CN108073782A - A kind of data assimilation method based on the equal weight particle filter of observation window - Google Patents
A kind of data assimilation method based on the equal weight particle filter of observation window Download PDFInfo
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Abstract
One kind the invention discloses the present invention is based on the equal weight particle filter data assimilation method of observation window, specifically includes following steps：(1) obtaining mode integration initial background field；(2) calculated according to observation information in observation window and propose density；(3) in given assimilation time, particle weights is calculated using equal weight method, adjust particle state；(4) using method for resampling, adjustment set particle maintains population to stablize；(5) the state posterior estimate after equal weight particle filter is calculated.The observation information that the present invention is stored using observation window, set particle state is adjusted before assimilation time by proposing density, further improve equal weight particle filter assimilation method, data assimilation quality under nonlinear model can be effectively improved using this method, can preferably be applied to current data and assimilate application field.
Description
Technical field
The present invention relates to air and oceanographic data to assimilate field, and in particular to one kind is based on the equal weight particle filter of observation window
The air of ripple and oceanographic data assimilation method.
Background technology
There are two types of mode, one kind is studied using numerical model for ocean dynamics research, another kind be to air with
Ocean is directly observed.With air and the rapid development of ocean related data information research, for air and oceanographic data
Requirement be also continuously improved, conventional numeric model can approx reflect ocean changing rule.And the number directly observed
According to although the reflection to time of day, due to the continuous variation of the limitation and observation point of equipment, observed result is inevitably deposited
In inevitable error and random error.The core of data assimilation is that new observation letter is constantly merged in kinetics model
To reach the purpose of improved mode initial value and the value of forecasting, sea is effectively improved in current research using data assimilation method for breath
The effect of ocean simulation reduces the uncertainty of ocean and climatic prediction primary condition, improves analysis of data again.
Data assimilation method is broadly divided into two classes：Variational method and sequential grammar, variational method include three-dimensional variation and four
Variation is tieed up, is mainly characterized by the minimization problem using numerical optimization Algorithm for Solving object function.Such as number of patent application
A kind of 201510047795.6 direct assimilation method of MODIS satellite datas of patent application, the patent are same using three-dimensional variation
Change method assimilation Satellite Observations improve numerical weather forecast precision.Sequential grammar is then the recurrence based on statistical estimate theory
Method the advantage is that the tangent linear and adjoint mode that need not develop thinking variational Assimilation algorithm needs, and can be explicitly
The initial disturbance of DATA PROCESSING IN ENSEMBLE PREDICTION SYSTEM is provided, therefore has been widely used with the improvement of design conditions, such as number of patent application
201010561909.6 patent application high rate observation data real time data quick ensemble Kalman filter assimilation method.This is specially
Profit improves the problem of deficiency is run in Ensemble Kalman Filter, but after Ensemble Kalman Filter is there are derivation pattern (DM) state
When testing average and variance, it is assumed that the prior distribution of mode state is in Gaussian, and this hypothesis is correct for linear model
, but all there are strong nonlinearities for most Atmosphere and Ocean numerical models in practice.Nonlinear Monte Carlo in recent years
Particle filter method is increasingly becoming the hot spot that data assimilation is studied in nonlinear model, such as number of patent application with being modeled as characteristic
201110090879.X conventional data assimilation method of the patent application based on Bayesian filter.The patent proposes a kind of combinations
The conventional data assimilation method of Ensemble Kalman Filter and particle filter.Since Kalman filtering uses just under non-Gaussian environment
There are certain deviation, non-linear in actual life and non-gaussian stochastic model has more common of meaning.Particle
Filtering method has the characteristics of its is exclusive, it can intactly solve the problems, such as that nonlinear data is assimilated first, secondly, particle filter
Using importance sampling method, it ensure that particle state is constant, so that pattern dynamic equilibrium is not destroyed.
The major advantage of particle filter data assimilation is sampled according to particle weights, is ensured that particle state is constant, is seen
Measurement information only changes the weight of particle in itself, so that pattern dynamic equilibrium is not destroyed.But particle filter data assimilation method
There is also shortcoming, although particle is according to the observation information constantly regulate weight of itself, single particle meeting in particle filter
Higher weight is obtained, other particles obtain low-down weight, so as to sample degeneracy phenomenon occur.For grain in particle filter
The problem of sub- degenerate problem, usually used method for resampling, single use method for resampling can cause sample degeneracy.
In current particle filtering research field for sample degeneracy problem, Britain's professor's VanLeeuwen detailed overviews height
Particle filter algorithm in dimensional pattern Data Assimilation, it is further proposed that weight particle filter algorithm, the main thought of this method
It is to realize that particle adjusts using proposal density, makes set particle close to observation information, improve effective particle in sampling process
Number makes each particle contribute maximum posterior probability density, so as to avoid the appearance of sample degeneracy and particle depletion issues.
This method reasonably adjusts particle state, i.e., it is hundreds of to thousands of that a small amount of particle can just meet traditional Resampling Particle Filter needs
The particle filter assimilation effect that particle reaches.In particle filter data assimilation, observation information plays the quality for assimilating result
Leading role, design is a kind of based on equal weight particle filter, efficiently using observation information based on the equal weight particle of observation window
Filtering data assimilation method has important application value.
The present invention proposes a kind of based on the equal weight particle filter data assimilation of observation window.With traditional particle filter
Data assimilation is compared, significant feature of the invention is that：It is not calculated in assimilation time according to observation singly and proposes density, adjustment
Particle position state.But the available observation information of acquisition is stored in observation window before assimilation time, according to storage
Observation information in the window calculates before assimilation time and proposes density, adjusts particle position close to observation information.Assimilating
Moment can effectively improve the number of effective particles of sampling according to the particle state after proposal density adjustment, then calculate sampling grain
Sub- weight selects optimal particle state to complete data assimilation according to weight.Method proposed by the present invention can overcome conventional particle
Sample degeneracy and particle depletion issues in filtering method, and equal weight particle filter method is improved in single assimilation time tune
The problem of whole state, particle filter assimilation result fluctuation is excessive.
The content of the invention
The content of the invention of the present invention is as follows：
A kind of data assimilation method based on the equal weight particle filter of observation window, which is characterized in that specifically include following
Step：
(1) obtaining mode integration initial background field；
(2) calculated according to observation information in observation window and propose density；
(3) in given assimilation time, particle weights is calculated using equal weight method, adjust particle state；
(4) using method for resampling, adjustment set particle maintains population to stablize；
(5) the state posterior estimate after equal weight particle filter is calculated.
The step (1) specifically includes：
Model equation is accumulated in input pattern equation initial fields, pattern variable is made to reach chaos state, is up to chaos shape
In this manner based on ambient field, particle filter set is obtained by Mode Equation as initial background field for the pattern variable of state
The initial value of particle.
The step (2) specifically includes：
(2.1) particle probabilities density is calculated according to observation information：
The probability density of use pattern equation calculation set particle is from the particle state at n-1 momentIt is integrated, is gathered
The probability density expression formula of particle is：
Represent the probability density from n-1 moment particle state to the n moment,Represent i-th of particle of n moment
State,It represents by n-1 moment Mode Equation integral results, Q intermediate scheme error covariance expression formulas are：
M represents set total number of particles,Represent set particle average；
(2.2) particle is asked for by probability density and proposes density：
Based on acquiring the proposal density in equal weight particle by the prior density calculated in (2.1) in Mode Equation,
It is asked for proposing that density expression formula is by prior density：
In formulaIt represents to observe proposal density of the y as target, KnRepresent its expression of relaxation stress matrix
Formula is Kn=QHT(HQHT+R)-1, KnIt is with mode error covariance Q and observation error covariance R expression formulas：
Represent the state of i-th of particle of n moment, H represents that the general value of Observation Operators represents observation information for 1, y；
Based in the equal weight particle filter of observation window, it is proposed that the calculating of density is not only confined to the observation information of assimilation time, carries
View density can use the observation information being stored in before assimilation time in observation window；
(2.3) optimal proposal density is selected, calculates particle weights：
Assuming thatThe weight of each particle is expressed as in set：
Wherein ωiRepresent the weight of i-th of particle in set,Represent the state of i-th of particle of n moment, ynWhen representing n
The observation information at quarter,Represent probability density,It represents to propose density,Represent the n moment
Particle state is for the probability density of n moment particles；According to the weight that particle in set is calculated.
The step (3) specifically includes：
(3.1) on the basis of agmiaated granules of density adjustment is proposed, the weight of particle is calculated：
It is calculated according to observation information in assimilation window and proposes particle position in density adjustment set, particle is made in assimilation time
Closer to observation information；Propose that density makes in set each particle close to observation information so that its obtain it is intimate identical
Weight；Each the weight expression formula of particle is：
WhereinWhat is represented is the priori weight of i-th of particle, and priori weight proposes each grain during density calculating
Sub- weight；Y represents observation vector；H represents observation projection operator, projection operator H=1 in simple mode；X represent state to
Amount；Subscript T representing matrix transposition；Q intermediate scheme error co-variance matrix；R represents observation error covariance matrix；Assuming that it obtains
Particle target weight C in seti, ensure CiMeet following expression：
For n moment statesMost particles keeps identical weight in set, and equal weight is not reaching to for some
Particle be adjusted by method for resampling；
(3.2) particle state is adjusted according to weight：
After set particle weights are obtained, particle state is adjusted according to weight, then the state at n moment can be expressed as：
Y represents observation vector in formula；H represents observation projection operator (H=1)；X represents state vector；K=QHT(HQHT
+R)-1, Q is mode error covariance, and R is observation error covariance, and particle weights approximation phase is gathered in equal weight particle filter
Whens waiting, for vector αiIt can be expressed as：
WhereinWithIn the two expression formulasC
Represent the target weight of selection,Represent the relative weighting in current time particle；In order to ensure particle state in set
Stochastic effects add in random entry, obtain final analysis equation, whereinWhat is represented is random error：
The step (4) specifically includes：
After equal weight particle filter is adjusted particle state, carried out most using method for resampling according to particle weights
Status adjustment afterwards maintains population to stablize.
The step (5) specifically includes：
According to the state of the set particle after resampling, the Posterior estimator set average of state is calculated：
Next step prediction and assimilation, weight are carried out using updated Posterior estimator set average as the initial value of analysis model
Multiple above-mentioned steps, in assimilation time, obtain final analysis field, data of the analysis field as reflection current ambient conditions
.
The advantage of the invention is that：
(1) it is improved for traditional particle filter method, is effectively improved the problem of sample degeneracy is with particle dilution, changes
Be apt to equal weight particle filter only makes assimilation result fluctuate the problem of excessive in assimilation time adjustment.
(2) before assimilation time, more reasonable distribution observation information, by proposing that density leans on particle to observation information
Closely, assimilation quality can be promoted, this method at all observation moment compared with carrying out assimilation, the calculating pressure of particle state adjustment
There is apparent reduction.
Description of the drawings
Fig. 1 is Atmosphere and Ocean data assimilation flow chart；
Fig. 2 is based on the improvement equal weight particle filter data assimilation flow chart of observation information.
Specific embodiment
It is illustrated with reference to specific embodiment.
The present invention provides one kind based on the equal weight particle filter data assimilation of observation window, specifically includes following
Step：
Step 1：Obtaining mode integration initial background field
The initial value of numerical model integration is selected, first integral mode equation makes Mode Equation reach chaos state and avoids out
The fluctuation of existing Mode Equation.Assimilation experiment model initial background field will be used as by mode state at this time, using ambient field as integration starting point,
Use pattern state equation is integrated.
Step 2：It is calculated according to observation information in observation window and proposes density
Observation information is stored according to window and set particle state solves and proposes density, and selects corresponding optimal proposal close
Degree further determines that the posterior probability density of set particle.
Step 3：In given assimilation time, particle weights are calculated using equal weight method, adjust particle state
According to the state after observation information and adjustment, by the weight of particle in weight calculation formula set of computations, ensure
Most particles can reach preferably weight in particle assembly, and particle state is adjusted according to weight.
Step 4：Using method for resampling, adjustment set particle maintains population to stablize
Using method for resampling, poor particle is showed to weight and is adjusted, for being unsatisfactory for the particle of weight requirement
It is rejected, replicates and show preferable particle, maintain the stabilization of set population.
Step 5：Calculate the state posterior estimate after equal weight particle filter
Particle after resampling is reintroduced back to Mode Equation, the renewal model integral process in assimilation process is formed most
Whole assimilation result.
In order to which simpler specific description is based on the equal weight particle filter assimilation method specific implementation step of observation window, with
Simple illustration is carried out exemplified by simple Lorenz-63 patterns.
(1) obtaining mode integration initial background field
Initial fields (0,1,0) input model equation is first integrated into 1,000,000 steps in Lorenz-63 patterns, makes pattern variable
Reach chaos state, pattern is avoided to introduce mode wave dynamic deviation in integral process is assimilated, the pattern for being up to chaos state becomes
In this manner based on ambient field, the initial of particle filter set particle is obtained by Mode Equation as initial background field for amount
Value.The reliability for the set particle state that Mode Equation obtains can be effectively improved.
(2) calculated according to observation information in observation window and propose density
In particle filter data assimilation, adjust and gather particle and the location status of observation information in assimilation process, adjustment
Particle in set is preferably close to observation information, improves number of effective particles in sampling process, and it is same to improve particle filter data
Sample degeneracy and the problem of particle dilution in change, precalculates and proposes density adjustment particle state.Circular is as follows：
(2.1) particle probabilities density is calculated according to observation information
For proposing the probability density for asking for needing first set of computations particle of density, using Lorenz-63 Mode Equations,
From the particle state at n-1 momentIt is integrated, the probability density expression formula for gathering particle is：
Represent the probability density from n-1 moment particle state to the n moment,Represent i-th of particle of n moment
State,It represents by n-1 moment Lorenz-63 Mode Equation integral results, the expression of Q intermediate schemes error covariance
Formula is：
M represents set total number of particles,Represent set particle average.The probability density of particle is calculated by the formula.
(2.2) particle is asked for by probability density and proposes density
It can further be acquired based on the prior density that previous step is calculated in 63 patterns of Lorenz of citing and weighed
Proposal density in heavy particle is asked for proposing that density expression formula is by prior density：
In formulaIt represents to observe proposal density of the y as target, KnRepresent its expression of relaxation stress matrix
Formula is Kn=QHT(HQHT+R)-1, KnIt is with mode error covariance Q and observation error covariance R expression formulas：
Represent the state of i-th of particle of n moment, H represents that the general value of Observation Operators represents observation information for 1, y.
Based in the equal weight particle filter of observation window, it is proposed that the calculating of density is not only confined to the observation information of assimilation time, carries
View density can use the observation information being stored in before assimilation time in observation window, it is therefore an objective to be adjusted before assimilation time
Set particle is located proximate to observation information.
(2.3) optimal proposal density is selected, calculates particle weights
The selection for proposing density is considered as to control particle and observation information position and the major criterion of granular Weights Computing,
The suitable optimal proposal density of selection can ensure the effectively quantity of particle, while ensure the reliability of particle weights in sampling,
And a most important part in weight particle filter, it finds optimal proposal density and assumesCollection
The weight of each particle is expressed as in conjunction：
Wherein ωiRepresent the weight of i-th of particle in set,Represent the state of i-th of particle of n moment, ynWhen representing n
The observation information at quarter,Represent probability density,It represents to propose density,Represent the n moment
Particle state is for the probability density of n moment particles.According to the weight that particle in set is calculated.
(3) in given assimilation time, particle weights is calculated using equal weight method, adjust particle state
(3.1) on the basis of agmiaated granules of density adjustment is proposed, the weight of particle is calculated
It is calculated according to observation information in assimilation window and proposes particle position in density adjustment set, make particle in assimilation time
Closer to observation information, ensure that most particle obtains equal power in the posterior probability density function of assimilation time
Weight.Propose that density makes in set each particle close to observation information, so that it obtains intimate identical weight.Each particle
Weight expression formula is：
WhereinWhat is represented is the priori weight of i-th of particle, and priori weight proposes each grain during density calculating
Sub- weight；Y represents observation vector；H represents observation projection operator, projection operator H=1 in simple mode；X represent state to
Amount；Subscript T representing matrix transposition；Q intermediate scheme error co-variance matrix；R represents observation error covariance matrix.Assuming that it obtains
Particle target weight C in seti, the particle to ensure in particle assembly 80% can reach the weight being calculated, avoid
The generation of sample degeneracy and depletion issues：
So as to find for n moment statesMost particles can keep identical weight in set, for
Some particles for being not reaching to equal weight can be adjusted by method for resampling.
(3.2) particle state is adjusted according to weight
After set particle weights are obtained, particle state is adjusted according to weight, then the state at n moment can be expressed as：
Y represents observation vector in formula；H represents observation projection operator (H=1)；X represents state vector；K=QHT(HQHT+
R)-1, Q is mode error covariance, and R is observation error covariance, and particle weights approximation phase is gathered in equal weight particle filter
Whens waiting, for vector αiIt can be expressed as：
InWithIn the two expression formulas
C represents the target weight of selection,Represent the relative weighting in current time particle.In order to ensure particle state in set
Stochastic effects add in random entry, obtain final analysis equation, whereinWhat is represented is random error：
(4) using method for resampling, adjustment set particle maintains population to stablize
The basic thought of resampling methods is the particle for removing small weight, replicates authority heavy particle.It is weighing in order to prevent
During weight, the particle for having only a few is not reaching to the requirement of equal equal weight, and particle state is carried out in equal weight particle filter
After adjustment, last status adjustment is carried out using method for resampling according to particle weights.
(5) the state posterior estimate after equal weight particle filter is calculated
According to the state of the set particle after resampling, the Posterior estimator set average of state is calculated：
Next step prediction and assimilation are carried out using updated Posterior estimator set average as the initial value of analysis model, repeats above-mentioned step
Suddenly, in assimilation time, final analysis field is obtained, data fields of the analysis field as reflection current ambient conditions.
Claims (6)
1. a kind of data assimilation method based on the equal weight particle filter of observation window, which is characterized in that specifically include following step
Suddenly：
(1) obtaining mode integration initial background field；
(2) calculated according to observation information in observation window and propose density；
(3) in given assimilation time, particle weights is calculated using equal weight method, adjust particle state；
(4) using method for resampling, adjustment set particle maintains population to stablize；
(5) the state posterior estimate after equal weight particle filter is calculated.
A kind of 2. data assimilation method based on the equal weight particle filter of observation window, which is characterized in that step (1) tool
Body includes：
Model equation is accumulated in input pattern equation initial fields, pattern variable is made to reach chaos state, is up to chaos state
In this manner based on ambient field, particle filter set particle is obtained by Mode Equation as initial background field for pattern variable
Initial value.
A kind of 3. data assimilation method based on the equal weight particle filter of observation window, which is characterized in that step (2) tool
Body includes：
(2.1) particle probabilities density is calculated according to observation information：
The probability density of use pattern equation calculation set particle is from the particle state at n-1 momentIt is integrated, gathers particle
Probability density expression formula be：
Represent the probability density from n-1 moment particle state to the n moment,Represent the shape of i-th of particle of n moment
State,It represents by n-1 moment Mode Equation integral results, Q intermediate scheme error covariance expression formulas are：
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Based on the proposal density in equal weight particle is acquired by the prior density calculated in (2.1) in Mode Equation, by elder generation
Density is tested to ask for proposing that density expression formula is：
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Represent the state of i-th of particle of n moment, H represents that the general value of Observation Operators represents observation information for 1, y；Based on
In the equal weight particle filter of observation window, it is proposed that the calculating of density is not only confined to the observation information of assimilation time, it is proposed that close
Degree can use the observation information being stored in before assimilation time in observation window；
(2.3) optimal proposal density is selected, calculates particle weights：
Assuming thatThe weight of each particle is expressed as in set：
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<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>|</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>|</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>|</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>,</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>|</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>;</mo>
</mrow>
Wherein ωiRepresent the weight of i-th of particle in set,Represent the state of i-th of particle of n moment, ynRepresent the n moment
Observation information,Represent probability density,It represents to propose density,Represent n moment particles
State is for the probability density of n moment particles；According to the weight that particle in set is calculated.
A kind of 4. data assimilation method based on the equal weight particle filter of observation window, which is characterized in that step (3) tool
Body includes：
(3.1) on the basis of agmiaated granules of density adjustment is proposed, the weight of particle is calculated：
It is calculated according to observation information in assimilation window and proposes particle position in density adjustment set, particle is made more in assimilation time
Close to observation information；Propose that density makes in set each particle close to observation information, so that it obtains intimate identical weight；
Each the weight expression formula of particle is：
<mrow>
<msub>
<mi>&omega;</mi>
<mi>i</mi>
</msub>
<mo>&Proportional;</mo>
<msubsup>
<mi>&omega;</mi>
<mi>i</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>s</mi>
<mi>t</mi>
</mrow>
</msubsup>
<mi>exp</mi>
<mo>{</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>-</mo>
<mi>f</mi>
<mo>(</mo>
<msup>
<mi>x</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<msup>
<mi>Q</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>-</mo>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<msup>
<mi>R</mi>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>}</mo>
<mo>;</mo>
</mrow>
WhereinWhat is represented is the priori weight of i-th of particle, and priori weight proposes that density calculates each particle power in the process
Weight；Y represents observation vector；H represents observation projection operator, projection operator H=1 in simple mode；X represents state vector；On
Mark T representing matrix transposition；Q intermediate scheme error co-variance matrix；R represents observation error covariance matrix；Assuming that gathered
Middle particle target weight Ci, ensure CiMeet following expression：
<mrow>
<msub>
<mi>C</mi>
<mi>i</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<msubsup>
<mi>log&omega;</mi>
<mi>i</mi>
<mrow>
<mi>r</mi>
<mi>e</mi>
<mi>s</mi>
<mi>t</mi>
</mrow>
</msubsup>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mi>T</mi>
</msup>
<msup>
<mrow>
<mo>(</mo>
<msup>
<mi>HQH</mi>
<mi>T</mi>
</msup>
<mo>+</mo>
<mi>R</mi>
<mo>)</mo>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msup>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
For n moment statesMost particles keeps identical weight in set, and the grain of equal weight is not reaching to for some
Son is adjusted by method for resampling；
(3.2) particle state is adjusted according to weight：
After set particle weights are obtained, particle state is adjusted according to weight, then the state at n moment can be expressed as：
<mrow>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<mi>K</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
Y represents observation vector in formula；H represents observation projection operator (H=1)；X represents state vector；K=QHT(HQHT+R)-1,
Q is mode error covariance, and R is observation error covariance, when gathering particle weights approximately equal in equal weight particle filter,
For vector αiIt can be expressed as：
<mrow>
<mi>&alpha;</mi>
<mo>=</mo>
<mn>1</mn>
<mo>-</mo>
<msqrt>
<mrow>
<mn>1</mn>
<mo>-</mo>
<msub>
<mi>b</mi>
<mi>i</mi>
</msub>
<mo>/</mo>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
</msqrt>
<mo>;</mo>
</mrow>
WhereinWithIn the two expression formulasC
Represent the target weight of selection,Represent the relative weighting in current time particle；In order to ensure set in particle state with
Machine effect adds in random entry, obtains final analysis equation, whereinWhat is represented is random error：
<mrow>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msub>
<mi>&alpha;</mi>
<mi>i</mi>
</msub>
<mi>K</mi>
<mrow>
<mo>(</mo>
<msup>
<mi>y</mi>
<mi>n</mi>
</msup>
<mo>-</mo>
<mi>H</mi>
<mi>f</mi>
<mo>(</mo>
<msubsup>
<mi>x</mi>
<mi>i</mi>
<mrow>
<mi>n</mi>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>)</mo>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>&beta;</mi>
<mi>i</mi>
<mi>n</mi>
</msubsup>
<mo>.</mo>
</mrow>
A kind of 5. data assimilation method based on the equal weight particle filter of observation window, which is characterized in that step (4) tool
Body includes：
It is last using method for resampling progress according to particle weights after equal weight particle filter is adjusted particle state
Status adjustment maintains population to stablize.
A kind of 6. data assimilation method based on the equal weight particle filter of observation window, which is characterized in that step (5) tool
Body includes：
According to the state of the set particle after resampling, the Posterior estimator set average of state is calculated：
<mrow>
<mover>
<mi>x</mi>
<mo>&OverBar;</mo>
</mover>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>M</mi>
</mfrac>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>i</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>M</mi>
</munderover>
<msub>
<mi>x</mi>
<mi>i</mi>
</msub>
<mo>;</mo>
</mrow>
Next step prediction and assimilation are carried out using updated Posterior estimator set average as the initial value of analysis model, in repetition
Step is stated, in assimilation time, obtains final analysis field, data fields of the analysis field as reflection current ambient conditions.
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