CN111239785B - Carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement - Google Patents

Carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement Download PDF

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CN111239785B
CN111239785B CN202010130759.7A CN202010130759A CN111239785B CN 111239785 B CN111239785 B CN 111239785B CN 202010130759 A CN202010130759 A CN 202010130759A CN 111239785 B CN111239785 B CN 111239785B
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张雷
吴新平
蒋玉东
欧冬秀
于钥
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Tongji University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/42—Determining position
    • G01S19/43—Determining position using carrier phase measurements, e.g. kinematic positioning; using long or short baseline interferometry
    • G01S19/44—Carrier phase ambiguity resolution; Floating ambiguity; LAMBDA [Least-squares AMBiguity Decorrelation Adjustment] method
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/53—Determining attitude
    • G01S19/54—Determining attitude using carrier phase measurements; using long or short baseline interferometry
    • G01S19/55—Carrier phase ambiguity resolution; Floating ambiguity; LAMBDA [Least-squares AMBiguity Decorrelation Adjustment] method

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Abstract

The invention discloses a carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement, which comprises the following steps: detecting cycle slip by using the MW combined observation value, detecting cycle slip of the observation value by using the TECR, and detecting and repairing the cycle slip and the three-frequency cycle slip of the original carrier phase observation value in a combined calculation manner; on the basis of the traditional cycle slip detection and repair method, MW combination and ionized layer TEC change rate (TECR) are comprehensively adopted to detect and repair cycle slip, a classical Turbo-Edit method suitable for a double-frequency observed value is comprehensively used for reference, an adaptive cycle slip detection threshold model is constructed, the method is expanded to a three-frequency situation to process three-frequency data, and therefore the overall performance of cycle slip detection is improved, and rapid and accurate cycle slip detection and repair of the three-frequency data are achieved.

Description

Carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement
Technical Field
The invention relates to the technical field of unmanned driving, in particular to a carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement.
Background
In recent years, with the continuous development of the field of intelligent automobiles and the continuous maturity of the technology in the field of automatic control, the research on the automatic control technology of automobiles is more and more concerned by people, in actual driving, an unmanned vehicle should have the capability of positioning and driving under different roads and meteorological conditions, and in GNSS relative positioning, centimeter-level high-precision positioning is realized by using a carrier phase observation value, wherein the detection and repair of cycle slip in the carrier phase observation value are key links for processing high-precision positioning data;
the complexity of the cycle slip detection and repair problem when the carrier is in motion is shown in the following aspects: the GNSS observation environment is changeable, the signal-to-noise ratio is low, the multipath influence is complex, and the cycle slip is more frequent; the complex change of the motion state of the carrier can cause the change rule of the phase observation value to be unobvious, so that the cycle slip detection method based on the geometric change rule of the satellite is not applicable any more; with the increase of the length of the base line between the stations under the dynamic condition, the ionospheric delay with strong spatial correlation is difficult to be well eliminated through double differences; under the high dynamic condition, the ionosphere delay error of an observed value is changed quickly, so that the cycle slip is more difficult to detect, and due to the complex conditions of cycle slip detection and repair under the dynamic condition, the problems of cycle slip missing judgment and misjudgment are easy to occur by adopting the traditional cycle slip detection and repair method, so that the invention provides the carrier phase cycle slip detection and repair method for unmanned positioning attitude measurement to solve the problems in the prior art.
Disclosure of Invention
In order to solve the problems, the invention aims to provide a carrier phase cycle slip detection and restoration method for unmanned positioning attitude measurement.
In order to realize the purpose of the invention, the invention is realized by the following technical scheme: a carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement comprises the following steps:
the method comprises the following steps: MW combined observed value detection cycle slip
And constructing MW combination through the pseudo range and the phase to obtain the following calculation formula of the widelane ambiguity:
Figure BDA0002395710630000021
in the formula:
Figure BDA0002395710630000022
and
Figure BDA0002395710630000023
phase observations at different frequencies (in cycles) respectively; rhoL1And ρL2Respectively pseudo-range observations (in m) at different frequencies; l isWLA wide lane combination (in m); lambda [ alpha ]WLWide lane wavelength;
then, the influence of noise is weakened, and the widelane ambiguity and the variance thereof are calculated by adopting the following smoothing algorithm:
Figure BDA0002395710630000024
in the formula:
Figure BDA0002395710630000025
the mean value of the widelane ambiguity; k and k-1 represent the current epoch and the previous epoch, respectively; sigma2The variance of the ambiguity of the wide lane is used, when the cycle slip occurs, the ambiguity of the wide lane jumps, and accordingly whether the cycle slip occurs can be judged, and the cycle slip judgment formula is as follows:
Figure BDA0002395710630000031
step two: cycle slip detection using TECR
Considering the ionospheric rate of change (TECR) value for a short time as a constant, the ionospheric TEC for epoch k-1 is calculated by:
TEC(k-1)＝f1 2((λ1φ1(k-1)-λ2φ2(k-1))-(λ1N1-λ2N2)-bi-Bp)/(4.3×1016(γ-1))
(6.4)
in the formula:
Figure BDA0002395710630000032
f1and f2Is the corresponding carrier frequency; biAnd BpThe deviation between the signal frequencies at the receiver and the satellite is considered as a constant in a period of time, and the TECR calculated value of epoch k is:
TECR(k)＝(TEC(k)-TEC(k-1))/Δt (6.5)
the ambiguity parameter and the frequency deviation parameter in the formula (6.5) are eliminated, and when the epoch k generates cycle skip, the calculation value is influenced by the cycle skip value;
then, the threshold value of the difference value between the calculated value of the TECR and the predicted value of the TECR is taken as 0.15TECU/s, the cycle slip of epoch k is obtained according to the calculated value of the TECR (k), and the calculation formula is
Figure BDA0002395710630000033
Step three: cycle slip for joint calculation of original carrier phase observations
Assuming that the cycle slip value of the wide lane combination obtained by MW combination detection is a, the cycle slip value lambda obtained by TECR detection1ΔN1(k)-λ2ΔN2(k) If b, the cycle slip calculation formula of the original observed value is as follows:
Figure BDA0002395710630000034
in the formula: a is an integer, b is a real number, and the real value Δ N obtained by the formula (6.9)1，ΔN2Removing the integer, respectively obtaining integer cycle slip values on two frequencies, and further repairing the observed value;
step four: three-frequency cycle slip detection and repair
Three mutually independent linear combinations are selected, wherein the linear combinations comprise a MW combination and two ionosphere residual combinations, and after the cycle slip of the three linear combinations is determined by the MW combination and the two ionosphere residual combinations, the cycle slip of the three original frequency observation values can be recovered, so that the original observation values are repaired.
The further improvement lies in that: in the step one, when N isWL(k) When equation (6.3) is satisfied, it is considered that cycle slip has occurred in epoch k.
The further improvement lies in that: in the second step, assuming that no cycle slip occurs in the phase observation value before epoch k, calculating the TECRs of all epochs before epoch k by using the formula (6.5), since the change of the ionosphere TECR is gradual in a short time, predicting the TECR of the current epoch by using the TECR information calculated by the previous epoch, assuming that the TECRs of k-2 and k-1 epochs are calculated, calculating the TECR predicted value TECR (k) of epoch k by using the following formula:
Figure BDA0002395710630000041
Figure BDA0002395710630000042
during actual calculation, the TECR (k-1) and the TECR (k-2) gradually carry out smooth calculation by using the phase observed value of the previous epoch so as to reduce the measurement noise of the TECR (k-1) and the TECR (k-2) and further obtain the TECR (k-1) and the TECR (k-2) with higher precision.
The further improvement lies in that: in the second step, according to the condition that the change of the TECR is relatively smooth in a short time and the cycle slip does not occur theoretically, the difference value between the calculated value of the TECR of the current epoch and the predicted value of the TECR is small, therefore, when the difference value of the calculated value of the TECR and the predicted value of the TECR exceeds a certain threshold value, the epoch is considered to have the cycle slip, and usually, the change rate of the TEC of the ionized layer is about 0.01TECU/s during the quiet period of the ionized layer; the ionosphere active period is more than 0.03TECU/s, and the dynamic characteristic and the data sampling interval of the station are considered, so the threshold value of the difference value between the TECR calculated value and the TECR predicted value is 0.15 TECU/s.
The further improvement lies in that: in the fourth step, for example, for BD2, the MW combination adopts (0,1, -1) (the combination coefficients of the frequency points B1, B2, and B3, respectively) ultra-wide lane combination, and detects and repairs the cycle slip by using a multi-epoch smoothing method, and at the same time, performs cycle slip detection by using an adaptive cycle slip detection threshold model, and the ionospheric residual combination first adopts (0,1, -1) and (1, -1,0) to form a first set of ionospheric residual combination observed values, and simultaneously adopts (1, -1,0) and (1,0,0) to form a second set of ionospheric residual combination observed values, and detects and repairs the cycle slip by using an inter-epoch difference method.
The invention has the beneficial effects that: on the basis of the traditional cycle slip detection and repair method, the invention comprehensively adopts MW combination and ionosphere TEC change rate (TECR) to detect and repair cycle slip, and extends the method to a three-frequency situation by using a classical Turbo-Edit method suitable for a dual-frequency observed value for reference so as to realize the rapid and accurate cycle slip detection and repair of three-frequency data, and the method specifically comprises the following steps: the method is characterized in that an atmospheric delay error, a station-satellite geometric distance, a satellite and receiver clock difference are eliminated by using a MW combination method, the cycle slip problem under a dynamic environment is processed through a longer wavelength, a continuous phase observation value without cycle slip is adopted to calculate a ionosphere change rate (TECR) to assist detection, the TECR is gently changed in a short time, and the smoothness of the calculated ionosphere TECR is damaged due to the cycle slip, so the cycle slip is detected according to the characteristic, the influence of the station-satellite geometric distance is eliminated, the influence of the ionosphere change is considered, and the method is suitable for cycle slip detection under the dynamic environment.
Drawings
FIG. 1 is a flow chart of TECR and MW wide lane combined observation cycle slip detection and repair in accordance with the present invention;
fig. 2 is a flow chart of the three-frequency cycle slip detection and repair according to the present invention.
Detailed Description
In order to further understand the present invention, the following detailed description will be made with reference to the following examples, which are only used for explaining the present invention and are not to be construed as limiting the scope of the present invention.
As shown in fig. 1 and 2, the present embodiment provides a carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement, including the following steps:
the method comprises the following steps: MW combined observed value detection cycle slip
And constructing MW combination through the pseudo range and the phase to obtain the following calculation formula of the widelane ambiguity:
Figure BDA0002395710630000061
in the formula:
Figure BDA0002395710630000062
and
Figure BDA0002395710630000063
phase observations at different frequencies (in cycles) respectively; rhoL1And ρL2Respectively pseudo-range observations (in m) at different frequencies; l isWLA wide lane combination (in m); lambda [ alpha ]WLWide lane wavelength;
then, the influence of noise is weakened, and the widelane ambiguity and the variance thereof are calculated by adopting the following smoothing algorithm:
Figure BDA0002395710630000064
in the formula:
Figure BDA0002395710630000065
the mean value of the widelane ambiguity; k and k-1 represent the current epoch and the previous epoch, respectively; sigma2The variance of the ambiguity of the wide lane is used, when the cycle slip occurs, the ambiguity of the wide lane jumps, and accordingly whether the cycle slip occurs can be judged, and the cycle slip judgment formula is as follows:
Figure BDA0002395710630000071
when N is presentWL(k) When the formula (6.3) is satisfied, the epoch k is considered to have cycle slip;
step two: cycle slip detection using TECR
Considering the ionospheric rate of change (TECR) value for a short time as a constant, the ionospheric TEC for epoch k-1 is calculated by:
TEC(k-1)＝f1 2((λ1φ1(k-1)-λ2φ2(k-1))-(λ1N1-λ2N2)-bi-Bp)/(4.3×1016(γ-1))
(6.4)
in the formula:
Figure BDA0002395710630000072
f1and f2Is the corresponding carrier frequency; biAnd BpThe deviation between the signal frequencies at the receiver and the satellite is considered as a constant in a period of time, and the TECR calculated value of epoch k is:
TECR(k)＝(TEC(k)-TEC(k-1))/Δt (6.5)
the ambiguity parameter and the frequency deviation parameter in the formula (6.5) are eliminated, and when the epoch k generates cycle skip, the calculation value is influenced by the cycle skip value;
calculating the TECR of all epochs before an epoch k by using an expression (6.5) on the assumption that the phase observation value before the epoch k does not generate cycle slip, predicting the TECR of the current epoch by using the TECR information calculated by the previous epoch because the change of the ionospheric TECR is smooth in a short time, and calculating the TECR predicted value TECR (k) of the epoch k by using the following expressions on the assumption that the TECR of k-2 and k-1 epochs are calculated:
Figure BDA0002395710630000073
Figure BDA0002395710630000074
during actual calculation, the TECR (k-1) and the TECR (k-2) gradually carry out smooth calculation by using the phase observed value of the previous epoch so as to reduce the measurement noise of the TECR (k-1) and the TECR (k-2) and further obtain the TECR (k-1) and the TECR (k-2) with higher precision;
according to the condition that the change of the TECR is relatively smooth in a short time and the cycle slip does not occur theoretically, the difference value between the TECR calculated value of the current epoch and the TECR predicted value is small, therefore, when the difference value of the two values exceeds a certain threshold value, the epoch is considered to have the cycle slip, and usually, the change rate of the TEC of the ionized layer is about 0.01TECU/s during the ionized layer quiet period; the ionosphere active period reaches more than 0.03TECU/s, the dynamic characteristic and the data sampling interval of the station are considered, the threshold value of the difference value between the calculated value of the TECR and the predicted value of the TECR is 0.15TECU/s, the cycle slip of epoch k is obtained according to the calculated value of the TECR (k), and the calculation formula is
Figure BDA0002395710630000081
Step three: cycle slip for joint calculation of original carrier phase observations
Assuming that the cycle slip value of the wide lane combination obtained by MW combination detection is a, the cycle slip value lambda obtained by TECR detection1ΔN1(k)-λ2ΔN2(k) If b, the cycle slip calculation formula of the original observed value is as follows:
Figure BDA0002395710630000082
in the formula: a is an integer, b is a real number, and the real value Δ N obtained by the formula (6.9)1，ΔN2Removing the integer, respectively obtaining integer cycle slip values on two frequencies, and further repairing the observed value;
step four: three-frequency cycle slip detection and repair
Selecting three mutually independent linear combinations comprising a MW combination and two ionospheric residual combinations to detect the cycle slip, for example, for BD2, the MW combination adopts (0,1, -1) (respectively, combination coefficients of B1, B2 and B3 frequency points) ultra-wide lane combination, and adopts a multi-epoch smoothing method to detect and repair the cycle slip, and simultaneously adopts an adaptive cycle slip detection threshold model to detect the cycle slip, the ionospheric residual combination firstly adopts (0,1, -1) and (1, -1,0) to form a first group of ionospheric residual combination observed values, and simultaneously adopts (1, -1,0) and (1,0,0) to form a second group of ionospheric residual combination observed values, and adopts an inter-epoch difference method to detect and repair the cycle slip, and after determining the cycle slip sizes of the three linear combinations by the MW combination and the two ionospheric residual combinations, the cycle slip of the three original frequency observed values can be recovered, and the original observed values are restored, as shown in fig. 2.
The carrier phase cycle slip detection and restoration method for unmanned positioning attitude measurement comprehensively adopts MW combination and ionized layer TEC change rate (TECR) to detect and restore cycle slip on the basis of the traditional cycle slip detection and restoration method, and extends the method to a three-frequency situation by using a classic Turbo-Edit method suitable for a dual-frequency observation value to realize rapid and accurate cycle slip detection and restoration of three-frequency data, and specifically comprises the following steps: the method is characterized in that an atmospheric delay error, a station-satellite geometric distance, a satellite and receiver clock difference are eliminated by using a MW combination method, the cycle slip problem under a dynamic environment is processed through a longer wavelength, a continuous phase observation value without cycle slip is adopted to calculate a ionosphere change rate (TECR) to assist detection, the TECR is gently changed in a short time, and the smoothness of the calculated ionosphere TECR is damaged due to the cycle slip, so the cycle slip is detected according to the characteristic, the influence of the station-satellite geometric distance is eliminated, the influence of the ionosphere change is considered, and the method is suitable for cycle slip detection under the dynamic environment.
The foregoing illustrates and describes the principles, general features, and advantages of the present invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, which are described in the specification and illustrated only to illustrate the principle of the present invention, but that various changes and modifications may be made therein without departing from the spirit and scope of the present invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (5)

1. A carrier phase cycle slip detection and restoration method for unmanned positioning and attitude measurement is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: MW combined observed value detection cycle slip
And constructing MW combination through the pseudo range and the phase to obtain the following calculation formula of the widelane ambiguity:
Figure FDA0002965378490000011
in the formula:
Figure FDA0002965378490000012
and
Figure FDA0002965378490000013
phase observed values at different frequencies respectively; rhoL1And ρL2Respectively are distance observed values at different frequencies; l isWLThe combination of the wide lanes is adopted; lambda [ alpha ]WLWide lane wavelength; then, the influence of noise is weakened, and the widelane ambiguity and the variance thereof are calculated by adopting the following smoothing algorithm:
Figure FDA0002965378490000014
Figure FDA0002965378490000015
in the formula:
Figure FDA0002965378490000016
the mean value of the widelane ambiguity; k and k-1 represent the current epoch and the previous epoch, respectively; sigma2The variance of the ambiguity of the wide lane is used, when the cycle slip occurs, the ambiguity of the wide lane jumps, and accordingly whether the cycle slip occurs can be judged, and the cycle slip judgment formula is as follows:
Figure FDA0002965378490000017
step two: cycle slip detection using TECR
Considering the ionospheric rate of change (TECR) value for a short time as a constant, the ionospheric TEC for epoch k-1 is calculated by: TEC (k-1) ═ f1 2((λ1φ1(k-1)-λ2φ2(k-1))-(λ1N1-λ2N2)-bi-Bp)/(4.3×1016(γ-1)) (6.4)
In the formula: y ═ f1 2/f2 2，f1And f2Is the corresponding carrier frequency; biAnd BpThe deviation between the signal frequencies at the receiver and the satellite is considered as a constant in a period of time, and the TECR calculated value of epoch k is:
TECR(k)＝(TEC(k)-TEC(k-1))/Δt (6.5)
the ambiguity parameter and the frequency deviation parameter in the formula (6.5) are eliminated, and when the epoch k generates cycle skip, the calculation value is influenced by the cycle skip value;
then, the threshold value of the difference value between the calculated value of the TECR and the predicted value of the TECR is taken as 0.15TECU/s, the cycle slip of epoch k is obtained according to the calculated value of the TECR (k), and the calculation formula is
Figure FDA0002965378490000021
Step three: cycle slip for joint calculation of original carrier phase observations
Assuming that the cycle slip value of the wide lane combination obtained by MW combination detection is a, the cycle slip value lambda obtained by TECR detection1ΔN1(k)-λ2ΔN2(k) If b, the cycle slip calculation formula of the original observed value is as follows:
Figure FDA0002965378490000022
in the formula: a is an integer, b is a real number, and the real value Δ N obtained by the formula (6.9)1，ΔN2Trimming to obtain 2 integer cycle slip values on two frequencies respectively, and further repairing the observed value;
step four: three-frequency cycle slip detection and repair
Three mutually independent linear combinations are selected, wherein the linear combinations comprise a MW combination and two ionosphere residual combinations, and after the cycle slip of the three linear combinations is determined by the MW combination and the two ionosphere residual combinations, the cycle slip of the three original frequency observation values can be recovered, so that the original observation values are repaired.
2. The method for detecting and repairing carrier phase cycle slip of unmanned positioning attitude measurement according to claim 1, wherein: in the step one, when N isWL(k) When equation (6.3) is satisfied, it is considered that cycle slip has occurred in epoch k.
3. The method for detecting and repairing carrier phase cycle slip of unmanned positioning attitude measurement according to claim 1, wherein: in the second step, assuming that no cycle slip occurs in the phase observation value before epoch k, calculating the TECRs of all epochs before epoch k by using the formula (6.5), since the change of the ionosphere TECR is gradual in a short time, predicting the TECR of the current epoch by using the TECR information calculated by the previous epoch, assuming that the TECRs of k-2 and k-1 epochs are calculated, calculating the TECR predicted value TECR (k) of epoch k by using the following formula:
Figure FDA0002965378490000031
Figure FDA0002965378490000032
during actual calculation, the TECR (k-1) and the TECR (k-2) gradually carry out smooth calculation by using the phase observed value of the previous epoch so as to reduce the measurement noise of the TECR (k-1) and the TECR (k-2) and further obtain the TECR (k-1) and the TECR (k-2) with higher precision.
4. The method for detecting and repairing carrier phase cycle slip of unmanned positioning attitude measurement according to claim 1, wherein: in the second step, according to the condition that the change of the TECR is relatively smooth in a short time and the cycle slip does not occur theoretically, the difference value between the calculated value of the TECR of the current epoch and the predicted value of the TECR is small, therefore, when the difference value of the calculated value of the TECR and the predicted value of the TECR exceeds a certain threshold value, the epoch is considered to have the cycle slip, and usually, the change rate of the TEC of the ionized layer is about 0.01TECU/s during the quiet period of the ionized layer; the ionosphere active period is more than 0.03TECU/s, and the dynamic characteristic and the data sampling interval of the station are considered, so the threshold value of the difference value between the TECR calculated value and the TECR predicted value is 0.15 TECU/s.
5. The method for detecting and repairing carrier phase cycle slip of unmanned positioning attitude measurement according to claim 1, wherein: in the fourth step, for a BD2, an ultra-wide lane combination is adopted for MW combination, the combination coefficient of a frequency point B1 of the ultra-wide lane combination is 0, the combination coefficient of a frequency point B2 is 1, the combination coefficient of a frequency point B3 is-1, cycle slip is detected and repaired by adopting a multi-epoch smoothing method, meanwhile, a self-adaptive cycle slip detection threshold model is adopted for cycle slip detection, ionospheric residual combination firstly adopts (0,1, -1) and (1, -1,0) to form a first group of ionospheric residual combination observed values, and simultaneously adopts (1, -1,0) and (1,0,0) to form a second group of ionospheric residual combination observed values, and cycle slip is detected and repaired by adopting an inter-epoch difference method.
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