The virtual test is adapted from a benchmark scenario describing isotope transport in an unsaturated soil column under non-isothermal conditions, which has been widely used for hydrological model validation studies (Fu et al., 2025; Zhou et al., 2021). To better reflect the complexity of land-atmosphere interactions, we designed two contrasting climatic regimes that generate distinct hydrological states and flux-partitioning behaviour: (i) an arid regime characterized by |E/P|>1 (absolute ratio of E to P is greater than 1), and (ii) a humid regime characterized by |E/P|<1 (absolute ratio of E to P is smaller than 1). For both regimes, MOIST outputs daily soil water and δ18O profiles as well as the evaporation and drainage (or non-evaporative) fluxes. These outputs are then used to evaluate SS, NSS, and ISONEVA by comparing their estimated E/P ratios against the benchmark value of E/P computed directly from MOIST-simulated fluxes. This virtual experiment serves as a controlled benchmark and it is designed to test our core hypothesis: by integrating both evaporative and non-evaporative fluxes, ISONEVA can estimate E/P more accurately than existing methods.

The simulated soil texture is Yolo light clay (Braud et al., 2005). The relationships between soil water content, pressure head, and unsaturated hydraulic conductivity for this soil type is described using the Brooks-Corey model (Brooks and Corey, 1964), with related parameters listed in Table 1.

The initial soil water content within this one-meter depth virtual column is uniformly distributed at 70 % saturated soil water content, while the initial isotope profile (δ18O) is uniformly distributed with a value of 0 ‰. The lower boundary is set to free drainage for both water and isotope transport, implying zero gradients in soil water potential and soil water isotopic composition at the bottom.

At the upper boundary, precipitation amount and the δ18O of rainfall are prescribed, while evaporation is controlled by potential evaporation (Ep) and atmospheric forcing. The isotopic composition of atmospheric vapor is set to δ18O = -14 ‰. Two climatic regimes are further defined to represent contrasting hydrological conditions:

Arid regime (|E/P|>1). Air temperature (Ta) and relative humidity (RH_a) vary diurnally to mimic realistic non-steady atmospheric control on evaporation. Ta is prescribed as a sinusoidal cycle (daily maximum at mid-afternoon and minimum near sunrise: Ta= 30 + 10 ⋅ sin(2πt)), while RH_a varies inversely with Ta (RHa= 0.5–0.3sin(2πt)). Potential evaporation Ep is set to 2 × 10⁻⁷ m s⁻¹. Rainfall occurs at low frequency (every 10 d, with each event lasting one day) with a flux of ϵ×3 × 10⁻⁷ m s⁻¹ per event, where ϵ is a random number between 0 and 1. This ensures total precipitation smaller than total evaporation, yielding |E/P|>1 during the simulated period. The isotopic signature (δ18O) of each rainfall event is randomly assigned within the range -50 ‰ to -10 ‰ using -50 +ϵ × 40 (‰), which is sufficient to encompass the natural variability of precipitation δ18O observed across a wide range of climatic conditions (Nelson et al., 2021).

The forward simulation of these two scenarios is conducted by MOIST model under various spatial resolutions (Δz= 0.2, 0.1, and 0.05 m) within a one-meter depth column, whose capability to accurately simulate isotope transport in soil has been demonstrated previously (Fu et al., 2025). MOIST output daily soil water content and soil isotope profiles. Additionally, the benchmark E/P ratio (and Q/P) can be calculated directly from the simulated evaporation (percolation) and precipitation fluxes provided by MOIST. These outputs from MOIST are used to evaluate backward calculation of E/P from SS (Eq. 11), NSS (Eq. 19), and ISONEVA (Eq. 23) methods. Note that in the humid regime, precipitation events are prescribed every 2 d, resulting in a denser sequence of wetting-drying cycles than in the arid regime (every 10 d). Therefore, a shorter simulation period (50 d) is sufficient to generate a comparable (and larger) number of rainfall events and evaluation windows for method assessment, while keeping the numerical experiment computationally tractable. By contrast, the arid regime requires a longer simulation (100 d) to include an adequate number of rainfall events and to span multiple multi-day aggregation windows under infrequent forcing. Consequently, the two regimes are configured with different simulation lengths to ensure comparable information content (event count and window samples) rather than identical duration.

To ensure consistency with typical field sampling practice, we evaluate multiple temporal windows (every 5 d in the humid regime and every 2 d in the arid regime) and three representative topsoil thicknesses (Δz= 0.05, 0.10, and 0.20 m), reflecting common sampling depths and frequencies (Dubbert et al., 2013; Shokri et al., 2008). Each temporal window is defined such that at least one rainfall event occurs within the interval. For a given temporal window, the initial and final soil water content and isotopic composition of the defined topsoil layer are extracted from MOIST outputs and used to estimate E/P over that period. For example, in a 5 d window, soil water content and isotopic composition on Day 1 and Day 5 are used to estimate the cumulative E/P for those five days. This procedure is repeated across all spatial resolutions to assess the sensitivity of SS, NSS, and ISONEVA to topsoil thickness and temporal aggregation.

Note that δ2H and δ18O in soil water can be strongly linearly correlated, resulting in near collinearity in isotope space. Consequently, they provide largely redundant rather than independent constraints on the unknown flux ratios and cannot be jointly used to uniquely constrain both x and y. In addition, δ18O generally exhibits smaller analytical uncertainty compared to δ2H, which reduces noise propagation during the inversion and improves numerical stability. Therefore, δ18O is selected as the representative tracer in this study.

Since SS and NSS contain only one unknown, which can be solved directly using output data from MOIST. By contrast, ISONEVA originally involves two unknowns, x= E/P and y= Q/P, but only one isotope-based equation, which makes the problem underdetermined if treated purely as a two-unknown inversion. To improve identifiability and enforce mass conservation more rigorously, we derived a water-balance constraint from the observed change in topsoil water storage over the evaluation window (Eq. 7): 24ΔVP=1+x-y where ΔV is the change in water storage of the topsoil layer and P is cumulative precipitation over the same interval. This constraint allows y to be eliminated in Eq. (23) and reduces the inversion to a one-dimensional optimization problem in x. By explicitly enforcing mass conservation, this reformulation removes the structural non-uniqueness associated with unconstrained (x, y) search.

Despite dimensionality reduction, the objective function remains highly nonlinear in x (Eq. 23) due to its ratio structure and exponential dependence, which may cause numerical instability under certain hydrological states. As a result, the objective function (Eq. 23) may exhibit strong curvature or local flat regions under certain hydrological states, potentially affecting numerical convergence and solution stability. We adopt the following optimization strategy to solve x:

Single-window inversion (default). For each evaluation window, x is estimated by minimizing the isotope residual objective with y analytically eliminated using the storage constraint (Eq. 24). The optimization is performed using a bounded one-dimensional search (fminbnd function in MATLAB), which ensures deterministic and reproducible solutions within physically meaningful limits.

To quantify uncertainty and avoid reliance on stochastic optimizer variability, measurement uncertainties are propagated through the inversion using Monte Carlo simulation. Specifically, we perturb (i) topsoil δ18O measurements and (ii) precipitation δ18O data by adding Gaussian noise (measurement error) with a standard deviation of 0.7 ‰ (von Freyberg et al., 2020), recompute x for each perturbed realization (1000 simulations per window), and report uncertainty as the standard deviation of the resulting x. This approach explicitly links reported uncertainty to observational error, thereby providing a physically interpretable confidence estimate.

The field experiment is conducted on continuously weighted soil lysimeters, situated at the École Polytechnique Fédérale de Lausanne (EPFL), in Switzerland (Nehemy et al., 2021). Lysimeters are exposed to atmospheric conditions and monitored for a period of 43 d after the application of an isotopically labelled irrigation event on the 16 May 2018, ending on the 29 June 2018. One bare lysimeter and one vegetated lysimeter are used to monitor evaporation and evapotranspiration, respectively.

Within the vegetated lysimeter, soil water content is measured at four depths (0.25, 0.75, 1.25, and 1.75 m) using frequency domain reflectometry probes (FDR; 5TM Devices Inc., USA), while soil water isotopic compositions are sampled at five depths (0.1, 0.25, 0.5, 0.8, and 1.5 m) with two replications at each depth (Fig. 2e) and analyzed at the Watershed Hydrology Lab at the University of Saskatchewan. To harmonize the spatial scales of these two datasets, we define 0–0.25 m as the topsoil layer. Details about the experiment and sample processing can be referred to Nehemy et al. (2021).

Since evaporation measurements from the neighbour bare lysimeter are only available between 4 and 29 June, thus, the field validation in this study is conducted over this period. Within this period, the daily evaporation rate (measured by the bare soil lysimeter) ranged from 0.97 to 2.27 mm d⁻¹ (Fig. 2a). Three precipitation events (including artificial irrigation) took place on 10, 14 and 26 June. The smallest daily input is 69.2 mm d⁻¹ (on 10 June), while the largest input is 193.5 mm d⁻¹ (on 26 June) (Fig. 2b). The input isotopic signals showed a gradual depletion as the precipitation amount increased (Fig. 2c). Under this water input pattern, soil water content in the topsoil layer (0–0.25 m) shows a “rise-decline-rise” trend (Fig. 2d).

Following the sampling frequency from Nehemy et al. (2021), several time intervals (4, 8, 12, 16, 20, and 24 d) are defined to estimate the E/P ratio for each period, starting from 4 June. Meanwhile, actual E/P ratios are calculated using evaporation data from the bare lysimeter, serving as a benchmark for evaluating the performance of SS, NSS, and ISONEVA.

The potential daily evaporation (Ep) is conservatively assumed to range between 0 and 10 mm d⁻¹, covering the plausible variability observed during the experimental period. Accordingly, the physically admissible bounds of E/P are constrained by the ratio between cumulative potential evaporation and cumulative precipitation within each evaluation interval, ensuring that the search space remains consistent with realistic hydrological limits.

Under the imposed mass conservation constraint (Eq. 24), the non-evaporative flux ratio Q/P (y) is analytically determined from the estimated x and the observed storage change, rather than independently optimized. This formulation guarantees internal consistency between water balance and isotope-based inversion.

Uncertainty in x is quantified by propagating isotopic measurement uncertainty through Monte Carlo simulation. Specifically, repeated isotope measurements are used to estimate the empirical standard deviation of δ18O. Gaussian perturbations based on this standard deviation are applied to the isotope observations, and the inversion is recomputed for each realization. The resulting distribution of E/P estimates (1000 realizations per interval) is summarized as mean ± standard deviation.

The isotopic composition of infiltration is set equal to that of rainfall, which is a standard measurement during field campaigns and has been justified in Fig. 1. For the non-evaporative flux, we assign the isotopic composition of the topsoil water. This is justified because (1) non-evaporative flux is expected to cause insignificant isotopic fractionation, and (2) the isotopic composition of topsoil water is directly measurable. Additionally, due to the absence of in situ measurements of isotopic compositions in atmospheric vapor, we adopt reference values of δ18O = -20 ‰ based on cold-trap measurements conducted in Vienna under comparable climatic and seasonal conditions (Kurita et al., 2012). Global water vapor isotope studies indicate that central European stations exhibit strong spatial coherence in vapor isotopic composition, with δ18O values typically clustering between -25 ‰ and -15 ‰ because of the dominant mid-latitude westerly circulation (Galewsky et al., 2016). Because the EPFL site in Lausanne is located within this same large-scale meteorological regime, its atmospheric δ18O is expected to fall within this characteristic range.

To account for uncertainty in atmospheric vapor isotopic composition, the inversion was repeated using three plausible δ18O values (-25 ‰, -20 ‰, and -15 ‰). Measurement uncertainty of isotopic compositions in soil water is further propagated through repeated sampling during the inversion procedure. The resulting flux estimates are reported as mean ± SD, which jointly reflect uncertainties arising from both atmospheric vapor isotopic composition and isotope measurement error.

In the ISONEVA framework, Q represents the total non-evaporative flux within the topsoil control volume and may include contributions from root water uptake, percolation, or other subsurface exchanges. The ratio derived from the ISONEVA framework is EPEP+QP=EE+Q, while the true ratio of E to ET is EE+Ttotal. The relationship between these two quantities depends on the relative magnitude of Q and Ttotal, where Ttotal is the total transpiration within the estimating time window.

Within the time interval that ISONEVA is applied, if the non-evaporative flux within the topsoil does not exceed Ttotal (Q≤Ttotal), then EE+Q≥EE+Ttotal and EE+Q from ISONEVA represents an upper bound of the true E/ET. By contrast, if additional subsurface fluxes within the topsoil become substantial (e.g., strong percolation or lateral flow), it is possible for Q to exceed total transpiration (Q>Ttotal). Then, the ratio EE+Q would underestimate the true value.

However, in the lysimeter validation used in this study, hydrometric observations (Benettin et al., 2021) provide evidence that the non-evaporative flux within the topsoil control volume is likely smaller than total plant transpiration during the experimental period. The lysimeter experiment is characterized by high evapotranspiration rates (5–20 mm d⁻¹) and mostly negligible bottom drainage. Tracer-based water balance analysis further showed that transpiration accounted for approximately 58 % of the exported water, whereas bottom drainage contributed only about 10.4 %. In addition, soil water observations from the lysimeter (Nehemy et al., 2021) indicate that no sustained downward drainage occurred during the experimental period. Together, these observations suggest that the non-evaporative flux is therefore likely smaller than total transpiration from the entire rooting zone (Q≤Ttotal). Consequently, the ratio derived from ISONEVA, EE+Q, represents a conservative upper bound of the true evaporation fraction EET.

In both the virtual and field validations, SS and NSS are applied using the same inputs, temporal resolution, and initial and final soil water and isotope profiles as the ISONEVA method. This ensures a fair comparison, removing the potential effects of data on the performance improvement of ISONEVA.

Accordingly, to quantify the accuracy of the average E/P value approximated by SS, NSS, and ISONEVA, we assess model performance using the mean absolute error (MAE): 25MAE=∑i=1NabsEPei-EPmiN where EPei and EPmi are estimated and measured (or estimated from MOIST in virtual tests) E/P values; N is the total number of measurements; i is the ith measurement.

The benchmark E/P values derived from MOIST simulations (black circles in Fig. 3) are consistently smaller than -1, indicating an evaporation-dominated water balance throughout the simulation period (arid regime). For E/P backward estimation, both SS and NSS show substantial deviations from the simulated values, with MAE values of 2.07, 1.73, and 1.38 for SS and 2.49, 2.08, and 1.76 for NSS under Δz= 0.2, 0.1, and 0.05 m, respectively (Fig. 3a–c). These discrepancies arise from the structural assumptions of the two approaches. The SS formulation assumes negligible changes in topsoil water storage, while NSS accounts for storage dynamics but neglects non-evaporative fluxes such as infiltration and percolation. Both assumptions are inconsistent with the simulated conditions, where topsoil water balance is simultaneously influenced by evaporation, precipitation, and vertical water exchange.

By contrast, ISONEVA produces E/P estimates that closely follow the simulated values across most time windows and spatial resolutions. Although noticeable deviations and relatively large uncertainty ranges occur during the earliest evaluation periods, the estimates rapidly converge toward the benchmark values as the temporal window increases (Fig. 3a–c). Compared with SS and NSS, the estimation accuracy of ISONEVA improves by 49.8 %–58.3 %, 74.7 %–79.0 %, and 65.4 %–72.9 % under Δz= 0.2, 0.1, and 0.05 m, respectively. This improved performance arises because ISONEVA explicitly resolves both evaporative and non-evaporative fluxes while simultaneously enforcing dynamic water storage constraints in the topsoil layer.

For Q/P estimation (Fig. 3d–f), NSS cannot provide estimates because non-evaporative fluxes are not included in its formulation. Under arid conditions, the simulated Q/P values are negative, indicating an upward compensating flux from deeper soil layers. SS provides relatively better approximations of Q/P than E/P. This behaviour likely results from error compensation associated with the steady-state assumption, where biases in E/P estimation partially propagate into the derived Q/P ratio. Nevertheless, ISONEVA still improves the accuracy of Q/P estimation compared with SS, with MAE reductions of 24.2 %, 30.7 %, and 65.0 % under Δz= 0.2, 0.1, and 0.05 m, respectively.

Uncertainty in ISONEVA estimates of E/P and Q/P ratios is relatively large during early evaluation windows but decreases as the temporal window expands. By contrast, SS and NSS show much narrower uncertainty ranges (Fig. 3). Although identical Monte Carlo perturbations (±0.7 ‰ for δ18O in soil water) are applied to all methods, the SS and NSS formulations rely on closed-form ratio expressions in which isotope values appear in both the numerator and denominator. As a result, small perturbations in isotope measurements tend to partially cancel, leading to limited propagation of measurement uncertainty. By contrast, ISONEVA estimates flux ratios through a nonlinear inversion that combines isotope mass balance with dynamic soil water storage constraints. Under short evaluation windows, changes in soil water storage and isotope composition are small, resulting in limited information content for constraining the inversion. In this situation, ISONEVA can result in large uncertainties from small soil water isotopic perturbations. As the temporal window increases, the inversion becomes progressively better constrained, and the associated uncertainty correspondingly decreases.

Under humid conditions, the benchmark E/P values derived from the MOIST simulations (black circles, Fig. 4) remain between -1 and 0 throughout the simulation period, indicating a precipitation-dominated hydrological regime. Compared with the arid regime, SS produces relatively reasonable E/P estimates, with MAE values of 0.41, 0.42, and 0.33 under Δz= 0.2, 0.1, and 0.05 m, respectively. Nevertheless, these errors remain larger than those obtained from ISONEVA (0.24, 0.15, and 0.10). Additionally, NSS continues to exhibit the largest deviations (MAE = 0.86, 0.84, and 0.78) in E/P estimates across all spatial resolutions.

The performance patterns differ for Q/P estimation. Under humid conditions, SS shows substantial bias in estimating Q/P with MAE values are 1.55 (Δz= 0.2 m), 1.50 (Δz= 0.1 m), 1.61 (Δz= 0.05 m), whereas ISONEVA continues to reproduce the simulated Q/P values with higher accuracy (Fig. 4d–f) and respective MAE are 0.18, 0.11, 0.09. This discrepancy arises because the SS formulation neglects temporal changes in soil water storage and therefore cannot correctly partition precipitation inputs between evaporation and non-evaporative fluxes. Under humid conditions, frequent precipitation events induce substantial changes in soil water storage, violating the steady-state assumption underlying SS. By contrast, ISONEVA provides more physically consistent estimates of Q/P by explicitly accounting for both storage dynamics and non-evaporative fluxes.

To synthesize the sensitivity of the inversion performance to hydrological regime and sampling configuration, Table 2 summarizes the mean absolute error (MAE) and the mean standard deviation (SD) of E/P estimates from ISONEVA under different topsoil thicknesses and associated multi-window settings.

Under arid conditions, estimation errors of E/P remain relatively large across all sampling depths, with MAE values of 1.04, 0.44, and 0.47 for Δz= 0.2, 0.1, and 0.05 m, respectively. In this regime, the inversion frequently relies on the multi-window strategy (up to three expanding windows) to obtain stable solutions. This behavior reflects the limited information content of individual evaluation windows under arid conditions. Because rainfall events occur infrequently, it introduces weak perturbations to topsoil water storage and isotopic composition within each window. Consequently, the isotope signal from a single window often provides insufficient constraints for resolving E/P, requiring the use of multiple expanding windows and resulting in larger estimation errors. Consistent with this pattern, the mean uncertainty is also higher under arid conditions, with SD values ranging from 0.21 to 0.42.

By contrast, under humid conditions the inversion becomes substantially more stable. The MAE values decrease to 0.24, 0.15, and 0.12 for Δz= 0.2, 0.1, and 0.05 m, respectively, and the optimal solutions are typically obtained using a single window. This improvement reflects the stronger storage signals and more frequent precipitation inputs in humid regimes, which increase the information content available for ISONEVA to estimate E/P. Correspondingly, the associated uncertainties are smaller and remain relatively consistent across sampling depths (SD ≈ 0.18–0.21). As a result, ISONEVA achieves higher accuracy and stability under humid than arid conditions.

Sampling depth also influences estimation performance of ISONEVA. Under arid conditions, MAE decreases markedly by about 58 % when the topsoil thickness is reduced from 0.2 to 0.1 m (from 1.04 to 0.44), whereas slightly increased 7 % from 0.1 to 0.05 m. A similar pattern is observed under humid conditions, where MAE decreases by about 38 % from 0.2 to 0.1 m (0.24 to 0.15), but 20 % from 0.1 to 0.05 m (0.15 to 0.12). This pattern reflects a trade-off between signal smoothing and measurement sensitivity. Thicker topsoil (0.2 m) tends to smooth isotope signals and reduce sensitivity to short-term flux dynamics, whereas thinner topsoil (0.05 m) may amplify short-term variability and become more sensitive to measurement noise. Overall, these results demonstrate that the identifiability of E/P using ISONEVA is strongly controlled by the information content of isotope and storage signals, which in turn depends on hydrological regime and topsoil thickness.

To evaluate the sensitivity of ISONEVA to the atmospheric isotopic composition, we repeated the inversion using four prescribed values of atmospheric δ18O (-10 ‰, -14 ‰, -20 ‰, and -30 ‰), where -14 ‰ corresponds to the value used in the MOIST simulations (also corresponds to the results reported in Figs. 3 and 4). The resulting MAE values of E/P estimates under different hydrological regimes and sampling depths are summarized in Table 3.

The performance of ISONEVA shows limited sensitivity to the prescribed atmospheric δ18O values. Across the tested range (-10 ‰ to -30 ‰), MAE values vary only moderately. Under arid conditions, MAE differences across atmospheric δ18O scenarios remain within approximately 0.08–0.12, depending on sampling depth. The coefficient of variation (CV) of MAE further indicates that the sensitivity of ISONEVA to atmospheric δ18O decreases with increasing topsoil thickness, with thicker topsoil layers exhibiting smaller CV values (Table 3).

A similar but even weaker sensitivity is observed under humid conditions. In this regime, the variation of MAE across atmospheric δ18O scenarios is considerably smaller, typically within 0.01–0.02 (Table 3). The corresponding CV values also show a decreasing trend with increasing topsoil thickness, indicating that E/P estimates from thicker topsoil is less sensitive to atmospheric isotopic compositions than the thinner one.

Overall, the relationship between model performance and topsoil thickness is not strictly monotonic. As shown in Tables 2 and 3, thicker topsoil layers tend to smooth short-term isotope fluctuations and reduce sensitivity to uncertainties in atmospheric δ18O, but excessive thickness (e.g., 0.2 m) can dilute isotope signals and lead to larger estimation errors (especially under the arid regime). Conversely, very thin topsoil (e.g., 0.05 m) preserves stronger isotope signals and can improve estimation accuracy, but it also becomes more sensitive to external parameters such as atmospheric isotope composition (especially under the humid regime). As a result, this trade-off also interacts with the hydrological regime, as precipitation frequency and storage dynamics influence the information content available for constraining the inversion. A topsoil of 0.1 m would provide the balance between estimation accuracy and sensitivity.