Ranging performance models based on negative-binomial (NB) distribution for photon-counting lidars

Song Li; Zhiyu Zhang; Yue Ma; Haomin Zeng; Pufan Zhao; Wenhao Zhang

doi:10.1364/OE.27.00A861

1. Introduction

Photon-counting lidars currently can benefit from sensitive sensors (e.g., Geiger mode avalanche photodiodes (GM-APD), photomultiplier tube (PMT), and superconducting nanowire single photon detectors (SNSPD)) to be used as ground-based space debris detection systems [1] and space-borne or airborne laser altimeter systems [2–4]. Many classic theoretical models that analyze the detection probability and ranging performance have been developed. Steven Johnson derived the discrete detection model for photon-counting ranging systems and analyzed the effects of noise [5]. In Godman’s research, the detected photon energy reflected by the target was strongly influenced by the target speckle, and the probability density function of signal photons satisfied a negative-binomial (NB) distribution [6]. In addition, he derived an approximate equation to calculate the speckle diversity by analyzing the spatial coherence function of the electric field over the receiving aperture [7]. Youmans’s research was based on Godman’s model and showed that the NB distribution can be simplified to be a Poisson distribution when the speckle diversity is much larger than the average signal’s photon count [8]. Based on the Poisson distribution, Fouche analyzed the effects of a long dead-time on the detection statistics of photon-counting ranging systems [9]. Based on the NB distribution, Gatt et al. derived detection statistics for photon-counting systems with arbitrary dead-times and verified the model using a Monte Carlo simulation [10].

The ranging performance is one of the most important indicators to evaluate a lidar, and its theoretical model can be used to optimize systematic parameters when a photon-counting lidar is designed. The dead-time of photon-counting detectors can introduce a few tens of centimeters of ranging bias into the ranging performance [11]. Many methods have been proposed to calibrate the range walk error based on the detection theory of photon-counting ranging systems. Oh et al. derived a method for reducing the range walk error based on a Poisson statistics theoretical model and achieved a greater than 2 cm standard deviation in 5000 repetitive measurements [12]. He et al. proposed a correction method by numerically fitting the function relationship between the range walk error and the laser pulse response rate [13]. Ye et al. proposed a method of unequal intensity dividing the echo pulses into dual GM-APDs, which reduced the range walk error by 86% [14]. Ma et al. developed a correction model for the range walk error that considered multiple detectors and reduced the error to approximately 1 cm in 1600 repetitive measurements [15].

However, the current ranging performance models are based on Poisson statistics, which are a special case of a NB distribution. In this paper, based on Goodman's theory on speckle diversity [7], a new ranging performance model of the NB distribution is derived that considers the effect of a target’s speckle, the dead-time, and noise. Next, the derived ranging performance model is verified by both an experiment based on a GM-APD lidar and a simulation using the recursive method. Third, the ranging performance model is implemented to analyze the effect of target speckle for two typical ranging systems aimed at different types of targets: the space-borne Ice, Cloud and land Elevation Satellite-2 (ICESat-2) for detecting Earth’s surface, and the ground-based laser ranging system for detecting space debris at Shanghai Astronomical Observatory (SHAO) station. The results indicate that for a space-borne or airborne lidar, the ranging performance model can be approximated to models that are based on Poisson statistics, but for a ground-based laser ranging system, the ranging performance model cannot be approximated to the Poisson statistics model.

2. Theoretical model

2.1. Detection probability of signal photon events

The detection probability P(K_s; N_s) of signal photon events (PEs) generated by a photon-counting detector follows the Poisson distribution [16]

P (K_{s}) = \frac{N_{s}^{K_{s}}}{K_{s}!} e^{- N_{s}},

where K_s is the number of signal PEs and N_s is the average number of signal photon counts. The average signal photon count N_s is related to the laser pulse energy received by a photon-counting detector and can be expressed as

N_{s} = \frac{η_{q}}{h f} W = \frac{η_{q}}{h f} \iint_{\sum_{r}} \int_{t}^{t + τ_{r}} I (x, y; t) d t d x d y,

where W is the received laser pulse energy of a single shot; η_q is the quantum efficiency of the detector; h is the Planck constant; f is the frequency of light; τ_r is the time duration of the laser pulse; ∑_r is the integral area of the receiving aperture; and I(x, y; t) is the spatial and temporal distribution of the laser pulse, which normally follows a two-dimensional Gaussian and a one-dimensional Gaussian distribution for a fundamental mode laser. According to Bayes’s theorem, the detection probability of K_s signal PEs generated by a photon-counting detector can be expressed as [17]

P (K_{s}) = \int_{0}^{\infty} P (K_{s} | W) p (W) d W,

where p(W) is the energy probability density function of the received laser pulse.

The intensity of a laser pulse received by a lidar is the interference result of the reflected electric fields of the small cells on a rough target surface, and each small cell on the rough surface contributes a random phase to the electric field over the receiving aperture. If the energy of the receiving signal is a speckle field with M degrees of freedom, i.e., the speckle diversity is M, the energy probability density function of a received laser pulse can be approximated by a Gamma probability density function with parameter M [18]

p (W) = {(\frac{M}{\bar{W}})}^{M} \frac{W^{M - 1} e^{- M \frac{W}{\bar{W}}}}{Γ (M)},

where

\bar{W}

is the average energy of the received laser pulse and Г() is the Gamma function. Substituting Eq. (1) and Eq. (4) into Eq. (3), the detection probability of K_s total PEs generated by a photon-counting detector with a speckle diversity of M can be expressed as

P (K_{s}) = \frac{Γ (K_{s} + M)}{Γ (K_{s} + 1) Γ (M)} {(\frac{N_{s}}{N_{s} + M})}^{K_{s}} {(\frac{M}{N_{s} + M})}^{M} .

If we use p instead of N_s/(N_s + M), Eq. (5) can be rewritten as

NB (K_{s}; N_{s}, M) = \frac{(K_{s} + M - 1)!}{K_{s}! (M - 1)!} p^{K_{s}} {(1 - p)}^{M} = (\begin{matrix} K_{s} + M - 1 \\ K_{s} \end{matrix}) p^{K_{s}} {(1 - p)}^{M} .

As shown in Eq. (6), the detection probability of K_s signal PEs follows the NB distribution.

2.2. Detection probability of signal PEs considering the noise and dead-time effects

The noise of a photon-counting lidar is composed of the dark noise of the photon-counting detectors and the background noise. The total noise PEs (including the dark and background noise) follow the Poisson distribution as P(K_n; N_n), where K_n is the total number of noise PEs and N_n is the average noise photon count. For each time bin of a lidar (determined by the time-to-digital converter (TDC) device), the variable N_n can be expressed as N_n = f_n·τ, where f_n is the noise rate and τ is the width of the time bins.

The number of total PEs generated by a photon-counting detector is K (i.e., K = K_n + K_s) and its detection probability P(K) can be derived by the convolution of the noise probability function with K_n and the signal probability function with K_s.

P (K) = \sum_{q = 0}^{K} \frac{N_{n}^{K - q}}{(K - q)!} e^{- N_{n}} \frac{(q + M - 1)!}{q! (M - 1)!} {(\frac{N_{s}}{N_{s} + M})}^{q} {(\frac{M}{N_{s} + M})}^{M}

Considering the dead-time of a photon-counting detector, the detection probability P_d within the time duration of (t, t + τ) can be expressed as

P_{d} (t, t + τ) = P_{nd} (t - t_{d}, t) \cdot P (t, t + τ; K > 0) = P_{nd} (t - t_{d}, t) \cdot [1 - e^{- f_{n} τ} {(\frac{M}{n_{s} (t, t + τ) + M})}^{M}] .

P(t, t + τ; K > 0) means that at least one PE is generated within the time duration of (t, t + τ), and it can be expressed as P(t, t + τ; K > 0) = 1 - P(t, t + τ; K = 0). n_s(t, t + τ) is the number of signal photon counts within the time duration of (t, t + τ), and it can be expressed as [19]

n_{s} (t, t + τ) = \int_{t}^{t + τ} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t,

where t_s is the centroid of the arriving time of the laser pulse and σ_s is the root-mean-square (RMS) pulse width. P_nd(t-t_d, t) means that no PE is generated within the dead-time before the time tag t, and t_d is the dead-time of the photon-counting detector. It should be noted that in Eq. (8), the time bin τ should satisfy the assumption of τ<<t_d. Normally, the time bin τ is a very short time interval that ranges from dozens of picoseconds to hundreds of picoseconds and is two or three orders of magnitudes smaller than the dead-time t_d of a single-photon detector, i.e., τ<<t_d. P_nd(t-t_d, t) can be expressed as

P_{nd} (t - t_{d}, t) =1 - P (t - t_{d}, t, K > 0) = e^{- f_{n} t_{d}} {(\frac{M}{n_{s} (t - t_{d}, t) + M})}^{M} .

In practice, the noise PEs and the signal PEs cannot be distinguished from each other. Normally, when the laser pulse arrives the energy within t_s ± 3σ_s occupies approximately 99.7% of the total laser pulse energy, and the PEs will be dominated by the signal PEs within t_s ± 3σ_s [20] (It should be noted that the systems ranging to retro-reflector equipped satellites and the retro-reflector arrays on the moon does not satisfy this condition). To derive an analytical model, we assume that all signal PEs of the laser pulse reflected from the target are included in the time duration of (t_s-3σ_s, t_s + 3σ_s); then, the detection probability P_d at different time tags t can be expressed as follows:

when the time tag t satisfies t < t_s - 3σ_s, the detection probability P_d can be expressed as [21]

P_{d} (t) = e^{- f_{n} t_{d}} (1 - e^{- f_{n} τ});

when the time tag t satisfies t_s - 3σ_s ≤ t ≤ t_s + 3σ_s, the detection probability P_d can be expressed as

\begin{array}{l} P_{d} (t) = e^{- f_{n} t_{d}} {(\frac{M}{\int_{t - t_{d}}^{t} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t + M})}^{M} \times [1 - e^{- f_{n} τ} {(\frac{M}{\int_{t}^{t + τ} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t + M})}^{M}]; \end{array}

when the time tag t satisfies t > t_s + 3σ_s, the detection probability can be expressed as

P_{d} (t) = e^{- f_{n} t_{d}} {(\frac{M}{N_{s} + M})}^{M} (1 - e^{- f_{n} τ}) .

2.3. The ranging performance model

Based on the classic laser ranging theorem, the ranging performance of a lidar consists of the ranging bias and the ranging precision (ranging uncertainty) [22]. For a laser ranging system based on the time of flight (TOF), the laser ranging precision has a linear relationship with the timing precision. The detectors for a photon-counting lidar usually experience a dead-time effect, typically ranging from a couple of nanoseconds to hundreds of nanoseconds, during which no further photons will be detected. Due to the existence of the dead-time, the centroid of the measured signal will drift from the true distance, which introduces a negative ranging bias to the measured distance (i.e., the distance will be underestimated). This ranging bias is normally called the range walk error.

The ranging bias (range walk error) is related to the first-order normalized moment of the received waveform for a linear lidar, or the probability density function f_s(t) of received PEs for a photon-counting lidar (i.e., Eq. (14)). The ranging precision is related to the second-order normalized moment (i.e., Eq. (15)), where Var denotes the variance. In fact, Eq. (14) and Eq. (15) represent the mean and variance of the time tags of the PEs, respectively.

\bar{t} = \frac{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} t f_{s} (t) d t}{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} f_{s} (t) d t}

V a r = \frac{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} {(t - \bar{t})}^{2} f_{s} (t)d t}{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} f_{s} (t)d t} = \frac{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} t^{2} f_{s} (t)d t}{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} f_{s} (t)d t} - {\bar{t}}^{2}

Based on the detection probability (probability distribution function) of a photon-counting lidar in Eq. (12), the probability density function f_s(t) can be solved by differentiating Eq. (12) over time t.

f_{s} (t) = \lim_{τ \to 0} \frac{P_{d} (t)}{τ} = (\frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} + f_{n}) \times e^{- f_{n} t_{d}} {(\frac{M}{\int_{t - t_{d}}^{t} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t + M})}^{M}

Next, substituting Eq. (16) into the mean time tags of the PEs (i.e., Eq. (14)) and the variance of the time tags (i.e., Eq. (15)) and multiplying by half of the velocity of light, Eq. (17) and Eq. (18) give expressions for the ranging bias and the ranging precision of a photon-counting lidar, respectively. c is the velocity of light.

R_{a} = \frac{c}{2} . (\frac{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} t [\frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} + η_{q} f_{n}] e^{- f_{n} t_{d}} {(\frac{M}{\frac{N_{s}}{2} [1 + erf (\frac{t - t_{s}}{\sqrt{2} σ_{s}})] + M})}^{M} d t}{e^{- f_{n} t_{d}} [1 - e^{- 6 f_{n} σ_{s}} {(\frac{M}{N_{s} + M})}^{M}]} - t_{s})

R_{p} = \frac{c}{2} . \sqrt{\frac{\int_{t_{s} - 3 σ_{s}}^{t_{s} + 3 σ_{s}} t^{2} [\frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} + f_{n}] e^{- f_{n} t_{d}} {(\frac{M}{\frac{N_{s}}{2} [1 + erf (\frac{t - t_{s}}{\sqrt{2} σ_{s}})] + M})}^{M} d t}{e^{- f_{n} t_{d}} [1 - e^{- 6 f_{n} σ_{s}} {(\frac{M}{N_{s} + M})}^{M}]} - {\frac{4 R_{a}}{c^{2}}}^{2}}

In the derivation process of Eqs. (17) and (18), an approximation is used as Eq. (19), where erf() is the error function.

\int_{t - t_{d}}^{t} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t \approx \int_{- \infty}^{t} \frac{N_{s}}{\sqrt{2 π} σ_{s}} e^{- \frac{{(t - t_{s})}^{2}}{2 σ_{s}^{2}}} d t \approx \frac{1}{2} N_{s} [1 + erf(\frac{t - t_{s}}{\sqrt{2} σ_{s}})]

When the system parameters of a lidar are known (i.e., the RMS laser pulse width σ_s, the average signal photon counts N_s, the dead-time of the detector t_d, the time bin width of the TDC τ, and the speckle diversity M) and the noise rate f_n can be estimated, the laser ranging performance (including the ranging bias and the ranging precision) can be evaluated based on Eq. (17) and Eq. (18).

The ranging performance model is significant in three main ways. (1) For the actual operation of a given lidar, the laser ranging performance model cannot influence the data processing, but it can predict or evaluate the ranging accuracy and ranging precision of the lidar that was used [16]. (2) For a photon-counting lidar, when the PEs are processed, the average signal photon counts N_s can be calculated; then, the range walk error (ranging bias) can be offset using the theoretical model of the range walk error [15]. (3) When a photon-counting lidar is designed, the system parameters can be optimized to achieve a balance between the detection probability and the range performance based on the ranging performance model.

3. Model verification

3.1. Verification by lidar ranging experiment

An experiment based on a GM-APD lidar is designed and implemented to verify the derived theoretical model of the range walk error, i.e., Eq. (17). Figure 1(a) shows a schematic of the photon-counting lidar; Fig. 1(b) shows a photograph of the photon-counting lidar; and Fig. 1(c) shows an infrared photograph of the target and the laser beam. In this photon-counting lidar, a semiconductor near-infrared laser of 905 nm (CETC PL-905) is used with a full width half maximum (FWHM) of 10.57 ns and a repetition frequency of 20 kHz. The laser beam is first shaped with a Thorlabs AYL-108B lens for the fast-axis and an OptoSigma CLB-1020-80IR lens for the slow-axis. Next, the laser is coupled into an optical fiber with a diameter of 200 μm and a numerical aperture of 0.22, and it is finally transmitted through an optical transmitting system with a diameter of 10 mm. The divergence of the transmitted laser beam is approximately 5 mrad.

Fig. 1 (a) Schematic of the photon-counting lidar; (b) a photograph of the photon-counting lidar; and (c) an infrared photograph of the target and the laser beam. The photon-counting lidar consists of a semiconductor laser, a power supply, transmitting and receiving optics, as well as a GM-APD detector and its TDC. The transmitted laser beam first illuminates the target, a circular paper of 52 mm diameter, then penetrates a glass door, and finally hits a wall.

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The diameter and field of view (FOV) of the receiver telescope are 35 mm and 10 mrad, respectively. The receiver telescope first focuses the received laser pulse into a fiber with a diameter of 1 mm and a numerical aperture of 0.22. The received laser pulse is then coupled into a Gm-APD detector (SPCM AQ4C) with a detection efficiency of approximately 27% at 905 nm, a dark noise rate of 500 Hz, and a dead-time of 50 ns. The time tag of the PE is measured by a TDC (MCS6A4T2), which can achieve a time bin of 200 ps and a time resolution of 45 ps. The circuit delay of this lidar (including the laser and detector response delays) is 96.92 ns and the speckle diversity M is equal to 25.98, which is calculated using the method in the Appendix. When the experiment is conducted, the total noise rate f_n is approximately 0.17 MHz.

The transmitted laser beam first illuminates the main target, a circular paper with a 52 mm diameter, then penetrates a glass door, and finally hits a wall. The true distance between the lidar and the paper target is 11.389 m, which is measured with a total station (Leica TS09 with a ranging accuracy ± 1mm). Four optical attenuators with different attenuation rates (Spiricon LDS-100, OD 0.4, OD 0.6, OD 1.3, OD 3) are used separately or integrally to measure eight groups of data and to set the average photon counts of the received laser pulse N_s from 0.05 to 10. Table 1 shows the detailed experimental results (including the used optical attenuators, attenuation rates, average signal photon counts, cumulative numbers (times of repetition), the measured range walk errors R_{a_mea}, and the theoretical range walk errors R_{a_the}) in the eight groups corresponding to the eight specific attenuation rates (i.e., corresponding to eight groups of specific average signal photon counts). Nos. 1 to 4 in Table 1 correspond to the attenuators OD 0.4, OD 0.6, OD 1.3, and OD 3, respectively.

Table 1. Experimental results including the used optical attenuators, attenuation rates, average signal photons, cumulative numbers (times of repetition), the measured range walk errors Ra_mea, and the theoretical range walk errors Ra_the with eight groups of specific attenuation rates.

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The theoretical range walk errors R_{a_the} shown in Table 1 are calculated by substituting the system parameters and the average signal photon counts N_s into Eq. (17). The measured distance between the paper target and the lidar in the eight groups is calculated by the time interval between the centroid of the cumulative histogram of the PEs (illustrated in Fig. 2(a)) and the centroid of the transmitted pulse. The measured range walk errors R_{a_mea} are calculated by subtracting the true distance (as obtained by the Leica total station) from the measured distances.

Fig. 2 (a) Cumulative histograms of the PEs in eight groups with different average signal photon counts. (b) The measured range walk error R_{a_mea} in the experiments (using red circles), the curve calculated by the theoretical model (using blue curve), and the bias between R_{a_mea} and R_{a_the} (using yellow filled circles).

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All eight experimental groups’ cumulative histograms of their received PEs are illustrated in Fig. 2(a). There are three peaks in the cumulative histograms of the PEs, which correspond to the paper target, the glass door, and the wall (from nearest to farthest). As the number of average signal photons N_s decreases from 8.46 to 0.059, the signal PEs in the cumulative histograms decrease correspondingly, whereas the noise PEs are nearly identical among the eight groups. Figure 2(b) illustrates the theoretical range walk error curve within a range of 0.1 to 10 signal photons (using Eq. (17)), the eight groups of measured range walk errors R_{a_mea} in the experiments (in Table 1), and the eight groups of biases between R_{a_the} and R_{a_mea}.

In Table 1, the mean bias between R_{a_mea} and R_{a_the} in all eight groups is −0.99 cm and the root mean square error (RMSE) is 3.43 cm. The RMSE is mainly due to the energy fluctuation effect of the transmitted laser device (the transmitted laser energy of the used semiconductor has an energy fluctuation of ± 5%). This result indicates that the theoretical range walk error R_{a_the} agrees well with the experiment results, which verifies the proposed theoretical model, i.e., Eq. (17).

3.2. Verification by the recursive method

In the experiment we have verified that the proposed theoretical model based on the NB distribution (Eq. (17)) agrees well with the experimental results. However, the photon-counting lidar only has one speckle diversity (M = 25.98). In this section, we again verify the proposed theoretical model (including Eqs. (17) and (18)) using the recursive method with multiple values of speckle diversity M.

Li et al. derived a recursive detection probability model based on the Poisson distribution and analyzed the effects of the dead-time, pulse width, and noise rate on the detection probability [23]. The recursive method calculates the detection probability of all time-bins one by one from the start of the range gate to the end. For a laser ranging system with a long range gate, e.g., the width of the range gate of the space-borne ICESat-2 is 10 μs [2], the simulation will require a long computation time. A recursive method based on the NB distribution is implemented to verify the derived ranging performance model (Eqs. (17) and (18)).

The detection probability of the first time bin P_dr(t₁) can be expressed as

P_{dr} (t_{1}) = 1 - P (t_{0}, t_{0} + τ; K > 0) = 1 - \exp (- f_{n} τ) {(\frac{M}{n_{s} (t_{0}, t_{0} + τ) + M})}^{M},

where t₁ is the time of the first time bin and t₀ is the start time of the range gate, which can be assumed to 0. The detection probability of the second time bin, P_dr(t₂), can be expressed as

P_{dr} (t_{2}) = [1 - P_{dr} (t_{1})] \cdot P (t_{1}, t_{1} + τ; K > 0) = [1 - P_{dr} (t_{1})] \cdot (1 - \exp (- f_{n} τ) {(\frac{M}{n_{s} (t_{1}, t_{1} + τ) + M})}^{M}),

where t₂ is the time of the second time bin and [1- P_dr(t₁)] is the probability that no PE occurs within the previous time bins. Similarly, the detection probability of the i-th time bin P_dr(t_i) can be expressed as

P_{dr} (t_{i}) = {\begin{matrix} [1 - \sum_{j = i - (n_{dn} - 1)}^{i - 1} P_{dr} (t_{j})] \cdot \begin{matrix} P (t_{i - 1}, t_{i - 1} + τ; K > 0) & (i > n_{dn}) \end{matrix} \\ [1 - \sum_{j = 1}^{i - 1} P_{dr} (t_{j})] \cdot P (t_{i - 1}, t_{i - 1} + τ; K > 0) \begin{matrix} \begin{matrix}  \end{matrix} & (i \leq n_{dn}) \end{matrix} \end{matrix} .

n_dn is the number of time bins that the dead-time of a photon-counting detector occupies, e.g., if the dead-time is 3.2 ns and the width of a time bin is 200 ps, n_dn is equal to 16. The mean and variance of the time tags of PEs in Eq. (14) and Eq. (15) can be calculated using the discrete detection probabilities of the time bins within the time duration of (t_s - 3σ_s, t_s + 3σ_s). Next, the range walk error and the ranging precision can be calculated using Eq. (17) and Eq. (18).

Ranging performance curves based on the theoretical model and the recursive method are illustrated in Fig. 3 with a typical noise rate of 5 MHz, an RMS pulse width of 0.65 ns, and a dead-time of 3.2 ns [2]. The speckle diversity M is set to 5 and 100. The solid blue curve and dashed red curve in Figs. 3(a) and 3(c) correspond to the range walk errors calculated by the proposed model in Eq. (17) and the recursive method, respectively. The vertical axes in Figs. 3(a) and 3(c) are inversely drawn because the range walk errors are negative values (underestimating the distance). Similarly, the solid blue curves and dashed red curves in Figs. 3(b) and 3(d) correspond to the ranging precisions simulated by the proposed model in Eq. (18) and the recursive method, respectively. The yellow filled circles in Figs. 3(a)-3(d) correspond to the bias between the results simulated by the theoretical model and recursive calculation.

Fig. 3 (a) Comparison between the range walk errors simulated by the theoretical model (the dashed red curve) and the recursive method (the solid blue curve), when the speckle diversity is M = 5. (b) Comparison between the ranging precisions simulated by the theoretical model (the dashed red curve) and the recursive method (the solid blue curve), when the speckle diversity is M = 5. (c) Comparison of the range walk errors when the speckle diversity is M = 100. (d) Comparison of the ranging precisions when the speckle diversity is M = 100. The yellow filled circles in all sub figures correspond to the bias between the results simulated by the theoretical model and the recursive method. The RMS pulse width is 0.65 ns, the noise rate is 5 MHz, and the dead-time is 3.2 ns.

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The comparisons in Fig. 3 indicate that the ranging performance simulated by the proposed theoretical model in Eq. (17) and Eq. (18) agrees well with the values calculated by the recursive method within an average signal photon count from 0 to 5 (which is the designed dynamic range of the ICESat-2) [2]. The maximum bias (between the theoretical and recursive values) of the range walk error and the ranging precision are 0.36 cm and 0.63 cm, respectively. With a small speckle diversity of M = 5 and a larger speckle diversity of M = 100, the proposed theoretical models (including Eqs. (17) and (18)) are again verified via the recursive method.

4. Discussion

4.1. Differences between the NB and Poison models

For the theoretical detection probability, based on Eq. (7), when at least one PE is generated within the range gate (i.e., K>0) and the average noise counts is assumed to be N_n = 1, the theoretical relationship between the detection probability P(K>0) and the average signal photon count N_s is illustrated in Fig. 4(a). The solid blue curve, the dashed curve, and the dash-dotted curve correspond to the results when the speckle diversities are M = 1, 5, and 100, respectively. Figure 4(b) illustrates the theoretical relationship between the detection probability and the average signal photon count N_s based on the Poisson distribution and the Bose-Einstein distribution. The solid red curve corresponds to the detection probability from the Poisson distribution, and the dashed yellow curve is based on the Bose-Einstein distribution. Comparing the dash-dotted curve in Fig. 4(a) and the solid red curve in Fig. 4(b) indicates that when the speckle diversity M is large enough (>100), the results can be well approximated using the Poisson distribution. In addition, when the speckle diversity is M = 1, the NB distribution degrades into the Bose-Einstein distribution [24].

Fig. 4 Theoretical relationship between the detection probability and the average signal photon counts when the average noise count N_n is equal to 1. (a) Detection probability of the NB distribution. The solid, dashed and dashed dotted curves correspond to a speckle diversity of M = 1, 5, and 100, respectively. (b) Detection probabilities of the Poisson distribution and the Bose-Einstein distribution. The solid red curve corresponds to the Poisson distribution and the dashed yellow curve corresponds to the Bose-Einstein distribution.

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For the theoretical ranging performance, the range walk errors of the NB distribution are illustrated in Fig. 5(a) using Eqs. (17) and (18). The range walk errors of the classic Poisson distribution using Eqs. (11) and (12) are also illustrated. The solid blue curve, the dashed red curve, and the dash-dotted yellow curve correspond to the NB distribution model when the speckle diversity M is equal to 100, 5, and 1, respectively. The blue filled circles are the results of the Poisson distribution model. Figure 5(b) illustrates the bias between the range walk errors (in Fig. 5(a)) calculated by the NB and Poisson distribution models when the speckle diversity M is equal to 100, 5, and 1, respectively.

Fig. 5 (a) Range walk error comparison between the results simulated by the NB distribution model (the curves show the speckle diversities of 1, 5, and 100) and the Poisson distribution (using dots); (b) the bias between the results simulated by the NB and the Poisson distribution models; (c) ranging precision comparison between the results simulated by the NB distribution model (the curves show the speckle diversities of 1, 5, and 100) and the Poisson distribution (using dots); and (d) the bias between the results simulated by the NB and Poisson distribution models.

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A comparison of the ranging precisions from the NB distribution and the Poisson distribution is illustrated in Fig. 5(c). The solid blue curve, the dashed red curve, and the dash-dotted yellow curve correspond to the NB distribution model when the speckle diversity M is equal to 100, 5, and 1, respectively. The blue filled circles are the results of the Poisson distribution. Figure 5(d) illustrates the bias between the ranging precisions (in Fig. 5(c)) calculated by the NB and Poisson distribution models.

Similar to the detection probability, when the speckle diversity M increases, the ranging performance from the NB distribution tends to be identical to the results from the classic Poisson distribution. The new proposed model is more universal and compatible with the classic model of the Poisson distribution, because when the speckle diversity is larger than 100, the results calculated from the new model will be identical to the classic model.

4.2. Different photon-counting lidars satisfying different models

In this section, typical photon-counting ranging systems with different speckle diversity values will be discussed. For a photon-counting lidar, the speckle diversity M is mainly determined by the receiving aperture and the speckle field on the target surface. The calculation of speckle diversity M can be divided into two different categories depending on the characteristics of the target. One is when the receiving aperture is larger than the size of speckle field, i.e., the whole speckle field is received by the detectors. The other is when the receiving aperture is smaller than the size of speckle field, i.e., only part of the speckle field is received by the detectors. In practice, these two conditions correspond to a point target and an area target, respectively. The speckle diversity M of both conditions is larger than 1, i.e., at least one speckle will influence the result. In the Appendix, methods for calculating the speckle diversity M are proposed for a given photon-counting lidar aimed at point targets and a lidar aimed at area targets.

Generally, the speckle diversity of a photon-counting ranging system that aims at area targets is very large. The ICESat-2 aboard the Advanced Topographic Laser Altimeter System (ATLAS) is a typical space-borne photon-counting laser ranging system aiming at area targets on the ground. The diameter of the effective receiving aperture of the ATLAS is 0.8 m, and the diameter of the laser beam on the transmitting aperture is 4.4 cm [2]. If the energy distribution of its laser pulse is assumed to Gaussian in shape and the altitude is 500 km, the speckle diversity of the ATLAS is approximately 1423.6 based on Eq. (33). Additionally, the result in Fig. 5 indicates that when the speckle diversity is M >100, the bias between the NB and Poisson models can be neglected, i.e., the detection probability and ranging performance model of a space-borne laser altimeter (e.g., the ICEsat-2 ATLAS) can be approximated by the classic Poisson distribution.

However, the speckle diversity of a photon-counting ranging system that aims at point targets at very long distances (hundreds of km) is normally small. Based on Eq. (29), the speckle diversity of the SHAO station (the space debris ranging system) can be calculated by inputting the system parameters. For the SHAO station, the diameter of the effective receiving aperture is 0.6 m [25]. When the diameter of the target is small (generally <1 m, it has successfully detected space debris with a size as small as 0.5 m at a 1000 km distance), the speckle diversity M is slightly larger than 1.

Normally, the speckle diversity of the ranging system at SHAO station varies from 1 to 10, according to the size of the target (space debris). As a result, if the speckle diversity is M = 1 and the average signal photons are N_s = 5, the range walk error simulated by the Poisson distribution is 0.7 cm larger than that of the NB distribution, but the ranging precision is 4.8 cm lower than that of the NB distribution. Therefore, to achieve a range accuracy of several millimeters, the ranging performance model of the ground-based ranging system that aim at space targets cannot be approximated by the Poisson distribution.

5. Conclusion

In this paper, theoretical models of the detection probability and ranging performance that consider the effects of target speckles, the noise, and the dead-time of photon-counting detectors are first derived based on the NB distribution. The proposed ranging performance model is verified by both an experiment using a GM-APD lidar and a simulation using the recursive method. The results of the theoretical model agree well with the results from the experiment and the recursive calculation, which proves that the proposed ranging performance model is reliable.

The ranging performance model is then used to analyze the effect of target speckles for two typical photon-counting ranging systems with different types of targets: the space-borne ICESat-2 for detecting the Earth’s surface, and the ground-based laser ranging system for detecting space debris at SHAO station. The results indicate that when the speckle diversity of the system is large (>100), e.g., the ICESat-2, its ranging performance model can be approximated to be the classic model based on the Poisson distribution. However, when the speckle diversity of the system is small (<10), e.g., the SHAO, the approximation will overestimate the range walk error by ~1 cm and underestimate the ranging precision by 4.8 cm. In practice, the ranging performance of a ranging system aimed at area targets, e.g., the space-borne (or airborne) laser altimeter can be well approximated to be the Poisson distribution model, whereas the ranging performance of a laser ranging system aimed at point targets cannot be simplified, e.g., the ground-based SHAO system. In addition, the analytic solution to the speckle diversity of a retro-reflector is difficult to derive, because the far-field diffraction of the retro-reflector is influenced by the dihedral angle and the incident angle of the laser. The detection model for systems ranging to retro-reflector equipped satellites is worth further studying.

The new ranging performance model based on the NB distribution is universal because it is compatible with the classic model of the Poisson distribution. When the speckle diversity is larger than 100, the results calculated from the new NB model are identical to the classic Poisson model. The proposed ranging performance model can be used to predict the ranging accuracy and precision of a photon-counting lidar, to correct the range walk error for a photon-counting lidar, and to optimize system parameters when a photon-counting lidar is designed.

6 Appendix: Calculation of the speckle diversity

According to Goodman’s statistical theory on speckle correlation, the speckle diversity M can be defined as [7]

M = {[\frac{1}{A_{D}^{2}} \int \int_{- \infty}^{\infty} K_{D} (Δ x, Δ y) {| μ_{A} (Δ x, Δ y) |}^{2} d Δ x d Δ y]}^{- 1},

where A_D is the double integral of the weight distribution function of the effective receiving aperture. It can be expressed as

A_{D} = \int \int_{- \infty}^{\infty} D (x_{r}, y_{r}) d x d y,

where D(x_r, y_r) is the weight distribution function of the effective receiving aperture of a lidar, and x_r and y_r represent the two axis on the receiving aperture. (Δx, Δy) is the difference in coordinates between two points on the receiving aperture, i.e., Δx = x_r1- x_r2, Δy = y_r2- y_r1. If the weight distribution function is uniform, A_D can be simplified to be the area of the effective receiving aperture.

K_D(Δx, Δy) is the self-correlation function of the receiving aperture and can be expressed as

K_{D} (Δ x, Δ y) = \int \int_{- \infty}^{\infty} D (x_{r}, y_{r}) D (x_{r} - Δ x, y_{r} - Δ y) d x_{r} d y_{r} .

μ_A(Δx, Δy) is the normalized covariance function of the received pulse intensity distribution on the receiving aperture. It is equivalent to the result of the Van Cittert-Zernike theorem and can be expressed as [26]

μ_{A} (Δ x, Δ y) = \frac{\int \int_{- \infty}^{\infty} I (u, v) e^{- j \frac{2 π}{λ z} [u Δ x + v Δ y]} d u d v}{\int \int_{- \infty}^{\infty} I (u, v) d u d v},

where I(u, v) is the intensity distribution of the speckle field on the target surface, (u, v) represents the coordinates of the rectangular coordinate on the target surface, λ is the laser wavelength, and z is the distance between the target and the receiving aperture. The speckle diversity M is mainly determined by the receiving aperture and the speckle field on the target surface. The calculation of the speckle diversity M of a lidar can be divided into point targets and area targets.

6.1. Speckle diversity for point targets

For point targets, the size of the target is normally smaller than that of the laser beam, and the pulse intensity distribution on the target surface can be assumed to uniform. If the target is circular with a diameter of D_target, μ_A(Δx, Δy) can be expressed as

\begin{array}{l} μ_{A} (Δ x, Δ y) = \frac{\int \int_{- \infty}^{\infty} circ (\frac{2 \sqrt{u^{2} + v^{2}}}{D_{target}}) e^{- j \frac{2 π}{λ z} (u Δ x + v Δ y)} d α d β}{\int \int_{- \infty}^{\infty} circ (\frac{2 \sqrt{u^{2} + v^{2}}}{D_{target}}) d α d β} \\ \begin{matrix} \begin{matrix} \begin{matrix} = \end{matrix} \end{matrix} \end{matrix} \frac{2 π \int_{0}^{D_{target} / 2} ρ J_{0} (2 π \frac{r}{λ z} ρ) d ρ}{π D_{target}^{2} / 4} = 2 \frac{J_{1} (\frac{π D_{target}}{λ z} r)}{\frac{π D_{target}}{λ z} r}, \end{array}

where circ() is the circle function, ρ represents the radius of the polar coordinates on the target surface, r = (Δx² + Δy²)^0.5 and J₀() and J₁() are the zero and first order Bessel function, respectively.

If the receiving aperture is also circular, K_D(Δx, Δy) can be expressed as

\begin{array}{l} K_{D} (Δ x, Δ y) = \int \int_{- \infty}^{\infty} circ (x_{r}, y_{r}) circ (x_{r} - Δ x, y_{r} - Δ y) d x d y \\ \begin{matrix} \begin{matrix} \begin{matrix} = A_{r} (\frac{2}{π}) \end{matrix} \end{matrix} \end{matrix} [arc \cos (\frac{r}{D_{r}}) - \frac{r}{D_{r}} \sqrt{1 - {(\frac{r}{D_{r}})}^{2}}], \end{array}

where the effective aperture area is A_r = πD_r²/4 and D_r is the diameter of the effective receiving aperture. The speckle diversity M of a point target can be derived by substituting Eq. (27) and Eq. (28) into Eq. (23)

M = {[\frac{16}{π} \int_{0}^{1} γ (arc \cos γ - γ \sqrt{1 - γ^{2}}) | 2 \frac{J_{1} (π β γ)}{π β γ} | d γ]}^{- 1},

with β = D_rD_target/λz and γ = r/D_r. Using Eq. (29), the speckle diversity of a lidar aiming at point targets can be calculated by inputting its system parameters.

6.2. Speckle diversity for area targets

When the size of the target is larger than the laser beam, the speckle field over the target is the result of the Fraunhofer diffraction from the transmitting aperture. The distribution of the laser pulse intensity on the target can be expressed as

I (u, v) = \frac{1}{λ^{2} z^{2}} {| FT [E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t})] |}^{2},

where D_t(x_t, y_t) is the transmitting aperture function, E(x_t, y_t) is the amplitude distribution on the transmitting aperture, (x_t, y_t) are the coordinates of a point on the transmitting aperture, and FT() is a two-dimensional Fourier transform. In this case, μ_A(Δx, Δy) can be expressed as

\begin{array}{l} μ_{A} (Δ x, Δ y) = \frac{\int \int_{- \infty}^{\infty} I (u, v) e^{- j \frac{2 π}{λ z} [u Δ x + v Δ y]} d u d v}{\int \int_{- \infty}^{\infty} I (u, v) d u d v} = \frac{FT {FT [{| E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t}) |}^{2}]}}{2 π \int \int_{- \infty}^{\infty} {| E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t}) |}^{2} d x_{t} d y_{t}} \\ \begin{matrix} \begin{matrix}  \end{matrix} & = \frac{FT {FT [E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t})] {FT}^{*} [E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t})]}}{2 π \int \int_{- \infty}^{\infty} {| E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t}) |}^{2} d x_{t} d y_{t}} \end{matrix} \\ \begin{matrix} \begin{matrix} = \frac{[E (- x_{t}, - y_{t}) D_{t} (- x_{t}, - y_{t})] [E^{*} (- x_{t}, - y_{t}) D_{t}^{*} (- x_{t}, - y_{t})]}{2 π \int \int_{- \infty}^{\infty} {| E (x_{t}, y_{t}) D_{t} (x_{t}, y_{t}) |}^{2} d x_{t} d y_{t}} \end{matrix} \end{matrix} . \end{array}

The symbol * represents the conjugation. If the transmitting aperture is circular, Eq. (31) can be further simplified to be

μ_{A} (Δ x, Δ y) = \frac{\int \int_{Σ_{t}} E (x_{t}, y_{t}) E^{*} (x_{t} - Δ x, y_{t} - Δ y) d x_{t} d y_{t}}{2 π \int \int_{Σ_{t}} {| E (x_{t}, y_{t}) |}^{2} d x_{t} d y_{t}},

where Σ_t is the integral area of the transmitting aperture. The speckle diversity M of an area target can be derived by substituting Eq. (32) and Eq. (28) into Eq. (23), then expressed as

M = {[\frac{2}{π A_{r}} \iint_{Σ_{r}} [arc \cos (ξ) - ξ \sqrt{1 - ξ^{2}}] {| μ_{A} |}^{2} d Δ x d Δ y]}^{- 1},

where Σ_r is the integral area of the receiving aperture and ξ = (Δx² + Δy²)^0.5/D_r. Using Eq. (33), the speckle diversity of a lidar aiming at area targets can be calculated by inputting its systematic parameters.

Funding

National Natural Science Foundation of China (41506210, 41801261); National Science and Technology Major Project (11-Y20A12-9001-17/18, 42-Y20A11-9001-17/18); Postdoctoral Science Foundation of China (2016M600612); Joint fund of Equipment pre-research for space technology (6141B06110102, 6141B06110103). State Key Laboratory of Geo-Information Engineering (SKLGIE2018-Z-3-1); Fundamental Research Funds for the Central Universities (2042019kf1001).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Experiment No.	Attenuators	Attenuation rates	Average signal photon counts	Cumulative numbers	R_{a_mea} (cm)	R_{a_the} (cm)
1	3	0.0501	8.46	10⁵	−93.92	−98.33
2	1, 3	0.0200	3.37	10⁵	−67.50	−59.71
3	2, 3	0.0126	2.13	10⁵	−41.54	−42.28
4	1, 2, 3	5.01 × 10⁻³	0.572	10⁵	−17.73	−14.21
5	4	1.00 × 10⁻³	0.456	2 × 10⁵	−14.28	−11.95
6	1, 4	6.18 × 10⁻⁴	0.221	2 × 10⁵	−6.65	−6.88
7	2, 4	2.59 × 10⁻⁴	0.164	3 × 10⁵	−6.41	−5.22
8	3, 4	5.42 × 10⁻⁵	0.059	3 × 10⁵	−0.89	−2.36

Ranging performance models based on negative-binomial (NB) distribution for photon-counting lidars

Abstract

1. Introduction

2. Theoretical model

2.1. Detection probability of signal photon events

2.2. Detection probability of signal PEs considering the noise and dead-time effects

2.3. The ranging performance model

3. Model verification

3.1. Verification by lidar ranging experiment

3.2. Verification by the recursive method

4. Discussion

4.1. Differences between the NB and Poison models

4.2. Different photon-counting lidars satisfying different models

5. Conclusion

6 Appendix: Calculation of the speckle diversity

6.1. Speckle diversity for point targets

6.2. Speckle diversity for area targets

Funding

Disclosures

References

Cited By

Figures (5)

Tables (1)

Equations (33)

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