CN113688503A - Single-site rainfall time sequence simulation method based on Markov chain and rainfall event - Google Patents

Single-site rainfall time sequence simulation method based on Markov chain and rainfall event Download PDF

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CN113688503A
CN113688503A CN202110829966.6A CN202110829966A CN113688503A CN 113688503 A CN113688503 A CN 113688503A CN 202110829966 A CN202110829966 A CN 202110829966A CN 113688503 A CN113688503 A CN 113688503A
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高超
唐雄朋
章四龙
刘磊
王晓艳
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Zhuhai Campus Of Beijing Normal University
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Abstract

The invention discloses a single-site rainfall time sequence simulation method based on a Markov chain and a rainfall event, which combines the Markov chain with the rainfall event simulation for the first time and expands the rainfall event simulation into random simulation of the rainfall time sequence, thereby not only simultaneously retaining the characteristics of rainfall on the time sequence and the event and improving the precision of the rainfall simulation, but also overcoming the limitation that the output of the rainfall event simulation method can only be applied to a hydrological model based on the input of the event sequence. The method can generate rainfall data of any number of times and any length, can be conveniently combined with a common hydrological model to carry out hydrological simulation, and can carry out accurate flood and drought risk assessment.

Description

Single-site rainfall time sequence simulation method based on Markov chain and rainfall event
Technical Field
The invention relates to the field of random rainfall simulation, in particular to a single-site rainfall time sequence simulation method based on a Markov chain and rainfall events.
Background
Rainfall is an important component of the hydrologic cycle, directly affects runoff formation, and is often used as an important input to hydrologic models for flood and drought risk assessment and the design of hydraulic buildings. However, in some areas, the rainfall observation sequence is short or even absent, which greatly affects the accuracy and reliability of the hydrological assessment.
In order to solve the problem of rainfall data shortage, rainfall random simulation is carried out, the rainfall random simulation can be used for generating long-sequence rainfall data and providing a plurality of possible rainfall sequences, and meanwhile rainfall sequences can be generated for areas without data through interpolation of parameters of adjacent stations. In addition, random rainfall simulation can even carry out downscaling on a global climate mode and a regional climate mode, and the rainfall simulation precision directly determines the accuracy of the results. At present, most of the simulation methods are developed on the basis of Markov chain, such as WGEN, CLIGEN and ClimGen, and the methods generate time sequence of wet day and dry day according to the Markov chain and then generate rainfall of the wet day. Although these simulation methods can effectively preserve the characteristics of the wet period and the dry period and the characteristics of the rainfall in time series, the dependence and distribution pattern of the rainfall for a plurality of consecutive days (i.e. rain type) are ignored, and the correlation between the total rainfall amount and duration of a rainfall is also ignored. These ignored characteristics all pertain to rainfall event characteristics (depth of rainfall, duration of rainfall and type of rainfall).
In recent years, more and more researchers pay attention to the importance of the characteristics of the rainfall events, such as the rainfall events can cause larger flood events due to the continuous accumulation of runoff, the correlation between the rainfall depth and the rainfall duration also influences the formation of the runoff, and the delayed rain type can generate larger flood peak than the front rain type, thereby inducing more serious flood, and the like. Therefore, simulations of these rainfall event characteristics should not be ignored in random rainfall simulations. At present, a few methods for directly and randomly simulating rainfall event characteristics exist, but the methods output a rainfall event, cannot be combined with a common hydrological model requiring rainfall time sequence input and carry out flood and drought risk assessment, and greatly limit the application of the random rainfall event simulation methods. In view of this, the invention provides a simple random rainfall simulation method coupling a Markov chain and a rainfall event, and aims to retain the characteristic of the rainfall event and the characteristic of a rainfall time sequence, improve the rainfall simulation precision and widen the application range of the rainfall simulation method.
Disclosure of Invention
The invention provides a single-site rainfall time sequence simulation method based on a Markov chain and a rainfall event, which is characterized in that the consideration and simulation of rainfall event characteristics are added into a traditional random rainfall model so as to achieve the purposes of simultaneously retaining the characteristics of rainfall on the time sequence and the event and improving the rainfall simulation precision.
In order to achieve the above object, the present invention adopts the following technical solutions,
a single-site rainfall time sequence simulation method based on a Markov chain and rainfall events comprises the following steps:
step 1: generating a time series of wet and dry day states using a Markov chain;
according to the definition of the wet day and the dry day, determining the wet day and the dry day state (the state value of the wet day is 1, and the state value of the dry day is 0) of the rainfall time sequence, statistically analyzing the transition probability of the wet day and the dry day, and randomly generating the time sequence of the wet day and the dry day state according to the transition probability. Wherein, the conversion probability calculation formula of the wet day and the dry day is as follows:
Figure BDA0003175152820000021
Figure BDA0003175152820000022
in the formula (I), the compound is shown in the specification,
Figure BDA0003175152820000023
the transition probability for day t, wet, depends mainly on the state of the first k days.
Figure BDA0003175152820000024
The transition probability is given as day t being dry day. Xt,Xt-1,Xt-2,……,Xt-kThe status values of the t-th day, the t-1 th day, the t-2 th day, … … and the t-k th day are shown.
Step 2: and (3) extracting rainfall events from the wet day and dry day time sequence generated in the step 1 according to the rainfall event definition, and determining the duration of each simulated rainfall event, which is also called the duration of rainfall.
And step 3: under the condition of a given rainfall calendar, a Copula function of the rainfall depth is constructed, and the rainfall depth of each rainfall event is simulated randomly, wherein the specific process comprises the following steps:
(1) according to the actual rainfall event, a Copula function is adopted to construct a joint distribution function of the actual rainfall depth and the rainfall duration, and the formula is as follows:
F(x1,x2)＝C(u,v)＝C(F(x1),F(x2))
(2) the rainfall depth accumulation probability of each rainfall event is randomly simulated by using a conditional Copula function under the given rainfall simulation duration, and the calculation formula is as follows:
Figure BDA0003175152820000025
(3) the inverse function of the cumulative probability distribution function of the rainfall depth is adopted to reversely deduce the simulated rainfall depth value, and the calculation formula is as follows:
x1＝F-1(u)
wherein x is1And x2Respectively the rainfall depth and the rainfall duration variable; u and v are each x1And x2Is accumulated inA probability distribution function; c is a Copula function; f (x)1,x2) Is x1And x2The joint distribution function of (1).
And 4, step 4: respectively determining the grade of each rainfall event according to the rainfall depth and the rainfall duration simulation value of each rainfall event;
and 5: and randomly simulating the rainfall type of each rainfall event according to the occurrence probability of different rainfall types under different rainfall event grades, and randomly generating a dimensionless rainfall time course distribution curve of the specified rainfall type. The calculation formula of the occurrence probability of different rain types is as follows:
Figure BDA0003175152820000031
in the formula, Pk,ijThe occurrence probability of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; n isk,ijThe occurrence times of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; p1,ij+P2,ij+…+PK,ij＝1.
Step 6: and (3) carrying out time course distribution on the rainfall duration and the rainfall depth of each rainfall event on a dimensionless rainfall time course distribution curve, and combining the rainfall events into a complete rainfall event. And when all rainfall events are combined, obtaining a complete rainfall simulation time sequence of a single site.
In the above technical solution, further, the rainfall event in step 2 is defined as: divided by two drought periods (more than or equal to 1 day) and continuously for 1 or more days.
Further, in the step 3, before the step (1) of fitting the cumulative probability distribution function of the rainfall depth, the rainfall depth is uniformly subtracted from the minimum value defined in the wet day, then the cumulative probability distribution function of the rainfall depth is fitted, and then the minimum value defined in the wet day is uniformly added, so as to ensure that the generated rainfall depth value is greater than or equal to the minimum value defined in the wet day.
Further, the grade of the rainfall depth in the step 4 is divided according to 30%, 60% and 90% of the rainfall depth distribution, and the grade is divided into 4 grades, namely a light rainfall event (less than or equal to 30%), a medium intensity rainfall event (30% -60%), a strong rainfall event (60% -90%) and an extreme rainfall event (> 90%). Similarly, the rainfall duration is also divided into 4 levels, short duration, medium duration, long duration and extreme duration events, respectively, according to the integers of the nearest rainfall duration distribution 30%, 60% and 90% quantites.
Further, the rain type in the step 5 is that the rainfall time interval distribution is divided into a forward type (Advanced) rain type, a Central-affected (Central-affected) rain type and a Delayed type (Delayed) rain type according to the fact that the rainfall peak value is located at the front 1/3, the middle 1/3 and the rear 1/3 of the rainfall process; when the rainfall intensity is uniformly distributed in the whole process, the rain model is a Uniform (Uniform) rain model. The four rain patterns are respectively marked as A, C, D and U rain pattern.
Further, the dimensionless rainfall time interval distribution curve in the step 5 is obtained by dividing the time interval distribution of the rainfall event by the duration of the rainfall in the transverse direction and by dividing the rainfall by the depth of the rainfall in the longitudinal direction.
By adopting the technical means, the invention achieves the following beneficial effects and advantages:
(1) the distribution characteristics of a wet period and a dry period are effectively kept by adopting a Markov chain;
(2) by adopting a conditional Copula function, the correlation between the duration of rainfall and the depth of rainfall is effectively kept;
(3) random simulation of the rain type probability is considered, and the occurrence probability of different rain types under different rain event levels is well reproduced;
(4) random simulation of dimensionless rainfall time interval distribution curves of different rainfall types well keeps the randomness of rainfall time interval distribution and is more consistent with the actual situation;
(5) the Markov chain is coupled with the rainfall event simulation, so that the rainfall time sequence characteristic and the rainfall event characteristic can be simultaneously reserved, and the rainfall characteristic simulation precision is improved.
The method provided by the invention is suitable for random rainfall simulation in all areas observing rainfall data, and can be combined with a hydrological model to carry out hydrological simulation so as to evaluate flooding and drought disasters.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
figure 2 is a schematic of a time series for generating wet and dry day states using a markov chain;
FIG. 3 is a schematic diagram of extracting rainfall events and determining the duration of rainfall for each rainfall event;
FIG. 4 is a schematic diagram of randomly generating a rainfall depth for each rainfall event according to a conditional Copula function;
FIG. 5 is a schematic view of a complete rainfall time series obtained by the method of the present invention.
Detailed Description
The technical solution of the present invention is further elaborated by the following examples in combination with the accompanying drawings.
As shown in fig. 1, the invention relates to a single-site rainfall time sequence simulation method based on a markov chain and a rainfall event, which specifically comprises the following steps:
step 1: a markov chain is used to generate a time series of wet and dry day states.
Firstly, the measured daily rainfall data of a certain site in 30 years of 1971-2000 is collected. For example, the wet day is defined as a daily rainfall of 0.1mm or more, and the dry day is defined as a daily rainfall<0.1 mm. The state values on wet and dry days were 1 and 0, respectively. According to the definition of wet day and dry day, a group of data sequences representing the states of the wet day and the dry day with only 1 and 0 values can be obtained from the rainfall sequence of the observation day. In the example, a 3-order Markov chain is adopted, and the conditional transition probability P of the wet day depending on the states of the previous three days can be obtained according to the formula (1)0001、P1001、P0101、P0011、P0111、P1011、P1101、P1111The probability of a transition between dry days is P according to the formula (2)0000、P1000、P0100、P0010、P0110、P1010、P1100、P1110Thereby using the transition probability to randomly generate a new set of time series of wet and dry day statesSuch as 30 years.
Figure BDA0003175152820000051
Figure BDA0003175152820000052
In the formula (I), the compound is shown in the specification,
Figure BDA0003175152820000053
the transition probability for day t, wet, depends mainly on the state of the first 3 days.
Figure BDA0003175152820000054
The transition probability is given as day t being dry day.
Step 2: and (3) extracting rainfall events from the simulated wet and dry day time sequence in the step 1 according to the definition of the rainfall events, and determining the rainfall duration of each rainfall event.
In this example, the rainfall event is defined as: (1) the rainfall in wet weather is more than or equal to 0.1mm or the state value in wet weather is 1; (2) the drought period is more than or equal to 1 day. According to the definition of the rainfall events, continuous wet-day events divided by the drought period are extracted, and the duration (rainfall duration) of each continuous wet day is determined, wherein the duration comprises 1 day and a plurality of days. 30 years of simulation sequence, and 1511 rainfall events are extracted.
And step 3: and constructing a condition Copula function of rainfall depth under the given rainfall calendar, and randomly simulating the rainfall depth of each rainfall event.
According to the definition of the rainfall events in the step 2, 1511 rainfall events are extracted from the observation daily rainfall sequence, so that the rainfall duration, rainfall depth and rainfall time interval distribution data of the 1511 rainfall events can be counted. The rainfall time interval distribution firstly carries out dimensionless formulation on the duration and depth of rainfall, and then the dimensionless rainfall time interval distribution is divided into a U rainfall type, an A rainfall type, a C rainfall type and a D rainfall type according to whether the rainfall intensity is uniformly distributed in the whole process and the distribution of strong rainfall in the front 1/3 part, the middle 1/3 part and the rear 1/3 part of the rainfall process.
In this example, when fitting the distribution of the rainfall depth, the rainfall depth data is subtracted by 0.1mm (minimum magnitude defined on wet days) before fitting the distribution. In the embodiment, a plurality of distribution functions such as single gamma distribution, exponential distribution, lognormal distribution, normal distribution, double gamma distribution, generalized extreme value distribution, Pearson type III distribution, three-parameter lognormal distribution, generalized pareto distribution and the like are adopted to fit the sequences of the rainfall depth and the rainfall duration, and finally, the double gamma distribution and the exponential distribution are selected to be respectively used as the optimal fitting distribution line type of the rainfall depth and the rainfall duration according to the minimum criterion of AIC (Akaike Information criterion) Information. Gumbel Copula, Clayton Copula and Frank Copula of Archimedes are adopted to construct a joint distribution function of rainfall duration and rainfall depth, and a conditional Copula function under the condition of the given rainfall duration is obtained according to a formula (3) and a formula (4). Also, according to the minimum criterion of the AIC information, a conditional Frank function is finally adopted. For each simulated rainfall event, the duration of rainfall is x2The method comprises the steps of obtaining the cumulative distribution probability v according to the exponential distribution, randomly generating a random number F, obtaining the cumulative distribution probability value u of the rainfall depth of the rainfall event according to the inverse function (such as a formula (5)) of the conditional Frank function, and obtaining the rainfall depth data x of the rainfall event according to the inverse function of the rainfall depth fitting distribution-double-gamma distribution1. By analogy, rainfall depth data of all 1511 simulated rainfall events can be obtained, and finally the simulated rainfall depth data is added back to 0.1 mm.
F(x1,x2)＝C(u,v)＝C(F(x1),F(x2)) (3)
Figure BDA0003175152820000061
u＝C-1(u|v)，x1＝F-1(u) (5)
In the formula, x1And x2Respectively the rainfall depth and the rainfall duration variable; u and v are each x1And x2The cumulative distribution function of; c is a Copula letterCounting; f (x)1,x2) Is x1And x2The joint distribution function of (1).
And 4, step 4: and determining the grade of each rainfall event according to the simulated rainfall depth and the rainfall duration of each rainfall event.
The rainfall depth was divided into four levels according to the 30%, 60% and 90% quantile point values (as in table 1) of the observed rainfall depth distribution, thereby determining the event level of each simulated rainfall depth. Similarly, the rainfall duration is divided into four levels according to the adjacent integers of 30%, 60% and 90% quantile points of the observed rainfall duration distribution, so as to judge the event level of each simulated rainfall duration.
TABLE 1 four classes according to the depth of rainfall and duration of rainfall
Figure BDA0003175152820000062
And 5: and randomly simulating the rainfall type of each rainfall event according to the occurrence probability of different rainfall types under different rainfall event grades, and randomly generating a dimensionless rainfall time course distribution curve of the specified rainfall type.
The probability of occurrence of four rain types (K ═ 4) in the observed rainfall event under different rainfall depths and rainfall duration levels is calculated according to the formula (6), and the calculation result is shown in the tabulated table 2. For a simulated rainfall event with known rainfall depth level and rainfall duration level, random rainfall patterns can be obtained according to the probability of four rainfall patterns, and the codes of the A rainfall pattern, the C rainfall pattern, the D rainfall pattern and the U rainfall pattern are 1, 2, 3 and 4 respectively. And the rest can be done until all the simulation of the rain type of all the 1151 simulated rainfall events is completed.
Figure BDA0003175152820000063
In the formula, Pk,ijThe occurrence probability of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; n isk,ijThe occurrence times of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; p1,ij+P2,ij+…+PK,ij＝1.
TABLE 2 probability of occurrence of different rain types under different rainfall event classes
Figure BDA0003175152820000071
A random simulation method for assigning dimensionless rainfall time interval of a designated rain type is disclosed in a patent 'a new random generation method of rainfall events' (patent number: ZL201710557548. X).
Step 6: and (3) carrying out time course distribution on the rainfall duration and the rainfall depth of each rainfall event on a dimensionless rainfall time course distribution curve, and combining the rainfall events into a complete rainfall event. And when all rainfall events are combined, obtaining a complete rainfall simulation time sequence of a single site.
The above example implementations are merely illustrative of the present invention and are not intended to be in any way limiting. The Markov chain has various orders, rainfall duration, rainfall depth fitting distribution and Copula function forms, and can be specifically formulated according to regions and requirements. However, any modification, equivalent replacement, improvement and the like made without departing from the contents of the claims of the present invention still belong to the protection scope of the present invention.

Claims (6)

1. A single-site rainfall time sequence simulation method based on a Markov chain and rainfall events is characterized by comprising the following steps:
step 1: generating a time sequence of wet day and dry day states by adopting a Markov chain, wherein the wet day state value is 1, and the dry day state value is 0;
step 2: according to the definition of the rainfall events, extracting the rainfall events from the wet day time sequence and the dry day time sequence generated in the step 1, and determining the duration of each simulated rainfall event, namely the duration of the rainfall;
and step 3: under the condition of a given rainfall calendar, a Copula function of the rainfall depth is constructed, and the rainfall depth of each rainfall event is simulated randomly, wherein the specific process comprises the following steps:
(1) according to the actual rainfall event, a Copula function is adopted to construct a joint distribution function of the actual rainfall depth and the rainfall duration, and the formula is as follows:
F(x1,x2)＝C(u,v)＝C(F(x1),F(x2))
(2) the rainfall depth accumulation probability of each rainfall event is randomly simulated by using a conditional Copula function under the given rainfall simulation duration, and the calculation formula is as follows:
Figure FDA0003175152810000011
u＝C-1(u|v)
(3) the inverse function of the cumulative probability distribution function of the rainfall depth is adopted to reversely deduce the simulated rainfall depth value, and the calculation formula is as follows:
x1＝F-1(u)
wherein x is1And x2Respectively the rainfall depth and the rainfall duration variable; u and v are each x1And x2A cumulative probability distribution function of; c is a Copula function; f (x)1,x2) Is x1And x2A joint distribution function of (a);
and 4, step 4: respectively determining the grade of each rainfall event according to the rainfall depth and the rainfall duration simulation value of each rainfall event;
and 5: randomly simulating the rainfall type of each rainfall event according to the occurrence probability of different rainfall types under different rainfall event grades, and randomly generating a dimensionless rainfall time course distribution curve of the specified rainfall type; the calculation formula of the occurrence probability of different rain types is as follows:
Figure FDA0003175152810000012
in the formula, Pk,ijThe occurrence probability of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; n isk,ijThe occurrence times of the kth rain type under the ith rainfall depth level and the jth rainfall duration level; p1,ij+P2,ij+…+PK,ij＝1；
Step 6: the rainfall duration and the rainfall depth of each rainfall event are distributed on a dimensionless rainfall schedule distribution curve, and a complete rainfall event can be combined; and when all rainfall events are combined, obtaining a complete rainfall simulation time sequence of a single site.
2. The method of simulating a single-site rainfall time series based on Markov chains and rainfall events of claim 1 wherein the rainfall events in step 2 are: the wet day is divided into at least 1 day by the two stages of drought periods, and the drought period is more than or equal to 1 day.
3. The method of claim 1, wherein in step 3, before the step (1) of fitting the cumulative probability distribution function of the rainfall depth, the rainfall depth is uniformly subtracted by the minimum value defined by the wet day, and then the minimum value defined by the wet day is uniformly added after the rainfall depth of the rainfall event is simulated, so as to ensure that the generated rainfall depth is greater than or equal to the minimum value defined by the wet day.
4. The method for single-site rainfall time series simulation based on Markov chain and rainfall events of claim 1 wherein the level of rainfall depth in step 4 is divided according to 30%, 60% and 90% quantiles of rainfall depth distribution into 4 levels, respectively light rainfall event ≦ 30%, medium intensity rainfall event 30% -60%, strong rainfall event 60% -90% and extreme rainfall event > 90%; similarly, the rainfall duration is also divided into 4 levels, respectively, less than or equal to 30% for short duration events, 30% -60% for medium duration events, 60% -90% for long duration events and > 90% for extreme duration events, according to the integers of the 30%, 60% and 90% quantites of the nearest rainfall duration distribution.
5. The method for simulating a single-site rainfall time series based on Markov chain and rainfall events as claimed in claim 1 wherein the rain profile in step 5 is a forward, middle and late rain profile, respectively designated as A, C, D rain profile, based on the rainfall peaks located at the front 1/3, middle 1/3 and rear 1/3 of a rainfall event; when the rainfall intensity is uniformly distributed in the whole process, the rain type is a uniform rain type and is marked as a U-rain type.
6. The method of claim 1, wherein the dimensionless rainfall schedule distribution curve in step 5 is obtained by dividing the rainfall schedule distribution of the rainfall events in the lateral direction by the duration of rainfall and in the longitudinal direction by the depth of rainfall.
CN202110829966.6A 2021-07-22 2021-07-22 Single-site rainfall time sequence simulation method based on Markov chain and rainfall event Pending CN113688503A (en)

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