across folds. This suggests that they may overfit specific subsets of the data and lack generalizability, and are therefore excluded from our analysis.

The segment length parameter N is a proxy of the average duration of surface flow, and larger values indicate surface flow of event sustain for longer time on average (Stoelzle et al., 2020). Our findings indicate that N is a power function of catchment area with exponent of 0.22–0.23 (F1, Table 2), indicating that larger catchments are characterized by longer average surface flow duration (Garzon et al., 2023; Mei and Anagnostou, 2015), and the increasing rate in duration slows down for larger As. This could be attributed to the longer averaged water paths for larger catchments (i.e., the Hack's Law), increasing the average traveling time for surface flow of event (Tarasova et al., 2024). Saturated hydraulic conductivity is another key factor in the predictions of N: N increases according to the power-law functions with exponents of 0.28–0.36 with respect to Ksat (F2 and F3 in Table 2). This indicates that higher Ksatvalues tend to increase the average duration of surface flow with decreasing rate. A possible explanation is that catchments with higher Ksat tend to promote infiltration, making more rainfall excess to route through the slow subsurface flow paths than the rapid overland ones (Nagy et al., 2024). F3 reveals that prolonged snow cover (higher fraction of days covered by snow) is linearly associated with longer surface flow duration. This is due to the snowpack acting as a seasonal storage that modulates streamflow timing (Stoelzle et al., 2020). The gradual melting of snowpack slowly released meltwater to the stream networks, increasing the time needed to leave the catchment (Noor et al., 2023; Barnhart et al., 2016; Godsey et al., 2014). Overall, the additive structure among A, Ksat, and fSWE reflect the separate contributions of different catchment-scale delay processes to the event flow through different pathways and at different time scales. Their additive combination therefore provides a parsimonious empirical approximation of the integrated flow-duration response.

Our experiments reveal that formulas for baseflow separation with higher structural complexity generally achieve better predictive performance. This is because more complex formulas can incorporate additional predictors for more detailed descriptions of the underlying hydrological processes. However, the predictive gain from increased complexity is not uniform across all hydro-climatic conditions. For example, when Ksat and fSWE take relatively small values (Ksat<25 mm h⁻¹ and fSWE<0.4), the performance differences among F1, F2, and F3 are minimal (Fig. 6). This is likely because small variations in Ksat and fSWE have limited impact to the prediction, allowing the regional calibration to partially compensate for the absence of these variables in the formulas. However, such compensation is limited for large Ksat and fSWE (Ksat>25 mm h⁻¹ and fSWE>0.4), where their influence becomes more pronounced (Jenicek and Ledvinka, 2020; Beven and Germann, 2013). Moreover, the effects of Ksat and fSWE are more evident when A is smaller than 100 km² (Fig. 6a). As catchment area increases (A>100 km²), their influence is outweighed by A, highlighting the dominant role of drainage area in shaping the streamflow response time (McGlynn et al., 2004; Sólyom and Tucker, 2007).