To determine toptimal of drought alleviation, we propose the following methodological framework that evaluates how early-stage hypothetical precipitation forcing and PDSI deviation magnitude jointly modulate the efficiency of alleviating grid-level MYD events. We calculate the PDSI using data from meteorological stations, identify representative grid-level MYD events which capture the key characteristics of persistent and severe droughts, and characterize them in terms of DS, DD, DI, Drought Development Period (DDP), and Drought Intensity Point (DIP, the PDSI value at the drought intensity point) (see Sect. 3.3 for identification criteria). Based on these identified grid-level MYDs, a series of numerical experiments is conducted by applying hypothetical precipitation increases along a prescribed gradient. The numerical experiment consists of the following three steps: (1) Validation of the nonlinear response of DS to incremental precipitation forcing applied along a gradient; (2) construction of a novel drought attenuation quantitative model (DAQM) to describe this response; (3) verification of the applicability of the key parameter derived from the DAQM, namely k, the MYD attenuation efficiency coefficient. k characterizes how rapidly additional precipitation is converted into reductions in DS. For each grid-level MYD, k is evaluated across different forcing timings, and its maximum value (kmax⁡) is identified. The toptimal is then identified as the timing corresponding to kmax⁡. In addition, we examine the relationship between DD, as a key event characteristic, and MYD attenuation efficiency under hypothetical precipitation forcing. A detailed description of each methodological component is provided in Sect. 3.2–3.4. For clarity, a consolidated table listing principal symbols, acronyms and their description is provided for reference (Table S1 in the Supplement). Figure 3 provides an overview of the workflow and methodological structure of the proposed framework.

In this study, PDSI was used to quantify the deviation between observed monthly precipitation and the required precipitation, which was estimated each month under Climatically Appropriate For Existing Conditions (CAFEC). The PDSI was calculated following the CAFEC formulation and calibrated at the grid level over the calibration period 1961–1990 to calculate CAFEC values and establish a fixed climatological reference (Karl, 1986). This ensured that PDSI thresholds remained unchanged throughout the analysis (Fig. S1). The calibration period was intentionally kept independent from the subsequent numerical experiments, which were conducted exclusively for 1991–2020, in order to avoid altering the reference climatology and to allow consistent evaluation of hypothetical precipitation forcing effects (Fig. S1a). During the calibration period, monthly climatological values of the water balance components, including precipitation, PET, recharge, runoff, and losses, were derived to compute CAFEC precipitation, calibration coefficients, and the corresponding Z-index and PDSI time series.

The PDSI formulation was based on a simple two-layer water balance model (Palmer, 1965), which considered both water supply and atmospheric evaporative demand. To ensure the credibility of PDSI results, we used the Penman-FAO56 method with a physically based formulation (Xu et al., 2022) to estimate PET (denoted as PET_pm), as shown in Eq. (1). Specifically, daily PET was first calculated from daily meteorological observations and then aggregated to monthly values for use in the PDSI calculation. 1PETpm=0.408⋅Δ⋅Rn-G+γ⋅900T+273⋅U2⋅es⋅1-Rh/100Δ+γ⋅1+0.34⋅U2 Where Δ is the slope of the vapor pressure curve, Rn is the net radiation (calculated using surface downwelling/upwelling shortwave/longwave radiation), G is the soil heat flux, which was assumed to be negligible (G≈0) at the daily timescale beneath the grass reference surface (Allen et al., 1998), γ is the psychometric constant, T is the monthly averaged air temperature, U2 is the wind speed at 2 m height, es is the saturation vapor pressure at a given air temperature, Rh is the relative humidity.

In this study, long-term PDSI series for the period 1961–2020 were calculated using the tool provided by the Palmer Drought Severity Index (Jacobi et al., 2013).

Run theory is a threshold-based method that defines drought events as temporally continuous periods during which PDSI remains below -1 (Yang et al., 2024), such that any month exceeding this threshold marks the termination of an event. According to this method, each identified drought event was further characterized using a set of event-based indicators, including DS, DD, DI, DDP, and DIP.

In this study, the identified grid-level drought events were further selected to capture the key characteristics of MYDs that are both persistent and sufficiently severe (hereafter referred to as typical MYDs), by retaining only those with DD>12 and DI<-3 (Brunner and Tallaksen, 2019; van Mourik et al., 2025; Chen et al., 2025). MYDs were required to be temporally continuous, with PDSI remaining below -1 throughout the entire event. Since the calibration period was set to 1961–1990 and the CAFEC values needed to remain fixed during numerical experiments, the final selection was restricted to grid-level MYD samples occurring during 1991–2020 (Fig. S1a). A total of 351 grid-level MYD samples meeting these criteria were identified. Each basin contained at least 11 grid-level MYD samples, thereby helping to ensure the statistical robustness of the analysis.

According to run theory, for a given drought event characterized by its baseline monthly PDSI series (Fig. S2a), a hypothetical precipitation increase applied at a specific month leads to a reduction in DS, assuming all other conditions remain unchanged. This provides a foundation for quantifying drought attenuation efficiency in terms of DS reduction. The numerical experiments therefore apply hypothetical precipitation increases to individual months of a drought event independently, keeping all other months unchanged, and compare the resulting DS responses across months. The timing of hypothetical precipitation increases (tforcing) is a key experimental parameter, representing the month at which the additional precipitation is applied within a grid-level MYD event.

Since this study focuses on drought alleviation, only hypothetical precipitation increase scenarios with incremental gradients are considered. The procedures of this numerical experiment were given as follows:

Focusing on drought alleviation, the numerical experiments were constrained to the tforcing range of DDP, with the minimum DDP among the selected MYD events being 8 months. Accordingly, t1, t2, …, and t8 represent the first to the eighth month of MYD. For each tforcing from t1 to t8, DS_rate values were calculated across hypothetical precipitation perturbations with pr_rate ranging from 0 to 1 at 0.05 increments. Figure 4a presents the empirical DS_rate responses derived directly from the numerical experiments, summarized as the mean ± standard error (SE) across MYD events under different precipitation perturbation scenarios, mainly to illustrate the overall attenuation behavior across the study area. The resulting responses were then fitted for each tforcing within each MYD event using the DAQM analytical form described in Sect. 3.4 (Eq. 5). The fitted relationships closely reproduced the numerical experiment outputs at the event scale with R2 values generally exceeding 0.90, confirming the suitability of the DAQM analytical form. This fitting yielded one theoretical attenuation efficiency parameter (k) for each tforcing, representing the drought attenuation efficiency of a given event at a specific timing. The corresponding k values are denoted as k1, k2, …, and k8, respectively.

To preserve regional differences, MYD events were identified and analyzed separately at each grid across the study area, rather than being calculated from regionally averaged conditions. By retaining individual grid-level responses, the analysis preserves spatial heterogeneity while comprehensively reflecting the overall MYD response characteristics across the study area. Based on these grid-level events, the changes in DS were not proportional along the increasing precipitation gradient. Within the same pr_rate range from 0 to 1, all timings from t1 to t8 exhibited a similar nonlinear diminishing-return response pattern (Fig. 4a). The overall responses for t1, t2, and t3 remained relatively close, whereas a clearer reduction emerged from t4 onward. Effect sizes between two groups were evaluated using Cohen's d, and the magnitude of differences was quantified using its absolute value (hereafter referred to as Cohen's d). Pairwise statistical comparisons further showed that although the differences among t1∼t3 were statistically significant (p<0.001), the corresponding effect sizes were extremely small (Cohen's d=0.002∼0.003) (Table S3). In contrast, comparisons involving t4 exhibited substantially larger effect sizes (Cohen's d=0.231∼0.342) (Table S3). These results indicate that, when aggregated across all MYD events over the study area, the overall drought attenuation efficiencies associated with t1∼t3 remained relatively close before a clearer decline emerges from t4 onward.

Since hypothetical precipitation inputs applied at a given tforcing can only influence the subsequent stage of MYD evolution, cumulative deficits were further calculated to quantify the drought impacts accumulated before each timing. Because PDSI values during MYD events remain below the drought threshold (PDSI<-1), cumulative absolute PDSI was used in the analysis. Based on this, the ratio of cumulative absolute PDSI from event onset to a given month within a single event relative to the DS of that event cumulativePDSIDS was defined to quantify the proportion of drought impacts accumulated before each tforcing. The ratio increased progressively with MYD development across all MYD events (Fig. 4b). The cumulativePDSIDS values corresponding to t1, t2, t3, and t4 were approximately 0.04, 0.10, 0.17, 0.25, respectively. These results indicate that a progressively larger fraction of DS had already accumulated by later tforcing during MYD development, thereby reducing the remaining alleviation potential.

Within the relatively early MYD stages (t1∼t3), further basin-scale analyses of pairwise comparisons among t1∼t3 showed that the distinctions among early tforcing became substantially more pronounced (Fig. S4). The median Cohen's d values for pairwise comparisons among t1∼t3 generally exceeded 0.80 in all ten river basins, corresponding to large effect sizes, while the minimum values were mostly greater than 0.50, indicating at least medium effect sizes. These results indicate that the differences among t1∼t3 remain distinguishable at the river basin scale, supporting further investigation of attenuation efficiency differences among early tforcing.

For each grid, the toptimal was first identified based on the DAQM-derived k. Figure 5a shows the spatial distribution of the corresponding toptimal across the study area. Overall, t1 dominates, accounting for 58.79 % of the grids, whereas t2 and t3 account for 22.11 % and 11.06 %, respectively. To further evaluate the robustness of the DAQM-derived toptimal, an independent estimate was obtained from DS_rate responses (Fig. S5). During the numerical experiments, it was found that the identified toptimal for a given MYD event was not always identical across different hypothetical precipitation perturbation scenarios, indicating that precipitation perturbation magnitude itself can influence the identified toptimal. To reduce the influence of perturbation magnitude on toptimal identification, 21 hypothetical precipitation perturbation scenarios (prrate=0, 0.05, 0.10, …, 1.00) were separately evaluated for each MYD event, thereby integrating both small and relatively large perturbations into a comprehensive evaluation method. For each scenario, the corresponding optimal timing was first determined according to the maximum DS_rate among different tforcing. The mode of the resulting 20 optimal timings was then used to represent toptimal across different precipitation perturbation magnitudes (within the mode-based identification process, scenario without additional precipitation input, i.e., prrate=0, was not included because identical DS_rate across all tforcing prevented a unique optimal timing from being identified), hereafter referred to as the mode-based approach. Because toptimal is expressed as a discrete positive integer, the mode was adopted instead of the mean or median to ensure that the identified toptimal remained physically interpretable as an integer month. Notably, the DAQM-based estimates are also derived from all hypothetical precipitation perturbation scenarios through least-squares fitting, making the mode-based and DAQM-derived results directly comparable. The mode-based approach yields proportions of t1 (67.84 %), t2 (12.06 %), and t3 (14.57 %) (Fig. S5). Comparison between the two approaches shows that 75.88 % of the grids exhibit identical toptimal (Fig. 6b). Further analyses focusing only on grids where the two approaches yield identical toptimal show proportions of t1 (72.84 %), t2 (11.92 %), and t3 (10.60 %) (Fig. S6).

According to the DAQM constructed in Sect. 3.4, as the increment in pr_rate approaches zero, the corresponding DS_rate also approaches zero. Under this condition, the estimated value of k approaches the actual k. Therefore, the actual values of k can be approximated using the numerical experiment controlling a small perturbation of a individual input variable (prrate=0.05) and its corresponding DS_rate. Based on this principle, actual values of k were calculated from the numerical experiments and compared with the theoretical k values derived from the DAQM developed in Sect. 3.4, at the grid scale. Figure 5c shows the spatial distribution of ϵ across all given grids, with 72.86 % of the grids exhibiting ϵ values below 5 %. Slightly larger deviations mainly occur in parts of northern and southern China, whereas semi-arid regions generally exhibit relatively stronger agreement between theoretical and actual estimates. Despite these regional differences, the overall ϵ remains modest, with most grids exhibiting values around 6 %, confirming the robustness of the DAQM across the study area (Figs. 5c and S4). To further evaluate whether ϵ influences toptimal, the small perturbation experiments (prrate=0.05) were also used to independently identify toptimal (Fig. S7). Comparison with the DAQM-derived toptimal shows agreement in 89.95 % of the grids (Fig. 5d), indicating that the ϵ does not substantially alter the spatial pattern of toptimal. To further evaluate the robustness of the DAQM framework at the basin scale, comparisons between theoretical (DAQM-derived) and actual k values (small perturbation-based) were conducted across the ten major river basins in China. A strong positive correlation was observed between the theoretical and actual values, with R2 values exceeding 0.90 for t1, t2, and t3 across the ten river basins in China (Fig. 5e–g). The comparison results for each month revealed no significant deviation from the 1:1 reference line, with slopes of 1.063, 1.060, and 1.037 for t1, t2, and t3, respectively (Fig. 5e–g).

Overall, these results demonstrate the robustness and applicability of the theoretical values at both the grid and river basin scales. For this reason, the DAQM was further employed to investigate drought attenuation efficiency at different tforcing. This quantitative model was also used to identify toptimal for precipitation input across t1, t2, and t3.

To identify toptimal in various river basins during t1, t2, and t3, a series of numerical experiments were conducted using the DAQM. Figure 6 presents the spatial distributions of the changes in k between t2 and t1 (Δk21=k2-k1), and between t3 and t1 (Δk31=k3-k1). As demonstrated by the validation results shown in Sect. 4.2, the DAQM framework performs robustly across most regions, supporting the interpretation of the following spatial patterns. Additional analyses based on small perturbation experiments (prrate=0.05) further confirm that the DAQM-derived results generally capture the spatial patterns (Fig. S8). Strong consistency was observed between the responses derived from the small perturbation experiments and those obtained from the full hypothetical precipitation-gradient framework, with correlation coefficients of R=0.957 for Δk21 (Figs. 6a and S8a) and R=0.955 for Δk31 (Figs. 6b and S8b) (p<0.001 in both cases). Therefore, the spatial patterns shown in Fig. 6 can be considered reliable for interpreting drought attenuation efficiency.

To enable a clearer comparison of magnitude among cases where Δk<0, absolute values were used. Notably, the larger absolute values of negative Δk (Δk<0) were mainly observed in the humid river basins of southern China, particularly in the Pearl River and Yangtze River basins (Fig. 6 and Table 1). By comparison, smaller absolute values of negative Δk were detected in the arid and semi-arid northern regions of China, particularly in the Liao River basin (Fig. 6 and Table 1). Moreover, in 118 out of 199 grids (59.30 %), the absolute values of negative Δk31 were larger than those of Δk21 (Fig. 6 and Table 1). This disparity indicates that, in more than half of MYD occurrences, the greatest attenuation efficiency tends to occur at earliest tforcing (t1). Interestingly, we found that 40 out of 199 grids (20.10 %) exhibited positive values in both Δk21 and Δk31 (Fig. 6). These results indicate that, in a non-negligible portion of grids, the greatest attenuation efficiency does not occur at t1.

These experiments also aimed to identify the key drought event characteristics responsible for the variation in tforcing. Since DS represents the cumulative water deficit over the entire DD, and k is derived from DS, DD constitutes an important factor influencing k in addition to the standardized characteristics of PDSI itself. Therefore, the spatial distribution and statistical characteristics of DD are examined to help interpret basin-scale differences in k (Fig. 7 and Table 1). We summarize the probability density function (PDF) of Δk21, and Δk31 for all of grid-level MYD events across 10 river basins in China (Fig. 7 and Table 1). The Δk values exhibit a roughly normal distribution across river basins (Fig. 7). The median values of both Δk21 and Δk31 are negative in 9 out of 10 basins, with the median of Δk31 being lower than that of Δk21 (Table 1). In most basins, the standard deviation of Δk31 is also larger than that of Δk21, indicating greater variability in the efficiency difference between t3 and t1 (Fig. 7). These results suggest an efficiency advantage of early precipitation input (t1>t2 and t1>t3). This advantage becomes more pronounced when comparing t1 and t3. However, positive Δk21 and Δk31 values were found in 33.67 % and 23.12 % of all grids, respectively (Fig. 6a and b). Similar results were also evident in the PDFs for all of China and each individual basin (Fig. 7). Consistent patterns are observed in both the DAQM-derived results (Fig. 7) and the small perturbation-based results (Fig. S9). These results suggest a non-negligible role of the higher sensitivity of normally distributed standardized drought indices to identical precipitation perturbations under more severe drought conditions. Longer DD tends to be associated with Δk values closer to or even above zero (Fig. 7 and Table 1). In particular, the Liao River, with a relatively long DD of 28 months (DDall China=21 months), shows median Δk values of 0.022 for Δk21 and 0.003 for Δk31 (Fig. 7 and Table 1). In contrast, the Pearl River, characterized by a shorter DD of 19 months, exhibits more negative median Δk values of -0.162 and -0.200 for Δk21 and Δk31 (Fig. 7 and Table 1), respectively.

To further investigate the influence of DD as an important event characteristic, an equal-width binning method was applied to examine the variation of Δk under different DD values. Specifically, the original DD values of the selected MYD events ranged from 13 to 51 months. These events were grouped into bins according to their DD values. To ensure the robustness of the results (each bin contained no fewer than 10 grid-level events), the DD range was limited to 13 to 40 months, and the bin width was set to 4 months. This setting yielded a total of seven bins (Fig. 8). We calculated the median values of both Δk and DD within each bin. As shown in Fig. 8, we found a positive correlation between DD and Δk, with R2 values of 0.526 and 0.578 for Δk21 and Δk31, respectively. Notably, when DD exceeds 29 months (as shown in Fig. 8), the Δk values approaching zero, indicating that the drought attenuation efficiencies of t2 and t3 become comparable to that of t1. In contrast, when DD is shorter (around 17 months, as shown in Fig. 8), t1 demonstrates notably higher drought attenuation efficiency than t2 and t3. These results suggest that DD is a key factor influencing toptimal.

To clarify the alleviation advantages of the toptimal, we consider both the cumulative temporal advantage of earlier precipitation input and the statistical properties of standardized drought indices. Figure 9 and Table 2 present a comparison between the drought attenuation efficiency of toptimal and those of t1 and tDIP (the month corresponding to DIP within a MYD event). The corresponding k at t1 (k1), tDIP (kDIP), and toptimal (kmax⁡) exhibit differences. For simplicity, two ratios, kmax⁡/k1 and kmax⁡/kDIP, are used to quantify the differences in k between toptimal and t1, and between toptimal and tDIP, respectively. In terms of magnitude, kmax⁡/k1 is typically close to 1, while kmax⁡/kDIP consistently exceeds 2 (Table 2). From the perspective of the cumulative temporal advantage of earlier precipitation input, t1 tends to dominate (Fig. 9a and Table 2). Nevertheless, the presence of grids with kmax⁡/k1>1 suggests toptimal>1 should not be overlooked (Fig. 9a). At the basin scale, t1dominates in most regions, except in the Liao River basin, where the median kmax⁡/k1 reaches 1.135 (Table 2). In contrast, toptimal offers a more substantial advantage in drought attenuation efficiency compared to tDIP (Fig. 9b and Table 2). This advantage is particularly pronounced in the Liao River and Songhua River basins, where the median kmax⁡/kDIP values reach 15.430 and 12.555, respectively. By contrast, the advantage of toptimal is less evident in the Southeastern River and Hai River basins, with median kmax⁡/kDIP values of 3.428 and 2.413, respectively.

Current studies on precipitation-based drought alleviation, referring to the dominant role of precipitation in driving drought alleviation, have primarily focused on the early stage of drought development (DeChant and Moradkhani, 2015; Livneh and Hoerling, 2016; Wu et al., 2018). Yang et al. (2017) integrated early precipitation input with drought-induced losses and concluded that earlier precipitation is more effective in alleviating drought impacts. This interpretation is consistent with the cumulative temporal advantage associated with earlier precipitation input considered in this study. However, due to the statistical properties of standardized drought indices, more severe drought conditions are often associated with higher attenuation efficiency under identical precipitation perturbations (Fig. 1). To further interpret the mechanisms controlling drought attenuation efficiency, two conceptual models were considered (Fig. S2). Conceptual Model 1 considers the same hypothetical precipitation increase applied at different tforcing within an MYD event (Fig. S2b). Because the precipitation perturbation can influence DS only from tforcing onward, earlier tforcing allows a longer period during which the drought process can be affected. Consequently, toptimal under Conceptual Model 1 tends to occur at the beginning of the event (t1), i.e., toptimal→t1. Conceptual Model 2 features the same precipitation forcing applied at the same tforcing, but under different PDSI values (Fig. S2c). As a standardized index, PDSI (similar to widely used normally distributed standardized indices such as SPEI and SPI) exhibits tail sensitivity, whereby values further below the drought threshold tend to experience larger numerical responses under identical precipitation perturbations. Because DS is calculated as the cumulative sum of monthly PDSI values over DD, this property of PDSI also affects event-level DS. This mechanism forms the basis of Conceptual Model 2 and favors toptimal toward the month of DIP occurrence (tDIP), i.e., toptimal→tDIP. The interaction between these two mechanisms suggests that toptimal should theoretically fall within the interval from t1 to tDIP, because tDIP of MYD events typically occurs much later than the onset (tDIP≫t1) (Fig. S2d). The nonlinear pattern in DS response to increased precipitation (Figs. 1 and 4) can be more comprehensively understood by jointly considering these two conceptual models. To further clarify this relationship, attenuation efficiencies at different tforcing were quantified using the DAQM developed in Sect. 3.4. Under the same hypothetical precipitation forcing, a higher value of k at a given tforcing indicates greater drought attenuation efficiency. The results further show that attenuation efficiencies among t1∼t3 remain relatively close, whereas clearer separation emerges after t4 (Fig. 4a and Table. S3). This pattern suggests that t1∼t3 constitute an important early-stage response window during which the effects described in Conceptual Models 1 and 2 partially offset each other. However, progressively increasing cumulativePDSIDS during drought development (e.g., approximately 0.04, 0.10, 0.17, and 0.25 at t1, t2, t3, and t4, respectively) indicate the continuous accumulation of drought impacts during MYD evolution. Once cumulativePDSIDS reaches a sufficiently high level (approximately 0.25 at t4), the cumulative temporal advantage associated with earlier precipitation input becomes dominant. These findings further reflect the complexity of drought alleviation processes and can be interpreted through the interaction between the two conceptual models.

Consistent with several existing studies (Pan et al., 2013; Haile et al., 2020), t1 remains the dominant toptimal across China. However, we noted that t2 and t3, while occurring later than t1, are still non-negligible in the early stage of MYD. Spatially, the grids where t2 and t3 are dominant account for 22.11 % and 11.06 % of all grids, respectively (Fig. 5a). According to the classification of climate zones based on average annual precipitation (Yang et al., 2022), compared with the humid river basins in southern China, the arid and semi-arid northern regions exhibited a higher probability that drought attenuation efficiency at t2 and t3 exceeds that at t1 (Figs. 6 and 9a). This interpretation is supported by the strong agreement between theoretical and actual k values shown in Fig. 5. We further noticed that the Liao River basin (semi-arid basin) and Pearl River basin (humid basin) exhibited relatively high and low MYD attenuation efficiency, respectively. This difference may be associated with the generally more severe drought conditions in the Liao River basin compared to those in the Pearl River basin (Huang et al., 2019; Lai et al., 2024; Zhang et al., 2025). Due to the statistical properties of standardized drought indices, increased precipitation tends to induce stronger attenuation responses under more severe drought conditions. Consequently, the influence of the second mechanism becomes relatively more evident in arid and semi-arid regions, increasing the probability that t2 or t3 may exceed t1 in attenuation efficiency within the early-stage response window. By contrast, in humid basins characterized by relatively milder drought conditions, the cumulative temporal advantage associated with earlier precipitation input remains dominant, resulting in a stronger preference for t1. This conclusion was consistent with previous findings (Wang et al., 2025), showing the importance of earlier precipitation input for alleviating MYD in the humid basins of China.

Overall, our results do not suggest that “later is better” for MYD alleviation. Instead, they reveal the existence of an early-stage response window during which t2 and t3 may still exhibit attenuation efficiencies comparable to or even exceeding that of t1. This finding is particularly important for MYDs because these events involve cumulative water deficits across seasons and years, making the timing of precipitation input especially critical for drought alleviation. Missing t1 does not necessarily imply that the opportunity for maximum attenuation efficiency has already been lost, provided that precipitation input still occurs within the early-stage window.

The basin-scale Cohen's d results indicate that the attenuation efficiency differences among t1∼t3 are distinguishable and cannot be attributed to random variability or noise (Fig. S4). Accordingly, the robustness of the DAQM-derived toptimal was evaluated through comparisons among the DAQM-derived results, the mode-based results, and the small perturbation-based results. Despite methodological differences among these three approaches, all consistently indicate that t1 dominates overall, while the possibility that t2 and t3 may exceed t1 within the early-stage response window remains non-negligible (Figs. 5a, S5, S6, and S7).

Comparison between the mode-based and DAQM-derived approaches shows that 75.88 % of grids exhibit identical toptimal (Fig. 5b). The distribution of timing differences is approximately symmetric and centered around zero, suggesting that the discrepancies are spatially random rather than systematically biased. In addition, Fig. S6 shows that, even when only grids with identical results are considered, t2 and t3 still account for a substantial proportion of toptimal (11.92 % and 10.60 %, respectively), confirming their non-negligible role within the early-stage response period. Generally, small perturbation experiments (prrate=0.05) represent more frequently occurring weak perturbation conditions in real-world situations. The comparison between small perturbation-based and DAQM-derived estimates further supports the robustness of the identified toptimal. Although relative errors in k exist across grids, the overall relative error magnitude remains small across China, with 72.86 % of grids showing relative errors in k below 5 % and most of the remaining grids around 6 %, indicating relatively limited deviations overall. Even though 27.14 % of grids exhibit relative errors exceeding 5 %, this has limited influence on the identification of toptimal, as 89.95 % of grids show identical results between the two approaches (Fig. 5d). The distribution of toptimal differences in Fig. 5d further indicates that the remaining discrepancies are not systematically biased. In addition, compared with the small perturbation-based results, the DAQM-derived probability distributions of Δk appear smoother (Figs. 7 and S9), reflecting that the DAQM integrates attenuation responses across the full precipitation-gradient experiment.

Overall, the DAQM-derived results demonstrate strong robustness across multiple validation approaches. The DAQM provides a theoretical basis that helps interpret and justify the patterns observed in the numerical experiments, thereby offering a more physically interpretable basis for attenuation efficiency and toptimal identification.

Recent studies have shown that droughts are increasing in both frequency and severity (Dai et al., 2020; Fischer et al., 2025), although substantial regional differences exist. Similarly, Chen et al. (2025) highlighted that MYDs are also becoming more frequent and severe at global scales. To further characterize the dynamic evolution of MYD, three commonly used features include DD, DS, and DI. Among these, DD is a key factor for identifying MYDs, as MYDs are typically defined by a duration exceeding 12 months (Massari et al., 2022). DS represents the overall impact of MYD, while DI is related to the onset timing of the recovery stage. In current studies, standardized drought indices are commonly used to quantify drought severity (Mishra and Singh, 2010). According to the statistical characteristics of the normal distribution (Dai, 2011), drought indices responses to identical precipitation inputs can vary considerably depending on the magnitude of index values below the drought threshold. This aligns with Conceptual Model 2 presented in Sect. 5.1, where small increases in precipitation have the potential to induce substantial shifts toward alleviation when the index deviates far below the drought threshold. Our findings further indicate that t1 and tDIP of MYD do not coincide, especially in longer-duration MYD, where tDIP tends to occur much later than t1. With that in mind, we examined the relationship between DD and k. Our results demonstrate a positive correlation between DD and k in MYD (Figs. 7 and 8). Specifically, basins in northern China, characterized by longer DD, tend to have a higher probability of t2 or t3 being the toptimal within the early-stage response window, in contrast to southern basins with shorter DD (Fig. 7 and Table 1).

Droughts are not only influenced by pr, but also by other non-precipitation factors such as wind speed, air temperature, radiation, and relative humidity, among others. These non-precipitation factors are widely used to estimate potential evapotranspiration. However, this study focuses solely on pr. This is primarily because drought alleviation mainly depends on the effective management and allocation of available water resources. In future studies, the framework could be further improved by considering these non-precipitation factors. Another limitation concerns the range of pr increase scenarios considered in our experiments. As shown in Fig. S3, the long-term mean and standard deviation of monthly pr are approximately comparable, indicating that the maximum increase corresponds approximately to one standard deviation. Increases beyond 100 % would therefore represent stronger anomalies that are less frequently encountered and were not considered. Future studies could assess the drought alleviation response under such conditions. It is also essential to recognize that our study is a retrospective analysis based on historical MYDs. This study provides a framework for exploring attenuation efficiency within the early-stage response window of MYD development. However, determining whether a drought event would develop into an MYD at an early stage remains a major challenge (Xu et al., 2021; Gumus, 2023). This is due to the complexity of the factors influencing the development of droughts into MYDs. Furthermore, this study currently focuses solely on meteorological drought. Research on other types of drought, such as agricultural, hydrological, and socio-economic droughts, will also be a key focus in future studies. Future research should evaluate the applicability of our framework to other types of drought. To this end, it is also necessary to further optimize the framework by fully considering additional influencing factors (Tian et al., 2022; Wu et al., 2022b; Meresa et al., 2023).