Method for measuring spin period of small asteroid with spectral timing correlation

Abstract

This invention discloses a method for measuring the rotation period of asteroids based on spectral time-series correlation. This method can measure the rotation period of asteroids by acquiring their spectra and utilizing the correlations between them, obtaining rotation period estimation results with lower signal-to-noise ratios and independent of shape model parameters, thus improving the measurement capability of asteroid rotation periods. The method includes the following steps: receiving asteroid echoes (using the principle of not receiving during transmission), calculating a spectrum segment within each receiving interval based on the radar echoes; calculating the correlation of each spectrum segment obtained in step one, correlating each spectrum segment with itself and the spectra of all subsequent time periods to obtain multiple sets of correlation values; averaging the correlation values; multiplying all correlation values ​​by a weighting coefficient to obtain a weighted average correlation value; and finally selecting the time corresponding to the maximum correlation value within the judgment range as the estimated rotation period.

Description

Technical Field

[0001] This invention relates to the field of inverse synthetic aperture radar technology, specifically to a method for measuring the rotation period of asteroids based on spectral time-series correlation. Background Technology

[0002] Asteroids are celestial bodies in the solar system that orbit the Sun but are much smaller than planets in size and volume. The vast majority of asteroids are located in the asteroid belt between Mars and Jupiter. Because asteroids have not undergone planetary evolution, they retain some original information and play an important role in studying the evolution of the solar system. At the same time, because asteroids themselves pose a risk of impacting Earth, calculating their motion and rotation parameters is particularly crucial for near-Earth defense.

[0003] Currently, there are two main methods for searching asteroid rotation parameters: optical and radar. Optical methods utilize optical telescopes to observe the photometric information of asteroids at different positions and attitudes, and then use photometric models to inversely solve for the asteroid's rotation axis direction and rotation period. However, optical methods are greatly affected by weather conditions and have limitations in obtaining information about distant targets. Radar, on the other hand, has the advantages of being available all day and in all weather conditions, and can generate high-precision two-dimensional images to obtain more information. The mainstream radar calculation method is currently the parameter inversion method based on SHAPE software, which uses radar two-dimensional images. This method generates delayed Doppler images by comparing and matching the similarity of the two images using a spherical harmonic function shape model constructed by the software, and then uses a least-squares method to compare and match the similarity of the two images to inversely solve for the asteroid's rotation parameters.

[0004] Current traditional algorithms require constructing a shape model and then inverting rotation parameters from multi-angle 2D images. Obtaining 2D images requires a high signal-to-noise ratio, making it difficult to handle ultra-distant targets like asteroids. Furthermore, because traditional algorithms incorporate rotation parameters into the shape model parameters for joint inversion, the final rotation parameter search results are correlated with the shape parameters, which is not conducive to quickly and accurately determining the rotation parameters individually.

[0005] Therefore, how to obtain rotation period measurement results independent of shape model parameters and improve the signal-to-noise ratio is an urgent problem to be solved. Summary of the Invention

[0006] In view of this, the present invention provides a method for measuring the rotation period of an asteroid based on spectral time-series correlation. This method can measure the rotation period of an asteroid by acquiring its spectrum and utilizing the correlation between the spectra, thereby obtaining a rotation period estimation result with a lower signal-to-noise ratio that is independent of shape model parameters, thus improving the ability to measure the rotation period of an asteroid.

[0007] To achieve the above objectives, the technical solution of the present invention includes the following steps:

[0008] Step 1: Receive asteroid echoes. When receiving radar echoes, the principle of not receiving during transmission is adopted. Based on the radar echoes, calculate a spectrum within each receiving interval.

[0009] Step 2: Use the spectrum obtained in Step 1 to calculate the correlation. Each spectrum segment is correlated with itself and the spectrum in all subsequent time periods to obtain multiple sets of correlation values.

[0010] Step 3: Average the relevant values ​​from each group;

[0011] Step 4: Multiply all relevant values ​​by a weighting factor to obtain a weighted average correlation value; finally, select the time corresponding to the largest correlation value within the judgment range as the estimated rotation period.

[0012] Furthermore, step one includes the following steps:

[0013] The asteroid's rotation period is calculated using radar echoes through a non-inversion method. During the process, based on certain prior information, at least two radar echoes are received.

[0014] The coordinates (x, y) of the scattering point p in the inertial frame p (t),y p (t),z p The asteroid's three-dimensional shape is described in the form of (t), and its slant distance from the radar is given at each slow time point t:

[0015]

[0016] Among them, (x ra (t),y ra (t),z ra (t) represents the radar position coordinates at this moment; using a linear frequency modulated signal as the radar transmission signal, the echo s(τ,t) of each scattering point in the range-azimuth direction is expressed as:

[0017]

[0018] Where τ is the fast time, σ p T represents the scattering coefficient at each scattering point. p k is the pulse width. r For a linear frequency modulated signal, where λ is the signal wavelength, after pulse compression of the received signal, the echo becomes:

[0019]

[0020] Among them, s c (τ,t) represents the signal after pulse compression, A mLet B be the signal amplitude after pulse compression, and B be the signal bandwidth. Then, in order to eliminate the influence of target translation on imaging and spectrum, translation compensation is performed on the signal. The compensated signal is represented as:

[0021]

[0022] in, The slope distance after translational compensation:

[0023]

[0024] R tr (t) represents the slant range of the radar relative to the asteroid's center of mass; Rewritten as follows:

[0025]

[0026] Among them, (a p r p Let θ(t) be the azimuth and range coordinates of the scattering point p in the ISAR equivalent turntable model, and let θ(t) be the angle rotated by the asteroid at time t, denoted by ωt, where ω is the rotation angular velocity. Perform an azimuth Fourier transform on the pulse-compressed signal to obtain a two-dimensional representation.

[0027]

[0028] Among them, T m For the coherent cumulative time in the azimuth direction, f c The carrier frequency is used; if the range resolution is limited to include all range imaging points of the scattering points, then the signal-containing portion of the two-dimensional image will eventually become a series of azimuth signals, i.e.

[0029]

[0030] This is the spectrum required within the current cumulative time.

[0031] When receiving radar echoes, the principle of not receiving during transmission is adopted, and a spectrum segment is calculated within each receiving interval.

[0032] Furthermore, step two specifically includes the following steps:

[0033] Step one yields a total of M spectrum segments, with each segment having a start time t. i As a subscript, S ti (f) indicates that the time difference between the subscripts of different spectra is denoted as τ. m Let m be the interval between two spectrum segments, m∈[0,M-1], so the two time intervals are τ. m The correlation coefficient between the spectra is

[0034]

[0035] Each spectrum segment must be correlated with itself and the spectrum of all subsequent time periods. To ensure that each correlation coefficient contains at least half of the spectrum, the maximum value of i is set to...

[0036] Furthermore, step three specifically involves:

[0037] To accumulate the true period, the correlation values ​​R of each group are... i (τ m Perform an averaging operation, i.e.

[0038]

[0039] The correlation values ​​for different groups have different lengths. It changes with the value of m

[0040]

[0041] Further, in step four, all relevant values ​​are multiplied by a weighting coefficient to obtain a weighted average correlation value; finally, the time corresponding to the maximum correlation value within the judgment range is selected as the estimated rotation period. The specific steps are as follows:

[0042] The period determination range is set to start from 300 seconds;

[0043] Multiply all relevant values ​​by a weighting factor, which varies with the number of average groups, from 1 to... Decreasing weighting coefficients; that is, the weighted average correlation value:

[0044]

[0045] The weighting coefficient exponent q is set to different values ​​depending on the situation, and the time corresponding to the maximum correlation value within the judgment range is finally selected as the estimated rotation period T.

[0046]

[0047] Beneficial effects:

[0048] This invention provides a novel method for estimating asteroid rotation periods by utilizing the correlation between asteroid spectra. First, the spectrum of the asteroid at various time points is calculated using radar echo signals. The calculated time-series spectra are then grouped according to different start times. Next, correlation coefficients are calculated between the spectra within each group to form multiple sets of correlation curves. Finally, the asteroid period is estimated based on the periodicity of the average of these correlation curves. This invention, by acquiring asteroid spectra and utilizing their correlations to measure the asteroid's rotation period, achieves a rotation parameter estimation method with a lower signal-to-noise ratio and independence from shape model parameters. Therefore, this invention's ability to measure asteroid rotation parameters is superior to the traditional SHAPE method. A comparison between this invention and the traditional SHAPE method using two-dimensional asteroid images is as follows:

[0049] 1) Compared to acquiring two-dimensional images of asteroids, spectrograms do not require multiple resolution cells in the range direction, and the radar cross-section of an asteroid represented by a single resolution cell in a spectrogram is larger than that in a two-dimensional image. Therefore, for the same target, the signal-to-noise ratio (SNR) presented by a spectrogram is higher than that of a two-dimensional radar image. When a two-dimensional image with sufficient SNR cannot be obtained, the spectral method can be used to measure the rotation period to reduce the SNR requirement.

[0050] 2) Traditional SHAPE methods for measuring asteroid parameters involve retrieving a three-dimensional shape model from a two-dimensional image of the asteroid, with rotation parameters being added later as a result. This method, which incorporates rotation parameters into the shape parameters, involves searching for numerous parameters and is time-consuming. Furthermore, since rotation parameters are correlated with shape parameters, obtaining accurate rotation parameters often requires a high-precision three-dimensional shape model. This invention eliminates the need to build a shape model throughout the process, using only the signal spectrum obtained from the asteroid itself to decouple rotation parameters from shape parameters. Attached Figure Description

[0051] Figure 1 This is the overall flowchart of the present invention;

[0052] Figure 2 This is a schematic diagram of the equivalent turntable model for ISAR imaging.

[0053] Figure 3 A time-period diagram for calculating the spectrum;

[0054] Figure 4 This is a schematic diagram showing the relative positions of the relevant values ​​in each group;

[0055] Figure 5 A schematic diagram showing the averaging of correlation values ​​for different lengths;

[0056] Figure 6 This is a schematic diagram of the periodicity determination boundary and weighting coefficients;

[0057] Figure 7A schematic diagram showing the periodic measurement error under different signal-to-noise ratios;

[0058] Figure 8 This diagram illustrates the period measurement error under different shapes and initial phases. Detailed Implementation

[0059] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0060] The flowchart of the spectral timing correlation asteroid period measurement method of the present invention is as follows: Figure 1 As shown, the specific steps include:

[0061] Step 1: Calculate the spectrum of radar echoes.

[0062] This invention calculates the asteroid's rotation period using radar echoes via a non-inversion method. During implementation, based on certain prior information, it requires receiving radar echoes for at least two periods. The coordinates (x, y) of the scattering point in the inertial frame are used as the basis for this calculation. p (t),y p (t),z p The asteroid's three-dimensional shape is described in the form of (t), and its slant distance from the radar is given at each slow time point t:

[0063]

[0064] Among them, (x ra (t),y ra (t),z ra (t) represents the radar position coordinates at this moment. Using a linear frequency modulated (LFM) signal as the radar's transmitted signal, the echoes from each scattering point in the range-azimuth direction are represented as:

[0065]

[0066] Where τ is the fast time, σ p T represents the scattering coefficient at each scattering point. p k is the pulse width. r Here, λ is the frequency modulation frequency of the linear frequency modulated signal, and λ is the signal wavelength. After pulse compression of the received signal, the echo becomes:

[0067]

[0068] Among them, A m Let B be the signal amplitude after pulse compression, and B be the signal bandwidth. Then, to eliminate the influence of target translation on imaging and the spectrum, translational compensation is performed on the signal. The compensated signal is represented as:

[0069]

[0070] in, The slope distance after translational compensation:

[0071]

[0072] R tr (t) represents the slant range of the radar relative to the asteroid's center of mass. It can be rewritten as follows:

[0073]

[0074] Among them, (a p r p Let ) represent the azimuth and range coordinates of the scattering point p in the ISAR equivalent turntable model, such as Figure 2 As shown. θ(t) is the angle the asteroid rotates through at time t, which can also be represented by ωt, where ω is the angular velocity of rotation. An azimuth-to-Fourier transform is performed on the pulse-compressed signal, and the final two-dimensional graph is shown below.

[0075]

[0076] Among them, T m For the coherent cumulative time in the azimuth direction, f c This is the carrier frequency. If the range resolution is limited to include all range imaging points of the scattering points, then the signal-containing portion of the two-dimensional image will ultimately become a series of azimuth signals, i.e.

[0077]

[0078] This is the spectrum required within the current cumulative time.

[0079] When receiving radar echoes, the principle of not receiving during transmission is adopted. A segment of the spectrum is calculated within each reception interval, such as... Figure 3 As shown, the higher part of the rectangle represents the period when signals are received, and the lower part represents the period when signals are not received.

[0080] Step 2: Calculate the correlation using each spectrum segment.

[0081] like Figure 3 As shown in the figure, the spectrum of the echoes within the receiving time periods ①②③④ and all subsequent time periods is calculated. Let M be the total number of spectrum segments, and let the start time t be the start time of each spectrum segment. i As a subscript, S ti (f) indicates

[0082] Let τ be the time difference between the subscripts of different spectra. m Let m be the interval between two spectrum segments, m∈[0,M-1], so the two time intervals are τ. m The correlation coefficient between the spectra is

[0083]

[0084] Each spectrum segment must be correlated with itself and the spectrum of all subsequent time periods. To ensure that each correlation coefficient contains at least half of the spectrum, the maximum value of i is set to... The starting positions of each related function are as follows: Figure 4 As shown

[0085] Step 3: Time-series correlation averaging.

[0086] To accumulate the true period, the correlation values ​​R of each group are... i (τ m Perform an averaging operation, i.e.

[0087]

[0088] The correlation values ​​for different groups have different lengths. It will change depending on the value of m.

[0089]

[0090] like Figure 5 As shown, in this case, only one set of related values ​​is used for averaging in segment (1), two sets of related values ​​are used for averaging in segment (2), and all sets of related values ​​are used for averaging in segment (3).

[0091] Step 4: Period Determination

[0092] To avoid large correlation values ​​in the spectrum with smaller turning angles, which could affect subsequent period determination, this invention assumes that asteroids with periods less than 300s are rare; therefore, the period determination range is set to start from 300s. Furthermore, to avoid the influence of correlation values ​​with a small average number of groups on the results, all correlation values ​​are multiplied by a factor that varies with the average number of groups, ranging from 1 to... Decreasing weighting coefficients. That is, the weighted average correlation value.

[0093]

[0094] The weighting coefficient exponent q can be set to different values ​​depending on different situations, and the weighting coefficient changes as follows: Figure 6 As shown. Finally, the time corresponding to the maximum correlation value within the judgment range is selected as the estimated rotation period T, i.e.

[0095]

[0096] To verify the feasibility of this method, a simulation experiment for estimating asteroid rotation parameters was conducted using the parameters in Table 1.

[0097] Table 1 Simulation Experiment Parameters

[0098] Simulation parameters Parameter values Pulse width 4μs Pulse repetition frequency 20Hz Total observation time 7320s Single-segment spectrum accumulation time 20s Inter-spectral time interval 40s

[0099] This experiment investigated tests under different signal-to-noise ratios, shapes, and orbits. First, tests were conducted under different signal-to-noise ratios, with the Bennu model fixed in shape and the orbit 2019vl5 near its perigee on November 14, 2025. The results of the signal-to-noise ratio and period measurement error (RMSE from 100 Monte Carlo simulations) under different pulse pressures are presented. Figure 7 As shown in the figure, the measurement error gradually decreases with the increase of the signal-to-noise ratio, dropping to around 1% error from about 0dB.

[0100] The second step involves fixing the signal-to-noise ratio after pulse compression to 50 dB, while maintaining the orbit of 2019vl5 near its perigee on November 14, 2025. The shape model and initial rotation phase are then modified to obtain the following result: Figure 8 The measurement error correspondence is shown. Under several shape models, the period measurement error varies little with phase and is all below 5%. The algorithm has a certain degree of robustness to different shape models.

[0101] Finally, the period measurement error under different orbits was analyzed, with the shape model fixed to Bennu, the initial phase fixed to 0, and the signal-to-noise ratio after pulse compression fixed to 50dB. The simulation test results are shown in Table 2. Under the same conditions, the measurement error can be basically less than 1% when the asteroid's orbit is about 100,000 km away.

[0102] Table 2 Test errors under different tracks

[0103] track distance Measurement error 2019vl5 will be near its perigee on November 14, 2025. Approximately 5.6 million km 0.0833％ 2022UX1 will be near its perigee on October 11, 2023. Approximately 1.2 million km 0 2021JW2 will be near its perigee on April 19, 2024. Approximately 620,000 km 0.2778％

[0104] It can be seen that this method can accurately measure the asteroid's period when the asteroid has certain shape characteristics, its orbital distance is greater than 100,000 km, and the signal-to-noise ratio after pulse compression reaches about 0 dB.

[0105] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for measuring the rotation period of asteroids based on spectral time-series correlation, characterized in that, include: Step 1: Receive asteroid echoes. When receiving radar echoes, the principle of not receiving during transmission is adopted. Based on the radar echoes, a spectrum segment is calculated within each receiving interval. Step 1 includes the following steps: The asteroid's rotation period is calculated using radar echoes through a non-inversion method. During the process, based on certain prior information, at least two radar echoes are received. With scattering point p Coordinates in an inertial frame ( The three-dimensional shape of the asteroid is described in the form of ) at each slow time point. Its slant distance to the radar is: （1） in,( This refers to the radar's position coordinates at this moment; using a linear frequency modulated signal as the radar's transmitted signal, the echoes from each scattering point in the range-azimuth direction are... Represented as: （2） in To save time, Here are the scattering coefficients at each scattering point. The pulse width. Frequency modulation for linear frequency modulation signals. Given the signal wavelength, the received signal is first pulse-compressed, and the echo becomes: （3） in, The signal after pulse compression. The signal amplitude after pulse compression. Let be the signal bandwidth; then, in order to eliminate the influence of target translation on imaging and the spectrum, translation compensation is performed on the signal, and the compensated signal is expressed as: （4） in, The slope distance after translational compensation: （5） The slant range is the radar's motion relative to the asteroid's center of mass. Rewritten as follows: （6） in,( , Let ) represent the azimuth and range coordinates of the scattering point p in the ISAR equivalent turntable model. for The angle through which the asteroid rotates is used. express, The rotational angular velocity is given; an azimuth-to-Fourier transform is performed on the pulse-compressed signal to obtain a two-dimensional representation. （7） in, For the coherent cumulative time in the azimuth direction, The carrier frequency is used; if the range resolution is limited to include all range imaging points of the scattering points, then the signal-containing portion of the two-dimensional image will eventually become a series of azimuth signals, i.e. （8） This is the spectrum required within the current cumulative time. When receiving radar echoes, the principle of not receiving during transmission is adopted, and a spectrum is calculated within each receiving interval; Step 2: Calculate the correlation of each spectrum segment obtained in Step 1. Correlate each spectrum segment with itself and the spectra of all subsequent time periods to obtain multiple sets of correlation values. Step 2 specifically includes the following steps: Step one yields a total of M spectrum segments, with each segment defined by its start time. As a subscript, This indicates that the time difference between the subscripts of different spectrum frequencies is denoted as . m is the interval between the two spectrum segments. Therefore, the two time intervals are The correlation value between the spectra is （9） Each spectrum segment must be correlated with itself and the spectrum of all subsequent time segments. To ensure that each correlation value contains at least half of the spectrum, the maximum value of i is set to... ; Step 3: Average the relevant values ​​from each group; Step 3 specifically involves: To accumulate the true period, the relevant values ​​of each group are... Perform an averaging operation, i.e. （10） The correlation values ​​for different groups have different lengths. It changes with the value of m （11）； Step 4: Multiply all the relevant values ​​obtained in Step 3 by the weighting coefficient to obtain the weighted average correlation value; finally, select the time corresponding to the largest weighted average correlation value within the judgment range as the estimated rotation period.

2. The method for measuring the rotation period of an asteroid based on spectral temporal correlation as described in claim 1, characterized in that, Step four involves multiplying all relevant values ​​by a weighting coefficient to obtain a weighted average correlation value. Finally, the time corresponding to the maximum correlation value within the judgment range is selected as the estimated rotation period. The specific steps are as follows: The period determination range is set to start from 300 seconds; Multiply all relevant values ​​by a weighting factor to obtain the weighted average correlation value: （12） The weighting coefficient exponent q is set to different values ​​depending on the situation, and the time corresponding to the maximum correlation value within the judgment range is finally selected as the estimated rotation period T. （13）。

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