CN107679614A - A kind of interval transit time real time extracting method based on particle group optimizing - Google Patents
A kind of interval transit time real time extracting method based on particle group optimizing Download PDFInfo
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Abstract
The invention provides a kind of interval transit time real time extracting method based on particle group optimizing, mainly solve the problems, such as that traditional STC algorithms ask for that interval transit time operand is big and precision is low.Its implementation process is：（1）Give the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time；（2）Initialize population；（3）Calculate the fitness value of particle；（4）The historical high fitness value of more new particle and the global history highest fitness value of population；（5）The position of more new particle；（6）Whether evaluation algorithm meets end condition, if satisfied, then stopping iteration, otherwise goes to step（3）.The present invention carries out interval transit time extraction using the particle swarm optimization algorithm based on population random search, the traversal search of time and slowness need not be carried out, and the algorithm is global optimization method, the operand of program is substantially reduced, can rapidly and accurately extract stratum interval transit time.
Description
Technical field
The present invention relates to oil drilling and well logging field, more specifically, is related to one kind and is adopted with sound wave underground signal is bored
Collection and the time difference real time extracting method of processing system.
Background technology
Acoustic logging while drilling technology grows up on the basis of wire logging techniques, compared with wireline logging, it
Data can effectively be gathered before F invades stratum, it is smaller by intrusion effect, can be more objective
Prime stratum situation is detected, formation information has higher researching value.The application of logging while drilling technology, drilling well and well logging
Two processes are combined into one, and well logging is completed during drilling well, improve drillng operation efficiency, greatly reduce well logging into
This.In addition, in some highly difficult logging operations, such as in the test of horizontal well, high angle hole, wireline logging can not be completed
Well logging, well logging can only be selected.
The amplitude of the fluid wave on stratum, compressional wave and shear wave, frequency and the time difference can be obtained using acoustic logging while drilling technology
Etc. relevant information, these parameters are to evaluate the important evidence of formation porosity, lithology and rock mechanics elasticity, by necessarily calculating
Formation rock can also be obtained and break pressure value, accurately react the hydrocarbon information of current formation, be the effective ways of hydrocarbon exploration.
Compared with other logging techniquies, the acoustic logging instrument developed using acoustic logging has that logging speed is fast, instruments weight
The advantages that light.But the data volume of acoustic logging while drilling technology collection is bigger, is transmitted using mud-pulse, data transfer effect
Rate is extremely low, it is impossible to the mass data collected is real-time transmitted to ground system, i.e., can not upload the sound wave measured in real time
Data, surface personnel can not determine whether instrument is operated in normal condition, so as to extract well from sonic data
The lower time difference will react bottom-up information.
Acoustic logging while drilling firsthand information can not be applied directly.Before application, it is necessary to firsthand information (Wave data)
Handled accordingly, extract drilling personnel, geological personnel, oil field development personnel and correlation engineering personnel etc. it is interested and
The information for easily being received and being applied.The original waveform data that conventional cable acoustic logging measures passes through cable transmission to ground
Handled accordingly again afterwards.It is different with the processing of normal cable acoustic logging original waveform, due to by well logging number
According to the limitation of transmission rate, directly original waveform data can not be transferred to after ground and be handled again, it is necessary in underground pair
Original waveform data is handled, then result is real-time transmitted into ground.
Slowness is extracted from sound wave, is link most basic in acoustic logging while drilling, it is real due to the complexity of subsurface picture
The quality of border well-log information is difficult to ensure that so that waveform processing work becomes considerably complicated and difficult.
It is currently known to be divided into three classes with brill interval transit time extractive technique, Mintrop wave probe method, STC methods, artificial intelligence
Can algorithm.
Mintrop wave probe method, method is simple, and it is convenient to realize, many loggers can be realized with hardware.But this
Kind method anti-noise ability is poor, for noise and sound wave waveform mixed in together, it is difficult to find head arrival accurately with it.
STC methods are a kind of a kind of methods that extraction acoustic slowness is handled using correlation that Kimbal et al. proposed in 1984, should
Method anti-noise ability is strong, and computational accuracy is high.But its amount of calculation is huge, technological difficulties are more, implement more difficult.Adopt at present
Intelligent algorithm includes simulated annealing method, genetic algorithm etc., due to the limitation of these algorithms in itself, result in very
Local minimum is easily trapped into, so that algorithm is possible to not restrain, the interval transit time so extracted is with regard to incorrect, this method
Although amount of calculation is smaller, computational accuracy is not high.
Under the pressure of the limitation with transmission conditions under the conditions of brill, the extraction of acoustic logging while drilling slowness can only be real by hardware in underground
Existing, above-mentioned Mintrop wave probe method precision is too low to be unsuitable for the time difference receiver under the conditions of brill.Although STC method anti-noise abilities are strong, meter
It is high to calculate precision, but amount of calculation is huge, technological difficulties are more, it is difficult to be realized in underground.According to the characteristics of acoustic logging while drilling, propose
A kind of interval transit time extracting method based on particle group optimizing.This method takes full advantage of the characteristics of particle swarm optimization algorithm,
Fast convergence rate and globe optimum can be found, can be used for Borehole Acoustic Waves compressional wave and the time difference receiver of quadrupole wavelet.
The content of the invention
It is an object of the invention to overcome the deficiencies of the prior art and provide a kind of interval transit time based on particle group optimizing is real
When extracting method, by the Related Computational Methods in the particle swarm optimization algorithm combination STC algorithms in intelligent algorithm, Ke Yishi
Global optimizing in present interval transit time extraction process, there is the characteristics of interval transit time extraction in stratum is fast, reliability is high.
For achieving the above object, the present invention proposes a kind of interval transit time extract real-time side based on particle group optimizing
Method, comprise the following steps：
(1) the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time, are given；
(2) population, is initialized；
(3) fitness value of particle, is calculated；
(4), the historical high fitness value of more new particle and the global history highest fitness value of population；
(5) particle rapidity and position, are updated；
(6), whether evaluation algorithm meets end condition, if satisfied, then stopping iteration, otherwise goes to step (3).
In step (1) as described above give a certain mode wave Mintrop wave arrival time and interval transit time scope use with
Lower step is realized：Geologic modeling is carried out according to regional earthquake data and offset well well-log information, log data first, inverting is to be drilled
The formation information of well, then the spread speed according to stratum data-evaluation sound wave in the stratum, finally launches with reference to receiver
Source between device away from and transmitter between the Mintrop wave arrival time of range estimation mode wave and the scope of interval transit time.
Initialization population includes herein below in step (2) as described above：
1) number of particles m in population, is initialized, then the molecular population of m grain is X={ X1,X2,…Xm}T；
2) iterations t, is initialized；
3) particle X, is initializediPosition be Xi=(xi1,xi2,…,xid)T, wherein d is search space dimension；
4) particle X, is initializediSpeed be Vi=(vi1,vi2,…,vid)T；
5) inertial factor w, is initialized；
6) Studying factors C, is initialized1,C2；
7) every one-dimensional speed V of particle, is initializeddScope be [- Vdmax,+Vdmax]。
The fitness value that particle is calculated in step (3) as described above is by carrying out correlation computations determination to data waveform
, conventional Related Computational Methods have：
1) correlation computations, are carried out to data waveform by waveform analogue method；
The similarity of multi-channel waveform is defined as：
Wherein, fkFor array Wave data, IW is length of window, and M is undulating path number.The scope of similar value is (complete from 0
Full negative correlation) to 1 (complete positive correlation)；Similar value between the incoherent noise data in M roads is it can easily be proven that be 1/M；
According to this computational methods, often a similarity factor is calculated to a window can determined.Therefore, pass through
Adjustment time parameter t and slowness parameter s is obtained with t with changing the location and shape of window and calculating its corresponding similarity
With a series of correlation function values that s is parameter；
2) correlation computations, are carried out to data waveform by cross-correlation method；
If xN(n)、yN(n) be that two row length are the N time serieses that contain similar waveform composition, then the correlation between them
Coefficient may be defined as：
Wherein, r2N(n) similarity factor of waveform in window when original position differs two of n point is represented, N represents that window is grown up
It is small, xNAnd y (i)N(i) respectively represent two time serieses when window in i-th of sample value；
3) correlation computations, are carried out to data waveform by n times root storehouse method；
N times root storehouse equation is：
Yi=Ri|Ri|N-1
Wherein, xi,jFor the ith sample point data of jth passage, 1≤i≤IW, 1≤j≤K, IW are each channel signal
Length (window length), K are overall channel number, and N is any positive integer (typically taking N >=4), GjIt is the gain of jth passage, G is to all
The gain of channel data, wjFor weighted factor, YiArray is exported for one-dimensional filtering, multi channel signals can export after the filtering of n times root
Similar part in multichannel signal.
The global history highest of the historical high fitness value of more new particle and population adapts in step (4) as described above
Angle value specifically refers to the adaptive value of the adaptive value and its experience desired positions of each particle of comparison, if more preferably, more new individual pole
Value point；Compare the adaptive value of each particle and the global adaptive value for undergoing desired positions, if more preferably, updating global extremum
Point.
Renewal particle rapidity and position equation are in step (5) as described above：
In formula,It is the speed that d is tieed up in particle ith iteration；rand1,2It is the random number between [0,1]；It is grain
The sub- i current locations that d is tieed up in kth time iteration；pbestidIt is particle i in the position of the d individual extreme points tieed up；gbest
It is whole group in the position of the d global extremum points tieed up.
A kind of interval transit time real time extracting method based on particle group optimizing of the present invention has an advantageous effect in that
Particle swarm optimization algorithm is based on population random search, is not influenceed by Mintrop wave arrival time step-length and slowness step sizes,
Interval transit time extraction is carried out using this method, it is only necessary to is done part calculating and is obtained with the given time difference, Mintrop wave arrival time
In the range of globally optimal solution.The traversal search of time and slowness need not be carried out, substantially reduces the operand of program, can
Rapidly and accurately extract stratum interval transit time.
Brief description of the drawings：
Fig. 1 is the interval transit time extract real-time flow chart based on particle group optimizing.
Fig. 2 is particle swarm optimization algorithm searching process.
Embodiment：
Below by drawings and examples, technical scheme is described in further detail.
Referring to the drawings 1, a kind of interval transit time real time extracting method based on particle group optimizing, comprise the following steps：
First step S101：Give the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time.
Giving the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time need to follow the steps below to realize：First
Geologic modeling need to be carried out according to regional earthquake data and offset well well-log information, log data, the formation information of drilling well is treated in inverting, so
Spread speed according to stratum data-evaluation sound wave in the stratum afterwards, finally with reference to the source between receiver transmitter away from and hair
The Mintrop wave arrival time of range estimation mode wave and the scope of interval transit time between emitter.
Second step S102：Initialize population.
Initialization population includes herein below：
1) number of particles m in population is initialized, then the molecular population of m grain is X={ X1,X2,…Xm}T；
2) iterations t is initialized；
3) particle X is initializediPosition be Xi=(xi1,xi2,…,xid)T, wherein d is search space dimension；
4) particle X is initializediSpeed be Vi=(vi1,vi2,…,vid)T；
5) inertial factor w is initialized；
6) Studying factors C is initialized1,C2；
7) every one-dimensional speed V of particle is initializeddScope be [- Vdmax,+Vdmax]。
3rd step S103：Calculate the fitness value of particle.
The fitness value for calculating particle is by carrying out correlation computations determination, conventional correlation computations side to data waveform
Method has：
1) correlation computations are carried out to data waveform by waveform analogue method；
The similarity of multi-channel waveform is defined as：
Wherein, fkFor array Wave data, IW is length of window, and M is undulating path number.The scope of similar value is (complete from 0
Full negative correlation) to 1 (complete positive correlation)；Similar value between the incoherent noise data in M roads is it can easily be proven that be 1/M；
According to this computational methods, often a similarity factor is calculated to a window can determined.Therefore, pass through
Adjustment time parameter t and slowness parameter s is obtained with t with changing the location and shape of window and calculating its corresponding similarity
With a series of correlation function values that s is parameter；
2) correlation computations are carried out to data waveform by cross-correlation method；
If xN(n)、yN(n) be that two row length are the N time serieses that contain similar waveform composition, then the correlation between them
Coefficient may be defined as：
Wherein, r2N(n) similarity factor of waveform in window when original position differs two of n point is represented, N represents that window is grown up
It is small, xNAnd y (i)N(i) respectively represent two time serieses when window in i-th of sample value；
3) correlation computations are carried out to data waveform by n times root storehouse method；
N times root storehouse equation is：
Yi=Ri|Ri|N-1
Wherein, xi,jFor the ith sample point data of jth passage, 1≤i≤IW, 1≤j≤K, IW are each channel signal
Length (window length), K are overall channel number, and N is any positive integer (typically taking N >=4), GjIt is the gain of jth passage, G is to all
The gain of channel data, wjFor weighted factor, YiArray is exported for one-dimensional filtering, multi channel signals can export after the filtering of n times root
Similar part in multichannel signal.
4th step S104：The historical high fitness value of more new particle and the global history highest fitness value of population.
It is every that the historical high fitness value of more new particle and the global history highest fitness value of population specifically refer to comparison
The adaptive value of individual particle undergoes the adaptive value of desired positions with it, if more preferably, more new individual extreme point；Compare each particle
Adaptive value and the global adaptive value for undergoing desired positions, if more preferably, updating global extremum point.
5th step S105：Update particle rapidity and position.
Renewal particle rapidity and position equation are：
In formula,It is the speed that d is tieed up in particle ith iteration；rand1,2It is the random number between [0,1]；It is grain
The sub- i current locations that d is tieed up in kth time iteration；pbestidIt is particle i in the position of the d individual extreme points tieed up；gbest
It is whole group in the position of the d global extremum points tieed up.
6th step S106：Whether evaluation algorithm meets end condition, if satisfied, then stopping iteration, otherwise goes to step
S103。
Finally it should be noted that：Various embodiments above is merely illustrative of the technical solution of the present invention, rather than its limitations；To the greatest extent
The present invention is described in detail with reference to foregoing embodiments for pipe, it will be understood by those within the art that：Its according to
The technical scheme described in foregoing embodiments can so be modified, either which part or all technical characteristic are entered
Row equivalent substitution；And these modifications or replacement, the essence of appropriate technical solution is departed from various embodiments of the present invention technology
The scope of scheme.
Claims (6)
- A kind of 1. interval transit time real time extracting method based on particle group optimizing, it is characterised in that：Comprise the following steps：(1) the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time are given；(2) population is initialized；(3) fitness value of particle is calculated；(4) the historical high fitness value of more new particle and the global history highest fitness value of population；(5) particle rapidity and position are updated；(6) whether evaluation algorithm meets end condition, if satisfied, then stopping iteration, otherwise goes to step (3).
- 2. the interval transit time real time extracting method according to claim 1 based on particle group optimizing, it is characterised in that：It is described The step of (1) in give the Mintrop wave arrival time of a certain mode wave and the scope of interval transit time is realized using following steps：It is first Geologic modeling is first carried out according to regional earthquake data and offset well well-log information, log data, the formation information of drilling well is treated in inverting, so Spread speed according to stratum data-evaluation sound wave in the stratum afterwards, finally with reference to the source between receiver transmitter away from and hair The Mintrop wave arrival time of range estimation mode wave and the scope of interval transit time between emitter.
- 3. the interval transit time real time extracting method according to claim 2 based on particle group optimizing, it is characterised in that：It is described The step of (2) in initialization population include herein below：(1) number of particles m in population is initialized, then the molecular population of m grain is X={ X1,X2,…Xm}T；(2) iterations t is initialized；(3) particle X is initializediPosition be Xi=(xi1,xi2,…,xid)T, wherein d is search space dimension；(4) particle X is initializediSpeed be Vi=(vi1,vi2,…,vid)T；(5) inertial factor w is initialized；(6) Studying factors C is initialized1,C2；(7) every one-dimensional speed V of particle is initializeddScope be [- Vdmax,+Vdmax]。
- 4. the interval transit time real time extracting method according to claim 3 based on particle group optimizing, it is characterised in that：It is described The step of (3) in calculate particle fitness value be by data waveform carry out correlation computations determination, Related Computational Methods Have：(1) correlation computations are carried out to data waveform by waveform analogue method；The similarity of multi-channel waveform is defined as：<mrow> <mi>s</mi> <mrow> <mo>(</mo> <mi>m</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>I</mi> <mi>W</mi> </mrow> </munderover> <msup> <mrow> <mo>&lsqb;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msub> <mi>f</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mn>2</mn> </msup> <mo>/</mo> <mi>M</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>I</mi> <mi>W</mi> </mrow> </munderover> <munderover> <mo>&Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>M</mi> </munderover> <msubsup> <mi>f</mi> <mi>k</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>Wherein, fkFor array Wave data, IW is length of window, and M is undulating path number, and the scope of similar value is from complete negative Correlation 0 arrives complete positive correlation 1；Similar value between the incoherent noise data in M roads is it can easily be proven that be 1/M；According to this computational methods, often the window calculation to a determination goes out a similarity factor, by adjustment time parameter t and Slowness parameter s obtains a system using t and s as parameter to change the location and shape of window and calculate its corresponding similarity Row correlation function value；(2) correlation computations are carried out to data waveform by cross-correlation method；If xN(n)、yN(n) be that two row length are the N time serieses that contain similar waveform composition, then the coefficient correlation between them It may be defined as：<mrow> <msub> <mi>r</mi> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>x</mi> <mi>N</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <msubsup> <mi>y</mi> <mi>N</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <mo>&lsqb;</mo> <mrow> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>x</mi> <mi>N</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>&CenterDot;</mo> <munderover> <mi>&Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>y</mi> <mi>N</mi> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> <mo>&rsqb;</mo> </mrow> </msqrt> </mfrac> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>...</mo> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>Wherein, r2N(n) similarity factor of waveform in window when original position differs two of n point is represented, N represents the long size of window, xNAnd y (i)N(i) respectively represent two time serieses when window in i-th of sample value；(3) correlation computations are carried out to data waveform by n times root storehouse method；N times root storehouse equation is：Yi=Ri|Ri|N-1<mrow> <msub> <mi>R</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <msub> <mi>w</mi> <mi>j</mi> </msub> </mrow> </mfrac> <munderover> <mo>&Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>K</mi> </munderover> <mfrac> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow> <mo>|</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>|</mo> </mrow> </mfrac> <mo>&CenterDot;</mo> <msup> <mrow> <mo>&lsqb;</mo> <mrow> <mfrac> <mrow> <msub> <mi>Gw</mi> <mi>j</mi> </msub> </mrow> <msub> <mi>G</mi> <mi>j</mi> </msub> </mfrac> <mo>|</mo> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>|</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mi>N</mi> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>Wherein, xi,jFor the ith sample point data of jth passage, 1≤i≤IW, 1≤j≤K, IW are the length of each channel signal (window length), K are overall channel number, and N is any positive integer, GjIt is the gain of jth passage, G is the gain to all channel datas, wj For weighted factor, YiArray is exported for one-dimensional filtering, multi channel signals can export similar in multichannel signal after the filtering of n times root Part.
- 5. the interval transit time real time extracting method according to claim 4 based on particle group optimizing, it is characterised in that：It is described The step of (4) in the historical high fitness value of more new particle and the global history highest fitness value of population specifically refer to compare The adaptive value of each particle undergoes the adaptive value of desired positions with it, if more preferably, more new individual extreme point；Compare each particle Adaptive value and the global adaptive value for undergoing desired positions, if more preferably, renewal global extremum point.
- 6. the interval transit time real time extracting method according to claim 5 based on particle group optimizing, it is characterised in that：It is described The step of (5) in renewal particle rapidity and position equation be：<mrow> <msubsup> <mi>V</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>wV</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msub> <mi>C</mi> <mn>1</mn> </msub> <msubsup> <mi>rand</mi> <mn>1</mn> <mi>k</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>pbest</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mn>2</mn> </msub> <msubsup> <mi>rand</mi> <mn>2</mn> <mi>k</mi> </msubsup> <mrow> <mo>(</mo> <msubsup> <mi>gbest</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>-</mo> <msubsup> <mi>x</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow><mrow> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <msubsup> <mi>X</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mi>k</mi> </msubsup> <mo>+</mo> <msubsup> <mi>V</mi> <mrow> <mi>i</mi> <mi>d</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>In formula,It is the speed that d is tieed up in particle ith iteration；rand1,2It is the random number between [0,1]；It is that particle i exists The current location that d is tieed up in kth time iteration；pbestidIt is particle i in the position of the d individual extreme points tieed up；Gbest is whole Group is in the position of the global extremum point of d dimensions.
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