Method for simulating time-domain characteristics of pulse laser in snowfall environment

Abstract

The invention discloses a method for simulating time-domain characteristics of pulse laser in a snowfall environment, which comprises the following steps of: simulating and counting an accumulative path of photons through a Monte Carlo multiple scattering transmission model based on collision probability, and fitting a photon motion time histogram through a time-domain analysis model based on EMG to obtain a fitted curve expression; calculating the full width at half maximum, peak time and amplitude photon number of the fitted curve; wherein the horizontal axis of the photon motion time histogram is the motion time range of received photons, the longitudinal axis is the number of photons in each time interval, and the number of groups is the number of time sub-boxes. According to the method, the time-domain characteristics of the pulse laser in the snowfall environment can be simulated, and a theoretical basis is provided for anti-snow interference design and detection algorithm optimization of the laser fuze.

Description

Technical Field

[0001] This invention belongs to the field of numerical simulation technology, specifically relating to a method for simulating the temporal characteristics of pulsed lasers in a snowfall environment. Background Technology

[0002] Laser fuses are widely used in various high-value munitions and some conventional munitions due to their advantages such as strong anti-electromagnetic interference capability, high ranging accuracy, and concentrated energy. However, their atmospheric penetration capability is insufficient, making them susceptible to scattering and absorption by aerosol particles, fog, smoke, dust, rain, and snow. The multipath effect caused by scattering leads to time-domain broadening of the received signal and pulse delay, thereby reducing the ranging accuracy, triggering reliability, and anti-interference capability of the laser fuse. Therefore, studying the transmission time-domain characteristics of pulsed lasers in complex atmospheres is of great significance for the anti-interference design and detection algorithm optimization of laser fuses. Existing technologies are mainly aimed at the transmission characteristics of pulsed lasers in seawater channels and dusty environments. For example, CN108023652A discloses a simulation method for laser transmission characteristics applied to seawater channels, which can make the simulation of underwater laser communication more accurate. CN113626997A discloses a simulation method for pulsed laser transmission characteristics in dusty environments, which can effectively simulate the detection characteristics of pulsed laser fuses in complex battlefield dusty environments, providing a theoretical basis for improving the detection performance of pulsed laser fuses. However, simulation methods and experimental measurement devices for the transmission time-domain characteristics of pulsed lasers in snowy environments are still lacking. Furthermore, in actual snowy environments, the size distribution of snow particles is not uniform, but rather a polydisperse medium with a certain scale distribution. The collision probability of photons with snow particles of different sizes is usually different. Currently, there is no effective method to simulate the time-domain characteristics of pulsed lasers by combining collision probability. Summary of the Invention

[0003] In view of the above-mentioned deficiencies in the existing technology and the need to improve the performance of laser fuses, the purpose of this invention is to provide a simulation method for the temporal characteristics of pulsed lasers in snowy environments, so as to fill the research gap in the transmission characteristics of pulsed lasers in snowy environments and provide key technical support for the reliable application of laser fuses under snowy conditions.

[0004] The technical solution to achieve the purpose of this invention is as follows:

[0005] A method for simulating the temporal characteristics of pulsed lasers in a snowfall environment includes:

[0006] The fitted curve expression is obtained by fitting the photon motion time histogram using the EMG model:

[0007]

[0008] Where t is the motion time of the received photon, y(t) is the number of photons at time t, erfc is the complementary error function, a is called the amplitude coefficient, μ = T′ is called the mean of the Gaussian component; σ g The standard deviation of the Gaussian component is called τ. e This is called the exponential decay time constant;

[0009] Calculate the full width at half maximum (FWHM), peak time, and amplitude photon count of the fitted curve;

[0010] In the photon motion time histogram, the horizontal axis represents the motion time range of the received photons, the vertical axis represents the number of photons in each time interval, and the number of groups represents the number of time bins.

[0011] The significant advantages of this invention compared to existing technologies are:

[0012] (1) This invention improves the traditional Monte Carlo method by establishing a Monte Carlo multiple scattering transmission model based on collision probability, making the simulation process closer to the transmission process of pulsed laser in polydisperse snow medium.

[0013] (2) This invention constructs a time-domain analysis model based on EMG (Exponentially Modified Gaussian), which can accurately describe the time distribution characteristics of the asymmetric tail caused by multiple scattering and precisely quantify parameters such as time-domain broadening and pulse delay. Attached Figure Description

[0014] Figure 1 This is a flowchart illustrating the simulation of the temporal characteristics of pulsed lasers in a snowfall environment according to the present invention.

[0015] Figure 2 A schematic diagram of photon propagation in a snowy environment;

[0016] Figure 3 The collision probability diagram of photons with particles of different diameter ranges under three snowfall conditions;

[0017] Figure 4 This is a schematic diagram illustrating the photon propagation trajectory of a pulsed laser in a snowy environment and the principle of propagation delay caused by multiple scattering.

[0018] Figure 5 The received pulse waveforms of a pulsed laser transmitted for 100m in three different snowfall environments;

[0019] Figure 6 This is a schematic diagram illustrating the experimental principle of the time-domain characteristics of pulsed laser transmission in a snowfall environment. Detailed Implementation

[0020] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0021] This invention provides a method for simulating the temporal characteristics of pulsed lasers in a snowfall environment. The method includes the construction of a Monte Carlo multiple scattering transmission model based on collision probability and the construction of a temporal analysis model based on EMG (Exponentially Modified Gaussian).

[0022] The temporal characteristics of pulsed laser propagation in three snowfall environments are analyzed as an example. The simulation steps of its multiple scattering propagation process in snowfall environments are shown in the appendix. Figure 1 As shown.

[0023] Step 1: Establish a coordinate system. With the center of the pulsed laser source as the origin, the initial propagation direction of the photons as the z-axis, and the xOy plane parallel to the receiving surface, establish the coordinate system as shown in the attached diagram. Figure 2 The right-handed Cartesian coordinate system O-xyz is shown.

[0024] Step 2: Set the physical parameters. Set the physical parameters for the Monte Carlo multiple scattering transport model based on collision probability as shown in Table 1.

[0025] Table 1 Model Physical Parameters

[0026]

[0027] Step 3: Define the laser emission pulse as a Gaussian pulse. Since Gaussian pulses are typical in theoretical research, the initial emission pulse is set as a time-domain Gaussian pulse, with the pulse width τ = 20 ns and the Gaussian standard deviation defined as follows: The time center is T0 = 0. A Box-Muller transform is used to generate a Gaussian-distributed random time offset t for each photon. offset :

[0028]

[0029] Where U1 and U2 are uniformly distributed random numbers.

[0030] Step 4: Initialize by emitting photons. The state of a photon is mainly determined by its spatial position (x, y, z) and the cosine of its motion direction (u). x ,u y ,u z The weights W of the photons are used to represent u. x u y and u z These are the cosines of the angles between the photon's motion direction vector and the x, y, and z axes, respectively. Each photon is initialized individually with an initial position of (x0, y0, z0) = (0, 0, 0) and an initial direction cosine of (u...). x0 ,u y0 ,u z0) = (0,0,1), with an initial weight of W0 = 1.

[0031] Step 5: Free-stroke sampling. When a photon propagates through a medium, it collides with particles in the medium. The random step size l between the (n-1)th and nth collisions is... n The survival probability of a photon is determined by the following formula:

[0032]

[0033] Where ξ1 is a random number between (0, 1), μ t The extinction coefficient for single scattering, l n The random step size is the distance between the (n-1)th collision and the nth collision.

[0034] To ensure the accuracy of the simulation, the measurement data during winter snowfall in a certain area from 2015 to 2019 were fitted using a three-parameter Gamma distribution, and the snow particle size distribution functions under three snowfall environments were obtained as follows:

[0035] 0 < SR ≤ 1: N(D) = 4897 × D 0.591 e -2.857D (m -3 mm -1 )

[0036] 1 < SR ≤ 2: N(D) = 8663 × D 0.669 e -2.459D (m -3 mm -1 )

[0037] 2<SR≤3: N(D)=10719×D 0.733 e -2.267D (m -3 mm -1 )

[0038] Wherein, SR is the snowfall rate in mm / h, and D is the equivalent volume sphere diameter of the snowflake, which is obtained by a two-dimensional video raindrop spectrometer (2DVD) in mm.

[0039] Based on Fraunhofer diffraction and X-ray scattering theory, the total scattering attenuation cross section of a single spherical snow particle at a wavelength of 905 nm is calculated to be 0.4857πD. 2 Then, combining the snow particle size distribution N(D), the scattering coefficient μ for the three snowfall environments was calculated. s Absorption coefficient μ a and single scattering extinction coefficient μ t They are respectively:

[0040] μ s=∫0.4857πD 2 N(D)dD

[0041]

[0042] μ t =μ s +μ a

[0043] Among them, absorption efficiency factor n r and n i These are the real and imaginary parts of the refractive index of ice, respectively.

[0044] The calculation results are shown in Table 2.

[0045] Table 2 Attenuation coefficients of 905nm laser in three snowfall environments

[0046]

[0047] Step 6: Update the position of photon scattering. Photon travel distance l n The position after (x) n-1 ,y n-1 ,z n-1 ) updated to (x n ,y n ,z n ):

[0048]

[0049] Among them, (x n-1 ,y n-1 ,z n-1 ) represents the spatial location of the (n-1)th collision point. Let x be the cosine of the motion direction after the (n-1)th collision, (x) n ,y n ,z n ) represents the spatial location of the nth collision point.

[0050] Step 7: Divide the snow particle diameter range. Given the complex distribution of snow particle sizes, the equivalent volume sphere diameter range of snow particles is divided into 10 intervals: 0–0.1mm, 0.1–0.3mm, 0.3–0.5mm, 0.5–0.7mm, 0.7–1mm, 1–1.5mm, 1.5–2mm, 2–3mm, 3–5mm, and 5–10mm (all including the right endpoint but excluding the left endpoint).

[0051] Step 8: Calculate the single albedo w of particles in each interval. k Asymmetric factor g k and collision probability P(Dk The optical parameters of the medium are important factors affecting the simulation results of the Monte Carlo method. For example, the single albedo affects the weight change of the particle after absorbing the photon, and the asymmetry factor affects the direction of the photon after it is scattered.

[0052] The asymmetry factor g for each interval was calculated using Mie scattering theory. k The single albedo of the particle in the k-th interval Where, μ s,k Let μ be the scattering coefficient in the k-th interval. t,k Let be the extinction coefficient for the k-th interval.

[0053] The probability of a photon colliding with a snow particle depends not only on the snow particle concentration per unit volume but also on the size of the snow particle. The probability P(D) of a photon colliding with a snow particle in the k-th diameter interval is... k ) is represented as:

[0054]

[0055] Among them, D min,k Let D be the minimum diameter of the particle within the k-th diameter interval. max,k Let M be the maximum diameter of the particle within the k-th diameter interval, and M be the total number of diameter intervals.

[0056] The collision probabilities of photons with particles of different diameters under three snowfall conditions are shown in the appendix. Figure 3 As shown.

[0057] Step 9: Determine the snow particle in the nth collision of the photon. Before simulating whether the photon is absorbed or scattered, it is necessary to select the interval particles based on the collision probability, and then determine the optical parameters (single albedo and asymmetry factor) for each collision. The cumulative collision probability CPDF(D) from the 1st diameter interval to the kth diameter interval is calculated. k ) is represented as:

[0058]

[0059] The interval particles that collide with the photon are determined by a random number ξ2 uniformly distributed between (0,1). If 0 < ξ2 ≤ CPDF(D1), then the snow particle that collides with the photon for the nth time is the snow particle of the first diameter interval; if CPDF(D1) ≤ CPDF(D2), then the snow particle that collides with the photon for the nth time is the snow particle of the first diameter interval. k-1 )<ξ2≤CPDF(D k If the snow particle that the photon collides with for the nth time is the snow particle of the kth diameter interval, then the snow particle that the photon collides with for the nth time is the snow particle of the kth diameter interval.

[0060] Step 10: Update the photon weights. After a photon moves one step, its energy decreases due to absorption by snow particles. Traditional Monte Carlo methods use the average single-pass albedo to calculate the weight change in each collision. This invention updates the photon weights based on the single-pass albedo of the particles in the k-th diameter interval selected in Step 9, changing the photon weight W after the nth collision. n Represented as:

[0061] W n =W n-1 w k

[0062] Among them, w k Let W be the single albedo of the particle in the k-th diameter range. n W represents the energy weight of the photons after the nth collision. n-1 The energy weight is the energy weight of the photon after the (n-1)th collision.

[0063] Step 11: Update the scattering angle and azimuth angle of the photon. Similar to the traditional Monte Carlo method, the azimuth angle of the photon after the collision... The scattering angle θ is obtained by sampling the HG scattering phase function, which follows a uniform distribution in the (0, 2π) region. However, the asymmetry factors of snow particles of different sizes are not entirely the same, resulting in different HG scattering phase functions. Therefore, this invention updates the scattering angle of the photon based on the asymmetry factor of the particles in the k-th diameter interval selected in step 9, and adjusts the azimuth angle of the photon after the nth collision. and scattering angle θ n They are represented as follows:

[0064]

[0065] Among them, g k Let ξ3 and ξ4 be the asymmetry factor of the particle in the k-th diameter interval, and let ξ3 and ξ4 be random numbers between (0,1).

[0066] The new propagation direction cosine after the nth collision of photons for:

[0067]

[0068] like New transmission direction cosine for:

[0069]

[0070] Step 12: Determine if photon motion has terminated. Whether photon motion terminates is primarily determined by the photon's energy weight, the maximum number of scattering events, and the geometric constraints of the target medium. Photon motion terminates when the photon's energy weight is less than the weight threshold set in Step 2; it terminates when the number of scattering events exceeds the set maximum number of scattering events; and it terminates when the photon's motion exceeds the geometric constraints of the target medium. The following are some situations where photon motion exceeds the geometric constraints of the target medium:

[0071] (1) The z-axis of photons n The coordinate is negative, i.e., z. n <0;

[0072] (2) The z-axis of photons n The coordinate is greater than the vertical transmission distance L between the pulsed laser and the receiving surface, i.e., z n >L;

[0073] (3) The distance between the photon position and the center of the receiving surface is greater than the radius R of the receiving surface, i.e.

[0074] (4) The photon's direction of motion exceeds the receiver's field of view (FOV), i.e.

[0075] When a photon satisfies condition (2) but not conditions (3) and (4), the photon is received. The cumulative path S of the photon from the emission point to the receiving surface is: S = l1 + l2 + l3 + ... + l n .

[0076] Among them, l n The random step size of the photon between the (n-1)th and nth collisions, and S is the cumulative path of the photon from emission to arrival at the receiving surface.

[0077] Step 13: Determine if it is the last photon. When the photon's motion is terminated or it reaches the receiving surface, it is necessary to determine if the photon is the last photon. If it is not the last photon, the simulation of the next photon begins; if it is the last photon, the loop is exited and the transmission simulation is completed.

[0078] The multiple scattering effect during pulsed laser transmission in a snowy environment not only leads to laser beam power attenuation and spot expansion, but the multipath effect caused by scattering also results in different arrival times of pulse signals from different paths at the receiving surface. This leads to time-domain broadening of the received signal and pulse delay. A schematic diagram illustrating the transmission trajectory and the transmission delay principle caused by multiple scattering in a snowy environment is attached. Figure 4 As shown.

[0079] This invention constructs a time-domain analysis model based on EMG, simulating the propagation process of all photons in a snowfall environment. The simulation and analysis steps are attached. Figure 1 As shown.

[0080] Step 14: Calculate the motion time of each received photon. For each photon arriving at the receiving surface, calculate its cumulative propagation path after multiple scatterings, and express the photon's motion time T(s) as:

[0081]

[0082] Where S is the cumulative path of the photon, c is the speed of light, and t is the velocity of light. offset This is a random time offset.

[0083] Step 15: Divide the time interval. This invention performs binning on the photon's motion time, dividing the continuous time axis into several equal-width intervals, and defining the time range as [T′-30, T′+40]. The number of time bins is n = 100, with a reference time base point. Where L is the vertical transmission distance between the pulsed laser and the receiving surface.

[0084] Step 16: Count the number of photons in each interval. Use the histcounts function to bin the motion time of all received photons and obtain the number of photons in each interval.

[0085] Step 17: Generate a time histogram. This invention uses a histogram to visualize the number of photons in each time interval, showing the concentration trend, broadening, and tailing phenomenon of photon motion time. The horizontal axis of the histogram represents the motion time range of received photons, the vertical axis represents the number of photons in each time interval, and the number of groups represents the number of time bins.

[0086] Step 18: EMG Fitting. When pulsed lasers propagate in a snowy environment, the path delay caused by multiple scattering effects leads to an asymmetric tail in the photon motion time distribution. Since the Gaussian distribution is symmetrical, it cannot describe the tailing phenomenon. Using Gaussian fitting would underestimate the width of the tail region, resulting in errors in pulse broadening calculations. Therefore, this invention uses an Exponentially Modified Gaussian (EMG) model to fit the histogram generated in step (17). The EMG model is a convolution of a Gaussian distribution and exponential decay, expressed as:

[0087]

[0088] Where y(t) is the number of photons at time t, t is the motion time of the received photons, and erfc is the complementary error function, used to describe the tailing behavior of the exponentially corrected Gaussian distribution. is called the amplitude coefficient, μ = T′, is called the mean value of the Gaussian component, σ g is called the standard deviation of the Gaussian component, which reflects the pulse width of the transmitted pulse and the broadening caused by scattering, and is obtained by non-linear least squares fitting, τ e is called the exponential decay time constant, which reflects the decay rate of the pulse tail caused by multiple scattering, and is obtained by non-linear least squares fitting.

[0089] Step 19: Calculate the full width at half maximum (FWHM), peak time and amplitude photon number of the fitting curve. The present invention uses the full width at half maximum to quantitatively describe the degree of time domain broadening, that is, the time difference corresponding to the two moments when the pulse peak is reduced to half. Due to the tailing phenomenon of the photon motion time, the traditional Gaussian full width at half maximum calculation formula is no longer applicable, and it is necessary to calculate the left half width and the right half width of the EMG fitting curve respectively, and take the average value of the two as the full width at half maximum of the fitting curve. According to the above simulation method of the time domain characteristics of pulsed lasers in a snowfall environment, the received pulse waveforms of pulsed lasers when transmitting 100 m in three snowfall environments of 0 < SR ≤ 1, 1 < SR ≤ 2 and 2 < SR ≤ 3 are obtained, as shown in the attached Figure 5 (a), (b) and (c) respectively.

[0090] The embodiment of the present invention also provides an experimental device for the time domain characteristics of pulsed lasers in a snowfall environment, as shown in the attached Figure 6 shown, including: a laser transmitting device, a laser receiving device and an auxiliary device.

[0091] The described laser transmitting device includes a driving power supply, a pulsed laser and a bracket. The function of the driving power supply is to provide energy and trigger signals for the pulsed laser; the function of the pulsed laser is to generate a laser pulse with a wavelength of 905 nm and a pulse width of 20 ns; the function of the bracket is to fix the pulsed laser and ensure the stability and consistency of the direction of the laser beam.

[0092] The described laser receiving device includes a photodetector, a storage oscilloscope and a bracket. The function of the photodetector is to convert the received optical signal into an electrical signal; the function of the storage oscilloscope is to capture, display and store the original pulse and the attenuated pulse signal output by the detector in real time; the function of the bracket is to fix the photodetector and ensure that the center of the photodetector is directly opposite to the center of the pulsed laser.

[0093] The described auxiliary device includes a weighing sensor, a tape measure and a distance benchmark. The function of the weighing sensor is to convert the collected precipitation weight electrical signal into precipitation and measure the snowfall intensity of the current environment in real time; the functions of the tape measure and the distance benchmark are to accurately control and mark the transmission distance between the pulsed laser and the photodetector.

[0094] This experimental setup quantitatively characterizes the effect of snowfall on the temporal properties of pulsed lasers by comparing the original, unattenuated waveform with the waveform after attenuation due to snowfall transmission. The experimental principle is as follows: Under certain receiving radius and field of view conditions, a pulsed laser emits a 905nm wavelength laser beam with a pulse width of 20ns, triggered by a driving power supply. The collimated laser beam propagates through a snowfall environment. The attenuated laser signal is received by a photodetector and converted into an electrical signal. An oscilloscope captures and stores the attenuated pulse signal in real time. By comparing the attenuated pulse waveform with the original, unattenuated waveform, the pulse delay and temporal broadening of the pulsed laser after snowfall attenuation can be obtained.

Claims

1. A method for simulating the temporal characteristics of pulsed lasers in a snowfall environment, characterized in that, include: The fitted curve expression is obtained by fitting the photon motion time histogram using the EMG model: Where t is the motion time of the received photon, y(t) is the number of photons at time t, erfc is the complementary error function, a is called the amplitude coefficient, μ = T′ is called the mean of the Gaussian component; σ g The standard deviation of the Gaussian component is called τ. e This is called the exponential decay time constant; Calculate the full width at half maximum (FWHM), peak time, and amplitude photon count of the fitted curve; In the photon motion time histogram, the horizontal axis represents the motion time range of the received photons, the vertical axis represents the number of photons in each time interval, and the number of groups represents the number of time bins.

2. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 1, characterized in that, The number of photons in each time interval is obtained by binning the photon motion time. The time interval is determined based on the time range and the number of bins, where the reference time base point for the time range is specified. Where L is the vertical transmission distance between the pulsed laser and the receiving surface, and c is the speed of light.

3. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 2, characterized in that, The photon's travel time is: S＝l1+l2+l3+…+l n Where S is the cumulative path distance of a photon from emission to arrival at the receiving surface, and t offset Let σ be the random time offset, σ be the Gaussian standard deviation of the transmitted pulse, U1 and U2 be uniformly distributed random numbers, and τ be the pulse width of the transmitted pulse. n Let be the random step size of the photon between the (n-1)th and nth collisions.

4. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 3, characterized in that, The criteria for determining whether a photon is received are: When the following condition is satisfied: photon z n The coordinates are greater than the vertical transmission distance L between the pulsed laser and the receiving surface, and do not satisfy the condition that the distance between the photon position and the center of the receiving surface is greater than the radius R of the receiving surface. It also does not satisfy the condition that the photon's direction of motion exceeds the receiver's field of view (FOV). Where the updated position of photon scattering (x) n ,y n ,z n ), in Let be the cosine of the motion direction after the (n-1)th collision, and let be the cosine of the motion direction after the nth collision. for: in and θ n These are the azimuth angle and scattering angle after the nth collision of the photon, respectively.

5. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 4, characterized in that, Azimuth angle after the nth collision of photons and scattering angle θ n They are respectively: Among them, g k Let ξ3 and ξ4 be the asymmetry factor of the particle in the k-th diameter interval, and let ξ3 and ξ4 be random numbers between (0,1).

6. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 5, characterized in that, The particle in the k-th diameter region that collides with the photon for the nth time passes through the cumulative collision probability CPDF(D). k To determine, if 0 < ξ2 ≤ CPDF(D1), then the snow particle in the nth collision of the photon is the snow particle in the first diameter interval; if CPDF(D k-1 )<ξ2≤CPDF(D k If the photon collides with the snow particle in the nth collision, then the snow particle in the kth diameter interval is the snow particle in the nth collision. Where ξ2 is a random number uniformly distributed between (0,1), and the cumulative collision probability CPDF(D) from the first diameter interval to the kth diameter interval is... k )for: Among them, P(D) k ) represents the probability of a photon colliding with a snow particle in the k-th diameter interval.

7. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 6, characterized in that, The probability P(D) of a photon colliding with a snow particle in the k-th diameter interval. k )for: Among them, D min,k Let D be the minimum diameter of the particle within the k-th diameter interval. max,k Let M be the maximum diameter of the particle within the k-th diameter interval, and M be the total number of diameter intervals.

8. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 3 or 4, characterized in that, The random step size l of the photon between the (n-1)th and nth collisions n for: Where ξ1 is a random number between (0, 1), μ t This is the extinction coefficient for single scattering.

9. The method for simulating the temporal characteristics of pulsed lasers in a snowfall environment according to claim 8, characterized in that, Single scattering extinction coefficient μ t for: m t =μ s +m a m s =∫0.4857πD 2 N(D)dD μ s and μ a Let n be the scattering coefficient and absorption coefficient of the snow particles. r and n i Let ξ be the real and imaginary parts of the refractive index of ice, respectively. p λ is the absorption efficiency factor, N(D) is the snow particle size distribution, D is the equivalent volume sphere diameter of the snowflake, and λ is the wavelength of the emitted pulse.

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