Uncertainty of Temporal and Spatial δ²H Interpolation on Young Water Fraction Estimates Using the StorAge Selection Function in Subtropical Mountain Catchments

Abstract

Water age reflects water sources, storage, and pathways, and regulates the solute retention and dissolution associated with biogeochemical processes, highlighting its hydrological and ecological importance. However, accurate water age estimation in tracer-aided models depends heavily on the quality and spatio-temporal resolution of precipitation isotopic signals. This study investigates how distributed rainfall δ²H signals affect the simulation of young water fraction (F_yw) via the Storage Age Selection (SAS) model in topographically complex subtropical mountain catchments. Eight precipitation δ²H scenarios were generated using two temporal approaches (stepwise and sinewave) and four spatial interpolation methods: (1) raw data, (2) reversed effective recharge elevation method (rERE), (3) linear regression with elevation (ER), and (4) regression-kriging (RK). Later on, the time-variant SAS model was calibrated against observed stream water δ²H collected from the year 2022 to the year 2024. Results show that the SAS model consistently produced similar F_yw estimates for catchments (8%~40%) across all eight scenarios, demonstrating strong robustness to input uncertainty and validating the dominant role of catchment characteristics in regulating water age. The combined stepwise temporal and rERE spatial approach provided better agreement with observed stream δ²H, particularly in the eastern, steeper catchments, yielding superior model efficiency along with better constrained uncertainty. This study highlights the sensitivity of age-tracking models to precipitation isotopic inputs and provides practical guidance for selecting an interpolation strategy in data-limited mountainous environments.

1. Introduction

Water age metrics, such as transit time and the young water fraction (F_yw), provide insights into the sources, storage, and flow pathways of catchment water, and play a critical role in regulating the retention and release of solutes involved in biogeochemical processes [1,2,3]. Solute attenuation and streamflow composition are tightly linked to the age distribution of outflowing water, which reflects mixing and storage processes within the catchment [4,5]. Consequently, transit time, the period between water input as precipitation and output as streamflow, has emerged as a key integrative descriptor of hydrological functioning and solute transport. Early studies often relied on simplified, steady-state representations that assumed time-invariant conditions [6,7,8]. While widely used because of their simplicity [9], they are inconsistent with observed hydrological variability and fail to account for the effects of short-term fluctuations in precipitation and storage dynamics on flow pathways [10,11,12] and neglect interactions between newly introduced water and older water retained in catchment storage [13]. To address these limitations, increasing attention has been placed on time-variant approaches [1,3,14,15,16]. Among these, the StorAge Selection (SAS) framework has become particularly influential [2,3]. SAS functions parameterize the preferential release of water of different ages from storage, thereby enabling a nonstationary and physically meaningful description of transit time dynamics. This approach has been successfully applied across a wide range of hydroclimatic and geomorphic settings (e.g., [15,17,18,19,20]).

Despite its strengths, the SAS framework remains subject to uncertainty related to model structure, parameterization, and input data [21,22]. Like other hydrologic models, SAS models require parameter calibration to constrain the model performance, but the calibration, in fact, strongly depends on the quality of continuous isotopic signals. In this context, precipitation isotopic signals play a vital role in constraining hydrological models and reducing predictive uncertainty in large mountainous basins, particularly in capturing streamflow seasonality and improving the physical realism of water source partitioning [23,24]. However, collecting rainfall isotope samples at high temporal and spatial resolution is costly and logistically challenging [25], particularly in mountainous watersheds. As a result, the sparse precipitation sampling network may lead to substantial data gaps [26] that inevitably necessitate interpolation methods [9,27]. Previous research has demonstrated that model structure and parameter choices can strongly influence estimated water age distributions [9,19]. However, relatively few studies have systematically examined how different temporal and spatial interpolation strategies for precipitation isotope inputs affect SAS model performance, parameterization, and derived water age metrics [19]. Temporally, commonly used interpolation schemes include sinewave fitting, stepwise assignment, and generalized additive models [11,19,28,29]. Spatially, many studies have relied on single-site inputs to represent entire catchments (e.g., [7,30]), which may lead to significant errors in mountainous regions with strong elevation effects [31,32,33]. To address this elevation-driven variability, some studies introduced the concept of effective recharge elevation (ERE), selecting rainfall isotope records that best match stream water signatures to represent dominant recharge zones [7,34,35]. When multiple stations are available, geostatistical approaches incorporating elevation, such as regression-based methods and kriging interpolation, have been used to better represent spatial patterns in precipitation isotopes (e.g., [19,36,37]). However, the sensitivity of SAS-derived water age metrics to the combined effects of these different interpolation strategies remains insufficiently studied.

Furthermore, the young water fraction (F_yw, [11]), which represents the proportion of streamflow younger than about two to three months, has emerged as a practical indicator of rapid hydrological dynamics and has been widely applied in catchment comparisons [7,9,20,33,36,37,38,39,40]. While numerous studies have investigated F_yw in temperate and high-latitude regions, comparatively few have focused on tropical mountain catchments [41,42,43]. Mountain regions serve as critical “water towers,” supplying essential water resources downstream [44]. Investigating how different tracer interpolation strategies influence SAS model performance and parameterization in these settings is especially valuable under data-scarce conditions.

In this study, we examine how tracer interpolation strategies influence SAS-based estimates of young water fraction and model parameterization in subtropical mountain catchments. We combine two temporal interpolation methods (sinewave fitting and stepwise assignment) with four spatial approaches (raw data, effective recharge elevation, simple elevation regression, and regression kriging), producing eight scenarios of spatiotemporal precipitation input signals. These signals are then applied onto the SAS model and a meso-scale mountainous watershed with twelve streamflow monitoring sites spanning a wide elevation range. We tested three hypotheses: first, we evaluated the applicability of the SAS-based modeling in the subtropical monsoon mountain region. Second, by analyzing twelve nested catchments spanning a wide range of elevations, we assess how topography influences the sensitivity of modeled water age to interpolation choices. Third, through a systematic comparison of interpolation schemes, we provide insights into the trade-off between methodological sophistication and the robustness of the resulting F_yw estimates. Together, these findings offer practical guidance for selecting appropriate temporal and spatial interpolation strategies when applying tracer-aided hydrological models in data-limited mountainous regions.

2. Materials and Methods

2.1. Study Site

The Zhuoshui River, originating from Mount Hehuan in central Taiwan, is the island’s longest river, extending 187 km and draining a basin area of approximately 3157 km² (Figure 1). The geological setting of Taiwan is highly dynamic: the active collision of the Luzon arc of the Philippine Sea Plate with the Eurasian continental margin, producing crustal shortening rates of about 82 mm/year [45], has resulted in rapid uplift and intense deformation across central Taiwan. As a result, the metamorphic grade of bedrock increases progressively from west to east [46], reflecting the island’s dynamic geological evolution. In the study area, the river system can be divided into two physiographic zones: the steep, mountainous upper basin and the relatively flat Zhuoshui alluvial fan in the downstream lowlands (Figure 1). The studied upstream region is characterized by sharp elevation gradients, rising from 100 m at the outlet to 3952 m above sea level over a horizontal distance of forty kilometers. These gradients yield average slopes exceeding 40% and locally reaching up to 87%, exerting strong controls on hydrological responses such as runoff generation, infiltration capacity, and flow path variability. Such pronounced tectonic and topographic conditions, particularly the large elevation gradients, make the Zhuoshui river catchments an ideal testbed for evaluating the sensitivity of water age estimates to spatial interpolation complexity in data-sparse, mountainous regions.

While the moisture sources from polar continental during the winter, and equatorial maritime and tropical maritime during the summer are the main parameters in controlling the precipitation’s isotope characteristics in Taiwan [47], the monsoon and the altitude effects are the two principal processes affecting δ values of inland precipitation in central Taiwan [48]. Annual precipitation averages 2459 mm and exhibits a highly seasonal pattern. The wet season, from May to October, is influenced by 3–5 typhoons per year, producing over 80% of the annual rainfall, while the dry season is markedly arid by comparison. Geologically, the basin transitions from young alluvial deposits in the west, through semi-consolidated and sedimentary rocks, to older, more deformed metamorphic rocks in the east [49]. Regolith thickness varies widely, from 0.6 m to over 50 m, with an average of around 15 m. Hydrogeological properties, including porosity and hydraulic conductivity, have been characterized via borehole drilling and in situ hydraulic testing [49,50], revealing substantial heterogeneity among rock types.

2.2. Hydrometrics of the Gauged and Ungauged Catchments

In the study area, daily weather data from the Central Weather Agency (CWA) and streamflow data from the Water Resources Agency (WRA) were used. The sixteen subcatchments studied were delineated based on sixteen stream water sampling sites (Figure 1). Given the lack of continuous discharge (Q) and actual evapotranspiration (AET) measurements at the outlets for the study period, these variables, essential for the subsequent water age modeling, were simulated through the Soil and Water Assessment Tool [51]. Previous applications of SWAT in the region have demonstrated its suitability [52,53,54]. Model calibration and validation followed the established procedures outlined by Abbaspour [55], with performance evaluated according to the statistical criteria of Moriasi et al. [56].

2.3. Stable Isotope Sampling and Data Collection

A network of eight precipitation collectors (

Table 1. Precipitation collectors and sample characteristics.

Site	Latitude	Longitude	Elevation (m asl)	δ2H (Weighted Mean) (‰)	n
P02	23.534	120.909	1919	−113 to −9 (−62)	35
P03	23.553	120.913	1440	−102 to −3 (−59)	35
P04	23.553	120.870	1077	−107 to −1 (−55)	34
P06	23.694	120.850	478	−110 to 7 (−43)	30
P07	23.843	120.866	361	−93 to −1 (−47)	27
P08	23.780	120.636	101	−68 to −1 (−40)	22
P10	23.630	120.629	433	−85 to 14 (−42)	32
P12	23.487	120.889	2619	−105 to −10 (−63)	12

) and sixteen stream water sampling sites (

Table 2. Catchment characteristics of stream sampling sites.

Stream	Site	Drainage Area (km2)	Mean Elevation (m asl)	Slope (%)	Mean δ2H (‰)	rERE (m asl)	Designated Raw Station
M	S01	1626	1915 (317 to 3819)	77	−78 (−88 to −64)	3305	P02
CYL	S02	90	1734 (774 to 2847)	73	−69 (−73 to −59)	2580	P02
CYL	S03	86	2229 (957 to 3855)	87	−80 (−84 to −71)	3492	P02
CYL	S04	15	1484 (757 to 2402)	72	−66 (−70 to −57)	2350	P03
CYL	S05	41	1926 (618 to 3250)	85	−72 (−75 to −64)	2809	P02
CYL	S06	364	1711 (472 to 3855)	75	−72 (−75 to −58)	2807	P02
SL	S07	43	744 (463 to 1341)	49	−52 (−58 to −44)	1244	P06
SL	S08	55	730 (360 to 1341)	43	−59 (−68 to −45)	1807	P06
SL	S09	82	697 (267 to 1341)	46	−65 (−71 to −58)	2276	P06
M	S10	2201	1772 (215 to 3855)	74	−73 (−84 to −62)	2873	P02
M	S11	2290	1724 (190 to 3855)	72	−70 (−83 to −46)	2678	P02
DPR	S12	73	866 (186 to 2024)	50	−49 (−55 to −43)	988	P04
CS	S13	259	1175 (214 to 2660)	65	−55 (−60 to −45)	1459	P03
CS	S14	87	1295 (198 to 2286)	67	−56 (−62 to −45)	1521	P04
CS	S15	406	1076 (117 to 2660)	61	−52 (−59 to −44)	1210	P04
M	S16	2893	1547 (89 to 3855)	68	−64 (−77 to −45)	2224	P03

Note: M: the main stream; CYL: the Chenyoulan river; SL: the Shuli creek; DPR: the Dongpurei creek; CS: the Chingshui river. rERE: reversed effective recharge elevation.

) were set to assess the input rainfall and the output stream isotope variations across the study catchments (Figure 1). The sixteen stream water sampling sites represent the outlets of the catchments used for modeling. These sites were strategically distributed across the main tributaries, the Chenyoulan (CYL), Shuli (SL), Chingshui (CS), and Dongpurei (DPR) creeks, and along the main stem of the Zhuoshui River to capture isotopic variation across elevation and scale (Figure 1). From the year 2022 to 2024, water samples were collected fortnightly. Rainwater samples represent volume-weighted averages over the preceding two weeks. All water samples were sealed in situ and stored at 4 °C until analysis back at the lab in the National Taiwan University, Taiwan. We conducted stable isotope analysis using a PICARRO L2130-i isotopic water analyzer (Picarro Inc., Santa Clara, CA, USA), a time-based measurement system that quantifies spectral features of gas-phase molecules in an optical cavity. Isotope signature was compared to VSMOW in ‰ with the precision of 0.1 ‰ [57]. δ²H was selected as the representative tracer for this study, as previous research indicates that δ²H and δ¹⁸O yield highly consistent SAS modeling results without providing divergent information [58,59] and that a single-isotope approach is sufficient for robust SAS modeling [34,39,60].

2.4. Precipitation δ²H Interpolation Methods

Accurate representation of the precipitation isotopic input is a critical prerequisite for tracer-aided modeling of catchment water transit time, as a representative input value must be determined for each catchment [27]. Given the limited spatial and temporal coverage of isotope measurements typical of data-sparse, mountainous regions, it is essential to systematically interpolate both the temporal variability and the spatial distribution of precipitation isotope compositions. In this study, we used two temporal methods combined with four spatial approaches to generate eight scenarios (

Table 3. The interpolation scenarios of rain δ²H and the water age model used.

Scenario	Spatial	Temporal	Storage Selection Function
ST1	Raw	Stepwise	Time variant fSAS (Beta distribution)
ST2	rERE
ST3	ER
ST4	RK
SW1	Raw	Sinewave
SW2	rERE
SW3	ER
SW4	RK

Note: Raw: taking raw data from the closest elevation collector; rERE: the reversed effective recharge elevation; ER: simple elevation regression; RK: Regression Kriging; fSAS: fractional SAS functions [19].

), aimed at representing the heterogeneity of precipitation isotope inputs across both time and space, thus providing a comprehensive set of boundary conditions for subsequent model calibration and simulation.

2.4.1. Spatial Interpolation and the Designation of Precipitation Isotopes to Catchments

In mountainous catchments such as the Zhuoshui river basin, the spatial variability of precipitation isotope composition is strongly influenced by elevation, orographic effects, and local meteorological conditions. Because rain collectors are limited in number and unevenly distributed, spatial interpolation is required to estimate representative δ²H inputs at the catchment scale. In this study, four approaches were applied to derive spatially distributed δ²H values: (1) selection of the closest elevation station (Raw), (2) estimation via effective recharge elevation (rERE), (3) simple δ²H–elevation regression (ER), and (4) regression kriging (RK). These approaches differ in complexity, assumptions, and spatial representativeness.

• The Closest Elevation Rain Collector Method (Raw)

This method assumes that the altitude effect is the dominant control on δ²H variation. Each catchment was assigned to the rain collector whose elevation most closely matched the mean elevation of the catchment. The original δ²H time series from the selected collector was used directly without interpolation. This point-based approach is computationally simple and preserves observed seasonal patterns but does not account for lateral spatial gradients or catchment heterogeneity.

2.

Reversed Effective Recharge Elevation (rERE)

Inspired by McGuire et al. [7], who calculated isotopic values based on the elevation difference between the effective recharge elevation and the reference precipitation station for each sampling period, and Peng et al. [48], who established the elevation-isotope lapse rate in the study area. We estimated the effective recharge elevation by reversing the δ²H–elevation relationship. The approach assumes that long-term streamflow integrates recharge from a characteristic elevation that can be inferred from the stream water δ²H signature. An annual weighted annual precipitation δ²H–elevation relationship is first established as:

$${{\delta }^2\mathrm{H}}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{m}}= a_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}}+ b_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}}·Z$$

(1)

where δ²H_annual is the annual weighted precipitation δ²H, a_annual and b_annual are regression coefficients, and Z denotes the elevation of precipitation collectors. By reversing the equation using the long-term mean stream water of δ²H (δ²H_stream), an effective recharge elevation Z_recharge is estimated for each catchment:

$$Z_{\mathrm{recharge}}={\displaystyle \frac{{{\delta }^2\mathrm{H}}_{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{m}}-a_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}}}{b_{\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{u}\mathrm{a}\mathrm{l}}}}$$

(2)

This inferred elevation is then used to reconstruct a representative time-varying precipitation isotope input by applying the precipitation isotope–elevation regression derived for each sampling period. The resulting δ²H input is assigned to the corresponding catchment. While this rERE method does not require spatial interpolation of precipitation isotopes, it incorporates both stream isotopic information and topographic context, offering a physically informed point-based estimate of precipitation input. However, it assumes that each stream’s δ²H signal is predominantly influenced by recharge from a specific elevation band and may not reflect complex flow path mixing.

3.

Simple Elevation-δ²H regression (ER)

This method uses a linear regression between observed δ²H values at precipitation collector sites and their elevations to estimate δ²H across the entire catchment, expressed as:

δ²H_precp = a + b · Z_average

(3)

where a is the intercept, b is the lapse rate (‰ per 100 m) of the δ²H on sample dates and Z_average is the average elevation of the catchment. By applying this equation to every grid cell within a catchment, δ²H values can be estimated directly from elevation data. This approach simplifies the derivation of catchment-average isotopic inputs without introducing additional spatial complexity or geostatistical assumptions. However, the validity of this method depends on the assumption that δ²H varies linearly with elevation and that this relationship holds uniformly across the entire catchment. The final isotopic input for each catchment was obtained by averaging the δ²H values of all grid cells within the catchment area.

4.

Regression Kriging (RK)

Regression Kriging combines a deterministic trend model with stochastic spatial interpolation of residuals via kriging. First, a regression model was developed to explain δ²H using elevation as a predictor (i.e., the Simple Elevation Regression (ER) model). Then, the spatially correlated residuals (observed minus predicted δ²H) were interpolated using ordinary kriging. The final δ²H surface was obtained by summing the regression prediction and kriged residuals at each grid cell. To obtain a representative input for each catchment, the mean δ²H value was calculated by averaging across all cells within that catchment. This hybrid approach captures both the large-scale topographic trend and local deviations, offering the most detailed spatial representation among the methods used.

The four spatial methods employed in this study offer a gradient of complexity, from simple point-based assignments to spatially distributed and geostatistical models. The closest elevation rain collector and the reversed effective recharge elevation methods provide straightforward, physically interpretable estimates but rely on strong assumptions of representativeness from individual points. In contrast, the simple δ²H–elevation regression and regression kriging methods account for spatial variability across the catchment, with RK offering the most detailed representation by combining elevation trends with local deviations. Together, these methods allow for a comprehensive evaluation of how different spatial interpolation strategies affect the isotopic input to catchments and, consequently, the simulation of water transit time.

2.4.2. Temporal Variation in Rainwater Isotopes

To bridge the gap between periodic sampling events and generate a continuous daily time series for each catchment, two distinct temporal interpolation techniques were employed. These methods allow for the characterization of both high-frequency variability and long-term seasonality.

• Stepwise Interpolation

The stepwise assignment approach was used to preserve the event-scale variability of rainwater isotopes. This method assumes that the δ²H value measured on a given sampling date represents the isotopic composition of precipitation for that day and remains representative of all rainfall until the subsequent sampling event [7,19]. Accordingly, the δ²H value assigned to each sampling date was held constant and extended forward in time until the subsequent sampling date, resulting in a staircase-like time series that reflects the temporal structure of sampling events.

This stepwise approach is particularly suitable for handling datasets with irregular or low-frequency sampling intervals, as it avoids introducing artificial trends between observations while retaining the observed isotopic variability. Moreover, by directly using observed values, the stepwise interpolation minimizes assumptions about seasonality or periodicity, which may be especially useful for capturing short-term fluctuations associated with individual precipitation events or seasonal shifts not fully resolved by the smoother sinewave fitting described in the next section.

2.

Sinewave Fitting

To characterize the long-term seasonal cycle, the δ²H_precp time series was modeled using sinewave fitting [28], a technique that is particularly effective for sparse and irregularly sampled datasets [11]. The seasonal variation is expressed as:

$${\delta }^2{\mathrm{H}}_{\mathrm{precp}}(\mathrm{t}) = {\mathrm{A}}_{\mathrm{precp}}\sin \left(2\pi f\mathrm{t} - {\phi }_{\mathrm{precp}}\right) + {\mathrm{k}}_{\mathrm{prep}}$$

(4)

where ${\delta }^2\mathrm{H}$ (‰) is the isotopic composition of precipitation sampled at time t (day), A (‰) is the amplitude of the seasonal isotope cycle, φ (in radians, with 2π rad = 1 year) is the phase, f (${\mathrm{yr}}^{-1}$) is the frequency, and k (‰) is the vertical offset of the seasonal isotope cycle. The subscript _precp denotes precipitation. The sinewave is fitted to isotopes measured in precipitation.

The seasonal wave can also be fitted by determining the cosine and sine coefficients a and b via a multiple regression model [61]:

$${\delta }^2{\mathrm{H}}_{\mathrm{precp}}(\mathrm{t}) = {\mathrm{a}}_{\mathrm{precp}}\cos \left(2\pi f\mathrm{t}\right) + {\mathrm{b}}_{\mathrm{precp}}\sin \left(2\pi f\mathrm{t}\right) + {\mathrm{k}}_{\mathrm{precp}}$$

(5)

and then calculating the amplitude and phase using the conventional identities:

$${\mathrm{A}}_{\mathrm{precp}} = \sqrt{{\mathrm{a}}_{\mathrm{precp}}^2 + {\mathrm{b}}_{\mathrm{precp}}^2}$$

(6)

$${\phi }_{\mathrm{precp}}=\arctan \left({\displaystyle \frac{{\mathrm{b}}_{\mathrm{precp}}}{{\mathrm{a}}_{\mathrm{precp}}}}\right)$$

(7)

The a_precp, b_precp, and K in Equation (5) are fitted on the isotope measurements using the Root Mean Square Error (RMSE) method, which results in estimates of the A and φ parameters.

2.5. StorAge Selection Function and Parameterization

The framework used in this study to estimate water age is proposed by Botter et al. [1], generalized by van der Velde et al. [14], and reformulated as the ranked StorAge Selection function (rSAS) by Harman [2]. Benettin and Bertuzzo [62] simplify the model structure and interpretation by expressing the conservation equation in a cumulative storage-based form and defining the outflow selection function, Ω_Q(S_T,t) over the domain S_T ∈ [0, 1], which represents the rank of storage from the youngest to the oldest. In this study, though other SAS functions such as power law [19,62,63] are proved to be powerful or satisfactory, we took Beta distribution [14,19,63] for the more flexible shape which may better correspond to the nature of the half-year long dry season in the study area, the beta distribution is defined as:

$${\omega }_Q(P_S(T,t),t) ={\displaystyle \frac{(P_S (T,t){)}^{ \alpha -1} · (1 - P_S (T,t){)}^{\beta -1}}{B(\alpha ,\beta )}}$$

(8)

where P_S(T,t) is the ranked cumulative storage at time t, i.e., the proportion of storage younger than age T, α, β are shape parameters of the Beta distribution, and β(α,β) is the Beta function. The term ω_Q represents the selection density function of streamflow, or the probability that outflow at time t originates from storage of age T.

In this implementation, the shape of the selection function varies dynamically with catchment wetness. The parameters Alpha (α) and Beta (β) are computed at each time step as functions of catchment wetness ws(t), following [60]:

$$\mathit{ws}(t) ={\displaystyle \frac{S(t)-S_{\min } }{{S_{\max }-S}_{\min }}}$$

(9)

where S(t) is the catchment storage at time t, and S_min and S_max are the minimum and maximum values of storage observed or estimated over the simulation period. This normalized wetness index is bounded between 0 (dry) and 1 (wet) and is used to adjust the shape of the selection function over time. The parameters of the Beta distribution, α and β, are computed at each time step as nonlinear functions of ws(t):

$$\Delta k=K_{\max }-K_{\min }$$

(10)

$${\mathit{Beta}=K}_{\min }+\Delta k · {\mathit{ws}(t)}^p$$

(11)

$${\mathit{Alpha}=K}_{\min }+\Delta k · {(1-\mathit{ws}(t))}^p$$

(12)

where K_min and K_max are the upper and lower bounds of shape parameters, p is a composite sensitivity exponent defined as p = θ · (Δk − 1). In this formulation, θ represents a shape controlling factor. This specific formulation was adopted to allow the age selection preference to scale dynamically with the catchment’s storage range, a modification tailored to the high-variability hydrologic response of these subtropical mountain systems. The parameters K, α and β control the catchment’s water-age preference for outflows. When α < 1 and β > 1, the system tends to discharge young water; when α > 1 and β < 1, old water is preferentially released. In the case of α = β = 1, the selection is uniform, representing complete water mixing [19].

The age selection of evapotranspiration (ET) is considered in this model. It is assumed to be time-invariant and is represented using a power-law selection function:

Ω_ET(t) = P_S (t)^ke

(13)

where Ps(t) is the ranked cumulative storage at time t, and k_e is a shape parameter governing the age preference of ET. A smaller k_e yields a gradually declining Ω_ET, indicating that ET is drawn from a wide range of storage ages. In contrast, a larger k_e produces a steeper decline, reflecting a preference for extracting predominantly younger water.

We also incorporate an isotopic fractionation parameter, hereafter denoted as α_frac(range: 0.9995–1). This parameter represents the vapor–liquid isotope fractionation factor during evaporation [64], defined as:

$${\alpha }_{frac}={\displaystyle \frac{R_{\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}}}{R_{\mathrm{v}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{r}}}}$$

(14)

where R is the isotope ratio (δ²H/δ¹H) in liquid and vapor phases. Values of α closer to unity indicate weaker isotopic fractionation, whereas smaller values enhance the enrichment effect (i.e., preferential loss of lighter isotopes and retention of heavier isotopes in residual water). The enrichment effect is implemented in a simplified but practical way as described in Appendix B in Benettin et al. [60].

2.6. Experimental Design

To assess the uncertainty associated with modeled streamflow δ²H and output young water fraction, we tested time-variant Beta-fSAS functions characterized by S₀ (initial storage), k_min, k_max, and θ (the fSAS function parameter), k_et (the evaporation selection function), and α_frac (the isotopic fractionation factor) (

Table 4. Summary of the SAS model parameters and their initial search ranges used for calibration.

Calibrated Parameter	Symbol (Unit)	Lower Bound	Upper Bound
initial storage	S0 (mm)	500	10,000
shape parameter	Kmin (-)	0.01	1
shape parameter	Kmax (-)	0.01	10
shape controlling factor	Θ (-)	0.01	10
ET selection parameter	Ket (-)	0.01	10
fractionation coefficient for evaporation	αfrac (-)	0.9995	1.0000

), under eight distinct δ²H input scenarios (Table 3). For each input configuration, a two-stage Monte Carlo experiment was conducted, with 10,000 parameter sets sampled in each stage.

The first stage was designed to constrain the initial catchment storage (S₀). This step is critical because S₀ dictates the overall mixing volume and exerts a first-order control on the partitioning between young and old water contributions [2]. As the initial storages were not available for the study catchments, they were considered a model calibration parameter [63]. We sampled 10,000 parameter sets across the full predefined ranges (Table 4) and identified the top 100 behavioral sets (1%) based on the Kling–Gupta Efficiency (KGE; [65]) of the modeled streamflow δ²H. The plausible range of the S₀ values from these 100 behavioral sets was then used as the constrained range for the second stage, providing a realistic baseline for the storage volume before refining the mixing and evaporation parameters. In the second stage, a new sampling of 10,000 parameter sets was performed but only within the constrained S₀ ranges defined by the first stage. This refinement aimed to improve calibration performance while maintaining plausible realism. Again, the resulting best one hundred sets, along with the parameter values, were collected for analysis.

To minimize the impact of model initialization and allow internal catchment storages to reach equilibrium, the observed two-year record of precipitation δ²H (biweekly from July 2022 to June 2024) was preceded by a five-year spin-up period. This was achieved by cycling the two-year observation sequence to initialize the model’s state. It is important to emphasize that this spin-up sequence was used solely for model initialization; all subsequent model calibration, performance evaluation, and water age estimations were conducted strictly on the original, unmodified 2-year period of actual observations. Initial δ²H of the catchment storages were set to the mean value of the observed stream δ²H during the calibration period for each catchment.

Since the objective of this study is to assess the influence of rainfall δ²H input scenarios on model outputs rather than to determine a single best parameterization, we adopted a fixed sample size of 100 parameter sets (1%) per catchment and scenario. Following the approach of Borriero et al. [19], this fixed-count strategy ensures the comparability of uncertainty envelopes across all setups and catchments. Maintaining a constant sample size prevents the result of the uncertainty analysis from being influenced by the number of behavior solutions, thereby allowing for a direct assessment of how different input signals affect the SAS model. Furthermore, sensitivity tests using up to 50,000 iterations confirmed that the F_yw distributions achieve numerical convergence at 10,000 iterations. This justifies our use of a stable, fixed-count threshold for consistent uncertainty comparison.

Model performance was evaluated using the KGE metric, and parameter sets that achieved KGE > 0 were classified as behavioral solutions, representing acceptable simulations for each δ²H input scenario. To quantify the range of behavioral solutions and the associated uncertainty, instead of computing the 95% prediction uncertainty (95-PPU) by calculating the 2.5th and 97.5th percentiles of the model outputs [66]. We computed the 100% prediction uncertainty derived from the 100 collected behavioral parameter sets from the second-stage Monte Carlo simulation. The quality of the uncertainty bounds was further assessed using the p-factor (percentage of observed data captured within the PPU) and the r-factor (thickness of the PPU band).

Our main diagnostic variables included the streamflow δ²H signature and the young water fraction (F_yw), which we extracted directly from the simulated age-ranked distributions. F_yw is used as an indicator of streamflow age because it is less sensitive to the poor identifiability of long tails in the transit time distribution [62,67] and is well-suited for inter-watershed comparisons [33]. We therefore used the KGE performance, p-factor, and r-factor to evaluate the robustness of simulated streamflow isotopes and F_yw across all δ²H input scenarios.

In this study, the calibration parameters (k_min, k_max, θ, kₑₜ and α_frac) were used solely as sampling variables to generate ensembles of model outputs. We deliberately did not interpret the parameter values themselves because SAS parameters are known to suffer from equifinality, where different parameter combinations can reproduce similar tracer dynamics [2,21]. For this reason, we focus our analysis on the simulated δ²H time series and water age indicators, assessing their uncertainty through ensemble statistics and confidence intervals, which provide more robust hydrological insights than direct parameter analysis.

The experimental design is largely inspired by Borriero et al. [19], who assessed uncertainty in tracer-based models. While our overall logic and ensemble-based approach are similar, we adapt the methodology to subtropical mountain catchments, extend δ²H input scenarios, take time-variable SAS, and evaluate model outputs using the young water fraction (F_yw) instead of Mean Transit Time (MTT). These modifications allow for a comparable, yet locally relevant, assessment of uncertainty across multiple catchments.

3. Results

3.1. Isotopic Patterns of Observed Precipitation and Streamflow

Seven precipitation and sixteen stream water collectors were installed across the catchments in the year 2022. One more precipitation collector (P12) was installed in late 2023 to improve coverage at higher elevations. All eight precipitation sampling stations had significant seasonal patterns of δ²H ranging from 7 ‰ to −112 ‰, with more depleted values observed in summer and enriched values in winter (Figure 2a, Table 1). Spatially, elevation effects for rainfall determined by regression analysis between δ²H and station elevations were for −1.20 ‰ per 100 m (R² = 0.88, p < 0.01), −1.0 ‰ per 100 m (R² = 0.89, p < 0.01), and −1.40 ‰ per 100 m (R² = 0.95, p < 0.01) for the annual, the dry season, and the wet seasons respectively. The annual weighted mean δ²H values are similar to those of Peng et al. [48] at the study area, while the lapse rate is also similar to that of Peng et al. [47] and Peng et al. [48].

While precipitation δ²H exhibited sharp and frequent fluctuations, stream water δ²H displayed a more dampened response with significantly lower temporal variability (Figure S1). The observed stream water δ²H values ranged from −88‰ to −43‰, from −84‰ to −57‰ for CYL, −71‰ to −44‰ for SL, −55‰ to −43‰ for DPR, −62‰ to −44‰ for CS, and −88‰ to −45‰ along the main stream (Table 2). This smoothing indicates mixing and storage processes within the catchments [48,68]. Spatially, stream stations at the eastern, higher elevation catchments, such as those in the S01 and CYL catchments (S02, S03, S04, S05, S06), generally exhibited more depleted δ²H values, consistent with the altitude effect.

3.2. SWAT Performance and Hydrometrics Output

The SWAT model was calibrated using observed daily streamflow data from five stream gauges, SG1 to SG5. Model performance was evaluated based on commonly used metrics, p-factor, r-factor, R², NSE, PBIAS, and KGE during the calibration period (2018–2020) and validation period (2021–2023) (

Table 5. SWAT model performance.

Station	Period	p-Factor	r-Factor	R2	NSE	PBIAS (%)	KGE	Mean Simulation (Mean Observation) (m3/s)
SG1	calibration	0.83	0.64	0.62	0.49	−19.8	0.68	19.69 (16.42)
	validation	0.58	0.56	0.67	0.60	26.8	0.57	16.71 (22.85)
SG2	calibration	0.71	0.52	0.59	0.54	−10.0	0.75	93.50 (84.97)
	validation	0.64	0.36	0.65	0.59	16.0	0.50	90.49 (107.70)
SG3	calibration	0.37	0.20	0.69	0.64	20.4	0.54	4.28 (5.37)
	validation	0.47	0.31	0.76	0.75	2.1	0.87	4.19 (4.28)
SG4	calibration	0.49	0.20	0.81	0.81	−16.1	0.81	24.03 (20.70)
	validation	0.47	0.23	0.69	0.64	−40.1	0.56	23.56 (16.82)
SG5	calibration	0.21	0.30	0.67	0.62	−50.2	0.46	140.31 (93.38)
	validation	0.48	0.37	0.72	0.68	−31.6	0.64	137.47 (104.46)

Note: Warm-up period: 2016–2017; calibration: 2018–2020; validation: 2021–2023; simulation: 2024.

). Overall, the model demonstrated acceptable to good performance at the calibration sites, with KGE values between 0.50 and 0.75 and NSE exceeding 0.50 at most stations. The r-factor generally remained below 0.7, indicating a relatively narrow uncertainty range in model predictions. However, several stations exhibited notable negative PBIAS values, such as SG1 (−19.8%) and SG4 (−50.6%), suggesting an overestimation of discharge. This bias likely results from anthropogenic water diversions not explicitly accounted for in the model. Specifically, at SG4, during high flow periods, up to 6.4 m³/s may be diverted toward the Hushan Reservoir, located northwest of the study basin. Additionally, at SG5, along with SG4, the Jiji Weir (Figure 1) can divert up to 160 m³/s downstream to the alluvial fan area for industrial and irrigation uses, substantially reducing observed discharge at this site. These unmodeled diversions likely contributed to the overestimation of flows in the SWAT simulations. Nevertheless, the model reasonably captured the temporal dynamics and flow magnitudes at the gauged sites. With the caution of water diversion downstream the Jiji weir, the calibrated model was applied to simulate daily streamflow for the 16 ungauged catchments (S01–S16) over the period 2016 to 2024, providing essential streamflow input (m³/s) for the subsequent transit time and isotope modeling.

3.3. Interpolated Rainfall δ²H Time Series

Figure 2b illustrates the rainfall δ²H time series of catchment S03 under eight interpolation methods (Figure S1 for all catchments, details in Tables S1 and S2). Clear differences appear in both magnitude and temporal structure between the stepwise (ST) and sinewave (SW) approaches. Both frameworks capture consistent seasonality, featuring enriched values in winter and more depleted values in summer. The stepwise methods exhibit discrete biweekly shifts that allow for more extreme isotopic spikes compared to the smoothed seasonal variations in the SW methods. Among them, the ST1 and ST3 methods frequently reach the highest enrichment peaks during the winter periods, occasionally exceeding −10‰ in 2024, while the ST2 method consistently produces the most depleted values across the time series, particularly during summer events where δ²H drops below −140‰. The sinewave models, SW2, provide the most depleted troughs, often reaching −110‰, whereas SW1 and SW3 remain more moderate.

Pairwise Pearson correlations were generally strong (r > 0.95), indicating consistent seasonal patterns (Figure S2) across methods, yet substantial differences in magnitude remain. Mean absolute difference (MAD) frequently exceeded 10‰, particularly between rERE and Raw (ST1–ST2, SW1–SW2) or RK (ST2–ST4, SW2–SW4) in S01 to S11, with statistically significant differences (p < 0.05) in most catchments and the largest discrepancies in S01, S03, S08, and S09. In contrast, rERE vs. RK (ST2–ST4, SW2–SW4) and ER vs. RK (ST3–ST4, SW3–SW4) showed lower MAD values (e.g., <5‰) in wester, lower elevation catchments (S12–S16), suggesting greater agreement.

Sinewave methods were broadly consistent with stepwise results but yielded slightly lower MAD values, especially in DPR and CSR. Overall, although all methods preserved similar temporal patterns, differences in magnitudes may affect SAS outputs, including young water fraction and parameter calibration, highlighting the importance of careful method selection.

3.4. SAS Model Performance with Different Interpolations

Of the 16 monitored catchments, four (S08, S09, S11, and S16) were excluded due to anthropogenic alteration. The Ming-Tan Pumped-Storage Hydropower Plant uses Sun Moon Lake as its upper reservoir and discharges water directly into Shuili Creek, affecting stations S08 and S09. This released water is a mixed and buffered artificial end-member, composed of diverted upstream water of S01, local precipitation, and long-residence-time reservoir water, resulting in a homogenized δ²H signal that does not reflect local natural processes (Figure S1). Similarly, flow diversion at the Jiji weir alters hydrological conditions at S11 and S16. These stations were therefore excluded to ensure that the remaining sites represent natural catchment hydrological and isotopic processes.

Figure 3 presents KGE values, p-factor, and r-factor of SAS simulations across 12 catchments. Under the stepwise scheme, S04, S07, S12, and S13 show good performance (KGE > 0.70, [19,69]), whereas S05 consistently exhibits only moderate fit [70]. The remaining catchments (S01 to S03, S06, S10, S14, S15) generally fall within an acceptable range (KGE = 0.50 to 0.70, Table S1). Differences among spatial methods are relatively small, indicating a stable model response. Under the sinewave scheme, overall performance is lower, with no catchment exceeding a KGE of 0.70 and most values remaining in the moderate range (0.50 to 0.60, Table S2). S07, S12, and S13 still perform best, and western, lower elevation catchments generally outperform eastern, higher elevation ones (S01 to S06). Spatial method performance is also more variable, especially in S01 to S04, suggesting reduced stability under sinewave interpolation.

Regarding uncertainty, the rERE method performs better, with the average p-factor values 0.89 for stepwise and 0.81 for sinewave, likely reflecting the benefit of elevation-weighted estimation. Other methods show low p-factor values (<0.2) in eastern catchments but higher values (>0.6) in western ones. The r-factor is similar between schemes, indicating comparable uncertainty spread.

3.5. The Initial Storage and the Storage Selection Behavior

The distribution of the initial storage parameter (S₀) and the storage selection parameters for all catchments are presented in Table S1 for the stepwise and Table S2 for sinewave temporal interpolation methods.

3.5.1. The Result of Initial Storage (S₀)

The nested structure is reflected in S₀ patterns (Figure S3). Catchments receiving upstream inflows tend to exhibit similar S₀ distributions. For example, S06 shows S₀ values comparable to its contributing catchments (S02 to S05), while S10, which is mainly composed of S01, shows values closer to S01. Under both temporal schemes, the CYL basin catchments (S02 to S06) generally display larger S₀ values, mostly exceeding 4000 mm, with S04 and S05 showing the highest means. In contrast, western lower elevation catchments (S12 to S15) typically have S₀ values below 4000 mm. Central catchments such as S07 and S12 show intermediate but relatively elevated values compared to S13 and S14, where S₀ commonly ranges between 2000 mm and 3000 mm.

Overall, the sinewave scheme produces greater variability among spatial interpolation methods than the stepwise scheme, particularly in the upper catchments (S01 to S06), while variations in western catchments (S12 to S15) remain relatively small.

3.5.2. Storage Selection Parameters: α and β

Although the selection function is defined by k_min, k_max, θ (together with catchment wetness, ws), direct interpretation of these parameters is not intuitive. Thus, they were transformed into the corresponding α and β parameters of the Beta distribution (Equations (11) and (12)). To illustrate wet condition behavior, boxplots correspond to ws = 0.95. Under very dry conditions (ws = 0.05), the relative magnitudes of α and β would be reversed.

Under stepwise interpolation (Figure 4a), both α and β exhibit relatively narrow interquartile ranges, indicating consistent selection behavior across spatial methods. In most catchments, α < β, reflecting a dominance of younger water contributions during wet conditions. The difference between α and β is smaller in eastern catchments (e.g., S02 to S05) but becomes more pronounced in western catchments (S12 to S15), suggesting stronger young water selection in western catchments during wet seasons. The sinewave scheme (Figure 4b) shows similar overall patterns but with greater variability in both parameters, likely due to smoother seasonal input signals allowing a wider range of acceptable parameter combinations. Despite this increased variability, both temporal schemes consistently indicate that streamflow under wet conditions is primarily composed of younger water, particularly in the western catchments.

3.5.3. Evaporation Parameters (k_et and α_frac)

The evaporation selection parameter (k_et) and the evaporative fractionation factor (α_frac) together describe how evaporation is partitioned within catchment storage and how strongly it alters isotopic composition. Across most catchments, mean k_et values range between 4 and 7 (Tables S1 and S2; Figure S4), indicating moderate preferential evaporation from storage. Higher values (>6) in S07, S12, S13 and S14 suggest a bias toward evaporation from younger, recently infiltrated water, likely associated with shallow or fast-responding soil layers. In contrast, lower k_et values in S02 to S06 imply a more mixed contribution of storage ages to evaporation.

The fractionation factor (α_frac) generally exhibits median values above 0.9985 (Figure S5), indicating weak overall evaporative enrichment. However, S13 to S15 show slightly lower medians and broader ranges, consistent with stronger and more spatially heterogeneous evaporative effects, possibly due to more exposed systems. Most eastern and central catchments (S01 to S12) display α values above 0.9999, suggesting limited evaporative influence, implying deeper, shaded environments.

Both stepwise and sinewave temporal schemes yield similar spatial patterns in k_et and α. However, under the sinewave approach, several catchments (S02 to S07 and S12) display greater k_et variability and α_frac values closer to 1.0000, reflecting a smoothing effect on isotopic variability that may partly mask evaporative signals. Differences among spatial interpolation methods are generally small, although the rERE approach (ST2, SW2) produces slightly lower α_frac values in eastern catchments (S01 to S06 and S10), while western catchments show consistent results across methods.

3.6. Results for Water Age Estimations (F_yw)

The young water fraction (F_yw), defined in this study as the portion of precipitation contributing to streamflow within 70 days, exhibited a distinct spatial pattern across the 12 catchments under both stepwise and sinewave interpolation methods (Tables S1 and S2, and Figure S5). Generally, mean F_yw values were lower in the eastern catchments (S02 to S06), mostly below 0.20, and higher in the western catchments (S13 to S15), where values generally approach or are over 0.30. This spatial trend was consistent across all interpolation methods (ST1 to ST4 and SW1 to SW4) although variation among methods was observed within specific catchments. In some catchments, such as S03, S04, S07 and S13, the distributions were relatively narrow, while others (e.g., S01 and S15) exhibited broader spreads.

4. Discussion

Rainfall δ²H serves as the input signal for the SAS (StorAge Selection) model. Given practical constraints [25] and tracer datasets are generally sparse in space and time [26], relying on temporal and spatial interpolations from scarce observations is necessary [9,27]. In this study, the model’s performance was evaluated using the Kling-Gupta Efficiency (KGE), the p-factor and the r-factor to assess model performance and uncertainty. Though McGuire et al. [7] emphasizes that no single interpolation approach is universally applicable, as the choice of temporal and spatial schemes can substantially influence δ²H inputs, and simulated transit times, we demonstrate that in high-relief tropical catchments, the choice of scheme significantly biases simulated water age.

4.1. Characterization of the Observed δ²H Signatures and Topographic Controls

The composition of precipitation and streamflow reflects a complex interplay between seasonal monsoon dynamics and extreme topographic relief. The pronounced seasonality summer depletion and winter enrichment, aligns with isotopic behaviors observed in other subtropical, monsoon regions [42,43,47,48]. During the summer, southwest monsoon air masses undergo progressively rain-out as they moved inland. This continental effect results in more depleted δ²H values in the eastern, high-elevation catchments, such as S01 and catchments along CYL as the heavier isotopes are preferentially removed during orographic lift (Figure 2a). Conversely, the winter signature is shaped by the Northeast monsoon and the resulting rain shadow effect.

While the absolute ranges and the values of δ²H observed in this study (−112‰ to 7‰) are comparable to those documented in various regions [7,19,29,30,34,36,41,42,43,71], a significant contribution of this work is the sampling density along a vertical gradient exceeding 2500 m (from 101 to 2619 m asl, Figure 1, Table 1). Most existing isotope studies in high-relief areas either rely on a limited number of representative stations [7,19,29,30,34,42,71] or cover elevation difference in less than 1000 m [36]. Our concurrent high-elevation sites allowed for robust lapse rate calculations (−1.20 to −1.40‰ per 100 m), reinforcing that our dataset captures the fundamental hydro-climatic processes of central Taiwan with high fidelity.

4.2. Variability of Temporal Interpolation Schemes

This study offers a direct comparison between the stepwise and sinewave interpolation methods which reveals arguments supporting the superior performance of the stepwise approach in the study area, particularly in the high-relief catchments.

The sinewave scheme is widely used to represent seasonal isotope variation in temperate latitudes [11,19,20,28,72], where catchments typically exhibit limited elevation ranges compared to the present study area. Consequently, the inherent smoothness of the sinewave interpolation is capable of satisfying short-term fluctuations and dampening seasonal rainfall isotope signals in low-relief areas. However, this characteristic may reduce its suitability in mountainous settings, where elevation strongly controls precipitation isotopes via steep lapse rates. Our study confirmed this by finding that the sinewave approach to be more capable in the western low relief catchments (DPR and CSR) where the slopes are generally less than 70%, but it fails to achieve good simulation coverage for catchments with slopes over 70% (CHY). The stepwise method, in contrast, inherently better preserves event-scale variability and sharp isotopic discontinuities, which is crucial for accurately simulating isotopic signals in high-relief environments [7,34,42]. Moreover, this preservation is particularly important for the light δ²H values contributed by tropical storm events, which constitute a main water source in the rain season.

The most direct evidence supporting the stepwise method is its significantly higher Kling-Gupta Efficiency (KGE) mean values compared to the sinewave method (Figure 3). As the KGE measures the goodness-of-fit for the simulated δ²H time series against the observations, the difference demonstrates the superior performance of the stepwise approach. Specifically, the stepwise interpolation, which assumes a constant isotopic composition over the interpolation interval, better reproduces the observed seasonal and event-based variability in the d²H tracer.

Second, the uncertainty metrics in Figure 3 also revealed that the p-factor and r-factor values for the stepwise and sinewave methods were generally not clearly distinguishable (Figures S6 and S7, Tables S1 and S2). It suggests that while both methods introduced a similar degree of model structural and parameter uncertainty, the stepwise method still achieved a substantially higher KGE. Therefore, the stepwise method improves model performance without increasing the overall uncertainty envelope of the simulation.

Third, the stepwise method in this study exhibited a greater and more consistent sensitivity to the choice of the spatial interpolation scheme (ST1 through ST4) in the high relief areas (catchment S01, and CYL catchments S02 to S06). Specifically, the stepwise performance showed ST2 (rERE) consistently yielding the highest KGE values. In contrast, the sinewave performance was relatively flat and uniformly poor across all four spatial methods. This result indicates that the stepwise interpolation may provide a better simulation foundation as a better model should be able to translate improvements in input quality into improvements in output performance (KGE), which the stepwise method successfully achieved, whereas the sinewave method masked these differences due to its inherently poorer temporal representation.

In addition, even though the stream water in the study area does not indicate extensive evaporation [48], but as small but significant effect of fractionation still being found in the energy-limited Scottish Highland [60], the fractionation parameter (α_frac) (Tables S1 and S2, Figure S5) may further support these contrasting behaviors. Under the sinewave scheme, α values tend to cluster near 1.000, indicating zero enrichment. This likely arises because the smoothed δ²H input suppresses short-term variability, diminishing the isotopic contrast between rainfall events. Consequently, the model compensates by reducing isotopic fractionation to reproduce the more depleted stream water signals observed in high-elevation catchments (S02 toS06). In contrast, the stepwise approach retains event-scale variability, allowing α to assume more realistic values (<1) across all spatial interpolation schemes. Under this condition, differences in α among spatial methods are relatively minor; however, the combination of stepwise and rERE interpolations yields higher KGE values, indicating improved consistency between simulated and observed isotope dynamics. This suggests that while the stepwise method effectively captures temporal fluctuations in rainfall δ²H, the rERE interpolation enhances the spatial realism of input isotopes, together leading to more physically meaningful simulations in steep mountain watersheds.

4.3. Variability of Spatial Interpolation Schemes

The spatial interpolation of δ²H precipitation is fundamental to model realism. An examination of the model performance across the four spatial schemes (ST1/SW1 to ST4/SW4) highlights the necessity of accounting for elevation gradients. Under the same temporal interpolation methods, the KGE performance of the various spatial interpolation schemes (ST1 to ST4) showed limited differentiation in the performance heatmaps (Figure 3), with only rERE (ST2/SW2) and RK (ST4/SW4) exhibiting slightly superior in the interior steeper catchments (e.g., CYL).

However, a further analysis incorporating the uncertainty metrics (Figure 5) reveals clearer distinctions regarding model reliability and precision. By partitioning the relationship between the p-factor (reliability) and r-factor (precision) into four quadrants, we can diagnose the specific modeling challenges across the different catchments:

• The lower-Right (high p-factor, low r-factor) quadrant represents the ideal zone where the model is both reliable and precise. Most western front catchments (gray outlines) using rERE and RK fall here, indicating the SAS model effectively captures isotopic variability within a narrow, physically realistic band.
• The lower-Left (low p-factor, low r-factor) quadrant represents an over-confident but biased model. RAW, ER, and RK points for eastern catchments (black outlines) frequently cluster here, particularly under sinewave interpolation. The model produces a tight uncertainty band that simply estimated over the observations, likely failing to account for high-altitude isotopic extremes (Figures S6 and S7). This may be caused by taking the raw data or the averaged δ²H value to represent the catchment, while stream water in the inner catchments is principally sourced from high-altitude upstream reaches rather than concurrent local precipitation as reported by Peng et al. [46].
• The upper-Right (high p-factor, high r-factor) quadrant is a conservative zone where reliability is high but precision is low. Several rERE points for steep catchments move into this quadrant. While the uncertainty band is wide, it successfully encompasses the stream observations, reflecting the high variability of source water in these catchments, which may need further study to constrain it.
• The upper-Left (low p-factor, high r-factor) points in this zone indicate a structural failure where even a wide uncertainty band cannot capture the stream signal. This suggests the SAS model’s mixing assumptions may not match the actual flow-path dynamics of those specific catchments, indicating the efficiency of the model input in the study area.

The widespread low p-factor for RAW, ER, and RK is likely related to their reliance on catchment average elevation or limited station data. Such methods introduce substantial uncertainty in mountainous basins where spatial variability is severe [31,32,33]. This is confirmed by the finding that these schemes performed better in the lower-gradient western catchments but failed to achieve higher p-factors in the steeper eastern catchments.

Our study utilized the rERE concept with elevation regression to estimate the δ²H value corresponding to the probable source elevation. This methodological shift is supported by the tracer-based findings of Peng et al. [48], who concluded that stream water in the inner catchments is principally sourced from high-altitude upstream reaches rather than concurrent local precipitation. By utilizing elevation lapse rates to target these upstream signatures, the rERE method explicitly incorporates the physical reality [23] of the catchment’s recharge dynamics. This explains why rERE is the only spatial scheme where points for the steep eastern catchments (black outline) move out of the over-confident lower-left and into the high-coverage quadrants (p-factor > 0.5).

Notably, while the p-factor and r-factor are widely applied in streamflow calibration [55], their interpretation in isotope-based modeling requires caution. The r-factor typically targets values to be less than 1.5; however, the high values found in the study of the steep eastern catchments are interpreted as a mathematical inflation due to the very small standard deviation of the damped isotope signal, rather than a definitive indication of poor model uncertainty banding. Thus, moderate mismatches (e.g., r-factor > 5) may still represent hydrologically consistent outcomes when KGE > 0.5, as isotope-based model evaluation should focus on reproducing realistic storage mixing behavior rather than perfect numerical agreement [62,73,74].

In summary, the evaluation of interpolation schemes demonstrates a clear hierarchy of realism. The stepwise temporal scheme is necessary for preserving event-scale variability, while the rERE spatial scheme is critical for accurately capturing the dominant elevation-driven isotopic gradient. The combination of stepwise and rERE represents the most robust and physically meaningful approach for hydrological modeling using stable isotopes in this mountainous region.

4.4. Water Age Estimation and the Controls

The young water fraction (F_yw) provides a critical benchmark for evaluating the physical realism of the StorAge Selection (SAS) model. It should be noted that while the choice of interpolation method strongly affects the simulated δ²H time series dynamic (KGE) and uncertainty metrics (p-factor and r-factor), the resulting F_yw values in this study were similar across all interpolation scenarios (ranging between 8 and 40%).

This convergence of aggregated F_yw estimates, despite varying input signals, points to the role of equifinality in the SAS calibration process. The model effectively compensates for the different variances in precipitation inputs, such as the over-smoothed sinewave versus the event-based stepwise schemes, by adjusting its selection parameters to match the observed damping in the streamflow isotopes. However, this robustness in the integrated output (F_yw) should not be confused with insensitivity to the input process. The clearly superior KGE and more realistic fractionation factors (α_frac) in the stepwise scenarios demonstrate that while equifinality can pull F_yw toward a hydrologically plausible range, only the stepwise approach captures the true high-frequency isotopic pulses essential for representing the flashy dynamics of these high-relief systems. Our results reveal a distinct spatial gradient: F_yw is significantly lower in the eastern, high-relief “Interior” catchments (S01–S06, typically <0.15) and higher in the western “Front” catchments (S13–S15, often >0.35). This trend is driven by a complex interplay of subsurface storage, climatic gradients, and internal selection mechanics.

4.4.1. Dominance of Subsurface Storage (S₀)

The initial storage volume (S₀), exhibiting a robust negative linear correlation to F_yw, may be the primary predictor across all simulation scenarios. The stepwise (ST) temporal scheme outperforms the sinewave (SW) scheme in capturing this relationship, yielding higher explanatory power (R², Figure 6a,b). This demonstrates that preserving event-scale variability is crucial for revealing fundamental storage-discharge laws.

The high elevation eastern catchments (S02–S05) cluster at the high storage end (S₀ > 5000 mm). As suggested by Hale et al. [34], the permeability distribution in the subsurface represents perhaps the most basic control on how water is stored and is a direct predictor of stream water age. These steeper eastern catchments are characterized by fractured shale geology [49,50], which supports higher permeability and larger active storage volumes. This fractured shale acts as a massive mixing tank that dilutes young water with vast quantities of old, deep-circulating water. In these catchments, the higher permeability of the fractured bedrock enhances deeper subsurface flow (as evidenced by higher S₀ in the CYL catchments), reducing the quick return of young water to the stream and subsequently lowering the F_yw. The fact that this negative correlation holds across all eight modeling scenarios demonstrates that the SAS model is capturing a fundamental physical process rather than a numerical artifact.

4.4.2. Eco Hydrological Gradient of k_et and α_frac

The transition from the western front to the high interior dictates how water is partitioned between the atmosphere and the stream. We found a positive correlation between k_et and F_yw (Figure 6c,d). Catchments like S13 and S14, located on the warm, high-energy mountain front, exhibit higher k_et values. In these shallow-storage systems, vegetation aggressively selects new water for evapotranspiration, yet the rapid turnover ensures the stream remains dominated by young water. Furthermore, a negative correlation exists between the fractionation factor (α_frac) and F_yw (Figure 6e,f). In the deep-flow eastern systems (S02, S03, S05), α remains near 1.000, indicating that water is shielded from evaporation within fractured bedrock or by the lower ambient temperatures of high altitudes. Conversely, the younger water in the western front undergoes more significant evaporative enrichment, as reflected by lower α_frac values, which may be due to the higher heat availability.

4.4.3. Selection Dynamics and the Old Water Paradox

Analyzing the selection preference of the β-α difference at a high wetness state (w = 0.95) reveals an internal mechanics paradox. We found a positive correlation between the β-α and F_yw (Figure 6g,h). particularly under rERE spatial schemes (R² = 0.47 for ST2; R² = 0.66 for SW2). The paradox is most evident in the inner, high-elevation catchments (S02–S05), which, during extreme wetness, show high β-α values. This indicates a strong preference for fresh water release that is comparable to the flashy western catchments. However, even with the strong fresh water preference during extreme wetness, their average F_yw remains remarkably low. This suggests that water age in these high relief environments is storage-limited rather than selection-limited, which aligns with the Old Water Paradox [10,75]. The preference for young water during storms is simply overwhelmed by the sheer volume of old water stored within the subsurface reservoirs. The fresh water selection only activates during rare peak wetness events, while the long-term annual signal is dominated by older water released during baseflow periods. Furthermore, the rapid storm infiltration may select new water while physically pushing pre-event old water into the stream via pressure-driven flow in fractured shale. This mechanism is supported by the slight upward peaks observed in the stream isotope signatures during storm events (Figure S1), indicating that even when the system attempts to release new water, the massive pre-existing storage maintains dominant control over the chemical and age signature of the discharge.

Together, these parameter characteristics define a topographic-geologic-climate interplay for the region. The western front catchments are high-energy systems defined by low storage, high evaporation potential, and aggressive water turnover. In contrast, the high interior catchments are buffered, energy-limited systems where deep storage and long residence times dominate, only yielding to the young behavior during the most extreme meteorological events.

4.5. Limitations and Recommendations for Future Work

This study is subject to several limitations. First, rainfall isotope sampling in the study was restricted to a limited number of stations, with sparse coverage in high-relief upstream catchments. This increases reliance on spatial interpolation and may introduce uncertainty, particularly where steep elevation gradients produce large spatial isotope contrasts. Second, as Benettin et al. [62] indicate, SAS model results are sensitive to the choice of the temporal resolution of input tracer data, and a finer resolution is generally recommended to achieve a satisfactory level of detail; the biweekly sampling interval may fail to capture short-lived isotopic variations from individual storm events. Third, the spatial interpolation methods used assume stable δ²H–elevation relationships over time, although these may vary seasonally or with shifts in moisture source regions [47], potentially leading to systematic input bias. Fourth, the SAS model structure adopts a fixed functional form for the storage–selection relationship, which may not fully represent complex flow paths and mixing processes in highly heterogeneous mountain catchments. Finally, the SAS simulations in this study depend on the SWAT-derived hydrological fluxes, meaning uncertainties in SWAT calibration and inputs can further propagate into the estimated transit time distributions.

Interpolation techniques present a trade-off between realism and cost. Future research should strategically address these limitations. Instead of pursuing a universally denser monitor network, efforts should prioritize expanding the rainfall isotope monitoring network, particularly along elevation gradients in mountainous areas, to establish a more robust, localized lapse rate. Concurrently increasing sampling frequency during the main events, such as typhoon events [57] to better capture transient signals. Incorporating dynamic, seasonally adjusted δ²H–elevation relationships could improve interpolation accuracy, while testing alternative SAS functional forms may better represent heterogeneous flow and storage dynamics. Furthermore, future applications should integrate uncertainty propagation from both hydrological modeling and isotope input estimation and extend the analysis to explore the sensitivity of water age and transit time to projected climate change scenarios. Finally, the interpolation schemes could be validated using a second, independent environmental tracer (e.g., EC) to constrain the model’s simulated storage mixing behavior. This multi-objective constraint may provide a much more robust test of the physical realism of the derived transit times.

5. Conclusions

This study addressed the critical challenge of generating realistic δ²H precipitation inputs for hydrological modeling in data-sparse, high-relief mountainous catchments by systematically comparing two temporal and four spatial interpolation schemes. The results confirm that the choice of interpolation method significantly influences the simulation performance and associated uncertainty.

The key finding is that the stepwise temporal interpolation method consistently yielded significantly higher KGE values and improved sensitivity to spatial inputs compared to the sinewave method. By preserving event-scale variability and sharp isotopic fluctuation, the stepwise approach captures the critical features of δ²H signals in tropical, high-elevation environments. Concurrently, the Reversed Effective Recharge elevation (rERE) scheme proved to be the most robust spatial approach. It was the only scheme to achieve both acceptable KGE and prediction uncertainty coverage (p-factor > 0.5) in the challenging, steep eastern catchments.

A diagnostic quadrant analysis of the p-factor and r-factor relationship revealed that while western mountain front catchments achieved an ideal balance of reliability and precision, the high-relief interior catchments were inherently more sensitive to the choice of spatial interpolation. The rERE method’s success in these steep terrains confirms that identifying the source-elevation isotopic signal is essential for accurately representing the recharge dynamics.

The simulated young water fraction (F_yw) exhibited a distinct and physically meaningful spatial trend (lower F_yw in the steeper eastern catchments and higher F_yw in the gentle western catchments) that remained consistent across all eight interpolation methods. This consistency highlights the robustness of the SAS model in capturing the underlying hydrological mixing characteristics dictated by topography and geology. Analysis of the selection preference (β-α) further suggests that the eastern catchments are storage-limited rather than selection-limited, as their massive subsurface storage (often >5000 mm) effectively buffers the stream from the young water signatures of individual storm events.

Critically, while the overall F_yw trend was preserved across all scenarios, the superior KGE performance and better constrained uncertainty of the stepwise-rERE combination confirm that this framework is essential for achieving the highest accuracy in tracer’s time-variance. This work may provide crucial guidance for isotope-based hydrological modeling in complex mountain catchments worldwide. In environments where precipitation isotopes are strongly controlled by elevation lapse rates, prioritizing event-scale temporal variability and incorporating the physical control of elevation are essential steps for reliable model calibration and the accurate estimation of water residence times under persistent data scarcity.