A novel Joint Probability Density Difference Approach for assessing the alteration of hydrologic regime

The construction and operation of upstream reservoirs have significantly altered downstream hydrologic regime. Appropriate and quantifiable assessment method for the alteration of hydrologic regime is considerably vital and emergent for ecological protection and restoration. The Range of Variability Approach (RVA) and modified RVA methods have been widely used in practice to assess the hydrological alteration. However, these methods have failed to concurrently describe the distribution of indicator and morphological features in detail, which might inevitably lead to the misjudgment of alteration. This paper proposes a Joint Probability Density Difference Approach (JPDDA) method to address the major drawbacks of these previous methods with the introduction of Gaussian Kernel Density Estimation (KDE) and copula function. The annual average flow is selected as the reference variable to construct a proper joint probability density function between itself and other hydrological alteration indicators. The JPDDA method could describe the marginal distribution in detail through Gaussian KDE and also link the morphological features with copula function. Along with pervious methods, the hydrological alterations at Yichang hydrological station, Yangtze River are estimated based on the measured flow from 1949 to 2022. It is shown that the hydrologic regime has suffered from a moderate or even heavy alteration under the influence of massive upstream cascade reservoirs, and the JPDDA outperforms the other methods in terms of rationality and stability for practical assessments. Thus, the proposed JPDDA method is strongly advised to handle the hydrological alteration and could provide a reasonable reference for ecological operation.


| INTRODUCTION
The hydrologic regime of water ecosystem has been deviated from its natural state due to global climate change and human activities (Huang et al., 2013;Zhang et al., 2016), especially the construction and operation of upstream reservoirs (Chen et al., 2010;Gao et al., 2012).As a result, the hydrological condition of the underlying surface has suffered potential alterations, and the habitats of downstream species have also been under risk and threat, indicating the significant impact on water ecosystem (Bunn & Arthington, 2002).
Therefore, the regulation of upstream reservoirs must attach great importance to ecological operation to mitigate negative impacts and modify natural hydrological regimes (Magilligan & Nislow, 2005;Souter & Schultz, 2014).However, the natural hydrologic regime contains variability of various hydrologic characteristics exhibited in rivers, lakes, and reservoirs, gathering all the complicated hydrological factors therein.For the convenience of reference and judgment, effective and quantifiable assessment of hydrologic regime and its alteration is essential in practice.Richter et al. (1996) proposed Indicators of Hydrological Alteration (IHA) based on measured flow data, which were later categorized into five groups of 33 IHA indicators by The Nature Conservancy, the United States, to abstract and quantify hydrologic regime (Duan et al., 2016;Gao et al., 2009).The IHA has been widely applied and proven to be convenient and practical for assessing hydrologic regime with the flow indicators of magnitude, timing, duration, frequency, and change rate (Costigan & Daniels, 2012;Zhou et al., 2020).Additionally, Richter et al. (1997) proposed the Range of Variability Approach (RVA) method, and defined the hydrologic alteration of IHAs by judging whether the actual measured number of samples falling within the RVA boundaries is close to the expected value in post-impact period, which is simple but classic, effective, and widely accepted (Irwin & Freeman, 2010).With the IHA-RVA method, the hydrologic regime and its alteration could be quantitatively analyzed and compared for reference (Babel et al., 2012;Eum et al., 2017).
However, the RVA method only considers the frequency within a single interval (i.e., 25%-75%) and misses the distribution and extreme values of samples outside the interval which is relatively primitive and rough (Shiau & Wu, 2008).Additionally, the RVA may lead to significantly unacceptable and irrational assessment in certain specific situation.For example, assuming that half of samples are within the RVA range, the other half are smaller than the lower bound, and none are greater than the upper bound, then the obvious decreasing alteration would be ignored with the unchanged alteration accessed by RVA method.Shiau and Wu (2008) proposed the Histogram Matching Approach (HMA) method to overcome these limitations of RVA method, which investigates the distribution of samples in a modified way through frequency histograms and contrasts the dissimilarity between histograms in the pre-and post-impact periods.Inspired by the HMA method, Huang et al. (2017) devised a Histogram Comparison Approach (HCA) method that deduces the hydrologic alteration from the similarity of the two histograms, which further takes both the intra-and cross-class information into account and addresses the shortcomings of the unstable similarity matrix in the HMA method.
Both of the HMA and HCA methods are based on the same assumption that the flow regime in pre-and post-impact periods would be similar if their histograms resembled each other.The introduction of frequency histogram has actually compensated for the limitations of RVA method to some extent, which could reveal the tail values of IHAs.The extreme conditions might deserve more attention in the evaluation of hydrological alteration.For example, extreme high discharge would even alter the shape of streamway, while extreme low discharge would threaten the survival of ecologic species.
Although the histograms could roughly reflect the distribution of extreme values rather than RVA method, the fundamental principle of the RVA, HMA, and HCA methods remains the same.The alteration of indicators is defined by comparing frequency of samples within a finite number of classes in pre-and post-impact periods, which still ignore the differences within any individual class and suffer from subjectivity in the division of classes in frequency histograms (Vahedberdi et al., 2022;Wglarczyk, 2018).Besides, as long as the finite classes are described in histograms, the sample points around the class boundaries will inevitably and severely influence the accuracy of alteration estimation (Kim & Billard, 2013).The aforementioned disadvantages could, of course, be mitigated by further subdividing the frequency classes, and then it is exactly the probability density curve that can be obtained with infinite subdivision if the sample size is sufficiently large.Once the density curve in post-impact period significantly distinguishes with that in pre-impact period, the IHAs and hydrologic regime would undoubtedly suffer violent alterations.Vahedberdi et al. (2022) proposed a Density Difference Approach (DDA) based on the difference of probability density functions in preand post-impact periods of IHAs, which was proven to be more accurate and comprehensive to estimate the alteration of hydrologic regime than the RVA, HMA, and HCA methods.The RRVA method focuses on the morphology of IHAs in time order, that is, how IHAs vary with time series.This is quite worth to concern in the analysis of hydrologic regime alteration, for example, continuous low flow will lead to an increase of algae in river reach (Mendoza-Lera et al., 2016;Oliver et al., 2014).Actually, if the flow process is viewed as a Markov process (Erkyihun et al., 2017;Zhou et al., 2013), the inter-annual changes of IHAs are relatively independent and random with respect to chronological order, whereas Hasse matrix places too much emphasis on the morphological alteration of IHAs over chronology, which would result in a relatively higher value of hydrologic alteration within the framework of RRVA method.To balance the correlation between IHAs and morphological features, a more reasonable reference variable should be chosen to describe their conjoint changes.Yin et al. (2015) proposed the HYT-RVA method, which the order of Hydrologic Year Type (HYT), namely, wet, average, or dry year, is considered in the modification of RVA method.As an important holistic feature, the natural HYT order will deeply affect riverine ecosystems and reflect significant flow events (i.e., floods or droughts), which disturb the local river ecology significantly (Lytle & Poff, 2004).However, the HYT-RVA method mainly pays attention to the reversal of HYT and ignores the features of annual average flow, even the importance of annual average flow has been acknowledged.This study will propose a novel Joint Probability Density Difference Approach (JPDDA) that combines the advantages of RRVA and DDA methods.The annual average flow is selected as the reference variable through the Principal Component Analysis (PCA) for all IHAs, while the copula function is introduced to describe the joint morphological differences and marginal distributions of both annual average flow and IHAs.The daily measured flow discharges data series at the Yichang hydrological station, Yangtze River is chosen for a case study.
The RVA, HMA, HCA, RRVA, HYT-RVA, DDA, and JPDDA methods are used to analyze the influence of the cascade reservoirs on downstream hydrologic regime, respectively.At the same time, a comparative study is conducted to evaluate the rationality and stability of the JPDDA method.
The rest of the paper is organized as follows: Section 2 briefly introduces these previous methods; Section 3 proposes a novel JPDDA method.The case study and result analysis are presented in Section 4, while the comparison and discussion of various methods are given in Section 5. Finally, Section 6 summarizes the main conclusions.

| Indicators of Hydrologic Alteration (IHA)
The indicators used in this paper to characterize hydrologic regime are based on the 32 indicators of IHAs originally proposed by Richter et al. (1996) and later reformed by The Nature Conservancy, the United States, which consist of 33 IHAs and are divided into five groups.Each indicator is numbered as listed in Table 1.

| Range of Variability Approach (RVA)
The lower and upper bounds of RVA target range (RVA lb , RVA ub ) are generally defined as the 25th and 75th percentiles of IHAs during natural or pre-impact period.Both a significant excess and absence of sample points within the RVA range might indicate that the hydrologic regime has altered in some way.Thus, the alteration of ith IHA in RVA method (D RVA , i ) is determined as follows (Richter et al., 1997): where N o,i denotes the actual number of years for ith IHA within the RVA range in post-impact period; N e denotes the theoretically expected number of years for ith IHA within the RVA range in postimpact period; N T denotes the total number of years in post-impact period; p denotes the difference in percentiles of RVA lb and RVA ub , which is correspondingly set as 50% in this paper.

| Histogram Matching Approach (HMA)
In the HMA framework, the difference of IHA in pre-and post-impact periods is quantified with the quadratic-form distance between their frequency histograms.The calculation procedure is introduced briefly as follows (Shiau & Wu, 2008): The number of classes in the frequency histograms (n c,i ) is given by T A B L E 1 Indicators of Hydrologic Alteration (IHAs).where n o denotes the total years of ith IHA in pre-and post-impact periods; R i denotes the difference between the maximum and minimum values in ith IHA; IQR i denotes the interquartile range of ith IHA.
Then, the dissimilarity between histograms is calculated as the quadratic-form distance in the two periods (d i ): where Ppre i composed of the frequencies of each class in the pre-and post-impact histograms, respectively; A = [a i,j ] denotes the similarity matrix, and the similarity between the ith and jth classes (a i,j ) is expressed as: where V i is the average of ith class.α[1, +∞] is the index of similarity, and is recommended to set as 1 (Shiau & Wu, 2008).
Finally, the alteration of ith IHA in HMA method (D HMA , i ) is normalized as follows: A greater quadratic-form distance indicates more significant difference between pre-and post-impact histograms in IHAs, then IHAs in post-impact period would deviate from that in pre-impact or natural period with larger probability and the hydrologic regime would suffer a heavy alteration.

| Histogram Comparison Approach (HCA)
Huang et al. ( 2017) pointed out that HMA only considers intra-class difference and ignores cross-class information, meanwhile, the similarity matrix A might significantly vary with fluctuations.Similar to the HMA method, the HCA method is also proposed based on the similarity of frequency histograms.The alteration of ith IHA in HCA (D HCA , i ) is determined by: where S1 i and S2 i denote the intra-class and cross-class similarity, respectively.The calculation of S2 i requires reordering Ppre i and Ppost i in magnitude of frequency with updated class numbers t after recording the original class numbers m, n.
The intra-class similarity can be clarified similarly to the quadratic-form distance in HMA, while the cross-class similarity might reflect the difference of IHA value in each frequency order.The latter is also important to hydro-ecology.For example, salmon migrate with the high discharge for reproduction in the upstream, thus whether the most frequent flow is high or small would deeply affect the survival of this species (Wang et al., 2014).Thus, hydrological alteration might be underestimated without cross-class similarity.

| Revised Range of Variability Approach (RRVA)
The influence of temporal morphological alteration is combined with the result of RVA by Hasse matrix in the RRVA framework, which characterizes the temporal order and the value of IHA (Zheng et al., 2021).To quantify the morphological characteristics of IHA from two perspectives of "positional variable" and "magnitude variable," Q(q 1 , q 2 , …, q n ) is set as a n-year sequence of IHA, in which ] serves as a two-dimensional vector, where q i (1) denotes the "positional variable," namely, the corresponding calendar year of indicators and q i (2) denotes the "magnitude variable" defined by the following equation: where x t denotes the value of IHA (t = 1, 2, …, n); then the element of Hasse matrix (h i,j ) could be expressed as: The morphological feature of IHA in pre-and post-impact periods can be solved by Equation ( 11) as H pre and H post , respectively.The Hasse distance D H between H pre and H post with the same dimensionality of matrix could be given by the following equations: where D D and D E denote the Hasse distances of diagonal and off-diagonal elements, respectively; D H , D D , and is the weight to balance the importance of "positional variable" and "magnitude variable." Since it is generally believed that both components are equally important, v could be set as 0.5 (Wang et al., 2023;Zheng et al., 2021).
If the two Hasse matrices are different in matrix dimensionality, then the higher one and the lower one could be redefined as a n 1 Â n 1 matrix H 1 and a n 2 Â n 2 matrix H 2 , respectively.By separating n 1n 2 + 1 n 2 -demensional submatrices from H 1 along the diagonal and calculating the Hasse distances between each submatrix and H 2 , the minimum distance is determined as D H of H 1 and H 2 .D H reflects the degree of similarity between the two Hasse matrices, and the larger the D H , the higher the similarity.Thus the alteration of ith IHA in RRVA (D RRVA , i ) is determined by: In Actually, the alteration of HYT order could also be understood as morphological difference, so Equation (15) and Equation ( 16) are mathematically quite similar.The frequent reversal of HYT can be a major cause of hydro-ecology alteration.Lytle and Merritt (2004) clearly argued that for some species, such as riparian cottonwood, the order of HYT was more important than those IHAs.

| Density Difference Approach (DDA)
Since the integral of overlap region between two probability density functions is statistically defined as the degree of similarity, then the integral of non-overlapping region can correspondingly be defined as the degree of dissimilarity, or rather the alteration (Cha, 2007).In the DDA framework, the difference between the probability density curves of IHA in the pre-and post-impact periods is integrated to estimate the alteration of IHA.Similar to histogram methods, the higher value of the integral indicates more significant difference in the pre-and post-impact periods, and connotes that the hydro-ecology has been deviated from natural condition.Meanwhile DDA method overcome the ecological information loss in histogram classes.As the integral of a density function across its overall feasible region equals 1, the integral of difference between two density functions is up to 2 if they are fully out of alignment.Hence, the alteration of ith IHA in DDA (D DDA , i ) can be normalized as follows to ensure that the defined value of alteration is not greater than 1: where f pre,i (x) and f post,i (x) are the probability density functions of ith IHA in pre-and post-impact periods, respectively.Since the IHAs cover a variety of indicators such as flow, duration, number, and rate, Vahedberdi et al. (2022) suggested applying Gaussian kernel function to simulate the density function of IHA based on the theory of Gaussian Kernel Density Estimation (KDE), which has well fitted the distribution of the indicators in practice (Wglarczyk, 2018).The Gaussian kernel function is given as follows (Sheather, 2004;Silverman, 1986): where n is the years of pre-or post-impact period; h is the bandwidth; b σ is the standard deviation of the samples; IQR denotes the interquartile range of IHA.

| JOINT PROBABILITY DENSITY DIFFERENCE APPROACH
The RVA, HMA, HCA, RRVA, HYT-RVA, and DDA methods have been used to assess hydrologic alterations, each of these methods has its own advantages and limitations in practical application.The where f pre,i (x i ,y) and f post,i (x i ,y) are the joint probability density functions of ith IHA and annual average flow in pre-and post-impact periods; c(u,v) is the mixed second partial derivative of copula function; f Xi (x i ) and f Y (y) are the marginal distributions of ith IHA and annual average flow, which can also be estimated by Equation ( 19).

| Copula function
Assuming there are two random variables X and Y with their marginal distribution functions respectively expressed as F X (X) and F Y (Y) and their joint distribution function expressed as H(x,y), then, there exists a function C satisfying the following equation according to Sklar's theorem (Sklar, 1959): where C is exactly the copula function.Equation ( 23) can be rewritten as follows by representing u and v with F X (X) and F Y (Y), namely, Hence, the joint probability density function of H can be expressed as (Salvadori et al., 2016;Salvadori & Michele, 2004): The two-dimensionality Archimedean copula cannot only describe the global correlation of variables, but also effectively reflect the tail relationship, which can represent the correlation in case of extreme probability events (Nelsen, 2006).In the hydrological analysis, it is practical and accurate to select an appropriate copula function based on the tail correlation.The Archimedean copula family, especially Gumbel copula (C G ), Clayton copula (C C ), and Frank copula (C F ), are commonly applied and can be expressed by following equations, respectively: where τ denotes Kendall's tau coefficient, θ denotes the parameter of copula function.

| Degree of hydrological alteration
The root mean square (RMS) of IHAs can ultimately be used to characterize the overall or group alteration of hydrologic regime by following Equation 29 (Huang et al., 2017;Vahedberdi et al., 2022): where D i denotes the quantified alteration of ith IHA; n 0 and n 1 are the starting and ending numbers of groups or overall IHAs, respectively.In addition, the alterations of IHAs are categorized as mild, moderate, and heavy degree according to 0 ≤ D < 33%, 33 ≤ D < 67%, and 67 ≤ D < 100%, respectively (Xue et al., 2017).

| The upper Yangtze River basin
The upper Yangtze River basin (defined as the catchment upstream from Three Gorges Dam at Yichang) is located in southwest China between 90 33 0 to 111 29 0 E and 24 30 0 to 35 45 0 N, with a watershed area of about 1,000,000 km 2 , and a river length of 4504 km.Its average flow is about 13,700 m 3 /s, with extensive hydropower resources.More than 50,000 reservoirs have been built there, with total storage capacity of

| Selection of reference variable
The selected reference variable is requested to have a strong relationship with all IHA indicators.Since plural reference variables are not advised, the PCA method is adopted to screen the dominant Principal Components (PC).Generally, the 33 IHAs are intercorrelated and could be reduced to remove the statistical redundancy with PCA (Gao et al., 2009).Yang et al. (2008)   From the perspectives of the physical mechanism and ecological relevance in hydrological processes, the annual average flow is recommended as the reference variable in the previous analysis to construct a joint distribution with IHAs.Generally, the annual average flow is a more noteworthy indicator than any single IHA which is able to consider the influence of human activities (e.g., the construction and operation of reservoirs) on the downstream hydrologic regime.Annual average flow would further affect the decision-making and directly or indirectly make a nonnegligible deviation from natural conditions (Yin et al., 2015).For example, the long-term impoundment of reservoirs in wet years would result in continuous submersion and high-water level in upstream, while the projected release discharge in dry years would help to relieve downstream drought.In addition, the annual average flow is not among the IHA indicators in Table 1, and is an ideal reference variable to construct joint distribution with IHA effectively.

| Alteration of hydrologic regime
The IHAs and the alteration of hydrologic regime at Yichang hydrolog- presents a dynamic relationship with precipitation in timing and magnitude (Wang et al., 2021).The hydrological alteration caused by upstream cascade reservoirs would concentrate more on the magnitude rather than timing of extreme since the outflow is directly  Morphologically, the JPDDA method would apply the copula function to tightly link IHAs and the reference variable, rather than discrete product correction like HYT-RVA and RRVA methods in Equations ( 15) and ( 16).Meanwhile, annual flow is a more rational reference variable than HYT and time order with strong ecological relevance between itself and IHAs.For example, the DDA method All of the above alteration assessment methods do not consider the morphological characteristics of IHAs which are crucial from the perspective of hydro-ecology.Zheng et al. (2021) connected the magnitude and location (i.e., time order) of the samples through the Hasse matrix, and proposed the Revised Range of Variability Approach (RRVA), which indicates the conjoint changes in temporal sequence and indicator frequency to assess hydrologic regime comprehensively.
morphology-based methods (RRVA and HYT-RVA) with the introduce additional reference variables such as time order or HYT order, and reveal how the alteration of IHAs would vary with selected reference variables.In other words, morphology-based methods focus on the conjoint change of reasonable reference variable and indicators(Arau'jo, 2010;Arau'jo et al., 2006), which is mathematically described as Hassen distance in RRVA and Euclidean distance of HYT in HYT-RVA method.We proposed a novel Joint Probability Density Difference Approach for assessing hydrologic alternations to combing the advantages of morphology-based methods and DDA method, in which the copula function is used to fit the joint distribution of each IHA and annual average flow.The annual average flow will not only have a direct influence on the hydrological flow indicators, but also indirectly influence the downstream hydrological regime through the decisionmaking of reservoir operation(Kangrang et al., 2017), especially when encountering consecutive dry (wet) hydrologic years or heavily dry (wet) hydrologic years.Since the probability density function is much accurate to describe the distribution and the DDA method performs much better than RVA, HMA, and HCA methods(Vahedberdi et al., 2022), the alteration of ith IHA in JPDDA (D JPDDA,i ) can be determined by Equations (21-22):

163. 3
billion m 3 , total flood prevention storage of 49.8 billion m 3 , total installed hydropower capacity of 11.424 GW, and total designed annual hydropower generation of more than 300 billion kWÁh.Three Gorges Reservoir (TGR), as the world's largest comprehensive hydro-junction project began to impound water in 2003, which has significantly altered the downstream hydrological regime(Cai et al., 2019;Cha, 2007).Four largescale cascade reservoirs (Wudongde, Baihetan, Xiluodu, and Xiangjiaba Reservoirs) in the upstream have successively started impoundment from 2012, which exacerbated the hydrological alteration (Guo et al., 2018; Zhang et al., 2021).Yichang hydrological station is the representative hydrological station of the upper Yangtze River basin and the TGR outflow control station.In this study, the flow data series of Yichang hydrological station from 1949 to 2022 are selected to assess hydrological alteration.Under the influence of the construction and operation of the upstream cascade reservoirs, the measured flow series of Yichang hydrological station from 1949 to 2022 is divided into 1949-2002 and 2003-2022 two flow series respectively.The former series represents the natural flow series, while the latter series represents the flow being influenced by upstream reservoir regulation.
have applied PCA to search the most representative hydrologic indicators as the ecologically relevant hydrologic indicators from IHAs.Gao et al. (2009) have dealt with 33 IHA indicators with PCA and traced the ecological relevance of those PCs for the selection of dominant IHA statistics.Thus, the PCA method is used to downscale the dimensionality of the 33 IHAs and identify the possibly correlated reference variable from a mathematical perspective (see Figure 1).

Figure 2
Figure 2 shows the PCA of 33 IHAs.The eigenvalues of PC1 and PC2 are dominant with the corresponding proportions up to 43% and 16%, respectively.PC1-8 is selected with their cumulative proportion up to 87%, indicating that most of the information ical station from 2003 to 2022 are calculated by RVA, HMA, HCA, RRVA, HYT-RVA, DDA, and JPDDA methods, respectively.The results are listed in Table 2 and shown in Figure 4.The monthly flow indicators suffer from various degree of alteration.On the whole, IHAs of monthly flow during the flood season (from June to September) present relatively mild hydrological alteration, while during the dry season especially from January to March show mediumheavy degrees of alteration.The assessment results would also be associated with the cardinal value of IHAs.For example, though the average of March monthly flow increased by about 2500 m 3 /s and that of July decreased by about 2700 m 3 /s, both of which have similar changes in magnitude, the quantified alteration of the former is much higher than the latter.Generally, the smaller the indicator cardinal, the smaller the range of its 25-75% RVA boundaries, which are [3900, 4600] m 3 /s for March and [26,100, 33,900] m 3 /s for July.Hence, similar difference in the magnitude of IHAs value might lead to heavier alterations on the indicators with smaller cardinals (Wang et al., 2023).The minimum flow indicators and base-flow index have been heavily altered from 2003 to 2022, while the alterations of F I G U R E 3 Scatter plot of Principal Components and annual average flow.maximum flow indicators seem to be influenced more mildly than other IHAs in group 2. There are numerous reservoirs in the main stream and tributaries in the upper Yangtze River basin, which form a complex pattern of mixed-connected cascade reservoirs.Hence, the above phenomenon would correspond to the regulation of the upstream reservoirs, namely, storing flood water to help meet future demands during the dry season.The difference degree of quantified alteration in minimum and maximum flow indicators might also be affected by cardinals.The timing of extreme flow shows the mildest change among five IHA groups, assessed as mild alteration by almost all methods except HYT-RVA.One-day minimum flow has occurred 10 days earlier on average and the 1-day maximum flow has been delayed by 11.8 days in post-impact period.Actually, the runoff in Yangtze River basin Figure 5d indicates max probability values in 2003-2011 and 2012-2022 are about 50% higher than that in 1949-2002.Internally compared with natural flow series, histograms estimate the distribution with higher probability, which contributes to a significant higher alteration in 2003-2011.Generally, the number and range of classes in histogram should have respectively been divided according to the differences between the maximum and minimum values and various for each series.But histogram-based methods unify IHA series in different periods into only one histogram with same class number and range.Then, for 2003-2011 series, whose differences between the extreme values are narrowed and the divided class range should have been reduced compared with 1949-2002 series, four of total nine years fall into the 7192-8345 m 3 /s with higher probability value of 0.44 as shown in Figure 5d.Thus, histogram-based methods are less accurate especially if the sample size is small.It comes to worse for range-based methods that only count 6 points in 25%-75% range, and 3 in 75-100% for 2003-2011 series.The density-based methods (DDA and JPDDA) can describe the distribution of IHA more precisely.

F
I G U R E 4 Group and overall alterations from 2003 to 2022 at Yichang hydrological station.[Color figure can be viewed at wileyonlinelibrary.com] recognizes the April monthly flow with mild alteration with a slight increase from 6600 to 7050 m 3 /s.However, the average of annual flow decreases from 13,900 to 12,400 m 3 /s, indicating that the proportion of April monthly flow in annual flow has increased from 3.9% to 4.7% as the upper-left-shifted distribution shown in Figure 5b.The HYT-RVA and RRVA methods failed to recognize the proportion change let alone other non-morphology methods.The JPDDA method could reflect how the operation of upstream reservoirs has altered the annual allocation of streamflow spatially and temporally and how annual flow indirectly influences IHAs through the reservoir release.To further compare and analyze all methods, Pearson correlation analysis is applied based on the assessed alteration of all IHAs from 2003 to 2022. Figure 7 shows Pearson correlation coefficient of various hydrological alteration assessment methods, on which the determined alterations indicate strong correlations.Unsurprisingly, the scatter distributions of RVA, RRVA, and HYT-RVA methods are almost linear, which is consistent with the previous analysis of Equations (15) and (16).In addition, the HMA and HCA, DDA and JPDDA methods are also linearly close to each other.The HMA and HCA methods are both based on the same histograms for The distributions of April monthly flow and annual flow at Yichang hydrological station.[Color figure can be viewed at wileyonlinelibrary.com] calculation and analysis.The JPDDA method is the improvement of DDA and contains information of DDA as determined in Equation (22).The application of the copula function connecting two variables to construct the joint distribution still demands the marginal distribution of the selected variables.It is worth mentioning that JPDDA method presents the lowest correlation with RVA, which has improved the RVA method as much as possible rather than other previous methods.6| CONCLUSIONSTo address the limitations of existing hydrologic alteration assessment methods, a novel JPDDA method that combines the advantages of morphology-based methods and DDA method is proposed and compared with previous methods.The alterations of hydrologic regime at Yichang hydrological station are estimated by the RVA, HMA, HCA, RRVA, HYT-RVA, DDA, and JPDDA methods, and the rationality and stability of the JPDDA method are analyzed and discussed.The main conclusions are summarized as follows: 1.The proposed JPDDA method has the following advantages: retain the distribution information of the samples to the maximum extent through the Gaussian KDE, reduce the influence of noise and boundary condition on alteration assessment, and take into account the morphology of the IHAs.2. With the construction and operation of the upstream cascade reservoirs, the hydrologic regime at the Yichang hydrological station has changed moderately from 2003 to 2022, and altered even severely from 2012 to 2022.The overall hydrological alterations in the two periods are 57.5% and 72.2%, respectively assessed by the JPDDA method.

F
I G U R E 7 Pearson correlation coefficient of various hydrological alteration assessment methods (the upper-right corner presents the Pearson correlation coefficient; the lower-left corner presents the scatter plots of each two methods.)F I G U R E 6 Comparative analysis for hydrologic alteration of April monthly flow with different methods and periods.[Color figure can be viewed at wileyonlinelibrary.com] Hasse matrix, diagonal elements indicate the magnitude and off- diagonal elements indicate the morphology of time order.The Hasse matrix could reflect how IHAs vary with time, and Hasse distances could quantify the morphological alteration of IHA.Assuming that the discharge during the dry season continues to rise with time order as those planned cascade reservoirs has impounded in the upstream and the hydro-ecology has obviously been influenced, but the magnitude distribution still happen to follow natural condition, then RVA, HMA, HCA, and DDA methods are unable to recognize this morphological increase, while RRVA could identify this abnormal alteration.2.6 | Hydrologic Year Type-Range of Variability ApproachYin et al.(2015)indicated that it is of necessity to incorporate annual average flow into hydrological alteration assessment which has a significant impact on hydrologic regime.Similar to Equation (10), the HYT order is divided into wet, average, and dry hydrologic years, denoted by 3, 2, and 1, respectively, based on the value of annual average flow and its 75% and 25% percentiles.The alteration of ith IHA in the HYT-RVA method (D HYT-RVA , i ) is determined with the Euclidean distance between HYT order vectors in pre-and postimpact periods, that is, HYT pre hytpre 1, hytpre 2,…,hytpre n pre ð Þ , HYT post hytpost 1, hytpost 2, …, hytpost n post ð Þ as shown in the following equations: Assessments of Indicators of Hydrologic Alteration and the alteration of hydrologic regime by various methods.