Method for calculating laser beam scattering based on jonswap spectrum sea surface modeling

Abstract

The application discloses a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling, and is implemented according to the following steps: setting a simulation scene, determining a JONSWAP spectrum and a direction distribution function, and calculating a two-dimensional sea wave frequency direction spectrum; based on the two-dimensional sea wave frequency direction spectrum, a complex sea surface model is established through a linear superposition method and is discretized to obtain a discretized model; according to the discretized model, a coordinate transformation formula between the sea surface and the laser beam is obtained based on coordinate transformation; a Kirchhoff approximation formula is derived, and the calculation of the scattering cross section of the sea surface to the laser beam is completed according to the coordinate transformation formula between the sea surface and the laser beam; the random complex sea surface model is obtained based on the JONSWAP spectrum and the linear superposition method, the natural development law of the sea surface is met, and the complex sea surface can be used to analyze the scattering cross section of a typical laser beam, especially the influence of the beam incidence angle, the beam waist radius and the parameters of the laser beam on the scattering cross section.

Description

Technical Field

[0001] This invention belongs to the technical field of marine microwave remote sensing methods, specifically relating to a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling. Background Technology

[0002] Compared to traditional radio guidance, laser guidance offers advantages such as high precision, good range cutoff characteristics, strong resistance to electromagnetic interference, and strong directional detection field of view. However, due to its operating wavelength, laser guidance technology is also significantly affected by meteorological conditions, especially complex marine environments. When a laser beam propagates in the marine atmosphere, atmospheric turbulence caused by random fluctuations in atmospheric refractive index leads to phenomena such as laser beam jitter, intensity fluctuations, spot drift, and beam expansion. These atmospheric turbulence effects cause laser signal attenuation. Furthermore, during laser beam transmission in the marine atmosphere, various gas molecules and marine aerosol particles absorb and scatter the laser beam, affecting its energy distribution. Among the factors causing laser attenuation, the scattering, absorption, and attenuation effects of marine aerosol particles have the greatest impact on energy loss and transmission characteristics. Additionally, the complex dynamic sea surface can lead to a decrease in the accuracy of target detection and identification based on laser guidance. In conclusion, the complex marine environment has become a bottleneck restricting the development and application of laser guidance technology.

[0003] In recent years, with the advancement of laser control technology, various laser beams with special amplitude, phase, and polarization state distributions have been proposed and realized. These typical laser beams exhibit a series of novel physical effects and phenomena. For example, laser beams with specific amplitudes can suppress the influence of ocean atmospheric turbulence on their transmission; laser beams with vortex phases carry orbital angular momentum, enabling the modulation of more information within the same frequency band; and vector beams with polarization state-dependent spatial distribution provide the possibility of improving the laser-guided identification capability of maritime targets. Due to the scattering effect of the polarization degree of the target scattered polarized light on the scattering medium and the sensitivity of the target material, the information provided by the polarization state can effectively distinguish scatterers of different materials and surface morphologies. Compared with traditional light intensity imaging, scattered polarized light imaging has the advantages of improving target contrast and distinguishing materials, greatly enhancing the ability to distinguish target information in all weather and complex environments.

[0004] Therefore, the study of electromagnetic scattering of plane waves by the sea surface has become a hot topic in physical oceanography; however, there is no existing technology for the scattering of laser beams by the sea surface, especially for the scattering of complex laser beams. Summary of the Invention

[0005] The purpose of this invention is to provide a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling. It has the characteristics of starting from the construction of a complex sea surface model, analyzing the reflection and scattering of the laser beam by the sea surface with the changes of beam incident height, polarization mode and beam waist radius parameters, and realizing the effective calculation of laser beam scattering by the sea surface.

[0006] The technical solution of this invention is a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling, which is implemented according to the following steps:

[0007] Step 1: Set up the simulation scene, determine the JONSWAP spectrum and directional distribution function, and calculate the two-dimensional wave frequency directional spectrum; Step 2: Based on the two-dimensional wave frequency directional spectrum, establish a complex sea surface model using the linear superposition method and discretize it to obtain the discretized model; Step 3: Based on the discretized model, derive the coordinate transformation formula between the sea surface and the laser beam based on coordinate transformation; Step 4: Derive the Kirchhoff approximation formula, and calculate the scattering cross section of the laser beam from the sea surface according to the coordinate transformation formula between the sea surface and the laser beam.

[0008] The invention is further characterized by the following steps in step 1: Step 1.1: Set up the simulation scene, select the JONSWAP spectrum for initial sea surface fitting, and the expression for the JONSWAP spectrum is...

[0009]

[0010] Where a is the dimensionless energy scale parameter, g is the local gravitational acceleration, and ω p γ is the peak frequency, γ is the spectral peak factor, F is the wind region, and U is the peak frequency. 10 σ represents the average wind speed at a distance of 10 meters above the sea surface, and σ is the peak shape parameter.

[0011] Step 1.2: Introduce the directional distribution function to describe the energy distribution of ocean waves relative to the directional angle. Combine different ocean spectral functions with the directional distribution function, i.e., the JONSWAP two-dimensional ocean spectrum is...

[0012] S(ω,θ)=S(ω)G(θ) (2)

[0013] Where G(θ) is the directional distribution function and S(ω,θ) is the directional spectrum function;

[0014] Step 1.3: For the JONSWAP two-dimensional ocean spectrum, the photoyiic distribution function is adopted, i.e.

[0015]

[0016] Where, θ sea The direction of wave propagation is the dominant direction, and D0(s) is a coefficient determined by the direction distribution concentration parameter s; the formula for calculating D0(s) is:

[0017]

[0018] Where Γ is the gamma function;

[0019] Step 1.4: According to formula (2), the two-dimensional wave frequency spectrum is obtained as follows:

[0020]

[0021] Step 2 is implemented in the following steps:

[0022] Step 2.1: Define the three-dimensional sea surface height using the linear superposition method, and based on the two-dimensional wave frequency spectrum S(ω,θ), use computer and MATLAB software to simulate and establish a complex sea surface model that effectively reflects the actual wave motion. The three-dimensional sea surface height is defined as...

[0023]

[0024] Where M and N are the discrete quantities of angular frequency and direction angle, respectively, ω is the angular frequency, and θ is the direction angle. n Let a be the nth direction angle. mn and b mn Let b be the amplitude and phase angle at the m-th angular frequency and the n-th direction angle. mn The wave phase angle;

[0025] Step 2.2: Based on the complex sea surface model, discretize it to reconstruct a sea surface model with smaller errors than the actual sea surface and capable of numerical calculation, i.e., the discretized model.

[0026] Step 2.2 is implemented according to the following steps:

[0027] Step 2.2.1: Divide the horizontal scale into equal parts, and label the points in the manner of first the horizontal axis and then the vertical axis. Record the spatial coordinate information of each point (x,y,z) in the following format.

[0028] Step 2.2.2: Every three points form a triangular element. Each element records the point numbers in counter-clockwise order. This operation can obtain two data files: one that records the node numbers of the triangular elements and the other that records the coordinate information of the nodes.

[0029] Step 2.2.3: Based on the two data files, reconstruct a sea surface model with small errors compared to the actual sea surface and capable of numerical calculations.

[0030] Step 3 is implemented in the following steps:

[0031] Step 3.1: Establish a sea surface coordinate system. The node numbers and coordinates of the triangular elements are obtained from the data file. Then, establish a local coordinate system to obtain the reflection of any triangular element under any incident ray.

[0032] Step 3.2: Introduce the interaction probability density function, which is defined as the product of the shadow function of whether the light source illuminates the surface element and the hidden function of whether the reflected light of the surface element can be received;

[0033] Step 3.3: Calculate the light field intensity of the beam emitted from the given laser position on the surface element, and perform coordinate transformation to satisfy the global coordinate system of the sea surface.

[0034] Step 3.1 The specific calculation formula is as follows: the three nodes of the triangular element are N1(x1,y1,z1), N2(x2,y2,z2), and N3(x3,y3,z3) in a counterclockwise order, U n (x n ,y n ,z n U is the normal unit vector of the surface element. i (x i ,y i ,z i () is the unit vector representing the incident direction of the light ray. U represents the zenith angle and azimuth angle of the incident light ray. r (x r ,y r ,z r () is the unit vector representing the direction of light reflection. θ represents the zenith angle and azimuth angle of the reflected light. ω The angle between the incident or reflected direction vector and the surface element normal vector is the incident angle or reflection angle; the parameters satisfy the following relationship:

[0035]

[0036]

[0037] A method for calculating the surface element normal vector and incident ray direction vector in the global coordinate system is presented. Using the specular reflection law, the corresponding reflected ray direction vector can be calculated. The specific formula is as follows:

[0038] U i =2(U r ·U n )U n -U r (9)

[0039]

[0040] Using the formula and the node coordinates of the triangular element, the reflection of any triangular element under any incident ray can be calculated.

[0041] Step 3.2 The specific calculation formula is as follows: the interaction probability density function Q is Q = S·H, and the specific expression of the function is:

[0042]

[0043]

[0044] Where S is the shadow function, U i U is the unit direction vector. n Let θ be the unit normal vector, A be the laser beam diffusion angle, B be the laser coordinates in the global coordinate system, C be the center coordinates of the surface element, A' be the position coordinates of the laser beam principal axis illuminating the sea surface, A' be the vertical projection of the laser onto the sea surface, and AC be the laser beam principal axis ray, with an angle θ between it and the z-axis. i The location of BC is at sea level, i.e., z b =z c =0, AB represents the diffuse ray, and the angle between AB and AC is the angle θ between the diffuse ray and the principal axis of the beam. c The coordinates of A and B are known quantities; the other parameters are as follows:

[0045] A'C = z a / tan(90°-θ i (13)

[0046]

[0047]

[0048]

[0049]

[0050]

[0051] cosθ c =(AC 2 +AB 2 -BC 2 ) / (2AC·AB) (19)

[0052] When θ c When θ < θ, the surface element is within the illumination range of the laser beam, therefore the form of function S is rewritten as follows:

[0053]

[0054] Step 3.3 The specific steps are as follows:

[0055] Step 3.3.1: Introduce a light field coordinate system Ou'v'w' with the same origin as the sea surface coordinate system. Rotate this coordinate system to make it coincide with the sea surface coordinate system, and define the rotation angle as the angle between the rotation and the positive direction of the coordinate axis.

[0056] Step 3.3.2: Rotate the axis of rotation around the w' axis by an angle α, so that the v' axis is in the xoy plane.

[0057] Step 3.3.3: Rotate the axis of rotation around the v' axis by an angle β, so that the w' axis coincides with the z axis.

[0058] Step 3.3.4: Rotate the coordinate system Ou'v'w' around the central axis by an angle γ, so that the Ou'v'w' coordinate system coincides with the Oxyz coordinate system. α, β, and γ are called the Euler angles of rotation. The transformation relationship between the Ou'v'w' coordinate system and the Oxyz coordinate system can be expressed in matrix form as follows:

[0059]

[0060] In equation (21), the transformation matrix T can be expressed in terms of Euler angles as follows:

[0061]

[0062] The transformation relationship between the Ouvw coordinate system and the Oxyz coordinate system satisfies

[0063]

[0064] Step 4 is as follows: Step 4.1, Introduce Kirchhoff's approximation and introduce Green's second theorem for vectors;

[0065]

[0066] Suppose that vector F exists in the following two forms: Substituting them into formula (24) and simplifying, we get:

[0067]

[0068] Let vector A be the electric field E, and vector B be any constant vector multiplied by the dyadic Green's function. Substituting the electric field and the dextral Green's function into formula (25), the result simplifies to...

[0069]

[0070] Step 4.2: Solve for the vertical and horizontal polarization components of the incident electromagnetic field in the local coordinate system;

[0071] make It is a unit vector perpendicular to the incident plane and pointing outwards. It is a unit vector parallel to the plane of incidence. The normal unit vector of the surface element. The unit vector of the incident direction. If the reflection direction is a unit vector, then and They form an orthogonal coordinate system, and satisfy the following relationship:

[0072]

[0073]

[0074]

[0075] Let the incident electric field be E i The incident magnetic field is H i If the polarization mode is arbitrary, the steps to solve the vertical polarization component of the incident electromagnetic field in the local coordinate system should be to first solve the polarization of the incident electric field in the global coordinate system, and then take the vertical component in the local coordinate system to solve the horizontal polarization component.

[0076] Let E ρ The incident electric field is a polarized electric field, and its perpendicular polarization component is... Its horizontal polarization component is The perpendicular polarization component of the incident magnetic field is Its horizontal polarization component is

[0077] Step 4.3: Based on the expressions for the vertical polarization components and the horizontal polarization components of the incident electric field and the reflected electric field, obtain the total tangential electromagnetic field;

[0078] Step 4.4: Based on the total tangential electromagnetic field, solve for the scattered field at any point above the sea surface. The mathematical expression for the scattering cross section in a specific direction is defined as follows:

[0079]

[0080] Step 4.3 The specific steps are as follows:

[0081] When specular reflection is satisfied, the incident direction With the direction of reflection Satisfying Relationship: The vertical polarization component of the reflected electric field is Its horizontal polarization component is The vertical polarization component of the reflected magnetic field is Its horizontal polarization component is

[0082] Based on the expressions for the vertical and horizontal polarization components of the incident and reflected electric fields, the total tangential electromagnetic field can be obtained as follows:

[0083]

[0084]

[0085]

[0086] The beneficial effects of this invention are as follows: Starting from the basic method of constructing a complex sea surface model, this invention uses the JONSWAP spectral function and the optically similar directional distribution function as models to simulate a two-dimensional random sea surface. Based on this, it provides the steps for establishing and discretizing the complex sea surface, employs the linear superposition method to discretize the interpolated two-dimensional random sea surface, and, based on coordinate transformation theory, gives the coordinate transformation formula between the sea surface and the laser beam. Simultaneously, it derives the Kirchhoff approximation formula, realizing the calculation of the scattering cross-section of the laser beam from the sea surface. The random complex sea surface model obtained by this invention based on the JONSWAP spectrum and the linear superposition method satisfies the natural development law of the sea surface and has physical feasibility. It can be used to analyze the scattering cross-section of typical laser beams from complex sea surfaces, especially the influence of beam incident angle, beam waist radius, and beam parameters on the scattering cross-section. The method of this invention for realizing the scattering of typical laser beams from complex sea surfaces has significant application value in fields such as marine remote sensing, marine resource exploration, and sea surface target detection and identification. Attached Figure Description

[0087] Figure 1 This is a flowchart of the laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling of the present invention;

[0088] Figure 2 This is a two-dimensional sea surface simulation image generated using the linear superposition method in the calculation method of this invention;

[0089] Figure 3 This is a schematic diagram of the discretization principle of the two-dimensional sea surface in the calculation method of this invention;

[0090] Figure 4 This is the data file format for discrete storage node information in the calculation method of this invention;

[0091] Figure 5 This is the data file format for discrete encoded surface element information in the calculation method of this invention;

[0092] Figure 6 This is a discrete graph of a random two-dimensional sea surface under different wind speeds in the calculation method of this invention, where the wind speed is U. 10 =5m / s;

[0093] Figure 7This is a discrete graph of a random two-dimensional sea surface under different wind speeds in the calculation method of this invention, where the wind speed is U. 10 =10m / s;

[0094] Figure 8 This is a schematic diagram of the triangular surface element coordinate system and the representation of each parameter in the calculation method of this invention;

[0095] Figure 9 This is a schematic diagram of the three-dimensional spatial distribution of the laser and surface elements in the calculation method of this invention;

[0096] Figure 10 This is a schematic diagram of the light field coordinate system in the calculation method of this invention;

[0097] Figure 11 This is a schematic diagram of the sea surface coordinate system in the calculation method of this invention;

[0098] Figure 12 This is a flowchart of the first step in the transformation of the coordinate system Ou'v'w' to Oxyz in the calculation method of this invention;

[0099] Figure 13 This is a flowchart of the second step in the calculation method of this invention, which transforms the coordinate system Ou'v'w' to Oxyz.

[0100] Figure 14 This is a flowchart of the third step in the calculation method of this invention, which transforms the coordinate system Ou'v'w' to Oxyz.

[0101] Figure 15 This is a flowchart of the fourth step in the calculation method of this invention, which transforms the coordinate system from Ou'v'w' to Oxyz.

[0102] Figure 16 This is a geometric schematic diagram of the sea surface coordinate system in the calculation method of this invention;

[0103] Figure 17 This is a scattering diagram of the fundamental mode Gaussian beam in the calculation method of this invention when the beam is sufficiently far from the sea surface;

[0104] Figure 18 This is a schematic diagram of the scattering cross-sections of the sea surface on the E-plane and H-plane of the beam in the calculation method of this invention;

[0105] Figure 19 The calculation method of this invention involves changing the incident zenith angle θ of the fundamental mode Gaussian beam. i Simulation diagram of the impact on sea surface scattering;

[0106] Figure 20 This is a simulation diagram showing the effect of changing the polarization of the fundamental mode Gaussian beam on sea surface scattering in the calculation method of this invention;

[0107] Figure 21Simulation diagram of the effect of changing the incident height h of the fundamental mode Gaussian beam on sea surface scattering in the calculation method of this invention;

[0108] Figure 22 Simulation diagram of the effect of changing the beam waist radius w0 of the fundamental mode Gaussian beam on sea surface scattering in the calculation method of this invention;

[0109] Figure 23 This is a simulation diagram of the scattering cross-section distribution of a Hermitian-Gaussian beam on a complex sea surface in the calculation method of this invention;

[0110] Figure 24 This is a simulation diagram of the scattering cross-section distribution of a Laguerre-Gaussian beam on a complex sea surface in the calculation method of this invention. Detailed Implementation

[0111] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0112] Example 1

[0113] like Figure 1 As shown, this invention provides a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling, which is implemented according to the following steps:

[0114] Step 1: Set up the simulation scene, determine the JONSWAP spectrum and directional distribution function, and calculate the two-dimensional wave frequency directional spectrum; Step 1.1: Set up the simulation scene, select the JONSWAP spectrum for initial sea surface fitting, the expression for the JONSWAP spectrum is,

[0115]

[0116] Where a is a dimensionless energy scale parameter, and g is the local gravitational acceleration, typically taken as 9.8 m / s². 2 ω p The peak frequency is γ, and the spectral peak factor is E, which is the peak value of the spectrum at the same wind speed. max Spectral peaks of PM spectrum The ratio between them, according to the data measured in the JONSWAP experiment, the value of γ ranges from 1.5 to 6.0, and the average value is usually taken as 3.3. ω p =22(U 10 F / g 2 ) -0.33 F represents the wind zone, referring to the area of ​​the sea surface over which the sea wind blows at a constant speed, measured in km. U 10 The average wind speed at a distance of 10 meters above the sea surface, measured in m / s. σ is a peak shape parameter, its value determined by ω and ω0. p Together, it is determined that when ω≤ω p When σ = 0.07, when ω > ω pAt that time, σ = 0.09.

[0117] Step 1.2: In reality, ocean waves are three-dimensional, multi-directional, irregular waves. A one-dimensional ocean spectrum cannot fully describe the geometric characteristics of ocean waves, so a directional distribution function is needed to describe the energy distribution of ocean waves relative to the azimuth angle. We assume that the effects of frequency and azimuth angle on energy distribution are independent; therefore, different ocean spectrum functions can be combined with the directional distribution function to describe the energy distribution of ocean waves relative to the azimuth angle. Combining different ocean spectrum functions with the directional distribution function, i.e., the JONSWAP two-dimensional ocean spectrum, is...

[0118] S(ω,θ)=S(ω)G(θ) (2)

[0119] Where G(θ) is the directional distribution function and S(ω,θ) is the directional spectrum function; both are determined by the angular frequency ω and the direction angle θ.

[0120] Step 1.3: For the JONSWAP two-dimensional ocean spectrum, the photoyiic distribution function is adopted, i.e.

[0121]

[0122] Where, θ sea The direction of wave propagation is the dominant direction, and D0(s) is a coefficient determined by the direction distribution concentration parameter s; the formula for calculating D0(s) is:

[0123]

[0124] Where Γ is the gamma function;

[0125] Step 1.4: According to formula (2), the two-dimensional wave frequency spectrum is obtained as follows:

[0126]

[0127] Step 2: Based on the two-dimensional wave frequency spectrum, a complex sea surface model is established and discretized using the linear superposition method to obtain the discretized model; Step 2.1: Establishing and discretizing the complex sea surface model. The sea surface exhibits randomness under stable sea conditions. A geometric model of the random sea surface can be established using a two-dimensional sea surface spectrum function. Through the linear superposition method, simulations are performed using a computer and MATLAB software to establish a sea surface model that effectively reflects the actual motion of the waves.

[0128] The linear superposition method, also known as the double superposition method, treats a random sea surface as the superposition of an infinite number of cosine waves with different heights, periods, phases, and directions of motion. Let H(x,y,t) be the sea surface height at any time t with coordinates (x,y). According to the principle of the linear superposition method, a sea surface with three-dimensional height can be defined as...

[0129]

[0130] Where M and N represent the discrete quantities of angular frequency and direction angle, respectively. For angular frequency ω, assume that the energy of the sea spectrum is mainly concentrated in [ω]. start ,ω end Within the range [ω], ignoring the high-frequency and low-frequency components with less energy, the interval [ω] is... start ,ω end Divide the space into M equal parts, with the width of each part being dω = (ω end -ω start ) / M; For the direction angle θ, the propagation will be in the principal direction θ sea The interval between -π / 2 and π / 2 on both sides [-π / 2+θ] sea ,π / 2+θ sea Divide the matrix into N equal parts, with each part having a width of dθ = π / N. m and ω m Let θ represent the wave number and frequency of the ocean wave at the m-th angular frequency. n Let a represent the nth direction angle. mn and b mn This represents the amplitude and phase angle at the m-th angular frequency and the n-th direction angle, with each parameter having the following specific forms: b mn =rand(0,2π). Wave phase angle b mn It is a random number that is uniformly distributed in the range [0, 2π], and can usually be implemented using the rand() function in MATLAB.

[0131] The above formulas and analysis illustrate the principle of generating a three-dimensional sea surface using the linear superposition method. Based on the adopted sea spectral function and directional distribution function, the height distribution data of a three-dimensional multidirectional irregular sea surface can be obtained by discretizing and simulating the relevant parameters using a computer. Using the JONSWAP spectrum and the optically similar distribution function, a lateral scale of 100m × 100m, a peak factor γ of 3.3, a wind zone of 20km, and a wind speed U... 10 A three-dimensional ocean wave with a speed of 5 m / s, a time t of 0, and a concentration parameter s of 25, such as Figure 2 As shown;

[0132] Step 2.2: Based on the complex sea surface model, discretize it to reconstruct a sea surface model with smaller errors than the actual sea surface and capable of numerical calculation, i.e., the discretized model.

[0133] Step 2.2.1: Divide the horizontal scale into equal parts, and label the points in the manner of first the horizontal axis and then the vertical axis. Record the spatial coordinate information of each point (x,y,z) in the following format.

[0134] Step 2.2.2: Every three points form a triangular element. Each element records the point numbers in counter-clockwise order. This operation can obtain two data files: one that records the node numbers of the triangular elements and the other that records the coordinate information of the nodes.

[0135] Step 2.2.3: Based on the two data files, reconstruct a sea surface model with small errors compared to the actual sea surface and capable of numerical calculations.

[0136] A realistic and effective random two-dimensional sea surface can be constructed using the linear superposition method. This invention studies laser scattering on the sea surface, exploring the influence of complex sea surfaces on laser beams based on numerical methods. It requires knowledge of the wave height at any (x,y) coordinate point on the sea surface; therefore, the interpolated and fitted two-dimensional random sea surface needs to be discretized. For example... Figure 3 , Figure 4 , Figure 5 As shown, taking a 100m × 100m sea area as an example, the horizontal scale is divided into 100 equal parts. Points are labeled first along the horizontal axis and then along the vertical axis. The spatial coordinates of each point are recorded in the format (x, y, z). Every three points form a triangular element, and the point numbers are recorded in counter-clockwise order within each element. This operation yields two files: one recording the node numbers of the triangular elements, and the other recording the coordinate information of the nodes. Based on these two data files, we can reconstruct a sea surface model with minimal error compared to the actual sea surface and suitable for numerical calculations.

[0137] like Figure 6 , Figure 7 As shown, the wind speed U at 10m above the sea surface is given. 10 Comparing the wind speeds of 5 m / s and 10 m / s, it is evident that the frequency of sea surface fluctuations increases dramatically with increasing wind speed. At lower wind speeds, the sea surface is relatively stable with small fluctuations, rapid local changes, short wave attenuation distances, and an overall height range below 2 meters. As wind speed increases, the vertical fluctuations of the sea surface become more pronounced, with a significantly higher number of large fluctuations compared to the lower wind speed scenario. Local changes are slower, wave attenuation distances are longer, and the overall height range remains below 5 meters. In conclusion, the random complex sea surface obtained using the JONSWAP spectrum and double superposition method conforms to the natural development laws of the sea surface, possesses physical feasibility, and provides a foundation for subsequent calculations of laser beam reflection and scattering on complex sea surfaces.

[0138] Step 3: Based on the discretized model, and using coordinate transformation, obtain the coordinate transformation formula between the sea surface and the laser beam;

[0139] To unify the beam coordinate system and the sea surface coordinate system, taking a fundamental Gaussian beam as an example, the beam is discretized into a large set of rays. Based on the geometric position of each ray propagating to the sea surface, the direction and intensity of the reflected light from the incident surface element are calculated using geometrical optics principles and Fresnel's law, establishing a mathematical model of the reflection of the fundamental Gaussian beam onto the complex sea surface. By statistically analyzing the direction and intensity of the reflected rays in three-dimensional space and superimposing them, the spatial distribution of reflected light above the complex sea surface can be obtained.

[0140] This invention discretizes the simulated random sea surface into many small triangular elements. When the elements are small enough, they can be regarded as ideal planes. Then, the sea surface reflection of the laser beam is calculated using these independent elements as reflecting surfaces.

[0141] Step 3.1: Establish a sea surface coordinate system. The node numbers and coordinates of the triangular elements are obtained from the data file. Then, establish a local coordinate system to obtain the reflection of any triangular element under any incident ray.

[0142] like Figure 7 As shown, the three nodes of the triangular element, arranged counterclockwise, are N1(x1,y1,z1), N2(x2,y2,z2), and N3(x3,y3,z3), U n (x n ,y n ,z n U is the normal unit vector of the surface element. i (x i ,y i ,z i () is the unit vector representing the incident direction of the light ray. U represents the zenith angle and azimuth angle of the incident light ray. r (x r ,y r ,z r () is the unit vector representing the direction of light reflection. θ represents the zenith angle and azimuth angle of the reflected light. ω The angle between the incident or reflected direction vector and the surface element normal vector is the incident angle or reflection angle; the parameters satisfy the following relationship:

[0143]

[0144]

[0145] A method for calculating the surface element normal vector and incident ray direction vector in the global coordinate system is presented. Using the specular reflection law, the corresponding reflected ray direction vector can be calculated. The specific formula is as follows:

[0146]

[0147] Using the formula and the node coordinates of the triangular element, the reflection of any triangular element under any incident ray can be calculated.

[0148] When a laser beam shines on a complex sea surface, it can only illuminate a portion of the discrete surface elements due to the beam divergence angle. In reality, due to the height difference between waves, there are situations where the incident light cannot reach the surface elements or the reflected light cannot propagate upwards. These triangular surface elements, which cannot contribute to the final three-dimensional light intensity distribution, should be discarded in the calculation. To address this, the interactive probability density function Q is introduced.

[0149] Step 3.2: Introduce the interaction probability density function, which is defined as the product of the shadow function of whether the light source illuminates the surface element and the hidden function of whether the reflected light of the surface element can be received;

[0150] The specific calculation formula is as follows: the interaction probability density function Q is Q = S·H, and the specific expression of the function is:

[0151]

[0152]

[0153] The shadow function S is not only the unit direction vector U of the incident ray. i The unit normal vector U of the surface element n The function is also related to the diffusion angle θ of the laser beam, which is used to determine whether the surface element is within the irradiation range of the laser beam. For example... Figure 9 As shown, the laser coordinates A, the center coordinates B, and the position coordinates C of the laser beam principal axis illuminating the sea surface in the global coordinate system form a triangle. A' is the vertical projection of the laser onto the sea surface, and AC is the laser beam principal axis ray, with an angle θ between it and the z-axis. i The location of BC is at sea level, i.e., z b =z c =0, AB represents the diffuse ray, and the angle between AB and AC is the angle θ between the diffuse ray and the principal axis of the beam. c The coordinates of A and B are known quantities; the specific forms of the remaining parameters are as follows:

[0154] A'C = z a / tan(90°-θ i (13)

[0155]

[0156]

[0157]

[0158]

[0159]

[0160]

[0161] When θ c When θ < θ, the surface element is within the illumination range of the laser beam, therefore the form of function S is rewritten as follows:

[0162]

[0163] In summary, the geometric ray reflection problem for any effective triangular element has been solved. The remaining task is to calculate the light field intensity of the fundamental mode Gaussian beam emitted from a given laser position on the element. The coordinate system of the fundamental mode Gaussian beam is its own light field coordinate system, and its propagation direction is along the positive z-axis. A coordinate transformation is required to satisfy the global coordinate system of the sea surface.

[0164] like Figure 10 As shown, the optical field coordinate system is Ouvw for the fundamental mode Gaussian beam, and the beam waist center is at coordinate O(x0, y0, z0) in the sea surface coordinate system. Figure 11 As shown, the sea surface coordinate system is Oxyz, with the center coordinates being O(0,0,0). Now, we need to convert the coordinates in any sea surface coordinate system to values ​​in the light field coordinate system.

[0165] Step 3.3: Calculate the light field intensity of the beam emitted from a given laser position on a surface element, and perform a coordinate transformation to satisfy the global coordinate system of the sea surface. For example... Figure 12 , Figure 13 , Figure 14 , Figure 15 As shown,

[0166] Step 3.3.1: Introduce a light field coordinate system Ou'v'w' with the same origin as the sea surface coordinate system. Rotate this coordinate system to make it coincide with the sea surface coordinate system, and define the rotation angle as the angle between the rotation and the positive direction of the coordinate axis.

[0167] Step 3.3.2: Rotate the axis of rotation around the w' axis by an angle α, so that the v' axis is in the xoy plane.

[0168] Step 3.3.3: Rotate the axis of rotation around the v' axis by an angle β, so that the w' axis coincides with the z axis.

[0169] Step 3.3.4: Rotate the coordinate system Ou'v'w' around the central axis by an angle γ, so that the Ou'v'w' coordinate system coincides with the Oxyz coordinate system. α, β, and γ are called the Euler angles of rotation. The transformation relationship between the Ou'v'w' coordinate system and the Oxyz coordinate system can be expressed in matrix form as follows:

[0170]

[0171] In equation (21), the transformation matrix T can be expressed in terms of Euler angles as follows:

[0172]

[0173] The transformation relationship between the Ouvw coordinate system and the Oxyz coordinate system satisfies

[0174]

[0175] Step 4: Derive the Kirchhoff approximation formula and calculate the scattering cross section of the laser beam on the sea surface based on the coordinate transformation formula between the sea surface and the laser beam.

[0176] Step 4.1: Introduce Kirchhoff's approximation and Green's second theorem for vectors;

[0177]

[0178] Suppose that vector F exists in the following two forms: Substituting them into formula (24) and simplifying, we get:

[0179]

[0180] Let vector A be the electric field E, and vector B be any constant vector multiplied by the dyadic Green's function. Substituting the electric field and the dextral Green's function into formula (25), the result simplifies to...

[0181]

[0182] Step 4.2: Solve for the vertical and horizontal polarization components of the incident electromagnetic field in the local coordinate system;

[0183] like Figure 16 As shown, let It is a unit vector perpendicular to the incident plane and pointing outwards. It is a unit vector parallel to the plane of incidence. The normal unit vector of the surface element. The unit vector of the incident direction. If the reflection direction is a unit vector, then and They form an orthogonal coordinate system, and satisfy the following relationship:

[0184]

[0185]

[0186]

[0187] Let the incident electric field be E i The incident magnetic field is H i If the polarization mode is arbitrary, the steps to solve the vertical polarization component of the incident electromagnetic field in the local coordinate system should be to first solve the polarization of the incident electric field in the global coordinate system, and then take the vertical component in the local coordinate system to solve the horizontal polarization component.

[0188] Let E ρ The incident electric field is a polarized electric field, and its perpendicular polarization component is... Its horizontal polarization component is The perpendicular polarization component of the incident magnetic field is Its horizontal polarization component is

[0189] Step 4.3: Based on the expressions for the vertical polarization components and the horizontal polarization components of the incident electric field and the reflected electric field, obtain the total tangential electromagnetic field;

[0190] When specular reflection is satisfied, the incident direction With the direction of reflection Satisfying Relationship: The vertical polarization component of the reflected electric field is Its horizontal polarization component is The vertical polarization component of the reflected magnetic field is Its horizontal polarization component is

[0191] Based on the expressions for the vertical and horizontal polarization components of the incident and reflected electric fields, the total tangential electromagnetic field can be obtained as follows:

[0192]

[0193]

[0194] Step 4.4: Based on the total tangential electromagnetic field, solve for the scattered field at any point above the sea surface. The mathematical expression for the scattering cross section in a specific direction is defined as follows:

[0195]

[0196] The advantage of this invention is that it studies the coordinate transformation between the sea surface and the laser beam. Based on the coordinate transformation method and the expression of the beam vector field, the scattering cross section of the laser beam on the complex sea surface is numerically calculated. The method first introduces the sea spectral function and directional spectral function for constructing the complex sea surface, then gives an effective method for establishing and discretizing the complex sea surface model, and then elaborates on the basic theory of mutual transformation between the sea surface coordinate system and the beam coordinate system, and derives the Kirchhoff approximation formula, which can facilitate the calculation of the scattering cross section of the laser beam on the sea surface and study the influence of beam incident angle, beam waist radius and beam parameters on the scattering cross section.

[0197] Example 2

[0198] This embodiment provides a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling, which is implemented according to the following steps:

[0199] Step 1: Set up the simulation scenario, determine the JONSWAP spectrum and directional distribution function, and calculate the two-dimensional wave frequency directional spectrum;

[0200] Step 2: Based on the two-dimensional ocean wave frequency direction spectrum, a complex sea surface model is established and discretized using the linear superposition method to obtain the discretized model;

[0201] Step 3: Based on the discretized model, and using coordinate transformation, obtain the coordinate transformation formula between the sea surface and the laser beam;

[0202] Step 4: Derive the Kirchhoff approximation formula and calculate the scattering cross section of the laser beam on the sea surface based on the coordinate transformation formula between the sea surface and the laser beam.

[0203] Example 3

[0204] This embodiment provides a laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling. In this embodiment, the parameters are set as follows: beam polarization is x-linear polarization, beam waist radius w0 is 2λ, incident height h is 100λ above the sea surface, and incident azimuth angle... incident zenith angle θ i =30°, calculate the scattering cross section of the E-plane. The sea surface model established in this invention is based on the JONSWAP spectrum, and all are under the condition of low wind speed and gentle sea surface undulation. The sea surface size is 20λ×20λ.

[0205] like Figure 17 , Figure 18 As shown, a simple analysis of the scattering cross section of a fundamental mode Gaussian beam on a complex sea surface is presented. Figure 17This is the scattering of a fundamental Gaussian beam at a sufficiently far distance from the sea surface. It is known that when the fundamental Gaussian beam is far enough from the target, it can be regarded as a plane wave. To verify the correctness of the program, a blue auxiliary line is added to the red curve obtained by numerical calculation to represent the trend of the scattering cross section with the scattering angle. After comparing with the literature, it can be seen that the trend of the blue curve is consistent with the curve of the sea surface scattering plane waves in the literature, which proves the correctness of the program. Figure 18 The scattering cross sections of the sea surface on the beam in the E-plane and H-plane were calculated. It is easy to see that for the x-linearly polarized beam, the scattering cross section in the E-plane has a certain regularity, while the scattering cross section in the H-plane, apart from the fluctuation of the value with the change of the scattering angle, basically does not show any other obvious regularity.

[0206] like Figure 19 , Figure 20 As shown, the effects of variations in the parameters of the fundamental mode Gaussian beam on sea surface scattering are presented. Figure 19 In the case of the incident zenith angle θ i For 0°, 30°, and 45° respectively, it can be seen that as θ... i The change in the scattering cross-section peak corresponds to the scattering angle θ. s Changes have also occurred, such as θ i When the value is 0°, the scattering angle θ corresponding to the peak value s Since the angle is 0°, and other incident zenith angles also follow this rule, it can be inferred that the backscattering cross section is the largest in the scattering of the fundamental mode Gaussian beam by the sea surface. Figure 20 In this process, the polarization of the laser beam is changed. It can be seen that at the scattering angle θ s At angles <50°, the scattering cross-sections of the fundamental mode Gaussian beam under linear polarization and left circular polarization are essentially the same on the sea surface, but at the scattering angle θ... s When the scattering angle is >50°, as the scattering angle θ s As the value increases, the difference in scattering cross-section between left circular polarization and x-linear polarization gradually increases.

[0207] like Figure 21 As shown, the effect of the beam incident height h on the scattering cross section was studied. It is easy to see that when the distance between the beam and the sea surface is relatively short, i.e., h is 10λ and 100λ respectively, the difference in the scattering cross section is not significant overall; only the curve for h = 100λ shows more dramatic fluctuations. However, when the beam is far from the sea surface, the value of the scattering cross section decreases overall. Figure 22 As shown, the effect of the beam waist radius w0 of the fundamental mode Gaussian beam on the scattering cross section is given. The results are obvious: the change of the beam waist radius does not affect the trend of the curve, but as the beam waist radius increases, the size of the scattering cross section will increase overall.

[0208] like Figure 23 , Figure 24As shown, the scattering cross-section distribution of Hermite-Gaussian beams and Laguerre-Gaussian beams on complex sea surfaces is presented. Figure 23 In the context, the incident zenith angle θ of the Hermitian-Gaussian beam i =30°, incident azimuth angle The beam waist radius is w0 = 2λ, and the polarization is x-linear polarization. We compared the scattering cross sections of the Hermitian-Gaussian beam with m and n values ​​of m = 1, n = 1, m = 0, n = 1, and m = 1, n = 0. The results are quite clear: the scattering cross section corresponding to m = 1, n = 0 is the largest. This is consistent with the results of the differential scattering cross section of particles on Hermitian-Gaussian beams in Chapter 4. It is worth noting that at the scattering angle θ... s At 30°, the value of the scattering cross section suddenly drops, which is different from the fundamental mode Gaussian beam. This is because the maximum light field intensity of the Hermite-Gaussian beam is not at the center of the beam. It is inferred that the Laguerre-Gaussian beam should also have this characteristic. Figure 24 In the middle, the incident zenith angle θ of the Laguerre-Gaussian beam i =30°, incident azimuth angle The beam waist radius is w0 = 2λ, and the polarization is x-linear polarization. Referring to the parameter values ​​of scattering of a Laguerre-Gaussian beams by marine aerosol particles, the scattering cross-section characteristics are analyzed for two cases: p = 1, l = 0 and p = 1, l = 2. It is evident that the scattering cross-section is larger overall for beams with larger orbital angular momentum l. Furthermore, as previously conjectured, the scattering cross-section corresponding to l = 2 has a larger θ... s The beam decreases when l=30°, but not when l=0. This is because the center of the Laguerre-Gaussian beam corresponding to l=0 has a bright spot, but the center of the beam corresponding to l=2 has a dark spot. This shows that the change of beam parameters plays a decisive role in the properties of the beam.

[0209] The above examples demonstrate that complex sea surfaces significantly influence the scattering cross-section of typical laser beams. The random complex sea surface obtained in this invention, based on the JONSWAP spectrum and double superposition method, conforms to the natural development laws of the sea surface and is physically feasible. It can be used to analyze the scattering cross-section of complex sea surfaces on typical laser beams, especially the influence of beam incident angle, beam waist radius, and beam parameters on the scattering cross-section. The method of this invention for analyzing the scattering of typical laser beams from complex sea surfaces has significant application value in fields such as marine remote sensing, marine resource exploration, and sea surface target detection and identification.

Claims

1. A laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling, characterized in that, The specific steps are as follows: Step 1: Set up the simulation scenario, determine the JONSWAP spectrum and directional distribution function, and calculate the two-dimensional wave frequency directional spectrum; Step 2: Based on the two-dimensional wave frequency direction spectrum, a complex sea surface model is established and discretized using the linear superposition method to obtain the discretized model; Step 3: Based on the discretized model, obtain the coordinate transformation formula between the sea surface and the laser beam based on coordinate transformation; Step 4: Derive the Kirchhoff approximation formula and calculate the scattering cross section of the laser beam on the sea surface based on the coordinate transformation formula between the sea surface and the laser beam. Step 1 is implemented in the following steps: Step 1.1: Set up the simulation scene and select the JONSWAP spectrum for initial sea surface fitting. The expression for the JONSWAP spectrum is: （1） in, For dimensionless energy scale parameters, For the local gravitational acceleration, Peak frequency, U is the peak factor, F is the wind region, and U is the peak factor. 10 The average wind speed at a distance of 10 meters above the sea surface. For peak shape parameters; Step 1.2: Introduce the directional distribution function to describe the energy distribution of ocean waves relative to the directional angle. Combine different ocean spectral functions with the directional distribution function, i.e., the JONSWAP two-dimensional ocean spectrum is... （2） in, Let be the directional distribution function. For directional spectrum functions; Step 1.3: For the JONSWAP two-dimensional ocean spectrum, the photoyiic distribution function is adopted, i.e. （3） in, It is the main direction of wave propagation. The concentration parameter of directional distribution A definite coefficient; The calculation formula is: （4） in, It is a gamma function; Step 1.4: According to formula (2), the two-dimensional ocean wave frequency spectrum is obtained as follows: （5）； Step 2 is implemented in the following steps: Step 2.1: Define the three-dimensional sea surface height using the linear superposition method, and based on the two-dimensional wave frequency direction spectrum... Using computer and MATLAB software, a complex sea surface model that effectively reflects the actual motion of ocean waves was established through simulation. The three-dimensional height of the sea surface was defined as... （6） in, and These are the discrete quantities of angular frequency and direction angle, respectively. Angular frequency, For the first One direction angle, and For the first The angular frequency, the first Amplitude and phase angle at each directional angle, The phase angle of the ocean wave; Step 2.2: Based on the complex sea surface model, discretize it to reconstruct a sea surface model with smaller errors than the actual sea surface and capable of numerical calculation, i.e., the discretized model.

2. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 1, characterized in that, Step 2.2 is implemented in the following steps: Step 2.2.1: Divide the horizontal dimension into equal parts, and label the points according to the method of first the horizontal axis and then the vertical axis. Record the spatial coordinate information of the points in the specified format; Step 2.2.2: Every three points form a triangular element. Each element records the point numbers in counter-clockwise order. This operation can obtain two data files: one that records the node numbers of the triangular elements and the other that records the coordinate information of the nodes. Step 2.2.3: Based on the two data files, reconstruct a sea surface model with small errors to the actual sea surface and capable of numerical calculation.

3. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 1, characterized in that, Step 3 is implemented in the following steps: Step 3.1: Establish a sea surface coordinate system. The node numbers and coordinates of the triangular elements are obtained from the data file. Then, establish a local coordinate system to obtain the reflection of any triangular element under any incident ray. Step 3.2: Introduce the interaction probability density function, which is defined as the product of the shadow function of whether the light source illuminates the surface element and the hidden function of whether the reflected light of the surface element can be received; Step 3.3: Calculate the light field intensity of the beam emitted from the given laser position on the surface element, and perform coordinate transformation to satisfy the global coordinate system of the sea surface.

4. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 3, characterized in that, The specific calculation formula for step 3.1 is as follows: the three nodes of the triangular element are arranged in a counterclockwise direction as follows: , and , The normal unit vector of the surface element. Let be the unit vector representing the incident direction of the light ray. Let be the zenith angle and azimuth angle of the incident light. The unit vector representing the direction of light reflection. The zenith angle and azimuth angle are the reflected light rays. The angle between the incident or reflected direction vector and the surface element normal vector is the incident angle or reflected angle; the parameters satisfy the following relationship: (7) (8) A method for calculating the surface element normal vector and incident ray direction vector in the global coordinate system is presented. Using the specular reflection law, the corresponding reflected ray direction vector can be calculated. The specific formula is as follows: (9) (10) Using the formula and the node coordinates of the triangular element, the reflection of any triangular element under any incident ray can be calculated.

5. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 3, characterized in that, The specific calculation formula for step 3.2 is the interaction probability density function. for The specific form of the function is: (11) (12) in, For the shadow function, It is a unit direction vector. The unit normal vector, The diffusion angle of the laser beam. The laser coordinates are in the global coordinate system. The center coordinates of the face element Here are the coordinates of the position where the laser beam's principal axis illuminates the sea surface. This is the vertical projection of the laser onto the sea surface. For the principal axis ray of the laser beam, and The included angle of the axis is , The location is at sea level, that is , Indicates diffused light. and The angle between them is the angle between the diffused ray and the principal axis of the beam. , and The coordinate information is known; the specific forms of the other parameters are as follows: (13) (14) (15) (16) (17) (18) (19) when At that time, the surface element is within the irradiation range of the laser beam, therefore for the function Rewritten in the form of, the corrected form is as follows: (20)。 6. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 3, characterized in that, The specific steps of step 3.3 are as follows: Step 3.3.1: Introduce a light field coordinate system with the same origin as the sea surface coordinate system. This coordinate system is rotated to coincide with the sea surface coordinate system, with the rotation angle defined as the angle between the rotation and the positive direction of the coordinate axes. Step 3.3.2, with Rotation angle of the axis as the center ,make The axis is in flat Step 3.3.3, with Rotation angle of the axis as the center ,make shaft and Axis coincidence Step 3.3.4, with Rotation angle around the central axis ,make coordinate system and The coordinate systems coincide. , and This is called the Euler angle of rotation. coordinate system and The transformation relationship between coordinate systems can be expressed in matrix form as follows: (21) In equation (21), the transformation matrix It can be expressed in Euler angles as (22) but coordinate system and The transformation relationship between coordinate systems satisfies (23)。 7. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 1, characterized in that, The specific steps of step 4 are as follows: Step 4.1: Introduce Kirchhoff's approximation and Green's second theorem for vectors; (24) Let vector There are two forms: , Substituting them into formula (24) and simplifying, we get... (25) Let vector electric field Vector The dot product of the dyadic Green's function and any constant vector Substituting the electric field and the dextral Green's function into formula (25), the result simplifies to... (26)； Step 4.2: Solve for the vertical and horizontal polarization components of the incident electromagnetic field in the local coordinate system; make It is a unit vector perpendicular to the incident plane and pointing outwards. It is a unit vector parallel to the plane of incidence. The normal unit vector of the surface element. The unit vector of the incident direction. If the reflection direction is a unit vector, then , and They form an orthogonal coordinate system, and satisfy the following relationship: (27) (28) (29) Let the incident electric field be The incident magnetic field is If the polarization mode is arbitrary, the steps to solve for the vertical polarization component of the incident electromagnetic field in the local coordinate system should be as follows: first, solve for the polarization of the incident electric field in the global coordinate system, then take the vertical component in the local coordinate system and solve for the horizontal polarization component. make The incident electric field is a polarized electric field, and its perpendicular polarization component is... Its horizontal polarization component is The perpendicular polarization component of the incident magnetic field is Its horizontal polarization component is ; Step 4.3: Based on the expressions for the vertical polarization components and the horizontal polarization components of the incident electric field and the reflected electric field, obtain the total tangential electromagnetic field; Step 4.4: Based on the total tangential electromagnetic field, solve for the scattered field at any point above the sea surface. The mathematical expression for the scattering cross section in a specific direction is defined as follows: (32)。 8. The laser beam scattering calculation method based on JONSWAP spectrum sea surface modeling according to claim 7, characterized in that, The specific steps of step 4.3 are as follows: When specular reflection is satisfied, the incident direction With the direction of reflection Satisfying Relationship: The vertical polarization component of the reflected electric field is Its horizontal polarization component is The vertical polarization component of the reflected magnetic field is Its horizontal polarization component is ; Based on the expressions for the vertical and horizontal polarization components of the incident and reflected electric fields, the total tangential electromagnetic field can be obtained as follows: (30) (31)。

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