CN102033851A - Hydrologic time series wavelet correlation analysis method - Google Patents

Abstract

The invention discloses a hydrologic time series wavelet correlation analysis method comprising the following steps of: firstly, selecting a wavelet function and a time scale range, then carrying out continuous wavelet transform analysis on hydrologic time series to be analyzed; calculating the wavelet cross covariance of the hydrologic time series under different time scales and time lags; and solving the wavelet correlation coefficient of two sequences under different time scales and time lags, calculating the wavelet correlation degree so as to describe the correlation degree of the two sequences on the whole time domain, and drawing an isopleth map of the wavelet correlation coefficient to realize time-frequency comprehensive analysis on the correlation between time sequences. The case analysis result shows the effectiveness and the superiority of the wavelet correlation analysis method which can analyze and quantificationally describe the correlation of a nonstationary time series under a special time scale and appointed time lags and overcome the limit of the traditional correlation analysis method, and is more flexible and applicable.

Description

A kind of Hydrological Time Series small echo cross-correlation analysis method

Technical field

The present invention relates to a kind of Hydrological Time Series analytical approach, specifically is a kind of Hydrological Time Series small echo cross-correlation analysis method.

Background technology

Wavelet analysis method (wavelet analysis, WA) have to nonstationary time series carry out the time-frequency analysis-by-synthesis ability (Cui Jintai. the wavelet analysis introduction. Xi'an: publishing house of Xi'an Communications University, 1995), therefore be suitable for studying the complicated hydrographic water resource (Wang Wensheng of system with dimensional variation characteristic of many time, the fourth crystalline substance, Li Yueqing. hydrology wavelet analysis. Beijing: Chemical Industry Press, 2005; Labat D.Recent advances in wavelet analyses:Part 1.A review of concepts.Journal of Hydrology, 2005,314:275-288).The needs that go deep into and solve practical problems along with theoretical research, the application of WA in hydrographic water resource is learned increasing (Wang Wensheng, fourth crystalline substance, Xiang Honglian. applied research and the prospect of wavelet analysis in hydrology. the hydroscience progress, 2002,13 (4): 515-520).Analysis-by-synthesis as can be seen, at present WA is mainly used in inner structure and the variation characteristic that discloses and describe hydrology system self.And disclose mutual relationship between each hydrographic features (as rainfall and runoff sequence, meteorological factor and rainfall and runoff sequence, water level and flow etc.) is the cyclic process of the understanding hydrology and another important channel of disclosing hydrology evolution mechanism.The tradition cross-correlation analysis method (comprising the cross-spectrum analytical approach) since exist following major defect (fourth crystalline substance, Deng Yuren. statistical hydrology. Chengdu: publishing house of Chengdu Univ. of Science ﹠ Technology, 1988): (1) is only applicable to steady ergodic sequence; (2) can't disclose the simple crosscorrelation relation of sequence in the different time range scale, make it that bigger limitation be arranged in actual applications.Small echo cross-correlation analysis method (wavelet cross-correlation, WCC) can realize the two nonstationary time series quantitative description that simple crosscorrelation concerns under special time yardstick and appointment time lag, therefore have bigger applicability and superiority, can perform well in studying and disclose the mutual relationship between each hydrographic features.Yet, because it is on the low side at the systematic study of WCC, lack unified solution formula and analytical approach at present, and theoretical system own is also very imperfect, only obtain certain application at aspects such as economics, signal Processing, clinical medicine, and the research in hydrographic water resource is learned and use considerably lessly and does not domesticly see that at present relevant report is arranged.

For this reason, the present invention is intended to inquire into small echo cross-correlation analysis theory and the method system that is suitable for studying hydrographic water resource knowledge topic.At first by analysis and arrangement, systematically introduce be used for hydrological sequence analysis based on continuous wavelet transform (continuous wavelet transform, small echo cross-correlation analysis method CWT); Defined small echo cross covariance and two quantitative targets of small echo cross correlation measure simultaneously, be used to describe the simple crosscorrelation relation of two time serieses on whole time domain based on CWT; Propose to draw the method for small echo cross-correlation coefficient isogram then, can reach the purpose of two time series simple crosscorrelation relation being carried out " time-frequency analysis-by-synthesis " by this equivalence line chart.In conjunction with instantiation brief analysis in addition, be used to study the applicability and the superiority of Hydrological Time Series at last to show small echo cross-correlation analysis method.

When the simple crosscorrelation between analysis elapsed time sequence x (t) and the y (t) concerned, following formula commonly used was found the solution cross-correlation coefficient in the reality.

$r_{\mathrm{xy}}\left(k\right)=C_{\mathrm{xy}}\left(k\right)/{\mathrm{\σ}}_x{\mathrm{\σ}}_y=\frac{\underset{t=1}{\overset{n-k}{\mathrm{\Σ}}}(x\left(t\right)-\stackrel{\‾}{x})(y(t+k)-\stackrel{\‾}{y})}{{\left[\underset{t=1}{\overset{n}{\mathrm{\Σ}}}{(x\left(t\right)-\stackrel{\‾}{x})}^2\underset{t=1}{\overset{n}{\mathrm{\Σ}}}{(y\left(t\right)-\stackrel{\‾}{y})}^2\right]}^{1/2}}$

In the formula, σ _x, σ _yThe mean square deviation of representing sequence x (t) and y (t) respectively, The average of representing x (t) and y (t) respectively, C _Xy(k) cross covariance of expression two time serieses under time lag k, r _Xy(k) be the cross-correlation coefficients of two time serieses under time lag (also claiming time shift) k, the length of n express time sequence.

In the reality, when needing that the simple crosscorrelation on certain special time yardstick concerns between each nonstationary time series of quantitative description, this formula is powerless.

Summary of the invention

The present invention is directed to the defective that conventional cross-correlation analysis method exists, and based on wavelet analysis method, a kind of Hydrological Time Series small echo cross-correlation analysis method has been proposed, and drawn small echo cross-correlation coefficient isogram, can reach the purpose of two time series simple crosscorrelation relation being carried out " time-frequency analysis-by-synthesis " by this equivalence line chart.

A kind of Hydrological Time Series small echo cross-correlation analysis method of the present invention is characterized in that may further comprise the steps:

(1) select wavelet function and time scale scope, then to Hydrological Time Series to be analyzed carry out the continuous wavelet transform analysis (continuous wavelet transform, CWT);

(2) calculate Hydrological Time Series at the small echo cross covariance that reaches on the different time yardstick under the different time lags;

(3), find the solution two sequences at the small echo cross-correlation coefficient that reaches on the different time yardstick under the different time lags according to small echo cross covariance result of calculation;

(4) try to achieve after the small echo cross-correlation coefficient that reaches on the different time yardstick under the different time lags, calculate the small echo cross correlation measure; To describe the simple crosscorrelation degree of two sequences on whole time domain;

(5) try to achieve after small echo cross-correlation coefficient and the small echo cross correlation measure, draw small echo cross-correlation coefficient isogram, and by labor small echo cross-correlation coefficient isogram, grasp is studied between the sequence and concerned to local simple crosscorrelation by integral body; Realization is carried out the time-frequency analysis-by-synthesis to simple crosscorrelation relation between the time series.

Above-mentioned steps (2) has two kinds of different situations of real part and mould according to the continuous wavelet transform analysis result, has defined the method for solving of small echo cross covariance respectively:

${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left({\mathrm{Wcov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(1\right)$

Or ${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(2\right)$

Wcov in the formula _Xy(a, k)=E[W _x(a, b) W _y(a, b+k)] (3)

In the following formula, W _x(a, b) and W _y(a, the continuous wavelet transform coefficient that obtains when b) representing analytical sequence x and y respectively, a express time scale factor, b express time location factor, k represents time lag.R () and I () represent to find the solution real and imaginary part in the parantheses respectively, and average is found the solution in E () expression.WC _Xy(a k) is illustrated in the small echo cross covariance of trying to achieve under time scale a and the time lag k.

Above-mentioned steps (3) has defined the method for solving of small echo cross-correlation coefficient respectively according to two kinds of the small echo cross covariance different situations:

${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{R\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}{\sqrt{R\left(W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right)R\left(W{\mathrm{cov}}_{\mathrm{yy}}(a,0)\right)}}---\left(4\right)$

Or ${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}}{\sqrt{\left|W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right|\left|W{\mathrm{cov}}_{\mathrm{yy}}(a,0)\right|}}---\left(5\right)$

In the following formula, WR _Xy(a k) is illustrated in the small echo cross-correlation coefficient of trying to achieve under time scale a and the time lag k, and ‖ represents to find the solution absolute value, and the remainder formula symbol is the same.

Above-mentioned steps (4) has defined the method for solving of small echo cross correlation measure according to small echo cross-correlation coefficient solving result, and to describe the simple crosscorrelation degree of two sequences on whole time domain, step is as follows:

(4.1) try to achieve two time serieses small echo cross-correlation coefficient WR under yardstick a and time lag k _Xy(a, on basis k), try to achieve the summation of the small echo simple crosscorrelation degree of two time serieses on corresponding whole time domain under the time lag k by integration:

WR _xy(k)＝∫WR _xy(a，k) ²da (6)

(4.2) then, find the solution small echo cross-correlation coefficient WR under each time scale a _Xy(a, weight coefficient k):

f(WR _xy(a，k))＝WR _xy(a，k) ²/WR _xy(k) (7)

(4.3) find the solution the small echo cross correlation measure of two time serieses under time lag k (wavelet cross-correlation degree WCCD) is:

WCCD _xy(k)＝∫WR _xy(a，k)f(WR _xy(a，k))da (8)

When step (5) is drawn small echo cross-correlation coefficient isogram, represent the value of time lag k with transverse axis, the value of longitudinal axis express time yardstick a, certain point value among the figure has characterized the size of the cross-correlation coefficient of two sequences under corresponding yardstick a and the time lag k.

Step (5) is drawn after the small echo cross-correlation coefficient isogram, and the relation of the simple crosscorrelation between two time serieses is carried out labor, and key step is as follows:

(5.1), analyze under fixed time lag the situation of change of two sequences simple crosscorrelation degree size on each time scale by small echo cross-correlation coefficient isogram is carried out vertical intercepting;

(5.2) by isogram is carried out taken transverse, analyze on the set time yardstick, two sequences are simple crosscorrelation degree change situation under each time lag;

(5.3) by analyzing the positive negativity of small echo cross correlation numerical value on each time scale, grasp the positive and negative situation of change of two sequences cross correlation on each time scale;

(5.4), discern and extract corresponding certain or the tangible time scale scope of several cross correlations by the size of small echo cross-correlation coefficient absolute value on each time scale of comparative analysis;

(5.5), identify the most significant time delay relation between two time serieses by the size of comparative analysis small echo cross correlation numerical value under each time lag.

The invention provides a kind of small echo cross-correlation analysis theory and method system that is suitable for studying hydrographic water resource knowledge topic.At first by analysis and arrangement, systematically proposed to be used for the small echo cross-correlation analysis method based on continuous wavelet transform (CWT) of hydrological sequence analysis; Defined small echo cross covariance and two quantitative targets of small echo cross correlation measure simultaneously, be used to describe the simple crosscorrelation relation of two time serieses on whole time domain based on CWT; And the method that proposes to draw small echo cross-correlation coefficient isogram, can reach the purpose of two time series simple crosscorrelation relation being carried out " time-frequency analysis-by-synthesis " by this equivalence line chart.The instance analysis result has shown the validity and the superiority of small echo cross-correlation analysis method, uses it and can realize by the purpose of integral body to local understanding time series simple crosscorrelation relation.Small echo cross-correlation analysis method can be analyzed and the simple crosscorrelation relation of quantitative description nonstationary time series under special time yardstick and appointment time lag, can overcome the limitation of traditional cross-correlation analysis method, has better dirigibility and applicability.Therefore, by using small echo cross-correlation analysis method each hydrographic features is analyzed and described, more help deep be familiar with complicated hydrology cyclic process and announcement hydrology evolution mechanism, also can promote the constantly perfect of small echo cross-correlation analysis self theory and method system simultaneously.

Description of drawings

Fig. 1 small echo cross-correlation coefficient isogram is drawn flow process.

Station, Fig. 2 Lijin and station, Huayuankou annual flow sequence small echo cross-correlation coefficient isogram (Fig. 2 a), (Fig. 2 c) small echo cross correlation number curve on (Fig. 2 b) and the different time yardstick under the different time lag.

Station, Fig. 3 Lijin and station, Huayuankou annual flow sequence small echo cross correlation measure curve and cross correlation number curve.

Embodiment

Below the concrete small echo cross-correlation analysis method of introducing based on continuous wavelet transform:

Make L ²(R) the expression quadractically integrable function space that is defined on the real axis, can surveys is then for time series x (t) ∈ L ²(R), its continuous wavelet transform (continuous wavelet transform, CWT) formula is:

$W_x(a,b)={\∫}_{-\∞}^{+\∞}x\left(t\right){{\mathrm{\ψ}}^{*}}_{a,b}\left(t\right)\mathrm{dt}$ ${\mathrm{\ψ}}_{a,b}\left(t\right)=|a{|}^{-1/2}\mathrm{\ψ}\left(\frac{t-b}{a}\right)$ a，b∈R，a≠0 (9)

In the formula, ψ ^*(t) be the complex conjugate function of ψ (t), W _x(a b) is the wavelet coefficient of x (t) behind continuous wavelet transform.A is a time scale factor, and b is the time location factor.ψ _{A, b}(t) being wavelet function, is to be obtained after the flexible and translation through yardstick by a wavelet mother function ψ (t) who satisfies " admissibility " condition (formula 10).In the formula (10), ψ ^*(ω) expression complex conjugate function ψ ^*(t) in the Fourier at frequencies omega place conversion.

$C_{\mathrm{\&psi;}}={\&Integral;}_{-\&infin;}^{+\&infin;}\frac{{\left|{\mathrm{\&psi;}}^{*}\left(\mathrm{\&omega;}\right)\right|}^2}{\left|\mathrm{\&omega;}\right|}\mathrm{d\&omega;}<\&infin;---\left(10\right)$

The essence of wavelet transformation is to adopt a kind of variable size, and the movable change window in position carries out analysis of spectrum to time series, therefore can satisfy the requirement that sequence time-frequency localization is analyzed.The wavelet coefficient that hydrology sequence is carried out obtaining behind the wavelet transformation be sequence in different time yardstick and the locational projection of different time, can be used to portray and describe the composition structure and the dimensional variation characteristic of many time of hydrology sequence.In addition, by studying the relation between the corresponding wavelet coefficient of two time serieses, can portray and describe the simple crosscorrelation relation between two sequences equally.

1, based on the small echo cross-correlation coefficient method for solving of CWT

Definition according to traditional time series cross covariance defines small echo cross covariance (wavelet cross-covariance) herein.Because some continuous wavelet transform coefficient results (for example using " Morlet " wavelet function to carry out wavelet transformation) has real part and two significant variables of mould, real part is represented the information of different characteristic time scale signal aspect locational distribution of different time and phase place two, the power of the main representation feature time scale signal of size of mould.Therefore, define corresponding small echo cross covariance WC herein respectively _Xy(a, k):

${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(1\right)$

Or ${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(2\right)$

Wcov in the formula _Xy(a, k)=E[W _x(a, b) W _y(a, b+k)] (3)

Wherein, W _x(a, b) and W _y(a, b) the continuous wavelet transform coefficient of difference express time sequence x (t) and y (t) correspondence under yardstick a, k represents time lag, R () and I () represent the real part (real part) and the imaginary part (imaginary part) of bracket internal variable respectively, E[] result's average in the expression square brackets, WC _Xy(a, k) the small echo cross covariance of expression sequence x (t) and y (t) correspondence under yardstick a and time lag k.

On the basis of trying to achieve the small echo cross covariance, (wavelet local correlation coefficient WLCC), is used to analyze the simple crosscorrelation relation of two sequences on special time yardstick and time location point at first to introduce the local cross-correlation coefficient of small echo.

${\mathrm{WLCC}}_{\mathrm{xy}}(a,b)=\frac{R\left(W_{\mathrm{xy}}(a,b)\right)}{\left|W_x(a,b)\right|\left|W_y(a,b)\right|}=\frac{R\left({W_x}^{*}(a,b)W_y(a,b)\right)}{\left|W_x(a,b)\right|\left|W_y(a,b)\right|}---\left(11\right)$

Then to time series x (t) and y (t) carry out the small echo cross-correlation analysis (wavelet cross-correlation, WCC).Equally at real part and two variablees of mould of continuous wavelet transform coefficient results, define corresponding small echo cross-correlation coefficient WR respectively herein _Xy(a, k):

${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{R\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}{\sqrt{R\left(W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right)R\left({\mathrm{Wcov}}_{\mathrm{yy}}(a,0)\right)}}---\left(4\right)$

Or ${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}}{\sqrt{\left|W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right|\left|{\mathrm{Wcov}}_{\mathrm{yy}}(a,0)\right|}}---\left(5\right)$

WR _Xy(a, k) quantitative description time series x (t) and y (t) in corresponding simple crosscorrelation degree on the time scale a and under the time lag k.

2, time series small echo cross correlation measure

The WCC method is mainly used in analyzes two time serieses in the simple crosscorrelation relation on the special time yardstick and under the appointment time lag.For ease of analyzing and describe the simple crosscorrelation degree on whole time domain between the time series, the author inquires into by analysis, time series small echo cross correlation measure (wavelet cross-correlation degree based on CWT is proposed herein, WCCD) notion is used to portray the simple crosscorrelation degree on the whole time domain between the time series.Specific as follows:

Try to achieve two time serieses small echo cross-correlation coefficient WR under yardstick a and time lag k _Xy(a, on basis k), try to achieve the summation of the small echo simple crosscorrelation degree of two time serieses on corresponding whole time domain under the time lag k by integration:

WR _xy(k)＝∫WR _xy(a，k) ²da (6)

Then, the small echo cross-correlation coefficient WR under each time scale a _Xy(a, weight coefficient k) may be defined as:

f(WR _xy(a，k))＝WR _xy(a，k) ²/WR _xy(k) (7)

Define two seasonal effect in time series small echo cross correlation measure (wavelet cross-correlation degree WCCD) is:

WCCD _xy(k)＝∫WR _xy(a，k)f(WR _xy(a，k))da (8)

WCCD _Xy(k) characterized the size of two time serieses simple crosscorrelation degree on whole time domain under the time lag k, its essence is under identical time lag k, simple crosscorrelation degree on each time scale is found the solution its weighting expectation value, but therefore concentrated expression two time serieses on each time scale about information aspect the size of cross correlation and the distribution two.

In addition, by drawing WCCD _Xy(k) with the change curve (WCCD of time lag k _Xy(k)～and the k curve), can analyze two sequences variation characteristics and rule of cross correlation during hysteresis at any time on whole time domain.

3, small echo cross-correlation coefficient isogram

Be similar to by draw the wavelet coefficient isogram can analysis time the structure and the dimensional variation characteristic of many time of sequence self, propose to draw the method for small echo cross-correlation coefficient isogram herein, and utilize small echo cross-correlation coefficient isogram to concern to the simple crosscorrelation under each time scale and each time lag the partial Quantitative Analysis time series by integral body, promptly reach the purpose of simple crosscorrelation relation between the time series being carried out " time-frequency analysis-by-synthesis ".

The method for drafting (Fig. 1) of small echo cross-correlation coefficient isogram is as follows: (1) at first selects rational wavelet function and time scale scope, and each time series of study is carried out continuous wavelet transform, obtains the wavelet coefficient result of each sequence correspondence; (2) utilize formula (4) (or formula (5) find the solution two time serieses at a time between the corresponding down small echo cross-correlation coefficient of yardstick a and time lag k; (3) get different a and k value successively, respectively two sequences are carried out the small echo cross-correlation analysis, finally obtain the small echo cross-correlation coefficient of two time serieses correspondence on each yardstick and each time lag; (4) draw small echo cross-correlation coefficient isogram, wherein transverse axis is represented the value of time lag k, the value of longitudinal axis express time yardstick a, and certain point value among the figure has characterized the size of the cross-correlation coefficient of two sequences under corresponding yardstick a and the time lag k; (5) by quantitative test small echo cross-correlation coefficient isogram, grasp is studied between the sequence and is concerned to local simple crosscorrelation by integral body; (6), grasp the simple crosscorrelation degree on whole time domain under the different time lags of two sequences by finding the solution WCCD.

Small echo cross-correlation coefficient isogram has the important use meaning to the mutual relationship of describing and portray between two time serieses.Mainly as follows: (1) can be analyzed under fixed time lag by small echo cross-correlation coefficient isogram is carried out vertical intercepting, the situation of change of two sequences simple crosscorrelation degree size on each time scale; (2) by isogram is carried out taken transverse, can analyze on the set time yardstick, two sequences are simple crosscorrelation degree change situation under each time lag; (3) by analyzing the positive negativity of small echo cross correlation numerical value on each time scale, can grasp the positive and negative situation of change of two sequences cross correlation on each time scale; (4), can discern and extract corresponding certain (or several) tangible time scale scope of cross correlation by the size of small echo cross-correlation coefficient absolute value on each time scale of comparative analysis; (5), can identify the most significant time delay relation etc. between two time serieses by the size of comparative analysis small echo cross correlation numerical value under each time lag.

Instance analysis

Selecting 54 years (1950-2003) annual flow sequences of station, Huayuankou, the Yellow River and station, Lijin two hydrology websites actual measurement is that example is analyzed.Study the mutual relationship between the two station annual flow sequences, runoff process and the hydrology variation characteristic of being familiar with the downstream, the Yellow River had great importance.

The small echo cross-correlation analysis method based on CWT of application of aforementioned is analysed in depth in the relation of the simple crosscorrelation between the two annual flow sequences on the different time yardstick and under the different time lag.Select for use formula (4) to find the solution small echo cross-correlation coefficient WR herein _Xy(a, k).In conjunction with document [Sang Yanfang, Wang Dong. the applied research [A] of continuous wavelet transform in the Huanghe River mouth district characteristic is analyzed. the 5th Chinese water forum's collection of thesis [C]. Beijing: Chinese Water Conservancy water power publishing house, 2007,766-770.] in the correlation analysis result, select for use " Morlet " wavelet function that the 54 years annual flow sequences in Huayuankou are carried out continuous wavelet transform herein equally.According to the CWT analysis result, find the solution the small echo cross-correlation coefficient under each time scale and each time lag respectively then, and (Fig. 2 a) to draw small echo cross-correlation coefficient isogram.Wherein, maximum time yardstick get 50a, maximum time lag is got 20a.For ease of analyzing explanation, draw time lag k=0,2 and 5 o'clock WR respectively _Xy(a, k)～a curve (Fig. 2 b) and time scale a=3,7,11 and 20 o'clock WR _Xy(a, k)～k curve (Fig. 2 c).Draw small echo cross correlation measure WCCD at last _Xy(k)～k curve and cross correlation number curve (Fig. 3).Tradition cross-correlation analysis method result of calculation shows (Fig. 3): the maximum cross correlation coefficient between the two annual flow sequences is r _Xy(0)=0.9655.

The analysis result of comprehensive above-mentioned two runoff sequence cross-correlation relation can obtain following Main Conclusions:

(1) small echo cross-correlation coefficient isogram is analyzed, can be identified two runoff sequences and mainly on four time scales, have comparatively significantly simple crosscorrelation relation: a=3,7,11 and 20.These four time scale correspondences also just in time be four obvious periodic values of each runoff sequence self.

(2) small echo cross-correlation coefficient isogram is carried out vertical intercepting, can understand two sequences relation of the simple crosscorrelation on each time scale under fixed time lag.Fig. 2 b shows that 1. under same time lag, two sequence cross-correlation degree varies in size on the different time yardstick; 2. the positive negative characteristic of two sequence cross-correlation is also inequality in the different time range scale.With k=5 is example, and the small echo cross-correlation coefficient on time scale 4a, 8a and 50a is respectively 0.58 ,-0.95 and 0.88; 3. along with time lag k increases to 5 by 0, two sequence cross-correlation relation has the trend that weakens gradually in time scale 5-50a scope, and the simple crosscorrelation relation during time lag 5 is more obvious than the simple crosscorrelation relation of time lag 2 in 1-5a time scale scope.

(3) small echo cross-correlation coefficient isogram is carried out taken transverse, can understand the simple crosscorrelation relation of two sequences under each time lag on the set time yardstick.Fig. 2 c shows, 1. on identical time scale, has fluctuation during the hysteresis at any time of the size of two runoff sequence cross-correlation, and the trend that exists correlativity to weaken gradually.With a=7 is example, and two sequences are at time lag k=0,4,9,14 and 19 o'clock, and the fluctuation of cross-correlation coefficient is changed to 0.94 ,-0.88,0.78 ,-0.59,0.58; 2. along with the increase gradually of time scale, two runoff sequence cross-correlation coefficients also increase at any time the cycle of fluctuation of hysteresisization gradually.

(4) small echo cross correlation measure analysis result Fig. 3 shows, the simple crosscorrelation degree difference on the whole time domain of two sequences under different time lags.The small echo cross correlation measure obtains maximal value 0.96 when time lag k=0, consistent with the result of traditional cross-correlation analysis method.The small echo cross correlation measure reduces gradually when increasing with time lag afterwards, and the small echo cross correlation measure reaches another maximum value 0.80 again when k=13.Contrast as can be seen with the cross correlation number curve, small echo cross correlation measure curve is Paint Gloss, fluctuation pattern is more obvious, this mainly is because small echo cross correlation measure curve is the weighted mean value of simple crosscorrelation relation on above-mentioned each time scale, the simple crosscorrelation relationship change situation in the time of therefore can more effectively describing two sequences and pass with time lag on whole time domain.And traditional cross-correlation analysis method can't be considered and quantitative description different time yardstick on difference between the sequence cross-correlation relation, and a plurality of peak points that obtain in the curve lack the actual physical meanings.

(5) analysis result of synthesizing map 2 and Fig. 3 as can be seen: 1. two at station, Lijin and station, Huayuankou actual measurement annual flow sequence is that 0 o'clock simple crosscorrelation relation is the most obvious in time lag, and this is consistent by the real time of station, Huayuankou to the station, Lijin with runoff; 2. the simple crosscorrelation relation of two annual flow sequences when time scale is 3a, 7a, 11a and 20a is the most obvious, shows promptly that in four obvious cycle transformation period range scales of sequence self simple crosscorrelation of two sequences relation is also the most obvious.(6) as can be seen by instance analysis, small echo cross-correlation analysis method can disclose with describe nonstationary time series on the different time yardstick with different time lags under simple crosscorrelation relation, therefore can overcome the limitation of traditional cross-correlation analysis method, help simple crosscorrelation relation between the time series is carried out careful quantitative test comprehensively.

Claims (6)

1. Hydrological Time Series small echo cross-correlation analysis method is characterized in that may further comprise the steps:

(1) selects wavelet function and time scale scope, then Hydrological Time Series to be analyzed is carried out the continuous wavelet transform analysis;

(2) calculate Hydrological Time Series at the small echo cross covariance that reaches on the different time yardstick under the different time lags;

(3), find the solution two sequences at the small echo cross-correlation coefficient that reaches on the different time yardstick under the different time lags according to small echo cross covariance result of calculation;

(4) try to achieve after the small echo cross-correlation coefficient that reaches on the different time yardstick under the different time lags, calculate the small echo cross correlation measure; To describe the simple crosscorrelation degree of two sequences on whole time domain;

(5) try to achieve after small echo cross-correlation coefficient and the small echo cross correlation measure, draw small echo cross-correlation coefficient isogram, and by labor small echo cross-correlation coefficient isogram, grasp is studied between the sequence and concerned to local simple crosscorrelation by integral body; Realization is carried out the time-frequency analysis-by-synthesis to simple crosscorrelation relation between the time series.

2. Hydrological Time Series small echo cross-correlation analysis method according to claim 1 is characterized in that step (2) has two kinds of different situations of real part and mould according to the continuous wavelet transform analysis result, has defined the method for solving of small echo cross covariance respectively:

${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left({\mathrm{Wcov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(1\right)$

Or ${\mathrm{WC}}_{\mathrm{xy}}(a,k)=\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}---\left(2\right)$

Wcov in the formula _Xy(a, k)=E[W _x(a, b) W _y(a, b+k)] (3)

In the following formula, W _x(a, b) and W _y(a, the continuous wavelet transform coefficient that obtains when b) representing analytical sequence x and y respectively, a express time scale factor, b express time location factor, k represents time lag, R () and I () represent to find the solution real and imaginary part in the parantheses respectively, average, WC are found the solution in E () expression _Xy(a k) is illustrated in the small echo cross covariance of trying to achieve under time scale a and the time lag k.

3. Hydrological Time Series small echo cross-correlation analysis method according to claim 1 and 2 is characterized in that step (3) according to two kinds of the small echo cross covariance different situations, has defined the method for solving of small echo cross-correlation coefficient respectively:

${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{R\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}{\sqrt{R\left(W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right)R\left(W{\mathrm{cov}}_{\mathrm{yy}}(a,0)\right)}}---\left(4\right)$

Or ${\mathrm{WR}}_{\mathrm{xy}}(a,k)=\frac{\sqrt{R{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2+I{\left(W{\mathrm{cov}}_{\mathrm{xy}}(a,k)\right)}^2}}{\sqrt{\left|W{\mathrm{cov}}_{\mathrm{xx}}(a,0)\right|\left|W{\mathrm{cov}}_{\mathrm{yy}}(a,0)\right|}}---\left(5\right)$

In the following formula, WR _Xy(a k) is illustrated in the small echo cross-correlation coefficient of trying to achieve under time scale a and the time lag k, and ‖ represents to find the solution absolute value, and the remainder formula symbol is the same.

4. Hydrological Time Series small echo cross-correlation analysis method according to claim 1 and 2, it is characterized in that step (4) is according to small echo cross-correlation coefficient solving result, defined the method for solving of small echo cross correlation measure, to describe the simple crosscorrelation degree of two sequences on whole time domain, step is as follows:

(4.1) try to achieve two time serieses small echo cross-correlation coefficient WR under yardstick a and time lag k _Xy(a on basis k), tries to achieve the summation WR of the small echo simple crosscorrelation degree of two time serieses on corresponding whole time domain under the time lag k by integration _Xy(k):

WR _xy(k)＝∫WR _xy(a，k) ²da (6)

(4.2) then, find the solution small echo cross-correlation coefficient WR under each time scale a _Xy(a, weight coefficient k):

f(WR _xy(a，k))＝WR _xy(a，k) ²/WR _xy(k) (7)

(4.3) finding the solution the small echo cross correlation measure of two time serieses under time lag k is:

WCCD _xy(k)＝∫WR _xy(a，k)f(WR _xy(a，k))da (8)

5. Hydrological Time Series small echo cross-correlation analysis method according to claim 1 and 2, when it is characterized in that step (5) is drawn small echo cross-correlation coefficient isogram, the value of representing time lag k with transverse axis, the value of longitudinal axis express time yardstick a, certain point value among the figure has characterized the size of the cross-correlation coefficient of two sequences under corresponding yardstick a and the time lag k.

6. Hydrological Time Series small echo cross-correlation analysis method according to claim 1 and 2 is characterized in that step (5) according to small echo cross-correlation coefficient isogram, and the relation of the simple crosscorrelation between two time serieses is carried out labor, and key step is as follows:

(6.1), analyze under fixed time lag the situation of change of two sequences simple crosscorrelation degree size on each time scale by small echo cross-correlation coefficient isogram is carried out vertical intercepting;

(6.2) by isogram is carried out taken transverse, analyze on the set time yardstick, two sequences are simple crosscorrelation degree change situation under each time lag;

(6.3) by analyzing the positive negativity of small echo cross correlation numerical value on each time scale, grasp the positive and negative situation of change of two sequences cross correlation on each time scale;

(6.4), discern and extract corresponding certain or the tangible time scale scope of several cross correlations by the size of small echo cross-correlation coefficient absolute value on each time scale of comparative analysis;

(6.5), identify the most significant time delay relation between two time serieses by the size of comparative analysis small echo cross correlation numerical value under each time lag.

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