CN104699961A - Method for calculating multiyear return period wave height of self-affine fractal on basis of Hurst rule - Google Patents

Abstract

The invention provides a method for calculating a multiyear return period wave height of a self-affine fractal on the basis of the Hurst rule. The method comprises the following steps: a, determining a period, periodically acquiring wave height extreme value data and analyzing the wave height extreme value data by utilizing a rescaled range analysis method, wherein the wave height extreme value data meets the Hurst rule; b, according to an analysis result, acquiring parameters of a self-affine fractal model by a least square method; c, calculating the multiyear return period wave height by utilizing the self-affine fractal model. Compared with a calculating result obtained by a conventional method, a calculating result obtained by the method provided by the invention has the advantage that when a designed wave height calculation return period is long, compared with a designed wave height calculated by the conventional method, the designed multiyear return period wave height calculated by the method is more accurate and reliable in obtained result.

Description

Based on the projectional technique of the Return period wave height of the self affine analysis of Hurst rule

Technical field

The present invention relates to marine field, particularly relate to a kind of projectional technique of Return period wave height of the self affine analysis based on Hurst rule.

Background technology

In ocean and coastal engineering and seashore are taken precautions against natural calamities, rationally selected marine environmental conditions design parameter exactly, inquires into ocean and coastal engineering design effective early warning that Return period reoccurrence period level and seashore take precautions against natural calamities significant.In recent years, people are generally such as: adopt year extremum method to inquire into Return period Design Wave in the engineerings such as oceanographic engineering, hydraulic engineering, seaside nuclear power station.The more Gumbel that has of domestic hydrology circle application distributes, Weibull distributes, Pearson-III type distribution etc., these methods have successfully applied in engineering construction, but they have a significant shortcoming: these methods are all priori, namely we meet our the artificial probability distribution curve selected by suitable collimation method supposition actual measurement annual extreme wave height in advance when using these methods, the distribution selected by us again carrys out matching annual extreme wave height data, then the cumulative distribution curve that matching obtains is extended and try to achieve Return period Design Wave, and can test of hypothesis be passed through in the ordinary course of things, but region is different residing for different sea area, marine environmental conditions is different, the reckoning of all sea areas marine environment design parameter can be suitable for without any a kind of distribution.

Wave is a kind of very complicated spontaneous phenomenon.The wave statistics theory that the random theory utilizing Longuet-Higgins to propose is set up is the main method of research wave.Through the development of many Chinese scholars, establish the classical theory about sea wave height sequence.Nineteen ninety-five Mei Liming analyzes sea wave height sequence, find that sea wave height sequence does not meet the hypothesis of this essence separate, wave height sequence has the fractal property of long-range dependence and statistics self affine, establishes Cauchy statistical model and the fraction statistical model of wave height sequence.But the spontaneous phenomenon of this complexity of wave is generally meet self similarity in statistical significance, its self-similarity and scaling invariance are only set up under statistical significance.Classic method is taked to suppose distribution in advance, and apriority, human factor are comparatively strong, sometimes on the low side when Design Wave, may bring certain hidden danger, therefore, need a kind of projectional technique of new Return period wave height to coastal engineering.

Summary of the invention

In view of this, the invention provides a kind of projectional technique of Return period wave height of the self affine analysis based on Hurst rule, to solve the problem.

The projectional technique of the Return period wave height of a kind of self affine analysis based on Hurst rule provided by the invention, comprises the steps

A. determine the cycle, and periodically gather Extreme Wave data and utilize Rescaled range analysis to Extreme Wave data analysis, described Extreme Wave data meet Hirst rule;

B. be obtained from the parameter of affine fractal model by least square method according to analysis result;

C. utilize self affine analysis model to calculate Return period wave height, described self affine analysis model is following formula:

$\frac{N(\≥r)}{N}=\frac{C}{r^D}$

Wherein, N (>=r) expression is more than or equal to the event of r or the number of set, total number of N presentation of events or set, represent cumulative percentage, C represents constant, and D represents point dimension of fractal sets;

The value of described parameter C, D is obtained by step b.

Further, the formula of described Hirst rule is:

$\frac{R}{S}~{\mathrm{\τ}}^H$

Wherein, H is positioned at the constant on [0,1] interval, i.e. Hurst Exponent; R is extreme difference; S is mean square deviation.

Further, described step a also comprises the data Extreme Wave data of collection being divided into many group constant durations, obtains the maximal value of wave height data in each time interval, calculates Hurst Exponent.

Further, described self affine analysis model is the annual extreme wave height variation model formed by fractional Brownian motion, and described fractional Brownian motion meets Hirst rule.

Further, described Hirst rule is the experimental formula according to the time series analysis statistics of wave height observation data for many years.

Further, described Hurst Exponent is more close to 1, and described seasonal effect in time series time-length interrelation is stronger, and wave height more has Long Memory.

Beneficial effect of the present invention: the present invention is different from the conventional method inquiring into Return period Design Wave, namely supposition wave height meets a certain distribution in advance, but on the basis of fractal theory, a kind of novel Return period wave height projectional technique of the self affine analysis dimensional formula based on Hurst rule is proposed.Reckoning result of the present invention is compared with the result calculated by classic method, the Return period Design Wave that Design Wave and its of the Return period that self affine analysis model is inquired into are asked is substantially identical, but along with the increase of reoccurrence period, the Return period Design Wave value that self affine analysis pattern is tried to achieve is larger, namely the Design Wave that application self affine analysis model is tried to achieve requires stricter, the result obtained is more reliable, avoids the hidden danger brought to coastal engineering.

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described:

Fig. 1 is principle process schematic diagram of the present invention.

Fig. 2 is the scatter diagram of the annual extreme wave height of the 1963-1989 of the embodiment of the present invention.

Fig. 3 is the experience accumulation function distribution histogram of the embodiment of the present invention.

Fig. 4 is the wave height data plot of the embodiment of the present invention.

Fig. 5 is the log-log plot of the wave height sequence of embodiment of the present invention 1963-1989 extreme value.

Fig. 6 is Extreme Wave Gumbel, Weibull, Pearson-III fitting of distribution function and empirical distribution function

Embodiment

Below in conjunction with drawings and Examples, the invention will be further described: Fig. 1 is principle schematic of the present invention.Fig. 2 is the scatter diagram of the annual extreme wave height of the 1963-1989 of the embodiment of the present invention, Fig. 3 is the experience accumulation function distribution histogram of the embodiment of the present invention, Fig. 4 is the wave height data plot of the embodiment of the present invention, Fig. 5 is the log-log plot of the wave height sequence of embodiment of the present invention 1963-1989 extreme value, Fig. 6 is Extreme Wave Gumbel, Weibull, Pearson-III fitting of distribution function and empirical distribution function.

As shown in Figure 1, the projectional technique of the Return period wave height of the self affine analysis based on Hurst rule of the present invention, comprises the steps

A. determine the cycle, and periodically gather Extreme Wave data and utilize Rescaled range analysis to Extreme Wave data analysis, described Extreme Wave data meet Hirst rule;

B. be obtained from the parameter of affine fractal model by least square method according to analysis result;

C. utilize self affine analysis model to calculate Return period wave height, described self affine analysis model is following formula:

$\frac{N(\≥r)}{N}=\frac{C}{r^D}$

Wherein, N (>=r) expression is more than or equal to the event of r or the number of set, total number of N presentation of events or set, represent cumulative percentage, C represents constant, and D represents point dimension of fractal sets;

The value of described parameter C, D is obtained by step b.

The formula of described Hirst rule is:

$\frac{R}{S}~{\mathrm{\τ}}^H$

Wherein, H is positioned at the constant on [0,1] interval, i.e. Hurst Exponent; R is extreme difference; S is mean square deviation.

In the present embodiment, the reckoning embodiment as Return period wave height enumerating the actual measurement wave height data of seaside nuclear power station 1963-1989 is described in detail, and seaside nuclear power station for many years average Extreme Wave is 3.727m.Make the scatter diagram of the annual extreme wave height of 1963-1989, as shown in Figure 2.Using 1963 as calculating zero point.From Fig. 2 and Fig. 3, we see the feature that the distribution of annual extreme wave height presents " the fat tail of spike ", have very big difference with normal distribution.

In the present embodiment, as shown in Figure 4, R/S method is adopted to analyze wave height data, R/S analytic approach also claims Rescaled range analysis (Rescaled Range Analysis), a kind of method that hydrologist Hurst proposes on the basis of a large amount of positive research, after through Mandelbrot (1972,1975), Mandelbrot, Wallis (1969), the effort of many people such as Lo (1991) is progressively able to perfect.In the present embodiment, the 1963-1989 data of 26 years are divided into equally spaced data: got maximal value (as shown in fig. 4 a) every 6 months; A maximal value (as shown in Figure 4 b) is got every 1 year; Got a maximal value (as illustrated in fig. 4 c) every 2 years, judge that wave height time series is completely random or there is tendency composition according to the size of the Hurst index of trying to achieve, as shown in table 1.

	Hurst index	c(t)	R 2
				Every 6 months	0.8369	0.5952	0.9745
Every 1 year	0.8033	0.5227	0.9824
				Every 2 years	0.8282	0.5761	0.9763

Table 1

As can be seen from Table 1, the fit solution of 3 groups of data is all better, and R ²all be greater than 0.95, illustrate that wave height time series data all meets Hurst rule.With

In the present embodiment, using 1963 as calculating zero point, obtain the time series x (t) of annual extreme wave height, obtained the value of R (τ), S (τ) and R (τ)/S (τ) by R/S analytical approach, as shown in table 2.

τ	Time	R(τ)	S(τ)	R(τ)/S(τ)
					1	1963
2	1964	7.0000	0.7000	10.0000
					3	1965	13.0667	10.3090	1.2675
4	1966	18.2500	10.3811	1.7580
					5	1967	24.4800	10.8277	2.2609
6	1968	29.9000	10.6780	2.8001
					7	1969	34.1143	10.4676	3.2590
8	1970	37.2750	10.1806	3.6614
					9	1971	39.7000	9.8669	4.0236
10	1972	41.6400	9.5595	4.3559
					11	1973	42.8455	9.2027	4.6558
12	1974	44.0750	8.9152	4.9438
					13	1975	45.0231	8.6351	5.2140
14	1977	45.1714	8.3229	5.4274
					15	1978	45.5200	8.0525	5.6530
16	1979	45.8063	7.8055	5.8685

17	1980	46.5176	7.6316	6.0954
					18	1981	46.7833	7.4256	6.3003
19	1982	47.3842	7.2773	6.5112
					20	1983	47.9100	7.1341	6.7156
21	1984	48.3857	6.9982	6.9140
					22	1985	48.8318	6.8711	7.1068
23	1986	48.6522	6.7260	7.2335
					24	1987	49.0000	6.6078	7.4155
25	1988	49.2000	6.4825	7.5897
					26	1989	49.2923	6.3585	7.7679

Table 2

In the present embodiment, according to Extreme Wave data, use least square method can obtain relational expression and the H value of R/S.Wherein H=0.8033, c (t)=0.5227, R ²=0.9824.

Hurst index is 0.8033, and according to H value more close to 1, c (t)=0.5227>0, seasonal effect in time series time-length interrelation is stronger, and the wave height that can obtain Chao Lian island has Long Memory.Track according to fractional Brownian motion displays lasting quality, and a rising tendency in Chao Lian island wave height past means a rising tendency in the future, and a minimizing trend in past means a minimizing trend in the future.Also show that Chao Lian island wave height tables of data reveals certain nonrandomness simultaneously.Coefficient R ²=0.9824 sum test statistics F=128.31 can find out R/S energy matching annual extreme wave height data preferably, as shown in Figure 5.

In the present embodiment, choose 1963-1989 totally 26 year year extreme value data and first 13 year year extreme value data; Choose 1963-1989 totally 26 years 7, year in August extreme value data and in first 13 years 7, the year extreme value data in August.By 26 year year extreme value data and first 13 years year parameter required by extreme value data in table 3; By 7 in 26 years, the extreme value data in August and first 13 years 7, parameter required by August data is in table 4:

Table 3

Table 4

Can be found out by table 3 and table 4, the model parameter that long term data and short-term data are obtained is more or less the same.7, August, extreme value data compared with parameter required by year extreme value data, and difference is no more than 7%, as seen can by observing 7, August wave height time series calculate annual extreme wave height data, new model has good stability when processing different data.

In the present embodiment, by Gumbel, Weibull, Pearson-III fitting of distribution wave height, its parameter and confidence level be 95% interval estimation as shown in table 5.

Table 5

As shown in Figure 6, Gumbel, Weibull, Pearson-III is not particularly preferred matching Extreme Wave data.With Gumbel, Weibull, Pearson-III distribute, self affine analysis model predict respectively 100,200,400,500,700,1000 years one meet wave height, as being shown in Table 6.

	Wave height (m)	Wave height (m)	Wave height (m)	Wave height (m)
					Reoccurrence period	Gumbel	Weibull	Pearson-III	Self affine analysis model
100	5.497	5.460	5.172	4.939
					200	5.625	5.616	5.349	5.261
300	5.692	5.701	5.449	5.458
					400	5.737	5.756	5.517	5.603
500	5.770	5.801	5.570	5.718
					700	5.818	5.862	5.647	5.896
1000	5.867	5.925	5.728	6.091

Table 6

Consider that in the present embodiment, season is on the impact of wave height, analysis 7 of Chao Lian island, the maximum value wave height sequence in summer in August, obtain the Design Wave in the summer of Return period, as shown in table 7:

	Wave height (m)	Wave height (m)	Wave height (m)	Wave height (m)
					Reoccurrence period	Gumbel	Weibull	Pearson-III	Self affine analysis model
100	5.577	5.566	5.316	5.023
					200	5.710	5.734	5.516	5.222
300	5.781	5.827	5.628	5.424

400	5.827	5.888	5.705	5.572
					500	5.862	5.935	5.764	5.690
700	5.912	6.002	5.852	5.872
					1000	5.963	6.070	5.943	6.072

In the present embodiment, from table 4, we see that the Return period Design Wave that the Design Wave of the Return period that self affine analysis model is inquired into and common distribution are inquired into is more or less the same.For Pearson-III, the Return period Design Wave asked with it of the Design Wave of the Return period that self affine analysis model is inquired into is maximum is no more than 0.294m, and the Return period Design Wave value of trying to achieve along with the increase self affine analysis pattern of reoccurrence period is larger, the Design Wave that this application self affine analysis model is tried to achieve requires stricter.Obtain in table 5 comparing Pearson-III, the Return period Design Wave error in summer that self affine analysis model is tried to achieve differs maximum and is no more than 5.5%.Can find out that three kinds of common distributions are not can good matching Extreme Wave data, application self affine analysis pattern to Return period Design Wave method more effective.

In the present embodiment, step a also comprises the data Extreme Wave data of collection being divided into many group constant durations, obtains the maximal value of wave height data in each time interval, calculates Hurst Exponent.Described self affine analysis model is the annual extreme wave height variation model formed by fractional Brownian motion, and described fractional Brownian motion meets Hirst rule.Described Hirst rule is the experimental formula according to the time series analysis statistics of wave height observation data for many years.Described Hurst Exponent is more close to 1, and described seasonal effect in time series time-length interrelation is stronger, and wave height more has Long Memory.

Hirst (Hurst) the Lv Shi Britain hydrologist Hurst adopted in the present embodiment finds in the research of a series of instance data sequence, and rule that most of spontaneous phenomenon is all followed " having inclined random walk ", namely a trend adds noise.He introduces a new statistic: Hurst index, and in order to the intensity of the trend of measuring and the level situation over time of noise, the analytical approach of use is exactly R/S analytic approach.Hurst index has purposes widely for all time serieses, because it is strong especially, it is little to the supposition required by studied system.The basic thought of this analytical approach comes from fractional Brownian motion and the TH rule of Mandelbrot proposition.A random series and a nonrandom sequences can make a distinction by R/S analytic approach, and are analyzed can also be carried out seeking of nonlinear system long-term memory process by R/S.Hurst proposes in statistical study the experimental formula that was called as Hurst rule afterwards after multiple spontaneous phenomenon observation data time series.Being expressed as follows of Hurst rule:

Consider a time series { x (t) }, t=1,2 ..., for arbitrary positive integer τ, have following formula

$\frac{R}{S}~{\mathrm{\τ}}^H$ Formula (1)

Wherein, H is positioned at the constant on [0,1] interval, is called Hurst index; R is extreme difference; S is respectively mean square deviation, and it is defined as:

R (τ)=maxD (t, τ)-minD (t, τ), 1≤t≤τ formula (2)

$S\left(\mathrm{\τ}\right)={\left\{\frac{1}{\mathrm{\τ}}{\mathrm{\Σ}}_{t=1}^{\mathrm{\τ}}{\{x\left(t\right)-\stackrel{\‾}{x\left(\mathrm{\τ}\right)}\}}^2\right\}}^{\frac{1}{2}}$ Formula (3)

Wherein, D (t, τ) is accumulated deviation:

$D(t,\mathrm{\τ})={\mathrm{\Σ}}_{\mathrm{\μ}=1}^t\{x\left(\mathrm{\μ}\right)-\stackrel{\‾}{x\left(\mathrm{\τ}\right)}\},1\≤t\≤\mathrm{\τ}$ Formula (4)

for equal value sequence:

$\stackrel{\‾}{x\left(\mathrm{\τ}\right)}=\frac{1}{\mathrm{\τ}}{\mathrm{\Σ}}_{t=1}^{\mathrm{\τ}}x\left(t\right),\mathrm{\τ}=\mathrm{1,2},.......$ Formula (5)

From formula (2) and (3), R/S is a zero dimension empirical statistics characteristic quantity, the observed result of Different time scales scope τ connects by the Hurst rule represented by formula (1), provides the possibility from small scale observed result extrapolation large scale statistical law.

Hurst index in the present embodiment has been widely used in measurement analyze serial correlation and trend intensity: when time, sequence belongs to random walk process; When with time, Sequence Trend is enhancing or contrary respectively, belongs to and has inclined random walk process.

In the present embodiment, to related function c (t) of the fractional Brownian motion easily increment of deriving over and following increment be

C (t)=2 ^2F-1-1 formula (6)

Wherein F is positioned at the constant on [0,1] interval, from formula (6) formula:

When time, c (t)=0, the increment in past is incoherent with following increment, and this has the feature of the Brownian movement of independent increment just; When time, c (t) ≠ 0, this is exactly just in time the feature of fractional Brownian motion, and works as time, c (t) > 0 He time, during c (t) < 0, show following increment and increment positive correlation or contrary in the past.Therefore, fractional Brownian motion meets the Hurst rule represented by formula (1).

In Fractals, self affine adds up fractal requirement:

F (br)=b ^hf (r) formula (7)

Wherein f (r) represents the feature of object.

In the present embodiment, fractional Brownian motion is self affine statistics fractal (Mandelbrot1985), so-called fractional Brownian motion, refers to the function of time B that in search time section [0, T] one is random _ht () has following statistical nature:

E(B _H(t))＝0

Var(B _H(t))～T ^2H

D(B _H(t))～T ^H

Wherein H is positioned at the constant on [0,1] interval, gets mean square deviation D (B _h(t)) as the statistical nature describing fractional Brownian motion:

F (bt)=D (B _h(t)) ~ b ^hf (t) formula (8)

Formula (8) and formula (7) equivalent, therefore fractional Brownian motion is that self affine statistics is fractal.Thus meet

P (r) ~ r ^-Dformula (9)

Wherein P (r) is statistical characteristic value, and r is the increment of variable, considers that P (r) represents the probability of stochastic variable x >=r, meet P (r)≤1, therefore power exponent gets negative sign.

In the present embodiment, can the approximate alternative P (r) of the frequency that occurs of use case.If N (>=r) represents be more than or equal to the event of r or the number of set, total number of N presentation of events or set, represent cumulative percentage, we as description wave height seasonal effect in time series statistical nature, so have:

$\frac{N(\&GreaterEqual;r)}{N}=\frac{C}{r^D}$ Formula (10)

Wherein C is constant, and D is point dimension of fractal sets.

Parameter C is extrapolated by least square method by annual extreme wave height data, the value of D, then by fixed parameter C, D brings formula (10) into, due to total number of N presentation of events or set, the value of N is brought in formula (10), calculate, obtain the numerical value of N (>=r), then Extreme Wave data are carried out matching and can obtain Return period wave height numerical value.

In the present embodiment, fractional Brownian motion is regarded as the model of annual extreme wave height change, thus Hurst should be met restrain and meet the fractal dimension formula represented by formula (9).

In the present embodiment, according to the difference in geographic position and the impact being subject to typhoon thereof, Various Seasonal is also different on the impact of wave height, by 7, the Extreme Wave in August and annual extreme wave height compare, the reoccurrence period higher than 100 years time, Return period Design Wave in the summer value calculated will be slightly less than Return period Design Wave, but the value deviation of Return period Design Wave in summer and annual extreme wave height is maximum is no more than 1.7%.This illustrates the actual measurement wave height paying close attention to summer 7, August from now on when observing long-term wave height data.

In the present embodiment, due to reckoning is Return period wave height, if when therefore Design Wave is less than real Return period wave height, quite serious impact can be caused, the Return period Design Wave value of trying to achieve along with the increase self affine analysis pattern of reoccurrence period when Return period Design Wave inquired into by the present embodiment application self affine analysis model is larger, for coastal engineering saves human and material resources and financial resources.

In the present embodiment, when applying self affine analysis model and inquiring into Return period Design Wave, be applied required by short-term data parameter be more or less the same compared with the application long term data parameter of trying to achieve, further illustrating these model treatment data of application has good stability.

What finally illustrate is, above embodiment is only in order to illustrate technical scheme of the present invention and unrestricted, although with reference to preferred embodiment to invention has been detailed description, those of ordinary skill in the art is to be understood that, can modify to technical scheme of the present invention or equivalent replacement, and not departing from aim and the scope of technical solution of the present invention, it all should be encompassed in the middle of right of the present invention.

Claims (6)

1., based on a projectional technique for the Return period wave height of the self affine analysis of Hurst rule, it is characterized in that: comprise the steps

A. determine the cycle, and periodically gather Extreme Wave data and utilize Rescaled range analysis to Extreme Wave data analysis, described Extreme Wave data meet Hirst rule;

B. be obtained from the parameter of affine fractal model by least square method according to analysis result;

C. utilize self affine analysis model to calculate Return period wave height, described self affine analysis model is following formula:

$\frac{N(\&GreaterEqual;r)}{N}=\frac{C}{r^D}$

Wherein, N (>=r) expression is more than or equal to the event of r or the number of set, total number of N presentation of events or set, represent cumulative percentage, C represents constant, and D represents point dimension of fractal sets;

The value of described parameter C, D is obtained by step b.

2. according to the projectional technique of the Return period wave height of the self affine analysis based on Hurst rule described in claim 1, it is characterized in that: the formula of described Hirst rule is:

$\frac{R}{S}~{\mathrm{\&tau;}}^H$

Wherein, H is positioned at the constant on [0,1] interval, i.e. Hurst Exponent; R is extreme difference; S is mean square deviation.

3. according to the projectional technique of the Return period wave height of the self affine analysis based on Hurst rule described in claim 2, it is characterized in that: described step a also comprises the data Extreme Wave data of collection being divided into many group constant durations, obtain the maximal value of wave height data in each time interval, calculate Hurst Exponent.

4. the projectional technique of the Return period wave height of the self affine analysis based on Hurst rule according to claim 3, it is characterized in that: described self affine analysis model is the annual extreme wave height variation model formed by fractional Brownian motion, described fractional Brownian motion meets Hirst rule.

5. according to the projectional technique of Return period wave height of the self affine analysis based on Hurst rule described in claim 4, it is characterized in that: described Hirst rule is the experimental formula according to the time series analysis statistics of wave height observation data for many years.

6., according to the projectional technique of the Return period wave height of the self affine analysis based on Hurst rule described in claim 5, it is characterized in that: described Hurst Exponent is more close to 1, and described seasonal effect in time series time-length interrelation is stronger, and wave height more has Long Memory.