CN111177211B - Runoff sequence variation characteristic analysis method based on pole symmetry modal decomposition - Google Patents
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Abstract
The invention relates to a runoff sequence variation characteristic analysis method based on pole symmetry modal decomposition. In view of the fact that the runoff sequence is a non-stationary sequence with multiple time scales and non-linear trends, the method utilizes a pole symmetric mode decomposition method which has the advantages of self-adaption, no basis, discrimination of large-scale circulation and non-linear trends of the runoff sequence, time-frequency analysis does not need to rely on integral transformation, the hydrologic characteristics and change rules contained in the runoff sequence are researched, and the upstream of the Yangtze river is taken as an example for application description. Firstly, the runoff sequence at the upstream of the Yangtze river is decomposed into stable modal components and trend remainder in different time scales step by step. And then, respectively utilizing the fast Fourier transform periodic chart, the trend remainder under the optimal self-adaptive global average line and the frequency and amplitude time-varying chart of the time-frequency analysis to obtain the periodic variation, the nonlinear trend variation and the mutation rule of the multiple time scales of the runoff sequence, and simultaneously grasping the variation characteristics of the runoff sequence in an omnibearing manner from the three aspects of the period, the trend and the mutation.
Description
Technical Field
The invention relates to the field of variation characteristic analysis of runoff time sequences, in particular to an analysis method of runoff sequence period, trend and mutation rule based on pole symmetry modal decomposition.
Background
The climate change is overlapped with the human activity which is continuously enhanced, so that a watershed water circulation system and an underlying surface are changed to different degrees, physical causes of watershed hydrologic circulation and water resources are obviously changed, further, different degrees of variation of runoff sequences are caused, extreme hydrologic events such as drought, flood and the like are frequently caused, and the sustainable utilization of the water resources is influenced. Therefore, the scientific understanding of the water circulation process taking runoff as a main indicator and the change thereof is a precondition of reasonably utilizing water resources, and has important value for grasping the change characteristics of runoff in a river basin. In recent years, the characteristics of river basin runoff sequence changes have been a great concern for climatists and hydroworkers. The study of the runoff sequence change characteristics can help people to know the time-space evolution law of the watershed water resource in a changed environment more deeply, and has important significance for comprehensive development and utilization, scientific management and optimal scheduling of the water resource.
Technical proposal of the prior art
The variation characteristics of the runoff sequence can be analyzed from three aspects of period, trend and mutation, and various methods such as power spectrum analysis, morlet wavelet transformation, mann-Kendall (M-K) trend test, sliding average method, mann-Kendall (M-K) mutation test, sliding T test, empirical mode decomposition (Empirical Mode Decomposition, EMD), ensemble empirical mode decomposition (Ensemble Empirical Mode Decomposition, EEMD) and the like are proposed and applied by the hydrologists at home and abroad around the variation characteristics of the hydrologic sequence. The power spectrum analysis method can reveal the periodicity of the discrete data sequence, and is commonly used for analyzing the periodicity rule of the runoff time sequence; the Morlet wavelet transformation has a multi-resolution function, and can clearly reveal periodic variation rules of different time scales hidden in runoff sequences; the M-K trend test method is a non-parametric test method, samples of which do not need to follow a certain specific distribution and are commonly applied to trend test of hydrologic time series; the sliding average method has the advantages of simplicity and intuitiveness, and is widely applied to hydrology; the M-K mutation detection is simple and convenient to calculate and wide in detection range, and is often applied to mutation point detection of hydrologic time sequences; the sliding T test is to consider two sub-sequences in the runoff sequence as two overall samples, and test whether the average value of the sub-sequences has significant difference; the EMD method is a novel nonlinear and non-stable time-frequency analysis method and has the characteristics of self-adaption, no basis and local change based on signals; the EEMD method is an improvement of the EMD method, not only maintains all advantages of the EMD method, but also obtains final modal components by adding different white noise to the original signal for multiple times to carry out EMD decomposition and averaging the multiple decomposition results, thereby solving the problem of modal aliasing in the EMD decomposition process. Zhong Yonghua et al analyze the periodic law of runoffs from above the cloud reservoir by means of continuous power spectrum and the like. Xu et al and Gu Lizi. Ai Ni Wal et al studied the periodic variation law of runoff sequences on multiple time scales by wavelet transformation, respectively.
Et al and->
The trend change rule of runoffs is studied by using Mann-Kendall (M-K) trend test and comprehensive Addition Wavelet Transform (AWT), M-K trend test and sequential M-K (SMK) trend test respectively. Nourani et al analyze the time scale characteristics and trend change rules of the runoff sequence using discrete wavelet transform and M-K trend test. Xie et al studied the trend law of annual runoffs in black river basin using M-K trend test. Wang Feng et al study the variation law of runoff in the Anhui section of Yangtze river by integrating methods such as M-K trend test and wavelet transformation. Chen Lihua et al study the trend change and mutation rule of rainfall runoff by comprehensive sliding average, M-K test and other methods. Yan Ming et al studied the mutation law of precipitation in the river basin of inkstone by using M-K mutation test and sliding T test. Ye et al studied the periodic variation law at the jing river Zhang Gushan station using the EMD method. Wang et al analyzed the variation law of runoff using EEMD method.
Disadvantages of the prior art
The above methods can analyze and master the variation characteristics of the runoff sequence from different sides, but have certain disadvantages: the power spectrum analysis is concentrated on directly carrying out periodic rule analysis of a single time scale on the runoff sequence, so that the periodic variation characteristics of multiple time scales in the runoff sequence evolution process are difficult to accurately reveal; the wavelet transformation result is affected by a preselected basis function, and the intrinsic law of runoff cannot be truly explored; the linear trend obtained by the M-K trend test can not reflect the change condition of each stage of the runoff sequence, and the rising or falling trend of the runoff sequence is partially exaggerated or reduced in the change degree; the application effect of the moving average method depends on the selection of parameters to a great extent, and certain subjectivity and randomness exist; the runoff mutation rule obtained by the M-K mutation test method may have false mutation years; the sliding T test mainly identifies and tests the mean variation, which may cause incomplete mutation results; the EMD method has the problem that different modal components may not be effectively separated according to time scale characteristics; the EEMD method has the problems that the decomposed trend remainder is rough and the screening times are difficult to determine. The runoff time sequence contains various frequency components, is a typical nonlinear trend and a non-stationary sequence with multi-time scale aliasing, and needs to adopt a time-frequency analysis method suitable for analyzing the nonlinear and non-stationary sequences.
Disclosure of Invention
Aiming at the defects existing in the prior art, the invention utilizes a pole symmetric modal decomposition (ESMD) method which is developed in recent years and is good at searching for the change trend from the observation sequence of years to study the change characteristics of the runoff sequence. The method is a data self-adaptive analysis method, does not need a priori basis function, has variable decomposition modes, frequency and amplitude, and is suitable for nonlinear and non-stationary time sequence analysis. The ESMD method comprises two parts of modal decomposition and time-frequency analysis. The modal decomposition utilizes the symmetric interpolation of internal poles, decomposes according to the scale characteristics of the data, gradually decomposes the runoff sequence into stable modal components and trend remainder of different time scales, and effectively discriminates the large-scale circulation and nonlinear trend of the runoff sequence. The trend remainders obtained by decomposition are optimal self-adaptive global average lines (Adaptive Global Mean curve, AGM) in the meaning of least squares, and the average lines accurately reflect the optimal change trend of each stage of the time sequence. The time-frequency analysis directly generates the instantaneous frequency from the discrete data by using a linear interpolation method, and does not need to carry out integral transformation on the data, thereby getting rid of the constraint of mathematical theory which is applied to the process of converting the discrete signal into an analytical function.
The ESMD method is a novel nonlinear and non-stable time sequence analysis method, and can simultaneously analyze period, trend and mutation rule by means of the internal pole symmetric interpolation of modal decomposition and the linear interpolation method of time-frequency analysis, so as to comprehensively study the change characteristics of the runoff sequence. Firstly, utilizing internal pole symmetric interpolation of ESMD modal decomposition, decomposing a runoff sequence step by step into stable modal components of different time scales and trend remainders according to time scale characteristics of data, and secondly, utilizing a Fast Fourier Transform (FFT) periodogram method to identify a periodic variation rule of multiple time scales in the runoff sequence and analyze trend variation characteristics of the runoff sequence based on optimal AGM. Finally, utilizing the time-varying chart of frequency and amplitude of ESMD time-frequency analysis to intuitively embody the mutation rule of the runoff sequence.
In order to achieve the purpose, the invention adopts the following specific technical scheme:
a radial flow sequence change characteristic analysis method based on pole symmetry modal decomposition utilizes a pole symmetry modal decomposition (ESMD) method to study the change characteristics of a radial flow sequence from three aspects of period, trend and mutation at the same time, and comprises the following steps:
step 1: modal decomposition, namely decomposing the runoff sequence step by step into steady modal components with different frequencies and trend remainder;
step 2: utilizing an optimal self-adaptive global average line (AGM) to master the general trend change of the runoff sequence;
step 3: utilizing a Fast Fourier Transform (FFT) periodic chart method to master the periodic variation rule of multiple time scales of the runoff sequence;
step 4: and (3) utilizing time-frequency analysis to master the mutation rule of the runoff sequence.
Step 1: modal decomposition, namely decomposing the runoff sequence step by step into steady modal components with different frequencies and trend remainder; the method specifically comprises the following calculation steps:
step 1-1: inputting a runoff sequence X (t), setting the maximum screening times K and the number of residual poles l, finding out all poles in the runoff sequence X (t), and marking as E i (1≤i≤n)，E i ＝(z i ，y i ) The method comprises the steps of carrying out a first treatment on the surface of the Connecting adjacent poles by line segments, and marking the points in the line segments as F in sequence i (1 is less than or equal to i is less than or equal to (n-1)); supplement left and right boundary midpoint F 0 ，F n ；
The 1 st and 2 nd maximum points are used as linear interpolation, and the 1 st and 2 nd minimum points are used as linear interpolation, so that two interpolation straight lines are respectively recorded as y1 (z) =p1z+b 1 And y 2 (z)＝p 2 z+b 2 The method comprises the steps of carrying out a first treatment on the surface of the Point 1 of the data is marked as Y 1 。
1) If b 2 ≤Y 1 ≤b 1 Will b 1 And b 2 Respectively defining a boundary maximum value point and a boundary minimum value point;
2) If b 1 ＜Y 1 ≤(3b 1 -b 2 ) And/2, Y 1 And b 2 Respectively defining a boundary maximum value point and a boundary minimum value point; if (3 b) 1 -b 2 )/2≤Y 1 ＜b 2 Will b 1 And Y 1 Respectively defining a boundary maximum value point and a boundary minimum value point;
3) If Y 1 ＞(3b 1 -b 2 ) And/2, Y 1 Is defined as the boundary maximum point, and the boundary minimum point is defined by a straight line drawn from the first minimum point, where the slope is determined by the distance between the left boundary point (0, Y 1 ) And a straight line of the first maximum point; if Y 1 ＜(3b 1 -b 2 ) And/2, Y 1 Is defined as a boundary minimum point, and a straight line drawn from the first minimum point is used to define a boundary or maximum point, where the magnitude of the slope is determined by the distance between the left boundary point (0, Y 1 ) And a straight line of the first maximum point;
step 1-2: two interpolation curves are respectively marked as L by the obtained (n+1) midpoints 1 And L 2 ；
L 1 Is generated by cubic spline interpolation for the midpoint of the odd ordinal, L 2 For the generation of the midpoint of even ordinal number by cubic spline interpolation, calculating a mean curve L * :
L * ＝(L 1 +L 2 )/2 (12)
Step 1-3: construction of the sequence (X (t) -L * ) For the sequence (X (t) -L * ) Repeating the steps 1-1 to 1-2 until the absolute value L * The number of screening times K is less than or equal to epsilon or reaches the set maximum screening times K, wherein epsilon is a preset allowable error, and the first empirical mode M is obtained by decomposition 1 (t)；
Step 1-4: construction of the sequence (X (t) -M 1 (t)) pair sequence(X(t)-M 1 (t)) repeating the steps 1-1 to 1-3 to obtain M in turn 2 (t)，M 3 (t)，…，M q (t) until the trend residual R (t) meets the number of residual pole points l preset in the step 1-1, wherein the trend residual R (t) is commonly called as AGM;
step 1-5: the value range of the given maximum screening times K is within an integer interval [ K ] min ，K max ]Calculating variance ratio G, recording radial flow sequence
Trend remainder->
Sum sigma 0 The relative standard deviation of X (t) -R (t) and the standard deviation of the runoff sequence X (t), respectively;
G＝σ/σ 0 (16)
in the middle of
Is the average value of the runoff sequence X (t). Where G is the smallest, this means that the sequence with the trend remainders R (t) removed and the runoff sequence X (t) are closest, i.e. the decomposition result is the best.
Step 1-6: drawing a variation graph of variance ratio with K, and selecting the maximum screening times K corresponding to the minimum variance ratio 0 When k=k 0 When R (t) is the best fit curve of the runoff sequence, at K 0 Repeating the steps 1-1 to 1-5 to obtain the most importantAnd (3) a best decomposition result, wherein the best decomposition result is a modal component and a trend remainder R (t) of different frequencies.
Reconstructing the modal components with different frequencies and the trend remainder R (t) obtained by decomposition to obtain a runoff sequence X (t), which can be expressed as:
step 2: obtaining the general trend change of the runoff sequence by utilizing the optimal self-adaptive global average line (AGM): the best adaptive global average line (AGM) curve is influenced by the number l of residual poles, and the larger l is, the larger the fluctuation of the best adaptive global average line (AGM) curve is, and the higher the fitting degree of the best adaptive global average line (AGM) curve with a runoff sequence is. However, an excessive increase in l may result in a decrease in the number of modal components obtained by decomposition, and thus the obtained runoff periodicity may not be comprehensive. Considering that the modal components obtained by the pole symmetric modal decomposition (ESMD) method reflect the change trend under different modes, the best adaptive global average line (AGM) reflects the general trend of the runoff sequence, and both reflect the change trend of the runoff sequence, so that the fitting degree of the best adaptive global average line (AGM) and the original runoff sequence is high without pursuing all. Therefore, under the condition of ensuring the obtained cycle rule to be comprehensive, the optimal self-adaptive global average line (AGM) can accurately reflect the overall trend change of the runoff sequence, the number of the remaining poles is continuously adjusted, and the optimal decomposition result is found. Then, based on the optimal self-adaptive global average line (AGM), the trend change rules of different stages of the runoff sequence are mastered.
Step 3: utilizing a Fast Fourier Transform (FFT) periodic chart method to master the periodic variation rule of multiple time scales of the runoff sequence;
the power spectrum of the modal components with different frequencies, which are obtained by decomposing by a pole symmetric modal decomposition (ESMD) method, is calculated by utilizing a Fast Fourier Transform (FFT) periodogram method, and the frequency of the radial flow signal is obtained according to the magnitude of the amplitude, so that the average period of each modal component is calculated.
Power spectrum available
Expressed as:
wherein: h is the total number of runoff samples, o H (v) For energy limited signal, O H (omega) is o H (v) The frequency domain value of the fourier transform, v is the random analog signal and ω is the frequency of the fourier transformed signal.
The modal components described in the above schemes are classified as internal odd-dipole point symmetry, external envelope symmetry or other symmetry.
Step 4: the mutation rule of the runoff sequence is mastered by utilizing time-frequency analysis, and the method specifically comprises the following calculation steps:
step 4-1: acquiring a time-amplitude variation curve: obtaining a time-amplitude change curve by measuring the upper envelope of the modal component of which the internal odd-dipole point symmetry or the external envelope symmetry is decomposed by an ESMD (object model description) method; other forms of symmetry are that absolute values are firstly taken for values corresponding to the model components, and then upper and lower envelopes are generated by maximum value points through interpolation, so that a time-amplitude change curve is obtained, wherein an amplitude function is recorded as A (t);
step 4-2: generating a phase angle: according to the pole E obtained in step 1-1 i (1. Ltoreq.i.ltoreq.n) the function value corresponding to the first point of the modal component
And instantaneous amplitude +.>
Taking the arcsine to generate the phase angle:
sequentially calculating t and E which are more than or equal to 2 and less than or equal to E according to a formula (20) 1 Is a phase angle of a point of (2);
similarly, every two poles are added, the phase is increased by 2π.
Step 4-3: acquiring a time-frequency change curve: calculating the instantaneous frequency in hertz (Hz) from the phase angle obtained in step 4-2 taking the center difference quotient with respect to 2 time steps deltac:
and supplementing left and right boundary values by a linear interpolation method:
f 1 ＝2f 2 -f 3 ， (24)
f H ＝2f H-1 -f H-2 (25)
a time-frequency variation curve is obtained from the instantaneous frequency.
Step 4-4: obtaining a time-varying graph of frequency and amplitude: and obtaining a time-varying graph of the frequency and the amplitude of each modal component according to the time-amplitude and time-frequency variation curve, and researching the mutation rule of the runoff sequence.
And (3) carrying out comprehensive runoff change characteristic analysis according to the general trend change of the runoff sequence obtained in the step (2), the periodic change rule obtained in the step (3) and the mutation rule obtained in the step (4).
The technical scheme of the invention has the beneficial effects that:
(1) And the internal pole symmetric interpolation of the ESMD method is utilized to discriminate the large-scale circulation and nonlinear trend of the runoff sequence. Furthermore, by utilizing the FFT periodogram method, the periodic variation rule of multiple time scales of the runoff sequence can be mastered;
(2) Based on the trend remainder under the optimal AGM, the trend change of different stages of the runoff sequence can be mastered;
(3) The mutation rule of the runoff sequence can be intuitively mastered by utilizing the time-varying graphs of the frequency and the amplitude of each mode obtained by ESMD time-frequency analysis;
(4) The ESMD method can analyze the period, trend and mutation rule simultaneously, achieves good effect in the runoff change rule analysis of 8 hydrologic stations of the main and branch flow at the upper stream of the Yangtze river, and provides basic support and decision basis for hydrologic forecasting and reasonable water resource allocation. Similarly, the method can be applied to runoff analysis and water resource development and utilization of other watershed hydrologic stations.
Technical key points and points to be protected of the application:
the ESMD method is applied to the variation characteristic analysis of the runoff sequence, and the FFT periodic chart method, the optimal AGM and the time-varying chart of frequency and amplitude are respectively utilized to obtain the periodic variation, the nonlinear trend variation and the mutation rule of the runoff sequence in multiple time scales, so that the variation characteristic of the runoff sequence is mastered in an omnibearing manner.
Drawings
The invention has the following drawings:
FIG. 1 is a diagram of a research framework of the present invention.
Fig. 2 is a schematic diagram of the geographic location of the main and branch stream 8 station upstream of the Yangtze river.
FIG. 3 shows the periodic variation law of runoff sequences of 8 stations of the dry tributary upstream of the Yangtze river at different time scales;
(a) ESMD is compared with Morlet wavelet transformation annual runoff period;
(b) The period of the runoff of the month and the day is regular.
FIG. 4 is a trend residual of the 8 station runoff sequence of the upstream main and branch stream of the Yangtze river obtained by ESMD decomposition;
(a) Annual runoff;
(b) Annual runoff;
(c) Annual runoff.
FIG. 5 is a time-varying plot of modal frequency (F) versus amplitude (A) of a Qingxi farm, wu Long station annual runoff sequence;
(a) A Qingxi field hydrologic station;
(b) Wu Long hydrologic station.
FIG. 6 shows the mutation time of 8 years and month runoff sequences of main and branch streams upstream of Yangtze river;
(a) Mountain-screening, high-field, zhu Tuo, north-medium hydrologic stations;
(b) Cun beach, wu Long, qingxi field, yichang hydrologic station.
Detailed Description
The present invention will be described in further detail with reference to fig. 1 to 6.
The ESMD method has the advantages of self adaption, no basis, capability of discriminating large-scale circulation and nonlinear trend of the runoff sequence, and no need of integral transformation in time-frequency analysis, and is used for comprehensively researching the change characteristics of the runoff sequence from three aspects of period, trend and mutation, and carrying out application description by taking 8 hydrologic stations of the main branch stream at the upstream of the Yangtze river as an example.
The Yangtze river basin (24-35 degrees North latitude and 90-122 degrees east longitude) is the third largest basin of the world, spans three economic areas of the eastern part, the middle part and the western part of China, and has a total area of 180 ten thousand km 2 Accounting for 18.8 percent of the area of China territories. The main flow of Yangtze river is upstream, 4504km in length and 70.4% of the whole length of Yangtze river, and the area of the river basin is controlled to be 100 ten thousand km 2 The main tributaries include Yahuliang, minjiang, tuojiang, jianlingjiang and Wujiang. The water resources in the Yangtze river basin are reasonably developed and utilized on the basis of mastering the runoff law, and the water resources have important significance for social and economic development. 8 hydrologic stations of the upstream dry tributary of the Yangtze river are selected as research objects, and the method comprises the following steps: mountain-screening hydrologic station, high-field hydrologic station, zhu Tuo hydrologic station, north-medium hydrologic station, cun beach hydrologic station, wu Long hydrologic station, qingxi field hydrologic station, yichang hydrologic station. The upstream runoff flow direction of the Yangtze river is as follows: the Jinshajiang mountain-screening station and the Minjiang high-field station are converged into a Yangtze river main flow Zhu Tuo station; continuing to sink to the lower reaches, wherein the Jiang river north medium station is converged into the Yangtze river dry flow beach station; thereafter, the station Wu Long of the Yangtze river is brought into the Qingxi station of the Yangtze river, and finally brought into the Yichang station of the Yangtze river. The geographical position of the main and branch flow 8 station on the upper stream of the Yangtze river is shown in figure 2The basic data are shown in Table 1.
TABLE 1 basic data for the upstream main and branch 8 stations of Yangtze river
Periodic variation law
The runoff sequence is affected by the combination of linear and nonlinear factors, the variation is very complex, and the fluctuation period is difficult to determine. The method is characterized in that the period and trend components of the runoff sequence can be effectively separated based on an ESMD method, the annual, monthly and daily runoff sequence of 8 stations of the upstream main and branch of the Yangtze river is decomposed by the ESMD method in a multi-time scale mode, the nonstationary runoff sequence is converted into stable modal components, and the average period of the modal components obtained by decomposing the annual, monthly and daily runoff sequence is calculated by an FFT periodogram method to obtain the periodic variation rule of the upstream runoff of the Yangtze river. The periodic variation rule of the runoff sequence of the main and branch stream 8 stations at the upstream of the Yangtze river is shown in fig. 3, and can be seen from fig. 3: the 8-station annual runoff sequence mainly has short period changes of 2-3 years, 5-7 years and 10-13 years, the 15-22-year long period changes, the month runoff sequence mainly has short period changes of 4-6 months, 1 year, 2-4 years, 6-9 years and 11-14 years, the 15-23-year long period changes, the day runoff sequence mainly has short period changes of 0.5-2 months, 6 months, 1 year, 2-4 years and 10-13 years, and the period characteristics of the long period changes of the upstream, month and day runoff sequences of Yangtze river are dominant respectively. Comparing the 8-station month and the daily runoff sequence of the dry tributary, the Zhu Tuo-station, the beach-station, the Qingxi-station, the Yichang-station and the daily runoff sequence have the periodic changes of 6 months and 1 year respectively, and the month and the daily runoff sequence of the north tributary, the north tributary and the daily runoff sequence have the periodic changes of 6 months and 1 year respectively. While the high-field station and the Wu Long station slightly change, and the Wu Long-month runoff sequence has a periodic change of 4 months. The periodic variation rules of the runoff sequences of the medium station in Jiang river and the medium station in Yangtze river are the same, and the high-field station in Min river and the Wu Long station in Wu river have respective periodic variation rules, so that the runoff relationship between Jiang river and Yangtze river is closer. Each hydrologic station year, month and day runoff sequence has short period change of 2-3 years and medium period change of 10-13 years, and each month and day runoff sequence has period change of 6 months and 1 year. Meanwhile, the runoff sequences of different time scales of each hydrologic station have respective periodic rules. The ESMD method does not need a basis function, but decomposes in a self-adaptive mode according to the data characteristics of the runoff sequences, so that the time scale and the period rule of the time scale obtained by decomposition are not completely the same, and the characteristics and the period change rule of different runoff sequences of each hydrologic station on the upper stream of the Yangtze river can be fully reflected.
In order to verify the feasibility and effectiveness of the ESMD method for analyzing the periodic rule, morlet wavelet transformation is adopted to carry out comparative analysis on the annual runoff sequence of the 8 stations of the main and branch stream at the upstream of the Yangtze river, as shown in fig. 3 (a). It can be seen from the figure that the ESMD method is similar to the periodic variation law obtained by the Morlet wavelet transform, but has certain difference, the wavelet transform is based on fourier transform, and is bound by aspects of wavelet basis function selection, constant multi-resolution and the like, while the ESMD method gets rid of the constraint of fourier transform, can decompose according to the time scale characteristics of the data, has stronger flexibility and adaptability, and is more beneficial to exploring the periodic law characteristics in the runoff sequence. Meanwhile, the ESMD method has higher cycle resolution, and can obtain a cycle rule with smaller runoff sequence.
Trend change law
The ESMD method decomposes the runoff sequence into modal components and trend remainder of different time scales, and discriminates large-scale circulation and nonlinear trends. Furthermore, the trend remainder is automatically optimized to be the optimal AGM by utilizing the principle of least squares, and the change trend of each stage of the runoff sequence is accurately reflected. And decomposing the year, month and day runoff sequence of the 8 stations of the main and branch stream at the upper stream of the Yangtze river by using an ESMD method, wherein the obtained trend remainder is shown in figure 4. As can be seen from fig. 4: the runoff amount of the dry flow hydrologic stations from top to bottom is gradually increased, the runoff amounts of the mountain-screening stations and the Zhu Tuo stations are in an ascending trend, and the runoffs of the rest hydrologic stations are in a descending trend. The trend changes of the annual, monthly and daily runoff sequences of all hydrologic stations are fluctuated for a plurality of times, the runoff of the Yangtze river main flow is in synchronous cyclic change of 'S-increase', the runoff of the Yangtze river main flow is only one tributary of the Yangtze river between the Yangtze river main flow and the Yangtze river station, the fluctuation trend of the runoff sequences of the Yangtze river Wu Long station is small, and the influence on the runoff change of the Yangtze river station is small. Thus, the runoff variation at the Yichang station is mainly affected by the runoff variation at the beach station. Overall, the upstream runoff of the Yangtze river has a reduced tendency to change. The change of the runoff quantity is influenced by the climate and the human activity, and the gold sand Jiang Wei is slightly influenced by the human activity in the western Sichuan, but the annual precipitation quantity of the river basin above the mountain-hold station is increased, and the reference evaporation quantity is reduced, so that the annual runoff quantity of the mountain-hold station tends to be slightly increased. The Jiang river and Min river are located at the eastern part and the middle part of Sichuan, and at present, the influence of human activities in these areas is very remarkable, and can be an important factor for reducing runoffs in high-field stations and north medium stations. Meanwhile, as the rainfall is obviously reduced in the years below the Jinshajiang mountain-screening station, the runoff of the hydrologic station below the mountain-screening station is basically reduced. In order to compare trend change conditions of different hydrologic stations, a study area is divided into 3 groups according to geographic positions, wherein the first group consists of a Jinshajiang mountain-screening station, a downstream length Jiang Ganliu Zhu Tuo station and two inter-region Min river high-field stations, the second group consists of a Yangtze river dry flow Zhu Tuo station, a downstream beach station and two inter-region Jianling Jiangbei station, and the third group consists of a Yangtze river dry flow beach station, a downstream Qingxi field station, a Yichang station and two inter-region Wujiang Wu Long stations. The first group shows a slightly increasing tendency of the mountain-screening station runoff, a slightly increasing tendency of the downstream Zhu Tuo station runoff, and a decreasing tendency of the high-station runoff between the two hydrologic stations, which indicates that the Minjiang high-station runoff is smaller than the main runoff, and the Zhu Tuo station runoff change is more influenced by the mountain-screening station. And the second group, the downstream radial flow of the base station is obviously reduced, and the north base station radial flow between the Zhu Tuo station and the base station is reduced, which shows that the change of the radial flow of the base station in Jiang river is more influenced on the base station than the change of the radial flow of the base station in Zhu Tuo. And the third group shows a slight downward trend in runoff at Wu Long stations between the beach station and the Qingxi station, and a downward trend in runoff at the downstream Qingxi station and Yichang station.
In order to verify the feasibility and effectiveness of the ESMD method for analyzing the trend change rule, the M-K trend test method is utilized for comparing and analyzing the trend change rule of 8-station annual runoffs of the main and branch flow at the upper stream of the Yangtze river. Meanwhile, the Hurst index is used for predicting the change trend of the runoff of the main and branch stream 8 station at the upstream of the Yangtze river in a future period. The variation trend of the annual runoff of the dry tributary 8 stations is compared with that of table 2, and can be seen from table 2: the M-K trend test shows that the annual runoff of the mountain-screening station and the Zhu Tuo station is slightly increased, the annual runoff of the Wu Long station is not obviously reduced, and the annual runoffs of other hydrologic stations are obviously reduced, which is basically consistent with the analysis result of the ESMD method. But the M-K trend test cannot discriminate large-scale circulation and trend change, the obtained linear trend cannot reflect the change condition of each stage of the runoff sequence, and the rising or falling trend of the runoff sequence is partially exaggerated or reduced in the change degree. The nonlinear trend obtained by the ESMD method can finely describe a specific change process, and the change trend of runoffs of various hydrologic stations on the upper stream of the Yangtze river can be reflected better. Hurst index analysis showed that: except Zhu Tuo stations and qingxi stations, the Hurst index of the annual runoff sequences of the rest hydrologic stations is greater than 0.5, which shows that the annual runoffs of Zhu Tuo stations and qingxi stations are reverse in duration, and the runoff sequences of the rest hydrologic stations are continuous. And combining the trend changes of the annual runoffs of the trunk and branch stream 8 stations, which are obtained through analysis, and in a period of time in the future, the annual runoffs of the mountain-holding stations and the Qingxi field stations are in an increasing trend, and the annual runoffs of the rest hydrologic stations are in a decreasing trend.
TABLE 2 comparison of trends of 8 station annual runoffs in the upstream main and branch streams of Yangtze river
Note that: the statistical value Z is positive, the sequence is represented as an ascending trend, and the statistical value Z is negative, the sequence is represented as a descending trend; given a significant level α=0.05, the statistic cut-off is ±1.96. The statistic 0 is less than or equal to H <0.5, which indicates that the future change trend is opposite to the past; h=0.5, indicating that the future change trend is irrelevant to the past; h is less than or equal to 1 and is 0.5, which indicates that the future change trend is consistent with the past.
Mutation law
The time-frequency analysis of ESMD utilizes a linear interpolation method to obtain a time-varying graph of each mode frequency and amplitude, and the runoff mutation rule of the main and branch flow 8 station on the upstream of the Yangtze river is analyzed and mastered by observing the time of low-frequency, large-amplitude or high-frequency and small-amplitude oscillation in the time-varying graph. For the sake of space, only a time-varying plot of the modal frequency (F) versus amplitude (a) of annual runoff sequences for mountain hold stations and tributary qing xi stations is given, as shown in fig. 5. As can be seen from fig. 5, the qing xi station: qingxi station: comparing F1 with A1, and generating low-frequency and large-amplitude oscillation in 2002; in contrast to F2 and A2, high frequency, small amplitude oscillations occurred in 1991. In contrast to F3 and A3, there is no significant moment of low frequency, large amplitude or high frequency, small amplitude oscillations. It shows that the Qingxi field station annual runoff sequence is mutated in 1991 and 2002. Wu Long station: in contrast to F1 and A1, high frequency, small amplitude oscillations occurred in 1972. In contrast to F2 and A2, F3 and A3, there is no significant moment of low frequency, large amplitude or high frequency, small amplitude oscillations. The Wu Long annual runoff sequence was mutated in 1972. The rest hydrologic stations are also obtained by adopting the same judging method, and the runoff sequences of the mountain-screening stations in 1955, 1980 and 2007 are mutated; the high-field station 1956 and 2004 runoff sequences were mutated; the runoff sequence at Zhu Tuo was not mutated. The runoff sequence was mutated in the beach 1960, 1996 and 2001. The runoff sequence was mutated in 1972 at Wu Long. The runoff sequences were mutated in Yichang stations 1995 and 2002.
Because the runoff sequence is complex, certain errors may exist in the identification methods of different variation points, in order to avoid the identification errors caused by a single method, the significance water alpha=0.05 is taken while the M-K mutation test method and the sliding T test are combined, the significance of the mutation year of the runoff sequence in the year and the month is tested by using the run-length test, and the obtained significant mutation year after the test is shown in fig. 6. As can be seen from fig. 6, the annual runoff sequences are compared: the mountain-screening station and Yichang station are mutated in 1994, the high-field station and the beach station are mutated in 1993, and the beach station and Yichang station are mutated in 1958, 1961 and 1963; comparing the moon runoff sequence: the high-field station and the Zhu Tuo station have more identical mutation moments, and the rest hydrologic stations have identical mutation rules. For example, the Zhu Tuo station and the beach station are mutated in 1975 and 9, and the beach station and the Qingxi station are mutated in 1991 and 8; the same mutation rule exists among stations of the main branch flow 8 on the upstream of the Yangtze river. However, as a plurality of reservoirs are built by the main and branch flows on the upper stream of the Yangtze river, a large amount of runoff is blocked, and the runoff is reduced. Meanwhile, rainfall below the mountain-screening station gradually decreases, and under the combined influence of climate change and intense human activity, the physical conditions of hydrologic cycle of the river basin are greatly changed, and the mutation time of each hydrologic station is not completely the same. The ESMD time-frequency analysis is adopted to obtain that the 8-station daily runoff sequence has more moments of low-frequency large amplitude or high-frequency small amplitude, and the method is limited to the space, is not listed one by one, and compares the mutation years of the 8-station daily runoff sequence of the main and branch flows at the upper stream of the Yangtze river: the daily runoffs of each hydrologic station also have the same time of mutation, and meanwhile, the time of mutation of the daily runoffs of the main and branch stream 8 stations is not completely the same due to the influence of factors such as climate change, human activities, branch inflow and the like.
The mutation year of the runoff sequence is judged by utilizing the frequency and amplitude time-varying diagram of ESMD, the method does not need a basis function, self-adaptive decomposition is carried out according to the scale characteristics of data, the corresponding time-frequency analysis does not need to convert discrete signals into an analytical function, the constraint of mathematical derivation is eliminated, and compared with the situation that more false mutation years (such as 1950 and 1954 of the annual runoffs of mountain-hold stations and 1969 of the mountain-hold beach stations) are obtained by M-K mutation inspection and mutation results possibly caused by sliding T inspection are incomplete (such as the non-mutation years of the annual runoffs of the mountain-hold stations and Wu Long stations inspected by the method), the mutation rule obtained by the ESMD method is relatively accurate and reliable.
Runoff sequence variation characteristics
And (3) researching the runoff sequence change characteristics of the main and branch flow 8 station at the upstream of the Yangtze river by adopting an ESMD method from three aspects of period, trend and mutation. In summary, it is possible to obtain: the 8 stations of the main and branch flow at the upstream of the Yangtze river mainly have short period change rules of 1 year, 2-3 years, 6-7 years, 9-10 years, 11 years and 14 years, and long period change rules of 15-17 years and 22-23 years. Except for the runoffs of the mountain-screening station and the Zhu Tuo station, the runoffs of the rest hydrologic stations are in a trend of decreasing. Climate change superimposes intense human activities, resulting in different times of mutation of the 8 stations of the main and branch stream upstream of the Yangtze river, but the same mutation time exists in each hydrologic station.
What is not described in detail in this specification is prior art known to those skilled in the art.
Claims (1)
1. A radial flow sequence change characteristic analysis method based on pole symmetry modal decomposition is characterized in that the radial flow sequence change characteristic is researched from three aspects of period, trend and mutation simultaneously by utilizing the pole symmetry modal decomposition method, and the method comprises the following steps:
step 1: modal decomposition, namely decomposing the runoff sequence step by step into modal components and trend remainder of different frequencies;
step 2: analyzing each modal component by using a fast Fourier transform periodogram method, and grasping a multi-time scale periodic variation rule of a runoff sequence;
step 3: utilizing the optimal self-adaptive global average line, namely a trend remainder, to master the overall trend change of the runoff sequence;
step 4: analyzing each modal component by using a time-frequency analysis method, and grasping a mutation rule of the runoff sequence;
the step 1 specifically comprises the following calculation steps:
step 1-1: inputting a runoff sequence X (t), setting the maximum screening times K and the number of residual poles l, finding out all poles in the runoff sequence X (t), and marking as E i (1≤i≤n)，E i ＝(z i ，y i ) The method comprises the steps of carrying out a first treatment on the surface of the Connecting adjacent poles by line segments, and marking the points in the line segments as F in sequence i (1 is less than or equal to i is less than or equal to (n-1)); supplement left and right boundary midpoint F 0 ，F n ；
The 1 st and the 2 nd maximum points are used as linear interpolation, the 1 st and the 2 nd minimum points are used as linear interpolation, and two interpolation lines are respectively recorded as y 1 (z)＝p 1 z+b 1 And y 2 (z)＝p 2 z+b 2 The method comprises the steps of carrying out a first treatment on the surface of the Point 1 of the data is marked as Y 1 ；
1) If b 2 ≤Y 1 ≤b 1 Will b 1 And b 2 Respectively defining a boundary maximum value point and a boundary minimum value point;
2) If b 1 ＜Y 1 ≤(3b 1 -b 2 ) And/2, Y 1 And b 2 Respectively defining a boundary maximum value point and a boundary minimum value point; if (3 b) 1 -b 2 )/2≤Y 1 ＜b 2 Will b 1 And Y 1 Respectively defining a boundary maximum value point and a boundary minimum value point;
3) If Y 1 ＞(3b 1 -b 2 ) And/2, Y 1 Is defined as the boundary maximum point, and the boundary minimum point is defined by a straight line drawn from the first minimum point, where the slope is determined by the distance between the left boundary point (0, Y 1 ) And a straight line of the first maximum point; if Y 1 ＜(3b 1 -b 2 ) And/2, Y 1 Is defined as a boundary minimum point, and the boundary maximum point is defined by a straight line drawn from the first minimum point, where the slope is determined by the distance between the left boundary point (0, Y 1 ) And a straight line of the first maximum point;
step 1-2: two interpolation curves are respectively marked as L by the obtained (n+1) midpoints 1 And L 2 ；
L 1 Is generated by cubic spline interpolation for the midpoint of the odd ordinal, L 2 For the generation of the midpoint of even ordinal number by cubic spline interpolation, calculating a mean curve L * :
L * ＝(L 1 +L 2 )/2 (12)
Step 1-3: construction of the sequence (X (t) -L * ) For the sequence (X (t) -L * ) Repeating the steps 1-1 to 1-2 until the absolute value L * Epsilon is less than or equal to epsilon or reaches the set maximum screening frequency K, wherein epsilon is a preset allowable error, and the first empirical mode M is obtained by decomposition 1 (t)；
Step 1-4: construction of the sequence (X (t) -M 1 (t)) for the sequence (X (t) -M 1 (t)) repeating the steps 1-1 to 1-3 to obtain sequentiallyM 2 (t)，M 3 (t)，…，M q (t) until the trend residual R (t) accords with the number l of residual poles preset in the step 1-1, wherein the trend residual R (t) is called the optimal self-adaptive global average;
step 1-5: the value range of the given maximum screening times K is within an integer interval [ K ] min ，K max ]Calculating variance ratio G, recording radial flow sequence
Trend remainder->
Sigma and sigma 0 The relative standard deviation of X (t) -R (t) and the standard deviation of the runoff sequence X (t), respectively;
G＝σ/σ 0 (16)
wherein:
is the average value of the runoff sequence X (t); when G is minimum, the sequence of removing the trend residual R (t) is closest to the runoff sequence X (t), and the decomposition result is best;
step 1-6: drawing a variation graph of variance ratio with K, and selecting the maximum screening times K corresponding to the minimum variance ratio 0 When k=k 0 When R (t) is the best fit curve of the runoff sequence, at K 0 Repeating the steps 1-1 to 1-5 to obtain the optimal decomposition result, wherein the optimal decomposition result is the modal components with different frequenciesAnd a trend remainder R (t);
reconstructing modal components and trend remainders R (t) of different frequencies obtained by decomposition to obtain a runoff sequence X (t), wherein the runoff sequence X (t) is expressed as:
the step 2 specifically comprises the following calculation steps:
step 2-1: calculating power spectrums of different frequency modal components obtained by pole symmetry modal decomposition, wherein the power spectrums can be used
Expressed as:
wherein: h is the total number of runoff samples, o H (v) For energy limited signal, O H (omega) is o H (v) The frequency domain value of the Fourier transform, v is a random analog signal, and ω is the signal frequency of the Fourier transform;
step 2-2: obtaining the frequency of the runoff signal according to the magnitude of the amplitude, so as to calculate the average period of each modal component;
the step 3 specifically comprises the following calculation steps:
step 3-1: the best self-adaptive global equalizing curve, namely, the trend remainder is influenced by the number l of residual poles, the larger l is, the larger the fluctuation of the best self-adaptive global equalizing curve is, and the higher the fitting degree between the best self-adaptive global equalizing curve and a runoff sequence is; under the condition of ensuring the obtained cycle rule to be comprehensive, ensuring that the optimal self-adaptive global homography can accurately reflect the overall trend change of the runoff sequence, continuously adjusting the number l of the residual poles, and finding out the optimal decomposition result;
step 3-2: based on the optimal self-adaptive global mean line, namely trend remainder, the trend change rule of different stages of the runoff sequence is mastered;
the step 4 specifically comprises the following calculation steps:
step 4-1: acquiring a time-amplitude variation curve: measuring the upper envelope of the modal components of which the internal odd-dipole points are symmetrical or the external envelope is symmetrical, which are decomposed by a pole symmetry modal decomposition method, so as to obtain a time-amplitude change curve; other forms of symmetry are that absolute values are firstly taken for values corresponding to the model components, and then upper and lower envelopes are generated by maximum value points through interpolation, so that a time-amplitude change curve is obtained, wherein an amplitude function is recorded as A (t);
step 4-2: generating a phase angle: according to the pole E obtained in step 1-1 i (1. Ltoreq.i.ltoreq.n) the function value corresponding to the first point of the modal component
And instantaneous amplitude +.>
Is taken to be the ratio of arcsine to generate the phase angle:>
sequentially calculating t and E which are more than or equal to 2 and less than or equal to E according to a formula (20) 1 Is a phase angle of a point of (2);
similarly, every two poles are added, the phase is increased by 2 pi;
step 4-3: acquiring a time-frequency change curve: calculating the instantaneous frequency in hertz from the phase angle obtained in step 4-2 by taking the center difference quotient with respect to 2 time steps deltac:
and supplementing left and right boundary values by a linear interpolation method:
f 1 ＝2f 2 -f 3 ， (24)
f H ＝2f H-1 -f H-2 (25)
obtaining a time-frequency change curve according to the instantaneous frequency;
step 4-4: obtaining a time-varying graph of frequency and amplitude: according to the time-amplitude and time-frequency change curves, a time-varying diagram of the frequency and the amplitude of each modal component is obtained, and the mutation rule of the runoff sequence is researched by observing the time of low-frequency, large-amplitude or high-frequency and small-amplitude oscillation in the time-varying diagram.
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