Off-axis holographic measurement method for planet-zoom based radio telescopes

Abstract

The off-focus holographic measurement method of the planet zoom-based radio telescope of the application carries out off-focus observation on a planet as a source, obtains an actual observed beam pattern of the planet through deconvolution, establishes a theoretical far-field model of the planet, compares the theoretical far-field model of the planet with the actually observed beam pattern to calculate a redundancy value, continuously updates coefficients of Zernike polynomials through minimization of the redundancy value and iteration until an optimal phase model is determined, and determines a phase error on an aperture plane according to the phase model. The off-focus holographic measurement method of the application introduces a planet as a radio source, not only improves measurement accuracy and stability, but also enhances real-time measurement effect and multi-frequency calibration capability, significantly improves calibration accuracy and overall performance of the Tianma radio telescope, and solves many limitations of traditional surface measurement methods, such as slow measurement speed, need of additional hardware devices, great influence of weather, limited application range and the like.

Description

Technical Field

[0001] The invention belongs to the field of radio astronomy, and in particular relates to a defocus holographic measurement method of a radio telescope based on planetary zoom. Background Art

[0002] The core indicators of large radio telescopes include sensitivity and resolution, both of which have a decisive impact on the quality and depth of astronomical research.

[0003] The sensitivity of a telescope is related to its effective receiving area. Specifically, the larger the antenna aperture, the smaller the minimum detectable flux, and the higher the sensitivity. This relationship shows that as the receiving area increases, the telescope's point source sensitivity significantly improves, and more electromagnetic waves can be collected, thus more effectively detecting weak celestial signals.

[0004] The resolution of a telescope is closely related to the beam width, aperture, and operating wavelength of the antenna. Specifically, the relationship between the resolution θ of a telescope, its primary mirror diameter D, and its operating wavelength λ is: , which shows that the larger the aperture and the shorter the working wavelength, the higher the imaging resolution of the telescope.

[0005] Therefore, sensitivity primarily measures an antenna's ability to receive weak signals, while resolution reflects its ability to detect target details. To improve antenna sensitivity and resolution, scientists are placing increasing demands on antenna aperture, operating frequency, and efficiency. This demand is driving research and development in antenna structure design, surface measurement methods, and active surface systems.

[0006] However, large telescopes are often limited by their size. This is primarily due to gravitational deformation caused by changes in the telescope's altitude, thermal deformation due to differential heating and cooling of the telescope structure, and inaccurate panel placement. When operating a radio telescope, errors caused by dynamic loads such as gravity, temperature, and wind load are unavoidable. Of these loads, gravity has the greatest impact on the radio telescope structure and the surface accuracy of the primary reflector, and gravitational deformation is often long-lasting. Temperature is second, and wind load is last.

[0007] Based on this, in the design and use of large radio telescopes, the surface measurement of the main reflector of the antenna is particularly important. Accurate surface measurement and adjustment can significantly improve the antenna's receiving efficiency and observation accuracy.

[0008] Out-of-Focus holography (OOF) is an effective surface measurement technique. Existing radio telescopes, such as the Tianma Telescope, typically use radio sources as their source for out-of-focus holography. Out-of-focus holography uses a linear combination of basis functions (Zernike polynomials) to represent aperture plane phase errors and introduces a defocus pattern, effectively solving the inverse problem of finding surface errors from the beam pattern. By comparing the theoretical far-field model with actual observations, OOF obtains information about antenna surface deformation. This information is then used to adjust the primary reflector system to compensate for this deformation, thereby improving antenna efficiency and helping astronomers understand the performance limitations, errors, and impacts of actual telescope systems.

[0009] The measurement principle of traditional defocus holographic measurement technology is as follows:

[0010] Traditional phase holographic measurement technology uses an auxiliary phase reference antenna to measure the amplitude and phase of the far-field beam pattern. After a simple Fourier inversion, the aperture function C(i,j) can be obtained, thereby understanding the deformation of the telescope. OOF (out-of-focus holography) technology utilizes a well-known relationship in antenna theory: the far-field pattern of a reflective antenna is the Fourier transform of the field distribution on the antenna aperture plane, which is applicable to small angles of the far-field pattern. The aperture plane is the effective receiving area of ​​the telescope. The far-field pattern of the reflective antenna and the aperture field distribution function are: This relationship (x,y) can be expressed as:

[0011] (1)

[0012] Where, is the far-field beam of the reflecting antenna in the direction cosine coordinate system; (x,y) is the aperture field distribution function of the reflector antenna in the aperture plane coordinate system, with the unit being w; λ is the observation wavelength.

[0013] In the defocus holographic measurement technology, according to the aperture model, the aperture field distribution function The relationship between (x,y) and aperture phase distribution is:

[0014] (2)

[0015] Where, is the obstruction distribution (such as truncation or occlusion distribution) on the aperture plane, in meters. is the illumination, δ(x,y,dz) is the additional term of the phase, that is, the additional phase caused by defocus (corresponding to the optical path difference), represents the aperture phase distribution, dz is the axial offset, and a positive value of dz corresponds to the movement of the sub-reflector away from the main reflector.

[0016] Unfortunately, defocus holography cannot directly measure the aperture phase distribution, only the power pattern, which means the problem is degenerate. Defocus holography aims to break this degeneracy by utilizing a series of beam patterns, one of which is in focus and the other two are defocused. The latter is achieved by adding an axial offset to the telescope's sub-reflector. This axial offset dz is known in advance.

[0017] In order to solve these problems, a convenient and efficient method is to use Zernike polynomials to parameterize the aperture phase distribution:

[0018] (3)

[0019] is the aperture phase distribution, are the coefficients to be determined, n is the radial order, l is the angular order, and n = 1 to max determines the total number of coefficients to be found by fitting [(n + 1)(n + 2) / 2]. Zernike polynomials are orthogonal on the unit circle. This strict orthogonality fails when applied to the Tianma Telescope (TMRT). Due to edge taper and central occlusion, the illumination is not uniform across the aperture, thus destroying the orthogonality. However, it is still approximately orthogonal, and some low-order Zernike polynomials are very similar to the aberrations that may exist in the optical system. In addition, low-order polynomials represent large-scale aberrations in the aperture plane.

[0020] Combining all the above situations, according to the theoretical far-field model, the far-field beam P(u,v) of the reflector antenna can be expressed as:

[0021] (4)

[0022] in, is the truncation or occlusion distribution on the aperture plane, is the illumination, δ(x,y,dz) is the additional term of the phase, that is, the additional phase caused by defocus (corresponding to the optical path difference), The unit of the aperture phase distribution and the far-field beam P(u,v) of the reflector antenna is w (watt).

[0023] For a standard Cassegrain telescope, the optical path difference δ(x,y,dz) is:

[0024] (5)

[0025] Among them, a and b are intermediate parameters, a=r / 2f, b=r / 2F, r is the cross-sectional radius of the telescope that changes with height, r= , unit is m; f is the focal length of the main reflector; F is the effective total focal length of the telescope at the Cassegrain focus; dz is the axial offset, and the positive value of dz corresponds to the movement of the sub-reflector away from the main reflector, unit is mm.

[0026] The amplitude model is a formula for calculating the amplitude of the illumination to be fitted. The amplitude model is:

[0027] (6)

[0028] Where R is the radius of the main reflector; (x0, y0) defines the center of illumination on the main reflector; σ r is the illumination cone, I0 is the amplitude of the illumination to be fitted, and I(x,y) is the illumination.

[0029] The optical path difference δ(x,y,dz) and the illuminance I(x,y) are both required to calculate the aperture model.

[0030] However, due to the shape of the Tianma Telescope (TMRT), formulas (5) and (6) cannot be used to calculate the additional phase and illumination. Otherwise, the final result will have a large error. Therefore, Dong Jian et al. proposed a new calculation method to calculate these two terms, which is defined by high-density points.

[0031] The design of the shaped double-reflecting optical system is based on the condition of equal optical distance, so the telescope of this shape can be regarded as a set of standard Cassegrain telescopes, and the focus of all Cassegrain telescopes is restricted to the optical axis. The following formula can be used to calculate the additional phase caused by defocus:

[0032] (7)

[0033] Where δ(r, dz) represents the additional phase caused by defocus; It represents the angle between the optical axis and the light from the focus to the sub-reflector; represents the angle between the optical axis and the light from the sub-reflector to the main reflector, r is the cross-sectional radius of the telescope that varies with altitude, r= , in m, dz is the axial offset, a positive value of dz corresponds to the movement of the sub-reflector away from the main reflector, in mm; λ is the wavelength, in mm.

[0034] The receiver feed pattern follows a Gaussian distribution, and the amplitude model is then:

[0035] (8)

[0036] Where I (r) is the illumination; I0 is the amplitude of the illumination to be fitted, in m; It represents the angle between the optical axis and the light from the focus to the sub-reflector, and the unit is rad. for The maximum value of σ r is the lighting cone.

[0037] Therefore, according to existing research, defocus holography has been applied to radio telescopes such as the Tianma Telescope. However, currently, there are relatively few radio sources available. For example, observations of 3C84 are easily affected by weather, making all-weather measurements impossible, resulting in low measurement accuracy, and poor real-time measurement performance. Summary of the Invention

[0038] The object of the present invention is to provide a defocus holographic measurement method for a radio telescope based on planetary zoom, so as to improve the calibration accuracy of the Tianma radio telescope.

[0039] In order to achieve the above object, the present invention provides a defocus holographic measurement method of a radio telescope based on planetary zoom.

[0040] Step S0: Set the resolution grid [xy] to achieve aperture discretization, set the iteration count value i=0, and establish the aperture model, amplitude model and phase model;

[0041] Step S1: Using the planet as the source, the sub-reflector is moved to perform defocus observation to obtain an observed power dataset, and the deconvolved observed power dataset, wavelength, and axial offset are used as input data for the model;

[0042] Step S2: According to the calculation formula and amplitude model of the additional phase generated by defocus, the additional phase matrix and illumination matrix are interpolated corresponding to the coordinates of each sampling point in the power data set;

[0043] Step S3: Initialize model parameters;

[0044] Step S4: Establishing a theoretical far-field model of the planet; determining the spectrum map of the planet based on the given planet map data, and performing frequency domain convolution based on the FFT transform to determine the beam output value of the theoretical far-field model of the planet;

[0045] Step S5: Compare the observed power data set with the beam output value of the theoretical far-field model to calculate the redundancy value;

[0046] Step S6: Determine whether the coefficients of the current Zernike polynomial are the optimal solution by using the current redundancy value and the least squares method. If so, proceed to step S9; otherwise, minimize the redundancy value to update the coefficients of the Zernike polynomial, and then proceed to step S7;

[0047] Step S7: If the current radial order n of the Zernike polynomial is less than the maximum radial order M, the radial order n and the iteration count value i are both incremented by 1, and step S8 is executed; otherwise, step S9 is executed;

[0048] Step S8: Obtain an updated phase model based on the coefficients of the Zernike polynomial, and then update the theoretical far-field model of the planet, and interpolate to obtain its beam output value; return to step S5;

[0049] Step S9: Determine the optimal phase model based on the coefficients of the Zernike polynomial, and determine the phase error on the aperture plane based on the phase model.

[0050] Preferably, in step S1, the observed power data set is obtained by performing flight scanning by moving the sub-reflector and undergoing preprocessing, and the preprocessing of the power data set includes:

[0051] Calculate the average value and center frequency of DIBAS data, obtain the original Tcal data, calculate the average Tcal data, average gain, system temperature and cal value to process the power data;

[0052] Process antenna data by interpolating azimuth, elevation, and time;

[0053] Fit the processed power data and antenna data and perform baseline processing;

[0054] Perform deconvolution processing to obtain a deconvolution-processed observed power data set.

[0055] Preferably, the deconvolution step includes:

[0056] Step B1: Build a convolution model of the observed image;

[0057] Step B2: Perform Fourier transform on the observed image and the point spread function of the telescope to obtain the spectrum values ​​of the observed image and the point spread function, and obtain the relationship between the two based on the convolution model of the observed image;

[0058] Step B3: Estimate the spectrum of the original image by inverse filtering based on the spectrum values ​​of the observed image and the point spread function and the relationship between the two;

[0059] Step B4: performing an inverse Fourier transform on the estimated spectrum of the original image to obtain a deconvolved image; or

[0060] The deconvolution algorithm used in the deconvolution process is Wiener filtering, Lucy-Richardson deconvolution or maximum entropy method.

[0061] Preferably, in step S0, the amplitude model is:

[0062] ,

[0063] where I (r) is the irradiance; I0is the amplitude of the light to be fitted, in m; denotes the angle between the optical axis and the light ray from the focal point to the sub-reflector, in rad, is the maximum of ; σ r is the illumination taper;

[0064] The phase model is:

[0065] ,

[0066] The aperture model is:

[0067] ,

[0068] (x,y) is the aperture field distribution function, in w, x, y are the coordinates of the points on the aperture plane, is the obstruction distribution on the aperture plane, in m, is the irradiance, is the aperture phase distribution, in rad, δ(x,y,dz) denotes the extra phase due to defocus, dz is the axial shift, in mm; λ is the wavelength, in mm, r is the telescope cross- section radius as a function of height, in m;

[0069] In the step S2, the formula for calculating the extra phase due to defocus is: ,

[0070] where r is the telescope cross-section radius as a function of height, in m; dz is the axial shift, in mm; λ is the wavelength, in mm, θ r is the angle between the telescope incident beam of the secondary reflector and the main axis of the telescope, φ r is the angle between the telescope incident beam of the primary reflector and the main axis of the telescope.

[0071] Preferably, in the step S4, the spectral map of the planet comprises a first spectral map and a second spectral map; the first spectral map and the second spectral map are obtained by experience of actually scanning the planet, such as the code planetFF in the virtual machine.

[0072] The frequency domain convolution comprises:

[0073] Step A1: first check whether the dimensions of the first spectral map and the second spectral map are the same, if the dimensions are not the same, end the process;

[0074] Step A2: extracting the amplitude distribution and phase distribution of the first spectrum map and the second spectrum map respectively, and then initializing the amplitude and phase to zero;

[0075] Step A3: transform the amplitude distribution and phase distribution using inverse Fourier transform;

[0076] Step A4: multiplying the transformed amplitude distribution and phase distribution to obtain a product result;

[0077] Step A5: Perform an inverse transform on the product result using a forward Fourier transform to obtain an inverse transform result;

[0078] Step A6: Divide the inverse transformation result by the size of the spectrum map and multiply it by the cosine value of the phase to obtain the spectrum map of the convolution result. The spectrum map of the convolution result is the beam output value of the far-field theoretical model.

[0079] Preferably, the step S3 specifically includes: setting the Zernike polynomial Z n,l The parameters (x, y) are the coordinates of the point on the aperture plane, n is the radial order, and l is the angular order; the initial value of the amplitude I0 of the illumination to be fitted in the amplitude model is set to 0; the preset threshold of the least squares algorithm is set to the default value;

[0080] Set the Zernike polynomial Z n,l The parameters of (x,y) specifically include: setting the Zernike polynomial Z n,l (x,y) the maximum radial order M; all the coefficients of the Zernike polynomial {a n,l} is set to 0; the current radial order n is initially set to 1.

[0081] Preferably, in step S6, the current redundancy value is input into a least squares algorithm to determine whether a preset threshold of the least squares algorithm is satisfied, thereby determining whether the coefficients of the current Zernike polynomial are an optimal solution.

[0082] Preferably, in step S8, the theoretical far-field model of the planet is updated, specifically comprising: interpolating the phase model to obtain three sets of aperture phase matrices; using the aperture model to combine the aperture phase matrix with the additional phase matrix δc(i,j,dz) and the illumination matrix Ic(i,j) to generate three sets of aperture field distribution functions, and then, respectively updating the three theoretical far-field models according to the three sets of aperture field distribution functions.

[0083] Compared with the existing defocus holographic measurement technology, the defocus holographic measurement method of the radio telescope based on planetary zoom of the present invention not only improves the measurement accuracy and stability, but also enhances the real-time measurement effect and multi-frequency calibration capability by introducing planets as radio sources, significantly improving the calibration accuracy and overall performance of the Tianma radio telescope, and solving many limitations of traditional surface measurement methods, such as slow measurement speed, the need for additional hardware equipment, great influence from weather, and limited scope of application. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 It is a flow chart of a defocus holographic measurement method of a radio telescope based on planetary zoom according to an embodiment of the present invention.

[0085] Figure 2 This is a structural diagram of a telescope involved in the defocus holographic measurement method of a radio telescope based on planetary zoom.

[0086] Figure 3 It is the function relationship diagram of the OOF preprocessor module.

[0087] Figure 4 It is a workflow diagram of the OOF pre-processor module and the OOF post-processor module. DETAILED DESCRIPTION

[0088] The preferred embodiments of the present invention are given below in conjunction with the accompanying drawings and described in detail.

[0089] The defocus holographic measurement method of the radio telescope based on planetary zoom of the present invention is used to receive the beam pattern and obtain the measurement of the antenna surface deformation through Fourier transformation. Figure 1 As shown, the defocus holographic measurement method of the radio telescope based on planetary zoom of the present invention determines the aperture phase error of the telescope through the defocus observation data, including the following steps:

[0090] Step S0: Set the resolution grid [xy] to achieve aperture discretization, set the iteration count value i=0, and establish the aperture model, amplitude model and phase model;

[0091] In the resolution grid [xy], [xy] represents the coordinates of the point on the aperture plane, and A is the grid size of the resolution grid.

[0092] In this embodiment, the amplitude model is determined by the receiver of the telescope. Due to the special geometric shape of the Tianma telescope, the amplitude model is:

[0093]

[0094] Where I (r) is the illumination; I0 is the amplitude of the illumination to be fitted, in m; It represents the angle between the optical axis and the light from the focus to the sub-reflector, and the unit is rad. for The maximum value of σ r is the lighting cone.

[0095] The phase model is:

[0096] ,

[0097] The aperture model consists of a phase model and an amplitude model, which is:

[0098] ,

[0099] (x,y) is the aperture field distribution function, the unit is w, x,y are the coordinates of the point on the aperture plane, is the blockage distribution on the aperture plane, in meters, is the illumination, is the aperture phase distribution, its unit is rad, δ(x,y,dz) represents the additional phase caused by defocus, and the calculation formula of the additional phase caused by defocus is It is expressed in the form of , where dz is the axial offset in mm, λ is the wavelength in mm, and r is the cross-sectional radius of the telescope that varies with altitude in m.

[0100] Based on the geometry of the Tianma telescope, the present invention replaces the traditional amplitude model I(x,y) with an improved amplitude model I(r), resulting in more accurate measurement results. In the present invention, the phase model and aperture model are given initial values ​​and then derived through formula iteration.

[0101] Step S1: Using the planet as the source, the sub-reflector is moved to perform defocus observation to obtain an observed power dataset, and the deconvolved observed power dataset, wavelength, and axial offset are used as input data for the model;

[0102] Existing technologies all use radio sources as the source for defocused holographic measurements. The far-field diagram based on the reflective antenna is the Fourier transform of the field distribution on the antenna aperture plane. Using planets as the source allows for all-weather measurements and a higher throughput.

[0103] The observed power dataset is obtained by performing flight scanning with a moving subreflector and undergoing preprocessing. The unprocessed power dataset is a fits file, which contains information such as power (W) and integration time (s). After preprocessing, the power dataset includes the far-field beam power and the sampling point coordinates on the aperture plane (including x- and y-axis offsets), represented as a power matrix.

[0104] In the present invention, power dataset preprocessing is performed using an OOF preprocessing module. The input to the OOF preprocessing module is a raw data fits file containing uncalibrated power, integration time, and other information. The output data is a fits file containing x-axis offset, y-axis offset, power, and time, representing the desired observed power dataset.

[0105] like Figure 3 As shown in Figure 1, the specific workflow of the OOF preprocessor module is as follows: It obtains raw data fits files from DIBAS, including power, time, azimuth, elevation, and displacement information for both focused and defocused data. The raw data is preprocessed using the OOF preprocessor module's rawBkDataReader.py function. Empty lists pData and integers are defined. The pyfits.open function is used to open the specified FITS data file, retrieve the data header information and the data itself. The TDIM keyword in the header is printed, and the TDIM3 value is output. Next, the data is processed line by line, extracting the timestamp and data. A series of calculations and processing are performed, and the results are stored in corresponding lists. Part of the integer list is printed out. The DIBAS data obtained from DIBAS is then preprocessed. This includes calculating the DIBAS data mean and center frequency, obtaining raw Tcal data, calculating average Tcal data, average gain, system temperature, and cal values ​​(i.e., calibration values) for power processing. Subsequently, rawTelDataReader.py is used to interpolate the azimuth, elevation, and time values ​​to process the antenna data. Then, the processed power data and antenna data are fitted through mergeData.py, baseline processing is performed through oof_bk.py, and finally, planet deconvolution is performed through deconvolve.py, and three fits files are finally obtained, including x-axis offset, y-axis offset, power and time, which serve as the observed power data set in the defocus holographic measurement method of the present invention and the input file of the OOF post-processing program module.

[0106] Different from the existing preprocessing process of power data of radio sources, since a planetary source is used in the present invention, the preprocessing includes the step of performing planetary deconvolution on the observed power data set.

[0107] Planetary Deconvolution: In radio astronomy, planets are extended sources with large angular diameters. Therefore, their signals are affected by the main beam of a radio telescope, causing the observed data to include the blurring effects of the instrument response function. Deconvolution is necessary to recover the true signal distribution of the planets. However, radio sources are typically point sources with angular diameters much smaller than the main beam of a telescope. Therefore, the observed signal can be approximated as the direct output of the instrument response function, eliminating the need for complex deconvolution.

[0108] Planetary deconvolution is an important method to recover the original signal from the observed image. There are several reasons for this:

[0109] 1. Improving image resolution: During observations, images are often blurred by the instrument's point spread function (PSF). Deconvolution technology can reduce this blur, improve image resolution, and make details on the planet's surface clearer.

[0110] 2. Reduce noise and artifacts: Deconvolution can effectively reduce noise and artifacts in images, making the observed data more accurate and reliable. This is very important for subsequent analysis and research.

[0111] 3. Enhance details and contrast: Deconvolution can enhance the details and contrast in images, making the features on the planet's surface more distinct and helping to identify and analyze subtle geological and atmospheric phenomena.

[0112] 4. Correcting the effects of the optical system: The optical properties of the telescope can affect image quality. Deconvolution technology can correct these effects and restore the true information of the image.

[0113] 5. Improve measurement accuracy: Accurate measurements are crucial for scientific research. Deconvolution can improve the measurement accuracy of planetary surface features and atmospheric phenomena, thereby providing more precise scientific data.

[0114] The steps of deconvolution are as follows:

[0115] Step B1: Build a convolution model of the observed image;

[0116] According to the convolution model, the observation image I obs is the original image I true Convolution with the telescope's point spread function (PSF), plus noise N.

[0117] I obs = I ture * PSF + N

[0118] Among them, represents the convolution operation.

[0119] Step B2: Observation image I obs The spectrum F(I obs ) and the value of the spectrum of the telescope's point spread function F (PSF), and obtain the relationship between the two based on the convolution model of the observed image;

[0120] Fourier transform: Convolution is difficult to handle in the spatial domain, but it becomes a simple multiplication in the frequency domain. Therefore, the observation image I can be obs The spectrum F(I obs ) and the spectrum of the telescope's point spread function F (PSF). The relationship between the two is:

[0121] F(I obs ) = F(I ture ) × F(PSF) + F(N)

[0122] Step B3: According to the spectrum of the observed image and the spectrum of the telescope's point spread function and the relationship between the two, inverse filtering is used to estimate the spectrum F(I ture ).

[0123] Inverse filtering: In the frequency domain, the spectrum F(I ture ).

[0124] The estimated spectrum of the original image F(I ture )for:

[0125] F(I ture )≈F(I obs ) / F(PSF)

[0126] It should be noted that direct inverse filtering tends to amplify noise, so regularization methods are usually used to suppress the influence of noise.

[0127] Regularization method: In step B3, in order to reduce the influence of noise, a regularization method such as Tikhonov regularization or constrained least squares method can be introduced.

[0128] After the regularization method is introduced, the estimated spectrum of the original image is:

[0129] F(I ture )=F(I obs )×F(PSF)* / [|F(PSF)| 2 +α]

[0130] where F(PSF)* is the complex conjugate of the PSF and α is the regularization parameter.

[0131] Step B4: inverse Fourier transform of the estimated spectrum of the original image F(I ture ), to obtain the deconvolved image (i.e. the original image I true ).

[0132] In this embodiment, the point spread function PSF of the telescope is known.

[0133] In other embodiments, the point spread function PSF of the telescope can be unknown, and accordingly, the deconvolution algorithm used in the deconvolution process can be a deconvolution algorithm commonly used in radio astronomy, such as Wiener filtering, Lucy-Richardson deconvolution, and maximum entropy method. Wiener filtering uses the signal-to-noise ratio in the frequency domain for deconvolution, has high computational efficiency, is suitable for images with known noise characteristics, but is highly dependent on the noise model and has limited processing capacity. Lucy-Richardson deconvolution is based on maximum likelihood estimation and can well restore the details of low signal-to-noise ratio data, but has high computational complexity, can produce artifacts, and is sensitive to parameter selection. The maximum method restores the image by maximizing the image entropy, is suitable for sparse or incomplete data, and can enhance details and contrast, but has complex calculation, relies on prior information, and may not be stable in convergence. The selection of a suitable algorithm needs to be based on the specific data characteristics and application scenarios.

[0134] Since the present application is a defocus holographic measurement method, the number of power data sets is 3 sets and is obtained by moving the secondary reflector for defocus observation, and the 3 sets of power data sets are obtained by moving the secondary reflector, including the positive defocus aperture plane moving towards the primary reflector and the negative defocus aperture plane moving away from the primary reflector, and the aperture plane located on the primary reflector.

[0135] In this embodiment, in this embodiment, the sampling interval of the aperture plane is a = DA x 3, where D is the diameter of the telescope, and A is the grid size (equal to 192). Therefore, the sampling point coordinates are (i, j), and the aperture plane sampling obtains a 192 x 192 power matrix A(i, j).

[0136] The wavelength is determined by selecting a specific waveband.

[0137] The axial offset is the defocus distance, which is freely assigned, and the physical quantity is length in units of cm.

[0138] Step S2: according to the calculation formula of the additional phase generated by defocus and the amplitude model , corresponding to each sampling point coordinate (i, j) of the power data set, the additional phase δc(r, dz) and illumination Ic(r) generated by defocus are interpolated to the aperture plane by linear interpolation method to obtain the additional phase matrix δc(i, j, dz) and illumination matrix Ic(i, j);

[0139] The subsequent calculation of the aperture field distribution function requires the coefficient a of the Zernike polynomial n,l , additional phase matrix δc(i,j,dz) and illumination matrix Ic(i,j). Therefore, we need to obtain the additional phase matrix δc(i,j,dz) and illumination matrix Ic(i,j). The units of additional phase δc(r,dz) and illumination Ic(r) are rad and W / m² respectively.

[0140] The calculation formula for the additional phase caused by defocus is: ,

[0141] Where r is the cross-sectional radius of the telescope that varies with altitude, in meters; dz is the axial offset, in millimeters; λ is the wavelength, in millimeters, and θ is the wavelength. r is the angle between the secondarily reflected telescope incident beam and the telescope axis, φ r is the angle between the incident beam of the telescope after one reflection and the main axis of the telescope. The specific positions of the two are as follows: Figure 2 shown.

[0142] Step S3: Initialize the model parameters, including: setting the Zernike polynomial Z n,l (x, y) parameters; set the initial value of the amplitude I0 of the illumination to be fitted in the amplitude model to 0; set the preset threshold of the least squares algorithm to the default value;

[0143] Among them, for the phase model formula, the Zernike polynomial Z n,l (x,y) represents a polynomial, x, y are the coordinates of the point on the aperture plane, n is the radial order, l is the angular order, a n,l is the coefficient of the Zernike polynomial, which is the key part to be solved. n,l will change.

[0144] Set the Zernike polynomial Z n,l The parameters of (x,y) specifically include: setting the Zernike polynomial Z n,l (x,y) the maximum radial order M; all the coefficients of the Zernike polynomial {a n,l} is set to 0; the current radial order n is initially set to 1.

[0145] The maximum radial order M is the maximum value of the radial order n. In this embodiment, the maximum radial order M of the Zernike polynomial is usually 5 or 6, depending on the signal-to-noise ratio.

[0146] The threshold of the least squares algorithm is used in subsequent steps.

[0147] Step S4: Start the first iteration; wherein, a theoretical far-field model of the planet is established, a spectrum map of the planet is constructed based on the given planet map data, and a frequency domain convolution based on the FFT transform is performed on the spectrum map to determine the beam output value of the theoretical far-field model of the planet;

[0148] In the present invention, the first and second spectrum maps of the planet are constructed according to the given planet map data (including the radius and wavelength of the planet), and finally the beam output value of the far-field theoretical model is obtained through the frequency domain convolution operation based on FFT transformation.

[0149] In this example, the first and second spectrum maps m1 and m2 are created directly based on the planet's properties using the mkModel function. The power of the planet is then extracted using the Power function. Both the first and second spectrum maps m1 and m2 are amplitude and phase spectra.

[0150] Among them, the input parameters of the frequency domain convolution operation include the first spectrum map m1 and the second spectrum map m2. The output of the frequency domain convolution operation is the spectrum map mres of the convolution result. The specific process of frequency domain convolution is as follows:

[0151] Step A1: First, check whether the dimensions of the first spectrum map m1 and the second spectrum map m2 are the same. If the dimensions are not the same, throw a std::runtime_error exception to end the process;

[0152] Step A2: extracting the amplitude distribution and phase distribution of the first spectrum map m1 and the second spectrum map m2 respectively, and then initializing the amplitude and phase to zero;

[0153] After extraction, the data is initialized to zero to facilitate subsequent data processing. The original amplitude and phase characteristics are not lost, but are initialized to zero here for subsequent operations.

[0154] Step A3: Use inverse Fourier transform (such as InverseFF function) to transform the amplitude distribution and phase distribution.

[0155] Step A4: Multiply the transformed amplitude distribution and phase distribution to obtain a product result, wherein the amplitudes are multiplied and the phases are added.

[0156] Step A5: Use forward Fourier transform (such as ForwardFF function) to perform inverse transform on the product result to obtain an inverse transform result.

[0157] Step A6: Divide the inverse transformation result by the size of the spectrum map and multiply it by the cosine value of the phase to obtain the spectrum map mres of the convolution result. The spectrum map mres of the convolution result is the beam output value of the far-field theoretical model.

[0158] For the second overloaded function FFTConvolve, a pointer to the spectrum map mres of the convolution result is returned, so the OFF preprocessing module can process it.

[0159] This code implements frequency domain convolution based on Fast Fourier Transform (FFT), which performs convolution operations by multiplication and phase adjustment between spectrum maps.

[0160] The theoretical far-field model obtained is:

[0161] ,

[0162] P(u,v) is the far-field beam of the reflecting antenna in the direction cosine coordinate system, (x, y) is the aperture field distribution function of the reflector antenna in the aperture plane coordinate system, with the unit being w; λ is the observation wavelength; u, v are the coordinates in the direction cosine coordinate system, and x, y are the coordinates of the point on the aperture plane, with the unit being m; is the blockage distribution on the aperture plane, in meters, is the illumination, δ(x,y,dz) is the additional phase caused by defocus, represents the aperture phase distribution, dz is the axial offset, and a positive value of dz corresponds to the movement of the sub-reflector away from the main reflector.

[0163] Step S5: Compare the observed power data set with the beam output value of the theoretical far-field model of the planet to calculate the redundancy value.

[0164] The redundancy value is the value of the actual observation data at the same coordinate position on the aperture plane minus the beam output value of the theoretical far-field model.

[0165] Step S6: Determine whether the coefficients of the current Zernike polynomial are the optimal solution using the current redundancy value and the least squares method. If so, proceed to step S9 (the coefficients of the Zernike polynomial output at this time can represent the phase model); otherwise, use the least squares method to minimize the redundancy value to update the coefficients of the Zernike polynomial, and then execute step S7 to continue iteration.

[0166] The current redundancy value is input into a least squares algorithm to determine whether it is less than a preset threshold value of the least squares algorithm, thereby determining whether the coefficients of the current Zernike polynomial are an optimal solution (if they are less than the preset threshold value, the coefficients of the current Zernike polynomial are an optimal solution). If the preset threshold value of the least squares algorithm is met, the process proceeds to step S9.

[0167] In this embodiment, the least square method used is the Levenberg-Marquardt (LM) algorithm.

[0168] Step S7: If the current radial order n is less than the maximum radial order M, then both the radial order n and the iteration count i are incremented by 1 (i=i+1 means n=n+1), and step S8 is executed to continue iteration; otherwise, step S9 is executed;

[0169] Step S8: According to the coefficients of all Zernike polynomials {a n,l} Obtain an updated phase model, then update the theoretical far-field model of the planet, and interpolate to obtain the beam output value of the theoretical far-field model; then return to step S5.

[0170] Update the theoretical far-field model of the planet, specifically including: interpolating the phase model to obtain three sets of aperture phase matrices; using the aperture model to combine the aperture phase matrix with the additional phase matrix δc(i,j,dz) and the illumination matrix Ic(i,j) to generate three sets of aperture field distribution functions, and then, respectively update the three theoretical far-field models according to the three sets of aperture field distribution functions. Among them, the aperture phase matrix corresponds to the aperture phase distribution function in the aperture model. , the additional phase matrix δc(i,j,dz) corresponds to the additional phase δ(x,y,dz) generated by the defocus in the aperture model, and the illumination matrix Ic(i,j) corresponds to the illumination in the aperture model .

[0171] When the three theoretical far-field model beam output values ​​are interpolated, these beam output values ​​are interpolated to the positions of the sampling points of each power data set through Gaussian function interpolation.

[0172] In this embodiment, each aperture phase matrix obtained by interpolation is a 192×192 matrix. Interpolation here means first flattening the aperture into a 192×192 matrix corresponding to the sampling points of the observed power data set on the aperture plane, and then interpolating the additional phase matrix and illumination matrix corresponding to the sampling points on the aperture plane.

[0173] The present invention only performs the calculation on the coefficients {a n,l} is updated without changing the remaining additional phase matrix δc(i,j,dz) and illumination matrix Ic(i,j).

[0174] Based on actual experience, the present invention can derive a theoretical far-field model of a planet (i.e., the first and second spectral maps), thereby inferring the aperture field distribution function. The aperture model and theoretical far-field model are initially determined. The theoretical far-field model is derived through actual planetary scanning experience, such as the code planetFF in a virtual machine. When the redundancy value obtained by subtracting the beam output value of the theoretical far-field model from the observed power data reaches a threshold, the Zernike polynomial can represent the aperture phase. When the redundancy value is large, the coefficients of the Zernike polynomial are reallocated, and the aperture model and the theoretical far-field model of the planet are subsequently modified until the redundancy value reaches the threshold.

[0175] Step S9: According to the coefficients of the current Zernike polynomial {a n,l}Determine the optimal phase model, determine the phase error on the aperture plane based on the phase model, and then determine the gravity deformation of the main reflector.

[0176] When the radial order n is greater than or equal to 2, the final Zernike polynomial coefficients {a n,l} Calculate the phase error on the aperture plane. Finally, calculate the gravity deformation of the main reflector through these phase errors.

[0177] According to the prior art, the calculation formula for phase error is:

[0178]

[0179] Where λ is the wavelength, x, y are the coordinates of the point on the aperture plane, F is the focal length of the main reflection surface, Z n,l (x,y) is a Zernike polynomial, a n,l are the coefficients of the Zernike polynomial, n is the radial order, l is the angular order, and phase{} represents the phase function, which returns the phase angle (angle) of a complex number, that is, the argument (arg) of the complex number. Phase is an angle that represents the direction of the complex number relative to the real axis in the complex plane.

[0180] The calculation formula for the gravity deformation of the main reflector is:

[0181]

[0182] Among them, a n,l (E) represents a Zerneike polynomial, a, b, and k are coefficients, and E represents the independent variable of the Zerneike polynomial.

[0183] Through the above steps, the planetary OOF algorithm can accurately calculate the phase error of the telescope aperture, providing an important basis for further optimizing telescope performance. This method achieves precise measurement and correction of telescope surface errors by improving the signal-to-noise ratio and optimizing the calculation process.

[0184] Experimental results:

[0185] The defocus holographic measurement method of the radio telescope based on planetary zoom of the present invention is implemented on a computer. In this experiment, the software system of the computer is composed of two program modules, namely the OOF pre-processing program module and the OOF post-processing program module. Figure 4 As shown, by inputting the original DIBAS data (fits file) and preprocessing it through the OOF preprocessing program module, three fits files are obtained, and the phase error map on the aperture plane is obtained through post-processing.

[0186] Among them, the input data of the OOF preprocessor module is a raw data fits file that includes uncalibrated power, integration time and other information, and the output data is a fits file that includes x-axis offset, y-axis offset, power, time, that is, the required observed power data set.

[0187] As described above, the OOF preprocessor module's workflow is as follows: It obtains raw data fits files from DIBAS, including power, time, azimuth, elevation, and displacement information for both focused and defocused data. The raw data is preprocessed using the OOF preprocessor module's rawBkDataReader.py function. Empty lists pData and integers are defined. The pyfits.open function is used to open the specified FITS data file, retrieve the data header information and the data itself. The TDIM keyword in the header is printed, and the TDIM3 value is output. Next, the data is processed line by line, extracting the timestamp and data. A series of calculations and processing are performed, the results are stored in the corresponding lists, and a portion of the integer list is printed. The DIBAS data obtained from DIBAS is then preprocessed. This includes calculating the DIBAS data mean and center frequency, obtaining raw Tcal data, calculating average Tcal data, average gain, system temperature, and the cal value (i.e., calibration value) for power processing. Subsequently, rawTelDataReader.py is used to interpolate the azimuth, elevation, and time values ​​to process the antenna data. Then, the processed power data and antenna data are fitted using mergeData.py, baseline processing is performed using oof_bk.py, and finally, planetary deconvolution is performed using deconvolve.py. Three fits files are ultimately obtained, including x-axis offset, y-axis offset, power, and time. These files serve as the observed power data set in the defocus holographic measurement method of the present invention and are also the input files for the OOF post-processing program module. The specific workflow of the OOF post-processing program module is as follows:

[0188] Step 2: Create a dataset mapres (pyoof.MapToResidualDS) based on the telescope information in the observation file, sky map sample, and interpolation parameters by using the MkMapResDS function with the pre-processed FITS file as the input file; output ps (pyoof.ObsDefocus) based on the telescope information in the observation file and aperture sampling by using the MkPhaseScreen function; add mapres and ps to oc (the object for comparing observations and far-field response models) by using the LoadOCData function; output FF (far-field object) (pyoof.PlanetFF) based on the telescope and chopthorw information in the observation file and the information of the telescope aperture plane electric field distribution according to the telescope name, receiver type, and observation source selection by using the MkFF function; generate the aperture distribution function (aperture) based on the observation file, image resolution, maximum Zernike order to be fitted, oversampling factor, fwhm, extent, and receiver type and observation number by using the MkObsCompare function; call the MkFF function to create the far-field antenna response object (ff) and output oc (for comparing observations and models) (pyoof.ObsCompare) based on aperture and ff; call the function (pybnmin1.LMMin) to fit the oc object using the least squares method LM, and save the fitting results to fitspars.txt; call the functions (pyoof.WriteAperture, pyoof.WriteBeams, pyoof.WriteAperture,) to write the aperture information (aperture), beam information (fitbeam.fits), and non-tilted aperture information (aperture-notilt.fits) in the oc object to the fits file and save them.

[0189] Finally, pass the oc object to Lmm, write the oc object containing perfect beam information to the (perfectbeams.fit) file by using (pyoof.WriteBeams), output fitpars.txt (the file of fitting parameters); and traverse the Zernike order by using the Red function, and use the highest order to fit and output.

[0190] The final output document of the program: This program finally gets the oofout file, each opened as a file from z1 to z6 (because the maximum Zernike order to be processed is 6) Each opened has (aperture.fits (aperture distribution), aperturenotilt.fits (aperture distribution of the telescope without tilt), fitsbeams.fits (beam information), fitpars.txt (file of fitting parameters), OOF phase error distribution.

[0191] The present invention's defocus holographic measurement method for a planetary zoom radio telescope uses deconvolution to obtain the beam pattern of an actual observed planet. Using the planet as a source, the defocus holographic measurement is performed and a theoretical model is established. The theoretical far-field model of the planet is compared with the observed beam pattern to determine the telescope's apparent error, which is then used to assess the calibration accuracy of the Tianma radio telescope. Planets are individual sources with sufficient flux and a high signal-to-noise ratio, which improves calibration accuracy. Furthermore, the precise location of the planetary source supports calibration across multiple frequency ranges, which is crucial for improving telescope performance.

[0192] Furthermore, because existing technologies lack data processing workflows for planetary sources, software for defocused holographic measurements faces difficulties in establishing theoretical far-field models using planetary sources, as well as convolution of observational data using planetary sources. This present invention optimizes the code for defocused holographic measurement software to establish a theoretical model using planetary sources.

[0193] Compared with the existing technology, the present invention has the following advantages:

[0194] 1) Higher measurement accuracy: Using planets as radio sources, which have high flux and high signal-to-noise ratio, significantly improves measurement accuracy, makes measurements faster, improves the efficiency and accuracy of astronomical observations, and provides more reliable data support for astronomical research.

[0195] 2) All-weather measurement capability: Planetary observations are less susceptible to weather, enabling all-weather measurement and improving measurement stability and reliability. Traditional methods often fail to provide high-precision and reliable measurement results under complex environmental conditions, which negatively impacts the efficiency and accuracy of astronomical observations.

[0196] 3) Calibration support for multiple frequency ranges: The precise location of planetary sources supports calibration in multiple frequency ranges, further enhancing the applicability and flexibility of the telescope in different observation missions.

[0197] 4) Improved real-time measurement: Using planets as radio sources improves real-time measurement, enabling high-quality data to be obtained even under dynamic conditions.

[0198] 5) Reduced Equipment Dependency: This technical solution eliminates the need for additional hardware, reducing equipment costs and maintenance difficulties, making it accessible to more research institutions. It also improves telescope calibration accuracy and stability, extending equipment life and further reducing long-term operating costs.

[0199] In summary, compared with the existing defocus holographic measurement technology, the defocus holographic measurement method of the radio telescope based on planetary zoom of the present invention not only improves the measurement accuracy and stability, but also enhances the real-time measurement effect and multi-frequency calibration capability by introducing planets as radio sources, significantly improving the calibration accuracy and overall performance of the Tianma radio telescope, and solving many limitations of traditional surface measurement methods, such as slow measurement speed, the need for additional hardware equipment, great influence from weather, and limited scope of application.

[0200] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of the present invention. Various modifications are possible. Any simple, equivalent changes and modifications made in accordance with the claims and description of the present invention are within the scope of protection of the patent claims. Anything not fully described in this invention is conventional technology.

Claims

1. A defocus holographic measurement method for a radio telescope based on planetary zoom, characterized in that: include: Step S0: Set the resolution grid [xy] to achieve aperture discretization, set the iteration count value i=0, and establish the aperture model, amplitude model and phase model; Step S1: Using the planet as the source, the sub-reflector is moved to perform defocus observation to obtain an observed power dataset, and the deconvolved observed power dataset, wavelength, and axial offset are used as input data for the model; Step S2: According to the calculation formula and amplitude model of the additional phase generated by defocus, the additional phase matrix and illumination matrix are interpolated corresponding to the coordinates of each sampling point in the power data set; Step S3: Initialize model parameters; Step S4: Establishing a theoretical far-field model of the planet; determining the spectrum map of the planet based on the given planet map data, and performing frequency domain convolution based on the FFT transform to determine the beam output value of the theoretical far-field model of the planet; Step S5: Compare the observed power data set with the beam output value of the theoretical far-field model to calculate the redundancy value; Step S6: Determine whether the coefficients of the current Zernike polynomial are the optimal solution by using the current redundancy value and the least square method. If so, proceed to step S9; Otherwise, minimize the redundancy value to update the coefficients of the Zernike polynomial, and then execute step S7; Step S7: If the current radial order n of the Zernike polynomial is less than the maximum radial order M, the radial order n and the iteration count value i are both incremented by 1, and step S8 is executed; otherwise, step S9 is executed; Step S8: Obtain an updated phase model based on the coefficients of the Zernike polynomials, and then update the theoretical far-field model of the planet, and interpolate to obtain its beam output value; Return to step S5; Step S9: determining an optimal phase model according to the coefficients of the Zernike polynomial, and determining the phase error on the aperture plane according to the phase model; In step S1, the observed power data set is obtained by performing flight scanning by moving the sub-reflector and undergoing preprocessing. The preprocessing of the power data set includes: Calculate the average value and center frequency of DIBAS data, obtain the original Tcal data, calculate the average Tcal data, average gain, system temperature and cal value to process the power data; Process antenna data by interpolating azimuth, elevation, and time; Fit the processed power data and antenna data and perform baseline processing; Performing deconvolution processing to obtain a deconvolution-processed observed power data set; In step S4, the spectrum map of the planet includes a first spectrum map and a second spectrum map; Frequency domain convolution includes: Step A1: First, check whether the dimensions of the first spectrum map and the second spectrum map are the same. If the dimensions are not the same, end the process; Step A2: extracting the amplitude distribution and phase distribution of the first spectrum map and the second spectrum map respectively, and then initializing the amplitude and phase to zero; Step A3: transform the amplitude distribution and phase distribution using inverse Fourier transform; Step A4: multiplying the transformed amplitude distribution and phase distribution to obtain a product result; Step A5: Perform an inverse transform on the product result using a forward Fourier transform to obtain an inverse transform result; Step A6: Divide the inverse transformation result by the size of the spectrum map and multiply it by the cosine value of the phase to obtain the spectrum map of the convolution result. The spectrum map of the convolution result is the beam output value of the far-field theoretical model.

2. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 1, characterized in that: The steps of deconvolution processing include: Step B1: Build a convolution model of the observed image; Step B2: Perform Fourier transform on the observed image and the point spread function of the telescope to obtain the spectrum values ​​of the observed image and the point spread function, and obtain the relationship between the two based on the convolution model of the observed image; Step B3: Estimate the spectrum of the original image by inverse filtering based on the spectrum values ​​of the observed image and the point spread function and the relationship between the two; Step B4: Perform inverse Fourier transform on the estimated spectrum of the original image to obtain a deconvolved image.

3. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 1, characterized in that: The deconvolution algorithm used in the deconvolution process is Wiener filtering, Lucy-Richardson deconvolution or maximum entropy method.

4. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 1, characterized in that: In step S0, the amplitude model is: ， Where I (r) is the illumination; I0 is the amplitude of the illumination to be fitted, in m; It represents the angle between the optical axis and the light from the focus to the sub-reflector, and the unit is rad. for The maximum value of σ r is the lighting cone; The phase model is: ， is the aperture phase distribution, is the coefficient to be determined, n is the radial order, l is the angular order, n=1~max, Z n,l (x,y) is a Zernike polynomial; The caliber model is: ， is the aperture field distribution function, the unit is w, x, y are the coordinates of the point on the aperture plane, is the blockage distribution on the aperture plane, in meters, is the illumination, is the aperture phase distribution, its unit is rad, δ(x,y,dz) represents the additional phase caused by defocus, dz is the axial offset, its unit is mm; In step S2, the calculation formula for the additional phase generated by defocus is: , Where r is the cross-sectional radius of the telescope that varies with altitude, in meters; dz is the axial offset, in millimeters; λ is the wavelength, in millimeters, and θ is the wavelength. r is the angle between the optical axis and the ray from the focus to the sub-reflector, It is the angle between the incident beam of the telescope after one reflection and the main axis of the telescope.

5. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 1, characterized in that: The step S3 specifically includes: setting the Zernike polynomial Z n,l The parameters (x, y) are the coordinates of the point on the aperture plane, n is the radial order, and l is the angular order; the initial value of the amplitude I0 of the illumination to be fitted in the amplitude model is set to 0; the preset threshold of the least squares algorithm is set to the default value; Set the Zernike polynomial Z n,l The parameters of (x,y) specifically include: setting the Zernike polynomial Z n,l (x,y) the maximum radial order M; all the coefficients of the Zernike polynomial {a n,l } is set to 0; the current radial order n is initially set to 1.

6. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 5, characterized in that: In step S6, the current redundancy value is input into the least squares algorithm to determine whether a preset threshold of the least squares algorithm is satisfied, thereby determining whether the coefficients of the current Zernike polynomial are the optimal solution.

7. The defocus holographic measurement method of a radio telescope based on planetary zoom according to claim 1, characterized in that: In step S8, the theoretical far-field model of the planet is updated, specifically including: interpolating the phase model to obtain three sets of aperture phase matrices; using the aperture model to combine the aperture phase matrix with the additional phase matrix δc(i,j,dz) and the illumination matrix Ic(i,j) to generate three sets of aperture field distribution functions; then, updating the three theoretical far-field models according to the three sets of aperture field distribution functions, where i and j are the sampling point coordinates and dz is the axial offset.

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