Research on the degree of non-stationarity in extreme precipitation in the continental United States

doi:10.21203/rs.3.rs-3208166/v1

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Research Article

Research on the degree of non-stationarity in extreme precipitation in the continental United States

https://doi.org/10.21203/rs.3.rs-3208166/v1

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published 07 Dec, 2023

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Identifying the non-stationarity of hydrological extreme value sequences is crucial for understanding the patterns of hydrological systems and conducting reliable risk assessments. Current research on detecting non-stationarity primarily focuses on determining whether the parameters of a sequence vary through statistical tests. In this study, we introduce the concept of distributional changes in the detection of non-stationarity and propose the Cumulative Distribution Function Change Index (CDCI) as a measure to quantify the degree of non-stationarity in extreme value sequences. To demonstrate the feasibility and effectiveness of CDCI in assessing non-stationarity, we apply it to hourly precipitation data from 102 meteorological stations in the United States for the period from 1975 to 2021. We compare CDCI with existing methods for detecting non-stationarity to validate its rationality. Additionally, we explore the relationship between distribution, non-stationarity, and return periods using CDCI. The findings of this research indicate that: (1) distributional changes can reflect the non-stationarity of the sequence, and CDCI is more sensitive in identifying non-stationarity compared to conventional methods; (2) by combining CDCI with changes in return periods, we can identify the response of the distribution to non-stationarity, including changes occurring predominantly in low or high percentiles, or changes occurring across all percentiles and their magnitude. Therefore, the results not only demonstrate the efficacy and validity of measuring the non-stationarity of hydrological extreme value sequences through distributional changes but also reveal possible relationships among non-stationarity, distributional changes, and changes in return periods.

degree of non-stationarity

cumulative distribution function

return period

extreme precipitation

optimal probability distribution

Extreme hydrological events, as one of the most common and destructive disasters in the realm of natural calamities, exert an impact on infrastructure development and socio-economic progress at a global scale (Salman and Li, 2018). In order to mitigate the consequences resulting from extreme hydrological events, individuals often resort to the utilization of historical and monitoring data to establish statistical models or meteorological and hydrological models for the purpose of analysis and prediction (Intergovernmental Panel on Climate, 2023). Currently, the analysis of extreme hydrological events predominantly relies on statistical models built upon historical data, with this approach mainly drawing inspiration from the theory of extremes (Katz et al., 2002).

In 1941, the statistician Gumbel first applied extreme value theory to calculate the return period of floods, which has since been widely developed and applied (Gumbel, 1941; Khaliq et al., 2006; Pan and Rahman, 2022). Currently, the application of extreme value theory mainly involves two steps: 1. Selecting extreme value sequences; 2. Finding the appropriate probability distribution (Gomes and Guillou, 2015). The most common methods for selecting extreme value sequences are Block Maxima (BM), which samples based on fixed intervals, and Peaks Over Thresholds (POT), in which sample values exceed a certain threshold. Although POT is more difficult to apply than BM, the results of POT are more reliable (Bezak et al., 2014; Nguyen et al., 2017; Tabari, 2021). Once the extreme value sequences are selected, distribution fitting can be performed. Then, the fitting results can be evaluated using non-parametric testing methods to determine the optimal distribution and establish a statistical model. In traditional extreme value theory, stationarity is generally regarded as a fundamental assumption. However, in recent years, scholars have raised doubts about the stationarity of hydrological extreme value sequences, considering factors such as climate change and human activities (Salas and Obeysekera, 2014).

In general, stationarity refers to the distribution and statistical parameters of a time series not changing over time, and it is difficult for actual sequences to strictly meet the requirements of stationarity. Broadly speaking, the stationarity of extreme hydrological events refers to the relatively stable range of distribution changes (Slater et al., 2021). The question of whether hydrological sequences are stationary has sparked widespread scientific debate since the assertion of "stationarity is dead" by Milly et al. in 2008 (Milly et al., 2008; Serinaldi and Kilsby, 2015; Salas et al., 2018). The current majority of studies indicate that when a sequence becomes non-stationary due to driving factors that can be reasonably explained by physical mechanisms, non-stationary methods are recommended for analysis; while in other cases, it is advisable to prioritize the use of stationary methods (Serinaldi and Kilsby, 2015; Koutsoyiannis and Montanari, 2015; Harrigan et al., 2018). Therefore, determining whether hydrological sequences exhibit non-stationary characteristics and providing a reasonable explanation based on physical mechanisms remains a significant challenge faced by hydrologists in the present era (Ribes et al., 2017; Slater et al., 2021; Yu et al., 2023).

Currently, commonly used methods for identifying non-stationarity primarily employ statistical tests to determine its presence, aiming to achieve this by detecting changes in statistical parameters (Rahmani et al., 2016; Marston and Ellis, 2021). This approach essentially transforms the identification of non-stationarity in a sequence into the detection of parameter changes within the sequence and indirectly infers non-stationarity through these changes. Research has indicated that employing parameter changes for identifying non-stationarity may introduce uncertainty and errors (Beven, 2016; Wang et al., 2020).

Many studies have examined the non-stationarity of extreme precipitation in the United States using point or gridded precipitation data (Jiang et al., 2016; Um et al., 2017; Swain et al., 2020; Davenport et al., 2021). These studies have indicated the potential non-stationarity of extreme precipitation at daily and sub-daily scales in the US region (Carvalho, 2020; Kirchmeier-Young and Zhang, 2020; Coelho et al., 2022). Currently, the prevailing research approach focuses on assessing the non-stationarity of extreme precipitation through changes in statistical parameters (Um et al., 2017; Alashan, 2018; Zhang et al., 2023). However, it is essential to note that the non-stationarity of extreme hydrological sequences fundamentally refers to the deviation from the assumption of a homogeneous distribution. Surprisingly, few scholars have explored non-stationarity from the perspective of distributional changes.

We propose a new method, the Cumulative Distribution Function Change Indicator (CDCI), to estimate the non-stationarity of hydrological extreme value sequences. We apply this method to precipitation data from 102 meteorological stations in the United States. Based on extreme value theory, we first use the Peaks Over Threshold (POT) method to select extreme precipitation sequences. Then, we assess the most suitable distribution for each station and calculate the distribution change by dividing the sequence into two segments (Rahmani et al., 2016; Iliopoulou and Koutsoyiannis, 2020; Coelho et al., 2022). To validate the reliability of our method, we conduct parallel experiments at 24-time scales. Finally, using CDCI, we investigate the relationship between the non-stationarity, distribution change, and return period variation of extreme precipitation. The main objective of this study is to explore the feasibility of using distribution change to measure sequence non-stationarity and investigate the response of sequence distribution to non-stationarity, providing a foundation for attributing non-stationarity to physical mechanisms.

2.1. Data

This study utilized the hourly precipitation dataset provided by the National Center for Environmental Information (NCEI, https://www.ncei.noaa.gov/) in the United States. The NCEI offers a comprehensive collection of meteorological data, including temperature, pressure, wind speed, wind direction, and precipitation, from over 20,000 ground-based weather stations, spanning from 1901 to the present (Smith et al., 2011). In order to ensure consistency, completeness, and accuracy of the precipitation series, a total of 102 airport weather stations were selected for investigation. The data was collected at an hourly resolution, covering the period from January 1, 1975, 00:00 to December 31, 2021, 23:00. The specific locations of the stations are depicted in Fig. 1. Due to the extended length of the station names, a renumbering was applied to these 102 stations, and the corresponding serial numbers and airport codes were presented in Table 1.

Table 1

Station Serial Number and Corresponding Airport Code
Station number	airport code	Station number	airport code	Station number	airport code	Station number	airport code	Station number	airport code	Station number	airport code
1	PAKN	18	KSAT	35	KTYS	52	KEVV	69	KBUF	86	KBTV
2	KEYW	19	KACT	36	KFSM	53	KSPI	70	KROC	87	KGRR
3	KMIA	20	KMAF	37	KSPS	54	KSGF	71	KORD	88	KAPN
4	KSAV	21	KABI	38	KABQ	55	KMCI	72	KPIA	89	KMKE
5	KCHS	22	KLBB	39	KFAT	56	KELY	73	KFWA	90	KMSN
6	KTPA	23	KELP	40	KRIC	57	KRNO	74	KSBN	91	KHON
7	KTLH	24	KPHX	41	KDCA	58	KSFO	75	KTOL	92	KMSP
8	KMCN	25	KSAN	42	KBWI	59	KEWR	76	KDTW	93	KBIL
9	KATL	26	KLAX	43	KACY	60	KLGA	77	KLAN	94	KBOI
10	KMOB	27	KRDU	44	KPHL	61	KPVD	78	KMLI	95	KSLE
11	KMGM	28	KORF	45	KROA	62	KBDL	79	KDSM	96	KPDX
12	KMSY	29	KGSP	46	KCRW	63	KAVP	80	KSUX	97	KMSO
13	KMEI	30	KCLT	47	KCVG	64	KABE	81	KBFF	98	KGEG
14	KIAH	31	KAVL	48	KLEX	65	KPIT	82	KCPR	99	KAST
15	KSHV	32	KGSO	49	KSDF	66	KCAK	83	KSLC	100	KSEA
16	KBRO	33	KHSV	50	KHTS	67	KCLE	84	KWMC	101	KUIL
17	KCRP	34	KCHA	51	KDAY	68	KYNG	85	KMFR	102	KJFK

Please note that airport codes can be used to access information about meteorological station stations at the following link: https://www.ncei.noaa.gov/maps/hourly/

2.2 The selection of extreme precipitation sequences

Due to variations in geography, climate, and environmental factors, different stations may experience varying frequencies and intensities of extreme precipitation. To provide a comprehensive and objective depiction of extreme precipitation at different stations, this study defines the threshold for POT as the minimum value in the annual maximum precipitation sequence (Agilan and Umamahesh, 2017). To ensure the independence of selected events, a time interval (e.g., one week) is specified based on the selected results, and the Durbin-Watson test (DW) is utilized to assess the absence of autocorrelation among the filtered events (Durbin and Watson, 1950; Arns et al., 2013). Research has indicated that shorter sequence lengths can impact the fitting results of probability distributions. To ensure the accuracy of the findings, a minimum of 30 years of data is employed for modeling (DeGaetano and Castellano, 2018; Li et al., 2018; Li et al., 2019). To facilitate further analysis, the hourly precipitation sequences are aggregated to form precipitation sequences from 1 to 24 hours. Subsequently, the precipitation sequences are divided into three time periods: 1975 to 2004, 1992 to 2021, and 1975 to 2021, to capture extreme precipitation at different time scales and intervals.

2.3. Frequency analysis under distribution optimization

Extreme precipitation sequences often exhibit characteristics of right-skewness and heavy tails. Therefore, a set of candidate distributions including Generalized Extreme Value (GEV), Generalized Pareto (GPA), Pearson type Ⅲ (PE3), Generalized Normal (GNO), Generalized Logistic (GLO), and Wakeby (WAK) distributions, are selected to describe these features (Esteves, 2013; Rahman et al., 2015; Nerantzaki and Papalexiou, 2022). In this study, we employ the method of linear moments proposed by Hosking in 1990 for hydrological frequency analysis (M., 1990). The linear moment method is commonly described as a linear combination of probability-weighted moments and is widely applied in hydrological frequency estimation due to its simplicity and accuracy. We calculate the first five-order linear moments of the extreme precipitation series and use these results to estimate the parameters of the six distributions. Following parameter estimation, we investigate the goodness-of-fit of the extreme precipitation series to the selected distributions using the Cramér-von Mises (CVM) test. Subsequently, we assess the root mean square error (RMSE), the Akaike Information Criterion (AIC), and the non-stationarity index for the validated distributions. Based on the ranking of these metrics, the optimal distribution for each time scale and station is determined. Finally, design values for the return periods of extreme precipitation under the chosen optimal distribution are derived.

The Cramer-von Mises (CVM) test is a nonparametric method that measures the differences between distributions based on the squared distance (Csorgo and Faraway, 1996). During application, we calculate the test statistic ω² and compare it with the critical value of the theoretical distribution with the same parameters. If the test statistic is greater than the critical value, it indicates that the sample does not conform to the theoretical distribution. This test considers all discrepancies between the sample and theoretical distribution, making it more rigorous in theory. The simplified formula for calculating ω² is shown in Eq. (1).

$${\omega ^2}=\frac{1}{{12n}}+\sum\limits_{{i=1}}^{n} {\left[ {\frac{{2i - 1}}{{2n}} - {F_\theta }\left( {{x_i}} \right)} \right]}$$

Where n is the number of extreme precipitation samples and F_θ(x) is the Cumulative Distribution Function (CDF) of the theoretical distribution under the same parameters.

The AIC criterion is widely employed as an evaluation index for assessing the goodness of fit of distribution results (Akaike, 1974). This criterion takes into account both the complexity of the distribution and the quality of data fitting in a comprehensive manner. It yields optimal results by considering the entropy perspective. A smaller value is indicative of a better fit (Laio et al., 2009). Formula (2) illustrates the calculation formula based on the sum of squared errors (SSE).

$$AIC=2k+N\ln \left( {SSE/n} \right)$$

Where k is the number of theoretical distribution parameters and SSE is the sum of squared errors of the sequence.

The term "return period" is commonly used to refer to the time interval at which extreme precipitation, equal to or exceeding a certain level, occurs within a specific period of historical data records. This time interval is used to calculate the average occurrence of such extreme precipitation. The calculation formula for the return period is shown in Eq. (3). Return period precipitation is frequently utilized in estimating flood design for water resources and hydropower engineering projects, and it also serves as an important objective in studying the degree of non-stationarity.

$${X_T}={F^{ - 1}}\left( {F\left( {{X_{\delta T}}} \right)} \right)$$

Where T is the return period, δ is the average number of extreme precipitation events per year, and F (X) is the CDF of the extreme precipitation series.

2.4. Non-stationarity Index

This study aims to evaluate the non-stationarity of extreme value sequences by segmenting time series and fitting distributions, considering the changes in cumulative distribution functions. The extreme value sequences are divided into two parts: the recent segment and the comparison segment. The recent segment describes the recent situation of extreme precipitation sequences, while the comparison segment describes the situation of extreme precipitation sequences in comparison. The non-stationarity index obtained can not only identify the response of distributions to non-stationarity but also provide support for studying the driving factors of non-stationarity in hydrological extreme value sequences and offer references for engineering design. It should be noted that segmenting time series does not imply the occurrence of abrupt changes or trend shifts. Instead, it is used to discuss the changes in cumulative distribution functions and measure and compare the non-stationarity levels of different sequences. When applying this method, it is important to ensure consistent time lengths and the relationship between the recent and comparison segments of each sequence, as well as to use the same method for filtering and modeling to eliminate possible interference or impacts from other factors.

Firstly, in order to calculate the indicators, the series of extreme values need to be divided into two segments. On the one hand, a sequence that is too short may result in a decrease in the accuracy of the fitting, while a sequence that is too long may increase the uncertainty of the fitting. Considering the limited length of the sequence, we divided the time period into two parts: from 1975 to 2004 representing the historical comparison segment, and from 1992 to 2021 representing the recent segment (referred to as the first 30 years and the past 30 years). We fit each of these two segments separately and explore the changes in the cumulative distribution curve. Although these two segments overlap in time, they have little impact on the inter-station analysis results. Additionally, to investigate the influence of overlapping time periods on the methodology, we also included a comparison of the entire historical period from 1975 to 2021 with the last 30 years segment (referred to as all years and the past 30 years). The computation formula for the Cumulative Distribution Function Change Indicator (CDCI) is shown in Eq. (4).

$$CDCI=\frac{{\int_{0}^{1} {\left| {F\left( {{X_1}} \right) - F\left( {{X_2}} \right)} \right|dF} }}{{\int_{0}^{1} {F\left( {{X_2}} \right)dF} }} \times 100\%$$

Where CDCI is the change index of the cumulative distribution function, F (X₁) is the theoretical cumulative distribution function of the extreme precipitation series in the comparison section, and F (X₂) is the theoretical cumulative distribution function of the recent extreme precipitation series.

During the computational process of CDCI, the most critical challenge lies in obtaining the analytical solution for the given distribution integral. In this study, we attempted to compute the analytical solution but encountered failure. Consequently, we calculated the precipitation values corresponding to ten thousand evenly spaced frequencies ranging from 0 to 1 based on the parameter estimation results. Finally, we employed the Newton integration method to calculate the numerical solution. The strengths of CDCI lie in quantifying the variations in theoretical distributions, enabling the discrimination of different sequence theoretical distribution changes in terms of their intensity and reflecting their respective non-stationarity. When compared to existing methods, CDCI demonstrates higher sensitivity and is capable of qualitatively distinguishing distribution changes through variations in the return period.

3.1. Screening extreme precipitation sequences

In our study, we examined precipitation data at different time scales (1–24 hours) and across three time periods (the first 30 years, the past 30 years, and all years). The minimum value of annual maximum precipitation was used as the threshold. Subsequently, we performed de-clustering on the resulting 7344 sets of extreme precipitation sequences using a minimum time interval of 100 hours and conducted a Durbin-Watson test (DW test). The results indicated that 7330 sets passed the test, with only 14 sets showing autocorrelation. This suggests that a minimum time interval of 100 hours is appropriate. Figure 2 displays the thresholds of extreme precipitation sequences at different time scales from 1975 to 2021. For most stations, the thresholds were below 2%. Moreover, as the time scale increased, the threshold for extreme precipitation gradually increased.

From the results of the screening of extreme precipitation sequences, it is observed that this screening method selects extreme precipitation values that are 2.5 to 8 times larger than the annual maximum (AM). Regardless of the station or the temporal scale, the threshold set by this method has statistical significance. Compared to methods that rely on quantiles and graphical judgment, this screening method is more objective. It allows for the maximum retention of extreme values and minimizes the inclusion of non-extreme values, making it preferable to the BM method.

3.2. Determining the optimal distribution

Extreme value theory suggests the asymptotic distribution for a given sequence, such as GPA for POT. However, in practical situations, extreme value sequences may not adhere to the assumptions of asymptotic distributions, which can be influenced by factors like climate change, human activities, and other phenomena. Hence, an asymptotic distribution may not necessarily be the optimal choice for real extreme value sequences. We calculated the parameters of alternative distributions for extreme precipitation sequences at different time scales and three time periods. We employed the CVM test to calculate the root mean square error, AIC, and CDCI for the first and past 30 years. We determined the optimal distribution for different time scales and stations through ranking. Most distributions for all stations and various time scales passed the CVM test. However, some stations may have different optimal distributions in different time periods. Despite minor discrepancies in the ranking results for different time periods, we only considered the optimal distributions obtained through ranking in our selection process, ignoring the variability in distributions across time periods. Figure 3 depicts the optimal distribution patterns at different time scales for each station. In the figure, variations exist in the optimal distribution patterns at different time scales and stations, with most stations showing a concentration of optimal distributions in GPA, PE3, and GNO. Demographic analysis revealed that, in the first 5 hours, GPA and GNO frequencies were higher than PE3. Between 6 and 11 hours, PE3 and GPA frequencies surpassed GNO. After 12 hours, PE3 and GNO frequencies exceeded GPA. Overall, PE3 distributions were slightly more prevalent than GNO, with GNO distributions slightly surpassing GPA.

Based on the optimal distribution results from different stations and temporal scales, the GPA, PE3, and GNO distributions are generally more suitable for describing extreme precipitation sequences at various stations. Among them, the PE3 distribution is slightly superior to the GNO distribution, while the GNO distribution is superior to the GPA distribution. Furthermore, each station exhibits an optimal distribution, and these optimal distributions display a concentrated trend in temporal scales.

3.3. The verification of the degree of non-stationarity

Mann-Kendall (MK) trend test and Pettitt change-point test are non-parametric methods for detecting trends/mutations based on mean changes (Mann, 1945; Pettitt, 1979). These methods do not require the assumption of a specific distribution for the original sequence, and they are not affected by a small number of outliers. They are widely used in identifying the non-stationarity of hydrological time series. By comparing the results of the CDCI with those of the MK trend test and the Pettitt change-point test, the rationality of the CDCI can be validated. We conducted the MK trend test and Pettitt change-point test on extreme precipitation sequences at various stations and time scales using a significance level of 0.05. It was found that significant trend changes or mutations in the means were detected at some stations. The high degree of overlap between the MK and Pettitt test results may be due to the fact that both methods are based on mean changes. It should be noted that the results for the same station at different time scales may vary. Please refer to Fig. 4 for specific results.

MK test indicates significant trend changes in the mean values of the sequence, while the Pettitt test detects significant mean shifts in the sequence. These changes in the sequence are reflected in the distribution curve, thereby providing a preliminary threshold for assessing the degree of non-stationarity. We separately calculate the CDCI values for each station over different time scales of the preceding and following 30 years and compare them with the results of MK and Pettitt tests. Statistically, using a threshold of 3, approximately 98% of stations that pass the MK test have a threshold value greater than 3, and approximately 90% of stations that pass the Pettitt test have a threshold value greater than 3. At this point, there is an average of 85 stations with a threshold value greater than 3 over all time scales. Using a threshold of 5.7, approximately 86% of stations that pass the MK test have a threshold value greater than 5.7, and approximately 70% of stations that pass the Pettitt test have a threshold value greater than 5.7. At this point, there is an average of 61 stations with a threshold value greater than 5.7 over all time scales. With the increase in the threshold value, the corresponding rates of significance for MK and Pettitt tests, as well as the number of stations exceeding the threshold, decrease.

The results indicate that CDCI is more sensitive in identifying the non-stationarity of the sequences compared to the MK and Pettitt tests. MK and Pettitt only describe the non-stationarity of the sequences from the perspective of the mean, failing to detect changes in other statistical measures. When the sequences exhibit non-stationarity, their distribution changes over time and CDCI can detect such changes. Comparing the results of MK, Pettitt, and CDCI, CDCI detects a significant number of stations, indicating its validity. However, the comparison results also reveal two issues. Firstly, there are occasional cases where MK or Pettitt tests pass while CDCI values are small (very rare). One possible reason for this is the irregular distribution of extreme values in MK and Pettitt tests, as they can only guarantee a minimum interval greater than 100 hours, which can impact the results. Another possible reason could be related to overlapping time periods, as there is a 30-year overlap due to the limitations in sequence length. If the time periods do not overlap, the results might be more accurate. The second issue is the high number of stations detected with CDCI values exceeding the threshold. This is mainly because MK and Pettitt's tests are based on changes in sequence means and overlook changes in other statistical parameters, resulting in changes in the distribution. Additionally, the use of POT to select extreme value sequences in this study may cause variations in the frequency of extreme precipitation in different time periods, which can also lead to changes in CDCI. Overall, CDCI can detect distributional changes that MK and Pettitt's tests fail to detect, thus reflecting the non-stationarity of extreme value sequences.

3.4. Distribution and recurrence period response under non-stationarity

Extreme value analysis is commonly used for risk assessment. In order to explore the relationship between non-stationarity and return periods, this study calculated the precipitation for return periods of 2 years, 5 years, 10 years, 20 years, 50 years, and 100 years over the preceding and following 30 years. The variation in return periods was quantified as a percentage, and it was found that the majority of the largest variations occurred at higher percentiles. The relationship between return periods and CDCI values for the preceding and following 30 years was established, as shown in Fig. 6, which illustrates the relationship between return periods and CDCI values for each station over 1 hour for both time periods. Overall, the relationship between the two variables appeared to be radial. When the degree of non-stationarity was low, the variation in return periods was also small, indicating a dense distribution of stations. However, as the degree of non-stationarity increased, the range of variation in return periods also significantly increased.

The analysis results indicate that there may be a certain relationship between non-stationarity and return period when the degree of non-stationarity is low. However, when the degree of non-stationarity is high, the uncertainty in the relationship with the return period significantly increases. Further analysis reveals that the uncertainty is primarily composed of two aspects: one is when the degree of non-stationarity is high but the return period changes minimally, and the other is when the degree of non-stationarity is low but the return period changes significantly. Taking the cumulative frequency curve of the 1-hour station 18 as an example, as shown in Fig. 7(a), there is high non-stationarity but a minimum change in the return period. On the other hand, Fig. 7(b) shows the cumulative frequency curve of the 1-hour station 42, illustrating the opposite scenario. It can be observed from the figures that the changes in the cumulative frequency distribution for scenario one mainly concentrate on the lower quantile section, whereas for scenario two, they mainly concentrate on the higher quantile section.

In summary, according to the position shown in Fig. 6, the closer the point is to the origin, the weaker the non-stationarity. On the other hand, when the point moves away from the origin and approaches the x-axis, it suggests that the distribution changes may mainly concentrate on lower quantiles. Conversely, when it moves towards the y-axis, it indicates that the distribution changes may concentrate at higher quantiles. Additionally, the CDCI quantification theory is used to describe the non-stationarity of the distribution, which not only manifests in changes in quantiles but also correlates with frequency changes. Further research is still needed to quantitatively distinguish the response of distributions to non-stationarity.

3.5. The impact of overlapping periods on CDCI

To investigate the impact of overlapping periods on the methodology, we calculated the CDCI values for each station based on the complete dataset and the last 30 years. These values were then compared to the CDCI values from the first 30 years and the next 30 years. The sorting of all stations was done according to the degree of non-stationarity. We found that when the CDCI values for the complete dataset and the last 30 years were less than 2, which accounted for around 25% of the total number of stations, there was considerable similarity in the sorting results with the first 30 years and the next 30 years. However, as the degree of non-stationarity increased, this correspondence declined. When the CDCI values for the complete dataset and the last 30 years were greater than 4, which accounted for approximately 40% of the total number of stations, this correspondence almost disappeared. A comparison was made between the results of the MK and Pettitt tests, revealing that around 91% of the MK tests had significant station thresholds greater than 2, and approximately 80% of the Pettitt tests had significant station thresholds greater than 2. In addition, around 70% of the MK tests had significant station thresholds greater than 4, while approximately 52% of the Pettitt tests had significant station thresholds greater than 4. The guarantee rates of the MK and Pettitt tests corresponding to the CDCI values for the complete dataset and the last 30 years were lower compared to the CDCI values for the first 30 years and the next 30 years. Differences were identified among the CDCI values for the complete dataset, the last 30 years, and the first 30 years at each station. To explore the reasons for these differences, we chose the cumulative frequency curves for the 1-hour station 18 and the 1-hour station 42 as examples (see Fig. 8). The figure clearly shows a significant "flattening phenomenon" in both scenario one and scenario two for extreme precipitation, which is more pronounced at lower quantiles while the changes at higher quantiles are not significant.

To examine the impact of "flattening phenomena" on the variability of return periods, we calculated the 2-year, 5-year, 10-year, 20-year, 50-year, and 100-year return periods for both the entire period and the subsequent 30 years. The changes in return periods were quantified as percentages. A comparison between the return periods of the previous and subsequent 30 years revealed no change in the trend. Furthermore, our analysis indicated a certain relationship between non-stationarity indicators and changes in return periods under this flattening scenario. Figure 9 illustrates the relationship between the CDCI (Climate Drought Condition Index) and return periods for all stations, for both the entire period and the subsequent 30 years. Overall, due to the flattening effect, a clear linear relationship exists between the CDCI and return periods for both periods. Additionally, at different time scales, when the CDCI for the entire period and subsequent 30 years is low (between 4 and 5), there is a noticeable concentration of non-stationarity in relation to changes in return periods. However, as the degree of non-stationarity increases, the concentration weakens.

In conclusion, overlapping the reference period completely with the recent period results in a flattened distribution, which weakens the impact of percentile changes on CDCI outcomes. This leads to a decrease in the corresponding MK and Pettitt assurance rates, but at the same time increases the significance of the relationship between CDCI values and changes in the recurrence period. Therefore, it may be more meaningful to compute the CDCI value by fully overlapping the reference period with the recent period. Additionally, Fig. 9 provides further evidence of the validity of the CDCI method.

The extreme precipitation series was filtered using the POT method in this study. The minimum value from the annual maximum series was set as the threshold, and then the data underwent a de-clustering procedure. After de-clustering, the results indicated that the magnitudes of extreme precipitation were amplified by a factor of 2.5 to 8 compared to the values obtained from the annual maximum method. This suggests that the results from the POT method are more reliable than the BM method since BM may select non-extreme values or result in omissions (Bezak et al., 2014; Nguyen et al., 2017; Tabari, 2021). It should be noted that determining the time interval for de-clustering is often challenging as it is influenced by various factors such as terrain and latitude. The interval for extreme precipitation patterns can differ among different regions and topography (Agilan et al., 2021). Additionally, the main focus of this study is not on de-clustering, but rather to ensure that the resulting extreme precipitation series is free from autocorrelation.

In order to obtain the optimal distribution of extreme value sequences, we computed the RMSE of distributions validated by the CVM test, the AIC values, as well as the CDCI for the first 30 years and last 30 years, and ranked them accordingly. The results revealed that, irrespective of the time scale, the majority of stations exhibited a concentrated optimal distribution, corresponding to the GPA, the PE3, and the GNO distribution. Furthermore, different asymptotic distributions were observed for the same station at different time scales. Although the optimal distribution for some stations may not be GPA at certain time scales, we still recommend choosing the GPA distribution due to its theoretical basis as the asymptotic distribution of POT events, provided that the differences between distributions are not significant (Nguyen et al., 2017). Additionally, during the selection process of distributions using CDCI, we found that for the same time scale and the same station, different distributions yielded similar values of CDCI, indirectly demonstrating the stability of this method.

We compared the results of the MK trend test, Pettitt's changepoint test, and the CDCI test in identifying non-stationarity in the sequences. The results indicate that CDCI is able to reflect the non-stationarity of extreme value sequences by quantifying changes in their distribution, and it outperforms the MK and Pettitt tests in terms of effectiveness. CDCI can effectively improve the under-detection of non-stationarity by MK and Pettitt tests. However, since CDCI uses POT to select extreme value sequences, it introduces some uncertainty as it not only reflects changes in the distribution of the sequences but also changes in their frequency. By comparing the CDCI, MK, and Pettitt test results for each station at different time scales (first 30 years, last 30 years, and overall), we can observe that CDCI can reflect the non-stationarity of the sequences by examining changes in their distribution.

The changes in precipitation for the first 30 years, the next 30 years, all years, and the following 30 years for the 2-, 5-, 10-, 20-, 50-, and 100-year return periods were computed. The relationship between return period changes and the Climate Distribution Consistency Index (CDCI) was analyzed, and the impact of temporal overlapping on CDCI was investigated. The results indicate that CDCI qualitatively reveals the response of the distribution to non-stationarity. Temporal overlapping of sequences leads to a flattening of the distribution, reducing its sensitivity to non-stationarity. Similar to the findings of Ballarin et al., the response of extreme precipitation distribution to non-stationarity may occur at higher or lower quantiles (Ballarin et al., 2022). In fact, CDCI quantitatively describes the changes in the probability distribution itself and the non-stationarity resulting from changes in frequency. Currently, CDCI allows for a qualitative analysis of the response of the distribution to non-stationarity, but further research is needed to develop quantitative analysis methods. When CDCI is small, non-stationarity can be ignored, and the sequence can be considered stationary. However, when CDCI is large, non-stationary models should be used for physical attribution analysis in order to obtain more accurate results.

The study focuses on precipitation data and analyzes the optimal distribution of extreme precipitation series at different time scales for different stations. Additionally, it proposes non-stationarity indicators to describe the degree of non-stationarity and examines their rationality. By exploring the relationships among non-stationarity, distribution, and return periods using the Climate Data Consistency Index (CDCI), this study has significant implications for infrastructure development, particularly in determining return periods. The findings of this research provide a scientific reference for the identification and attribution of non-stationarity in extreme hydrological events. The following are the conclusions of this paper:

(1) Changes in distribution can reflect the non-stationarity of a sequence. Compared to conventional non-stationarity identification methods, CDCI exhibits greater sensitivity in identifying these changes.

(2) CDCI can reflect the degree of non-stationarity in a sequence. By combining changes in recurrent periods, it can qualitatively identify the main locations and extent of distribution changes caused by non-stationarity.

(3) Overlapping time periods can lead to a phenomenon called "flattening" of the distribution. This phenomenon weakens the impact of quantile changes on CDCI results, thus reducing the precision of CDCI in describing non-stationarity. However, it also significantly increases the correlation between CDCI and changes in recurrent periods, which may have certain practical significance in engineering applications.

Ethical Approval

This manuscript does not involve ethical matters.

Consent to Participate

All authors agree to participate in the research in this manuscript.

Consent to Publish

All authors agree to publish the research findings in this manuscript.

Competing Interests

The authors declare no conflicts of interest relevant to this study.

Funding

This research was supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), the Graduate Research and Innovation Projects of Jiangsu Province (No.KYCX23_3547) and the Science and Technology Innovation Fund of Yangzhou University (No.135030607).

Authors Contributions

All authors contributed to the design and research of the manuscript. Junbo Shao and Wenyue Wang were responsible for the data collecting and made a lot of work for the compiling of tables and plotting of graphs. Junbo Shao and Jingcai Wang carried out the data analysis and prepared the draft of the manuscript. Fan Li and Chen Wu participated in the research work and offered many important suggestions for the manuscript structure. All authors commented on previous versions of the manuscript.

Availability of data and materials

The research data obtained by the NCEI and the authors would like to express much gratitude to the data provider. Matters related to data analysis in the research are available from the first or corresponding author upon request.

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Research on the degree of non-stationarity in extreme precipitation in the continental United States

Status:

Journal Publication

Version 1

Abstract

Figures

1. Introduction

2. Materials and Methods

2.1. Data

2.2 The selection of extreme precipitation sequences

2.3. Frequency analysis under distribution optimization

2.4. Non-stationarity Index

3. Results

3.1. Screening extreme precipitation sequences

3.2. Determining the optimal distribution

3.3. The verification of the degree of non-stationarity

3.4. Distribution and recurrence period response under non-stationarity

3.5. The impact of overlapping periods on CDCI

4. Discussion

5. Conclusion

Declarations

Ethical Approval

Competing Interests

Funding

Authors Contributions

Availability of data and materials

References

Status:

Journal Publication

Version 1