CN104965102A - Method of effectively improving flow direction measurement precision of acoustic doppler current profiler - Google Patents

Method of effectively improving flow direction measurement precision of acoustic doppler current profiler Download PDF

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CN104965102A
CN104965102A CN201510270429.7A CN201510270429A CN104965102A CN 104965102 A CN104965102 A CN 104965102A CN 201510270429 A CN201510270429 A CN 201510270429A CN 104965102 A CN104965102 A CN 104965102A
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CN104965102B (en
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戴昊
商少平
贺志刚
刘轲
魏国妹
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Xiamen University
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Abstract

Provided is a method of effectively improving the flow direction measurement precision of an acoustic doppler current profiler (ADCP), relating to an ADCP. The method comprises: firstly, establishing a mathematic model, analyzing the relation between magnetic north angle change and final flow direction change, and obtaining that an angle variable quantity is equal to a flow direction variable quantity when only the magnetic north angle change exists; and based on the above, using a least square method to fit errors of a flow direction influenced by magnetism, and using a fitting coefficient to compensate for ADCP actual measurement flow direction output. Compared with a standard flow direction, the root-mean-square value of the errors of a flow direction influenced by magnetism can be reduced by 70%. The method can effectively compensate for the flow direction error of an ADCP internal magnetic compass caused by environmental magnetic field interference in an azimuth scope of 0-360 DEG, thereby substantially improve flow direction measurement precision in an omnibearing scope, and have engineering guidance value for long-term fixed point flow measurement in sea observation.

Description

A kind of method of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision
Technical field
The present invention relates to acoustic Doppler fluid velocity profile instrument (Acoustic Doppler Current Profiler, ADCP), belong to oceanographic observation Instrument technology field, especially relate to the method flowing to a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision of error correction for ADCP actual measurement.
Background technology
ADCP is a kind of instrument utilizing acoustic Doppler principle to measure flow rate of water flow and flow.Utilize the skew existed between the echo frequency of ADCP receive MUT and transmission frequency, calculate the speed of particle in water, the movement velocity of current, and then in known certain hour, flow through the discharge in cross section [1].ADCP has become current measuring instrument important in the water body such as ocean, river.
During actual use, there is multiple error source to affect the flow measurement precision of ADCP, make the measurement of flow velocity and exact value there is significantly deviation, have a strong impact on observed result and follow-up data process&analysis.The error affecting ADCP output is broadly divided into sound velocity error in frequency error measurement, water, alignment error and environmental factor error etc., ocean research worker has carried out large quantity research to error source, adopts diverse ways to carry out analyzing and compensating for different errors.
Arrowband ADCP from [2], to pulse coherence ADCP thereafter, then broadband A/D CP up till now [3], frequency error measurement constantly suppresses; The sonic velocity change caused by temperature, salinity, the degree of depth directly can cause measuring errors, and modern ADCP tends to use phased-array technique to maintain the constant of the velocity of sound more, eliminates this type of error [4]; The existence of ADCP alignment error can make the wave beam velocity vector recorded occur deviation when converting final flow rate to and flowing to [5]; Environmental factor source of error is in multiple aspect: near ADCP, flow disturbance, rolling moving platform, river bed/seabed Bed Load Movement etc. all can form measuring error [6,7,8,9].
In many Analysis of error source and solve scheme, seldom relate to and cause magnetic north angle output error about ADCP inner magnet compass by all kinds of influence of magnetic field in environment---the discussion of what magnetic deviation brought flow to error problem.For the suppression flowing to error that magnetic deviation causes, conventional method is solved by external import compass or GPS [10,11], the mathematical analysis that exports and affect by magnetic deviation is flowed on ADCP and mathematics correction there is not yet relevant report.Given this, inventor furthers investigate the problems referred to above, then has this case to produce.
List of references:
[1]Gordon R L.Acoustic Doppler Current Profile Principles of Operation,APractical Primer.San Diego,California USA by RD Instruments.1996,January 8,7PP.
[2]Roger L,Robert S.Pulse-to-pulse coherent doppler sonar signal processingtechnology[J].Journal of Atmosphere and Oceanic Technology,1984,1(4):293-308.
[3]Blair H,Brumley,Ramon G.Cabrera,Kent L.Deines,and Eugene A.Terray.Performance of a broad-band acoustic doppler current profiler[J].Journal of OceanicEngineering,1991,16(4):402-407.
[4]JU¨RGEN FISCHER,PETER BRANDT,MARCUS DENGLER,AND MARIO MU¨LLER Surveyingthe Upper Ocean with the Ocean Surveyor:A New Phased Array Doppler CurrentProfiler[J].Journal of Atmosphere and Oceanic Technology,2003,20:742-751.
[5]Terrence M.JOYCE.On in situ“calibration”of Shipboard ADCPs[J].Journalof Atmospheric and Oceanic Technology,1989,(6):169-172.
[6]David S.Mueller,Jorge D.Abad,Carlos M.García,Jeffery W.Gartner,MarceloH.García,and Kevin A.Oberg.Errors in Acoustic Doppler Profiler VelocityMeasurements Caused by Flow Disturbance[J].Journal of Hydraulic Engineering,2007,1411-1420.
[7]Marian Muste,Dongsu Kim and Juan A.González-Castro.Near-Transducer Errorsin ADCP Measurements:Experimental Findings[J].Journal of Hydraulic Engineering,2010,275-289.
[8] Yu Yide, Xiong Ying. under rolling moving platform, ADCP measuring errors is analyzed and corrects [J]. marine environment science .2012,31 (2): 246-249.
[9] Xie Bo, Tian Yueming, Ye Jianhong, Wu builds the test of flourish .ADCP river discharge and error analysis [J] thereof. water resource research .2007,28 (4): 34-36.
[10] Chen Liping. irony is surveyed improve one's methods [J] of ADCP flow directional detection error on ship. water conservancy and hydropower bulletin .2001,22 (18): 22-23.
[11] Chen Shourong, fragrant Tianyuan, Jiang Jianping .ADCP external equipment is to the investigation [J] of discharge measurement Accuracy. people the Changjiang river .2010,41 (1): 29-34.
Summary of the invention
The object of the present invention is to provide a kind of method of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision.
The present invention includes following steps:
1), when utilizing the platforms such as ADCP lift-launch buoy to carry out long-term anchored-position observation current, because ADCP may be positioned near ferrous material or motor, its measurement flowed to easily is subject to environmental magnetic field impact; Start flow measurement, record [0,360 °) N number of position within the scope of four quadrants in orientation (N>=4, and N is integer) ADCP inner magnet compass of affecting by environmental magnetic field exports, i.e. magnetic north angle (i=1 ... N), in this, as actual measurement magnetic north angle;
2) in step 1) in when carrying out anchored-position observation current, under using outside high precision magnetic compass to obtain same orientation condition, not by the ADCP magnetic north angle that environmental magnetic field affects (i=1 ... N), in this, as magnetic north angle standard value;
3) mathematical derivation establishes magnetic north angle output error (i.e. magnetic deviation) and its respective streams caused that ADCP inner magnet compass causes by influence of magnetic field all kinds of in environment to the quantitative relationship between error;
4) according to magnetic deviation mathematical model, to step 1) and 2) in the actual measurement magnetic north angle of N number of position that obtains and the accurate Value Data of magnetic north footmark, use least square fitting penalty coefficient;
5) step 4 is used) the middle penalty coefficient obtained, the flow direction (what namely magnetic deviation caused flows to error) the Δ ψ that in calculated for subsequent anchored-position observation current, each actual measurement magnetic north angle should compensate, flows to ψ after obtaining compensation.
In step 1) in, for making the penalty coefficient of least square fitting accurate as far as possible, N number of measuring position of getting should cover as far as possible [0,360 °) azimuthal four quadrants; The time of described beginning flow measurement can be in 24h.
In step 3) in, described magnetic north angle output error (i.e. magnetic deviation) and its respective streams caused have to the quantitative relationship of error: wherein, represent magnetic deviation, Δ ψ represents and flows to error accordingly.
In step 4) in, described magnetic deviation mathematical model is:
Wherein, represent the actual measurement magnetic north angle affected by environmental magnetic field, represent magnetic north angle standard value, A, B, C, D, E are penalty coefficient.
In step 5) in, flow to ψ=ψ '-Δ ψ after compensation, wherein ψ ' represents that actual measurement flows to, and Δ ψ represents the flow direction that should compensate.
The present invention utilizes [0,360 °) magnetic north angle standard value that the high precision magnetic compass of actual measurement magnetic north angle that N number of position ADCP of choosing in azimuth coverage exports and outer setting obtains, with least square fitting coefficient, flows direction compensation is carried out to long-term anchored-position observation current.
The present invention establishes output error and corresponding A/D CP output stream that ADCP inner magnet compass causes by magnetic influence to the quantitative relationship between error, sets up ADCP and flows to the Fundamentals of Mathematics exporting and affect error compensation by environmental magnetic field; Proposing a kind ofly can affect the method causing and flow to error by effective compensation ADCP inner magnet compass by environmental magnetic field, improve ADCP flow directional detection precision.
Beneficial effect of the present invention is as follows:
First the ADCP inner magnet compass output error that causes is affected on environmental magnetic field and respective streams has carried out mathematical derivation to the quantitative relationship between error, make mathematically to flow to error revised to this kind, and in practical engineering application, the long-term flow directional detection precision of effective raising ADCP when being subject to environmental magnetic field and affecting, promotes its environmental suitability.
Accompanying drawing explanation
Fig. 1 is the transition diagram that beam coordinate system arrives carrier coordinate system;
Fig. 2 is the transition diagram that geographic coordinate is tied to carrier coordinate system;
Fig. 3 is the magnetic deviation matched curve that before compensating in embodiment, magnetic north angle error and environmental magnetic field produce;
Fig. 4 flows to error and magnetic deviation matched curve (during depth of water 4m) before and after compensating in follow-up flow measurement in embodiment;
Fig. 5 flows to error and magnetic deviation matched curve (during depth of water 5m) before and after compensating in follow-up flow measurement in embodiment;
Fig. 6 flows to error and magnetic deviation matched curve (during depth of water 6m) before and after compensating in follow-up flow measurement in embodiment.
Embodiment
Following examples will the present invention is further illustrated by reference to the accompanying drawings.
(1) at waters of Zhangzhou, Fujian, carry certain unit buoy, use the ADCP of Nortek company of Norway to carry out long-term anchored-position observation current.This ADCP transmission frequency 1MHz, arranges blind area degree of depth 0.2m, thickness 1m, and Transmission Time Interval is 1s, and averaging time is 100s.Because float portion composition is containing ferrous material, can have an impact to ADCP flow directional detection.One day 12 time to next day 13 time, [0 °, 360 °) measure 150 orientation in azimuthal scope, obtained the magnetic north angle measured value affected by environmental magnetic field by ADCP inner magnet compass;
(2) demarcate the output relative datum of outside high precision magnetic compass, make to be consistent with the output relative datum of ADCP inner magnet compass, utilize outside magnetic compass to obtain magnetic north angle standard value under same orientation condition;
(3) mathematical derivation establishes environmental magnetic field affects the ADCP magnetic north angle output error and respective streams that cause to the quantitative relationship between error, and detailed process is as follows:
Due to the actual flow velocity used in geographic coordinate system, the wave beam flow velocity that therefore transducer in ADCP will be recorded is transformed into the flow velocity in geographic coordinate system, detailed process is: the wave beam flow velocity that in ADCP, transducer records, by twice coordinate transform: ADCP transducer beams coordinate system (hereinafter referred to as " beam coordinate system ") → ADCP carrier coordinate system (hereinafter referred to as " carrier coordinate system ") → geographic coordinate system, the flow velocity that can obtain finally exports (ADCP Coordinate Transformation Formulasand Calculations.San Diego, California USA by RD Instruments.1998, July).
The process that beam coordinate system is transformed into carrier coordinate system is: set ADCP flow measurement structure as four wave beam Janus type structure maximum in actual use, four wave beams with form beam coordinate system, as shown in Figure 1.Each wave beam and horizontal plane angle are α, the projection in the horizontal plane of each wave beam and y bthe angle of axle is β, and the flow velocity so on four beam directions can be expressed as in carrier coordinate system:
v n 1 v n 2 v n 3 v n 4 = - cos &alpha; sin &beta; cos &alpha; cos &beta; - sin &alpha; cos &alpha; sin &beta; - cos &alpha; cos &beta; - sin &alpha; cos &alpha; sin &beta; cos &alpha; cos &beta; - sin &alpha; - cos &alpha; sin &beta; - cos &alpha; cos &beta; - sin &alpha; &CenterDot; v bx v by v bz - - - < 1 >
Wherein, v ni(i=1,2,3,4) are the flow velocitys that four beam directions record, v bj(j=x, y, z) is carrier coordinate system x b, y b, z bflow velocity on direction of principal axis.
Geographic coordinate is tied to the conversion of carrier coordinate system, has following process: as shown in Figure 2, suppose that ADCP carrier coordinate system is Ox by bz b, be connected on instrument, its axle Ox b, Oy b, Oz bthe transverse axis of instrument respectively, the longitudinal axis and vertical pivot, meet " right-front-on " right hand rule, geographic coordinate is Ox ty tz t, be chosen in " east-north-sky ".
The rotation of coordinate system completes (Fig. 2) according to following order: first fix z taxle, rotates counterclockwise course angle φ; Secondly fixing y paxle, rotates counterclockwise pitching angle theta; Finally fix x p' axle (i.e. x baxle), rotate counterclockwise roll angle γ, that is: to rotate counterclockwise direction for positive dirction, rotate course angle φ, pitching angle theta, roll angle γ successively, can by geographic coordinate system Ox ty tz tbe transformed into carrier coordinate system Ox by bz b.So, the transition matrix of carrier coordinate system is tied to by geographic coordinate can be expressed as follows:
C t b = cos &gamma; 0 - sin &gamma; 0 1 0 sin &gamma; 0 cos &gamma; 1 0 0 0 cos &theta; sin &theta; 0 - sin &theta; cos &theta; cos &phi; sin &phi; 0 - sin &phi; cos &phi; 0 0 0 1 = cos &gamma; cos &phi; - sin &theta; sin &gamma; sin &phi; cos &gamma; sin &phi; + sin &theta; sin &gamma; cos &phi; - cos &theta; sin &gamma; - cos &theta; sin &phi; cos &theta; cos &phi; sin &theta; sin &gamma; cos &phi; + sin &theta; cos &gamma; sin &phi; sin &gamma; sin &phi; - sin &theta; cos &gamma; cos &phi; cos &theta; cos &gamma; - - - < 2 >
Because this transition matrix is orthogonal matrix, so by the transition matrix of carrier coordinate system to geographic coordinate system for:
C b t = cos &gamma; cos &phi; - sin &theta; sin &gamma; sin &phi; - cos &theta; sin &phi; sin &gamma; cos &phi; + sin &theta; cos &gamma; sin &phi; cos &gamma; sin &phi; + sin &theta; sin &gamma; cos &phi; cos &theta; cos &phi; sin &gamma; sin &phi; - sin &theta; cos &gamma; cos &phi; - cos &theta; sin &gamma; sin &theta; cos &theta; cos &gamma; - - - < 3 >
If the velocity vector that ADCP records is expressed as v in carrier coordinate system b=[v bxv byv bz] ', in geographic coordinate system, be expressed as v t=[v txv tyv tz] ', so in geographic coordinate system, can be expressed as:
v t = C b t v b = cos &gamma; cos &phi; - sin &theta; sin &gamma; sin &phi; - cos &theta; sin &phi; sin &gamma; cos &phi; + sin &theta; cos &gamma; sin &phi; cos &gamma; sin &phi; + sin &theta; sin &gamma; cos &phi; cos &theta; cos &phi; sin &gamma; sin &phi; - sin &theta; cos &gamma; cos &phi; - cos &theta; sin &gamma; sin &theta; cos &theta; cos &gamma; v bx v by v bz - - - < 4 >
That is:
v E = v tx = ( cos &gamma; cos &phi; - sin &theta; sin &gamma; sin &phi; ) v bx - cos &theta; sin &phi; v by + ( sin &gamma; cos &phi; + sin &theta; cos &gamma; sin &phi; ) v bz v N = v ty = ( cos &gamma; sin &phi; + sin &theta; sin &gamma; sin &phi; ) v bx + cos &theta; cos &phi; v by + ( sin &gamma; sin &phi; - sin &theta; cos &gamma; cos &phi; ) v bz v U = v tz = - cos &theta; sin &gamma; v bx + sin &theta; v by + cos &theta; cos &gamma; v bz - - - < 5 >
Wherein, v e, v n, v urepresent the east orientation of velocity vector in geographic coordinate system that ADCP records respectively, north orientation and sky are to component.
Final output flow velocity size v and flow to ψ and be respectively:
v = v E 2 + v N 2 - - - < 6 >
Suppose that course angle standard value is φ, flow to standard value accordingly for-ψ flow to(arrive because the flow direction is defined as geographic north the angle formed between final flow rate v clockwise, so place has " ﹣ " number), course angle measured value is φ ', flowing to measured value so is accordingly-ψ ', then course angle changes delta φ=φ '-φ, changes delta ψ=(-ψ ' the)-(-ψ of the flow direction flow to), then have:
Wherein,
F = ( cos 2 &gamma; + sin 2 &theta; sin 2 &gamma; ) v bx 2 + cos 2 &theta; v by 2 + ( sin 2 &gamma; + sin 2 &theta; cos 2 &gamma; ) v bz 2 + 2 sin &theta; cos &theta; sin &gamma; v bx v by - 2 sin &theta; cos &theta; cos &gamma; v by v bz + 2 sin &gamma; cos &gamma; cos 2 &theta; v bx v bz - - - < 9 >
From formula <8>, Δ ψ=Δ φ.That is: when course angle exists error delta φ, respective streams is to error delta ψ=Δ φ.Due to the built-in magnetic compass output of ADCP is magnetic north angle, the i.e. angle of compass bearing and magnetic north, with course angle---the angle of compass bearing and north geographic pole, differ a geomagnetic declination, though this drift angle according to specific time and site and different, but being in the same localities is fixed value in a short time, can to inquire about in advance or on-site measurement obtains.So there is magnetic deviation in the magnetic north angle exported when the built-in magnetic compass of ADCP time, identical error can be caused to course angle and the flow direction, namely
(4) according to magnetic deviation mathematical model (Ma Chengyao, Qian Jinwu, Shen Linyong, Zhang Yanan. the error analysis of flat electronic compass and compensation [J]. Shanghai University's journal (natural science edition) .2009,15 (2): 186-190):
Wherein, represent the actual measurement magnetic north angle affected by environmental magnetic field, represent magnetic north angle standard value, A, B, C, D, E are penalty coefficient.
The actual measurement magnetic north angle that N number of position is obtained and the accurate Value Data of magnetic north footmark, use least square fitting penalty coefficient, detailed process is as follows:
Suppose [0 °, 360 °) choose N number of measurement orientation in interval the result being recorded this N number of orientation by magnetic compass is respectively so by formula <10>, can be expressed as with matrix:
If Y = A B C D E ,
Then formula <11> can abbreviation be:
Y=(X'X) -1X'Z <12>
Penalty coefficient A can be obtained by formula <12>, B, C, D, E, should compensate under then can calculating each compass actual measurement magnetic north angle according to formula <10> and then show that angle that the flow direction corresponding to each compass actual measurement magnetic north angle should compensate (namely ).Can the penalty coefficient simulated be set in upper computer software, with facilitate to subsequent flows to measured result compensate.
150 the magnetic north angle measured values obtained in (1) and (2) and corresponding magnetic north angle standard value are carried out matching by formula <10>, as shown in Figure 3, fitting coefficient is shown in formula in figure in the magnetic deviation matched curve that before compensating, magnetic north angle error and environmental magnetic field produce.As seen from the figure, these 150 magnetic north angles have covered four quadrants, the tendency component contained in magnetic north angle error before the magnetic deviation matched curve that environmental magnetic field produces better can describe and compensate.
(5) according to formula <10>, the penalty coefficient obtained in (4) is used, the flow direction that in calculated for subsequent anchored-position observation current, each actual measurement magnetic north angle should compensate.In order to compensation effect is described, by arrange seat bottom type ADCP obtain normal stream to, and with flowing to mean square of error root to weigh.Fig. 4, Fig. 5 and Fig. 6 are respectively depth of water 4m, and under 5m and 6m situation, use the penalty coefficient in (4), i.e. A=-13.863, B=-24.132, C=3.375, D=-4.624, E=16.790, to the result that follow-up flow measurement compensates.Can find out, the depth of water be respectively 4,5 and 6m time, flow to after compensation error comparatively compensate before be the straight line of 0 closer to value; Also can find out from table 1, no matter which kind of depth of water, the root-mean-square error of the flow direction is improved all about 70%, and illustrate that flowing to error significantly reduces, compensation effect is obvious.
Table 1 flows to the contrast of error mean square root before and after compensating
The present invention proposes a kind of method effectively improving acoustic Doppler fluid velocity profile instrument (Acoustic Doppler CurrentProfiler, ADCP) flow directional detection precision.First by founding mathematical models, the change of analysis of magnetic Beijiao and the final relation flowed between change, draw: when only depositing the change of magnetic north angle, and this angle variable quantity is with to flow to variable quantity equal.On this basis, use least square fitting flows to the error by magnetic influence, and is used for by fitting coefficient compensating the output of the ADCP actual measurement flow direction.With normal stream to compared with, the present invention can finally realize flow to by magnetic influence mean square of error root reduce about 70%.The present invention can [0,360 °) in azimuth coverage effective compensation ADCP inner magnet compass bring by environmental magnetic field interference flow to error, thus all can significantly improve flow directional detection precision in comprehensive scope.The present invention has guidance value to the long-term anchored-position observation current in oceanographic observation.

Claims (6)

1. effectively improve a method for acoustic Doppler fluid velocity profile instrument flow directional detection precision, it is characterized in that comprising the following steps:
1), when utilizing the platforms such as ADCP lift-launch buoy to carry out long-term anchored-position observation current, because ADCP may be positioned near ferrous material or motor, its measurement flowed to easily is subject to environmental magnetic field impact; Start flow measurement, record [0,360 °) N number of position affects by environmental magnetic field within the scope of four quadrants in orientation ADCP inner magnet compass exports, i.e. magnetic north angle i=1 ... N, in this, as actual measurement magnetic north angle; Wherein, N>=4, and N is integer;
2) in step 1) in when carrying out anchored-position observation current, under using outside high precision magnetic compass to obtain same orientation condition, not by the ADCP magnetic north angle that environmental magnetic field affects i=1 ... N, in this, as magnetic north angle standard value;
3) mathematical derivation establishes magnetic north angle output error and its respective streams caused that ADCP inner magnet compass causes by influence of magnetic field all kinds of in environment to the quantitative relationship between error; The magnetic north angle output error that described all kinds of influence of magnetic field causes and magnetic deviation;
4) according to magnetic deviation mathematical model, to step 1) and 2) in the actual measurement magnetic north angle of N number of position that obtains and the accurate Value Data of magnetic north footmark, use least square fitting penalty coefficient;
5) step 4 is used) the middle penalty coefficient obtained, what in calculated for subsequent anchored-position observation current, each actual measurement magnetic north angle should compensate flows to error delta ψ, and what namely magnetic deviation caused flows to error, flows to ψ after obtaining compensation.
2. the method for a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision as claimed in claim 1, it is characterized in that in step 1) in, for making the penalty coefficient of least square fitting accurate as far as possible, N number of measuring position of getting should cover as far as possible [0,360 °) azimuthal four quadrants.
3. the method for a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision as claimed in claim 1, is characterized in that in step 1) in, the time of described beginning flow measurement is in 24h.
4. the method for a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision as claimed in claim 1, is characterized in that in step 3) in, described magnetic north angle output error and its respective streams caused have to the quantitative relationship of error: wherein, represent magnetic deviation, Δ ψ represents and flows to error accordingly.
5. the method for a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision as claimed in claim 1, is characterized in that in step 4) in, described magnetic deviation mathematical model is:
Wherein, represent the actual measurement magnetic north angle affected by environmental magnetic field, represent magnetic north angle standard value, A, B, C, D, E are penalty coefficient.
6. the method for a kind of effective raising acoustic Doppler fluid velocity profile instrument flow directional detection precision as claimed in claim 1, is characterized in that in step 5) in, flow to ψ=ψ '-Δ ψ after compensation, wherein ψ ' represents that actual measurement flows to, and Δ ψ represents the flow direction that should compensate.
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