Ranging performance is the key indicator for a ranging lidar system, and a theoretical ranging performance model can be used for optimizing systematic parameters when designing a lidar system. The fluctuation of the signal photon numbers introduces range walk error to photon-counting lidars that can be as high as tens of centimeters. In this paper, based on the lidar equation and the statistical property of photon-counting detectors, a theoretical ranging performance model for photon-counting lidars with multiple detectors is first derived. Next, a theoretical correction method for offsetting range walk error is proposed, as verified by experiments using a Gm-APD (Geiger mode avalanche photodiode) lidar system. The results indicate that multiple detectors are very useful to maintain a consistent ranging precision when the mean received signal photons are variable. With a 1600 repetitive ranging measurement, the new method can achieve a centimeter ranging accuracy, with a mean and standard deviation of the residual errors of 1.14 cm and 1.23 cm, respectively. This new method is potentially suitable for a satellite photon-counting lidar system such as ICESat-2, as only a certain number of repetitive measurements, the time tag, and a certain number of triggered detectors are required to offset the range walk error.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
A photon-counting lidar system equipped with Geiger mode avalanche photodiodes (Gm-APDs) or Photomuitpliers (PMTs) can only respond to the presence of return signals and record the time tags with the output of 0 or 1, but cannot record the return waveforms [1-2]. Benefiting from the photon-counting detector with higher detection sensitivity and lower dark noise, the energy of transmitted laser pulses is much lower (normally 2 or 3 orders of magnitude) compared to that of a conventional lidar system. A lower transmitted laser energy allows a smaller laser divergence and a higher repetitive frequency to be achieved, thereby improving the resolution and density of the laser point cloud. With these advantages, photon-counting lidars have been used to acquire 3D depth-resolved images [3-4], along-track laser footprints (borne on an aircraft or satellite) [5-6], etc.
Ranging performance is the key indicator for a ranging lidar system, and a theoretical ranging performance model can be used for optimizing systematic parameters when designing a lidar system. Classical theoretical ranging performance models are mostly concentrated upon lidars with a single detector [7-8]; however, the most advanced photon-counting lidar borne on the ICESat-2 (Ice, Cloud, and land Elevation Satellite-2)  and its airborne testing lidar system MABEL (Multiple Altimeter Beam Experimental Lidar) are both equipped with multiple detectors [5, 10]. In addition, a photon-counting lidar faces the range walk error caused by the fluctuation of the signal photon numbers, which can introduce tens of centimeters of ranging bias . Oh et al.  derived a correction method based on the theoretical detection probability and achieved a standard deviation of residual errors of over 2 cm by using a Gm-APD lidar with a 5000 measurement accumulation. Xu et al.  proposed a correction method based on Gaussian function fitting for a 3D imaging photon-counting lidar that can achieve a ranging accuracy of 1.2 cm with 12000 repetitive measurements. He et al.  and Ye et al.  derived correction methods; however, a priori model should be established or a dual detection system is required, thereby limiting the use of these methods.
The fiber of the MABEL was damaged and thus the mean received signal photons decreased to less than 0.2 (corresponding to a very weak range walk error) ; as a result, the MABEL does not require offsetting the range walk error. The future ATLAS (Advanced Topographic Laser Altimeter System) borne on ICESat-2 must offset the range walk error because the designed mean signal photons of the ATLAS register from 0.1 to 10. However, the current correction methods mentioned above cannot be used to offset the range walk error because the ATLAS will only aggregate 100 successive measurements along-track and some priori cannot be accepted for a satellite lidar.
Based on the lidar equation and the statistical property of photon-counting detectors, this paper first derived a theoretical ranging performance model for photon-counting lidars with multiple detectors and then, quantified the effects (e.g., number of detectors, mean received signal photons, and mean pulse width) that influence the ranging performance. Next, a theoretical correction method for offsetting the range walk error was proposed, which was verified by a Gm-APD lidar system to achieve a centimeter ranging accuracy with only 1600 accumulations (corresponding to 100 successive measurements of ICESat-2).
2. Detection probability with multiple detectors
For a satellite lidar system, the transmitted laser pulse undergoes Fresnel diffraction during propagation and then, is reflected by the Earth's surface. After another Fresnel diffraction process, the laser pulse, which is incident on the telescope objective, is focused onto the photoelectric detector. For a typical flat target with slope and roughness, if the waveform of the transmitted laser pulse is Gaussian in shape, then the received signal photons S(t) is also Gaussian in shape and is expressed as Eq. (2) ; fn is the total noise rate (normally in units of MHz); and σs is the width of the received laser pulse.19], and the relationship is fn = fd + ηqfb.
A photon-counting detector can only respond to the presence of signals when a single observation is operated. According to Fouche's theory , the Poisson distribution can well represent the response of a photon-counting detector, and the detection probability Psn can be expressed in Eq. (3).Eq. (3), the mean noise photons before the arrival of the laser signal photons are the product of the total noise rate fn and the dead-time τd of the photon-counting detector. In Eq. (3), the detection probability Psn corresponds to the case that the signal photons are captured within range gate and no noise photon is captured within the dead-time before the signal arriving. The total noise rate is normally 0~5 MHz for satellite lidars and the dead-time of photon-counting detectors is from a few nanoseconds to many tens of nanoseconds. For a short dead-time Gm-APD or PMT (several nanoseconds), the first exponential function in Eq. (3) is nearly equal to 1, but for a long dead-time Gm-APD (many tens of nanoseconds), the first exponential function is not equal to 1 when the noise rate is several MHz.
If a lidar system has multiple detectors (normally a detector array), e.g., the ATLAS systems, then each detector can only capture the photons locating within its corresponding sub-FOV (Field of View). For a fundamental mode laser, the laser cross section of the transmitted laser pulse is assumed to be Gaussian in shape and can be expressed as Eq. (4). If the intensity reflection coefficient βr is identical within the laser footprint, then the energy fraction Pij corresponding to the i-th row and j-th column detector can be expressed as Eq. (5) .Eq. (6) can be simplified asEq. (6)), multiple detectors share the total photons and the detection probability of each detector decreases. When the detector number n is equal to 1, Eq. (7) is identical with Eq. (3). Equation (7) is a more universal expression. For short dead-time detectors (e.g., PMTs used in the ATLAS system) or a very weak noise ratio (e.g., measuring at night), the first exponential function in Eq. (7) is nearly equal to 1 and can be ignored.
3. Ranging performance model and range walk error correction
3.1 Ranging performance model for multiple detectors
The PDF (Probability Density Function) of a photon-counting lidar with multiple detectors fp(t) can be obtained by differencing Eq. (7) as given by Eq. (8). Moreover, when the detector number n is equal to 1, Eq. (8) can represent the fp(t) corresponding to a lidar system with a single detector. When a photon-counting lidar system performs a repetitive measurement, the PDF fp(t) represents the time tag distribution of all photon events that is similar with the waveform of a conventional lidar. Figure 1 illustrates the detection probability density of a photon-counting lidar, when the signal photons are Ns = 1, 2, 3, and 4, respectively; the detector number is n = 1; the received width is σs = 2ns; and the noise is ignored.
From Fig. 1, when the number of mean signal photons increases, the signal photons are more probably concentrated to the centroid of the curve of the detection probability density, indicating a better ranging precision; however, a more distinct systematic ranging bias (called range walk error) emerges. The range walk error is always a minus value, i.e., the range is underestimated. When the noise effect is ignored (e.g., measuring at night and fn = 0), the mean time tags of the photon events can be calculated as Eq. (9), and the variance of time tags is given by Eq. (10), where Var denotes the variance.
When the laser ranging is based on the time of flight, the ranging systematic bias (range walk error) arising from the laser itself and the target will be the product of the mean time tags of the photon events and half of the light velocity, i.e., , where c is the light velocity. The ranging precision will be the product of the time tags' standard deviation and half of the light velocity, i.e.,. Based on the law of propagation of errors, if a target is independently measured by multiple detectors of a lidar system (equivalent to a repetitive measurement), then the accumulated ranging precision will be improved by dividing the root of the repetition number n, i.e.,. Equations (11) and (12) give the expression of the range walk error and the ranging precision, respectively.
Using Eq. (11) and Eq. (12), Fig. 2 illustrates the theoretical relationship between the ranging performance and the photon number with different widths and a single detector. Note that, the range walk error is always a minus value as mentioned above and the absolute values of the range walk errors are used in Fig. 2(a). Figure 2 indicates the following: (1) both the range walk error and ranging precision will be benefitted from decreasing the laser received width, and (2) when the expected signal photons increase, the ranging precision will be better, but the range walk error will increase.
According to Eq. (7) in , Brenner et al. indicates that, the received width σs is mainly influenced by the transmitted width σf, the divergence angle θT, the altitude z, and the roughness and slope of the target surface. For a typical target surface with a 0.3 m roughness and 1° slope, when the transmitted width is 0.65 ns, the divergence angle is 0.031 mrad, the altitude is 500 km (all close to ICESat-2's parameters) , and the received width will be approximately 2 ns.
When the received width is σs = 2ns, Fig. 3(a) illustrates the theoretical relationship between the range walk error and the photon number for different detector numbers, and Fig. 3(b) illustrates the relationship between the ranging precision and the photon number. Figure 3 indicates the following: (1) when multiple detectors are used, the total signal photons are shared and the signal photons for each detector are decreased, thereby decreasing the range walk error; (2) although the ranging precision will be worse when the mean signal photons for each detector are decreased, multiple detectors give repetitive measurements for a homogeneous target surface (such as ice sheets), thereby improving the ranging precision, and when the mean signal photons are in the range of 0.1 through 10 (very close to ICESat-2's designed signal photons), the ranging precision will be better if multiple detectors are used because the effect of repetitive measurements are the dominate factor; (3) multiple detectors are very useful to maintain a consistent ranging precision when the mean signal photons change from 0.1 to 10; (4) the mean received signal photons of the ICESat-2's lidar are up to 10 corresponding to more than 5 cm range walk error, even though using a 4 × 4 detector array, which cannot be neglected in its error budget; (5) for a typical target surface with a roughness of 0.3 m and slope of 1°, the ICESat-2 lidar can achieve an approximately 7.5 cm ranging precision after discarding the background noise photons. Currently, the advanced signal processing algorithms for photon-counting lidars are able to discard most of the noise photons [20–22].
3.2 Range walk error correction
When a lidar is performing a measurement, the detection probability of each detector within the range gate is expressed as Psn_i = ndet_i/nshot, where ndet_i is the signal and noise photons captured by the i-th detector, and nshot is the laser shot numbers. As mentioned above, after the signal processing of the data from photon-counting lidars, most of the noise photons can be discarded, and ndet_i will be mainly signal photons. Substituting this relationship into Eq. (7), we can obtain the expression of the observed averaged signal photons as Eq. (13).
Substituting Eq. (13) into Eq. (11), we can estimate the range walk error Ra_est using Eq. (14). If we discard the noise photons and calculate the number of ndet_i and nshot, Eq. (14) can be used as an offset to calibrate the range walk error.16]), then the range walk error will be more accurately estimated because of the ergodicity property in statistics.
A Gm-APD photon-counting lidar system is established to verify the derived ranging performance model and the correction method of the range walk error; Figs. 4(a) and 4(b) show a schematic of the lidar system and a photograph of the lidar system, respectively. An OSRAM semiconductor laser (SPL90_3) with a laser wavelength of 905 nm was used; the laser was driven by a power supply (LDP AV40-70) to produce laser pulses with a full width at half maxima (FWHM) of 7 ns (corresponding to a laser width of 3.2 ns) and a pulse repetition frequency of 50 kHz. After beam shaping by two lenses (Thorlabs AYL-108B for the fast-axis and OptoSigma CLB-1020-80IR for the slow-axis), the cross section of the laser is nearly Gaussian in shape, with a laser divergence of 4.2 mrad. The aperture diameter of the telescope is 35 mm, and the FOV is 10 mrad. A Gm-APD (SPCM AQ4C) was used to capture the photons; this detector is the same as the detector of MABEL and has a dead-time of 50 ns, a dark noise rate of 500 Hz, and a photon detection efficiency of approximately 27% at 905 nm. The signal is finally recorded by a TOF digitizer (MCS6A4T2), which can achieve a time bin of 200 ps and no dead-time effect.
Five different attenuators were used to capture 5 groups of experimental data under different mean numbers of signal photons. In each group, the ranging measurement is repeated 10000 times, and ndet is the number when the detector is triggered. The 'true range' between the lidar system and the target was 49.620 m, which was measured by a Leica TS09 total station with a ranging accuracy of millimeters. Table 1 shows the experiment results in detail. In addition, as shown in Fig. 4(b), the experiment was performed at night, and background noise rate was approximately 500 Hz (10−7 counts per a 200 ps time bin); thus, compared to the signal rate, the noise rate fn can be ignored in this experiment. In Table 1, in the repeated measurement of 10000 times, the actual detection probability Psn and the averaged signal photons Ns were calculated using Eq. (13), with the results ranging from 0.156 to 4.335 photons.
The actual ranging results Runc were equal to the mean of the time tags of the captured signal photons, the actual range walk error Ra_mea is the difference value between the actual ranging results Runc and the true range Rtruth (measured by the total station), and the theoretical range walk error Ra (when the width is σs = 3 ns) is calculated by using Eq. (11). The actual range walk errors Ra_mea are illustrated in Fig. 2(a) using green filled circles and the data are in accordance with the theoretical range walk errors based on Eq. (11). The results show that the mean and standard deviation of the differences between actual and theoretical range walk errors (Ra_mea and Ra) are 0.28 cm and 1.38 cm, respectively, thus verifying the derived theoretical range error model.
The correction of range walk error Ra_est is calculated based on Eq. (14) and used as offsets to correct the Runc. Rcor is the corrected range. The residual error is the difference between the corrected range Rcor and the truth range Rtruth. The results show that the mean and standard deviation of the residual errors is 1.14 cm and 1.23 cm, respectively, thus verifying that the derived Eq. (14) can be used to effectively correct range walk errors.
The correction results in Table 1 were obtained when the measurement was repeated 10000 times, but only successive 100 measurements will be aggregated for the ICESat-2 photon-counting lidar. To verify the correction of range walk error using Eq. (14) when the number of repeated measurements is small, new experiments were performed for the attenuation rates of 1/70 and 1/35, which correspond to the mean signal photons Ns of approximately 0.70 and 1.44, respectively, in Table 1. Note that, in the new experiments, the mean signal photons Ns is an unknown value that should be calculated based on Eq. (13). In each attenuation rate, 30 groups data were captured when the number of repeated measurements was 160 times, and another 30 groups data were captured when the number of repeated measurements was 1600 times. Because the lidar borne on the ICESat-2 has 16 photon-counting detectors, 160 times corresponds to 10 successive along-track measurements (only 1/10 of the planned 100 successive measurements), whereas 1600 times corresponds to 100 successive measurements.
When the attenuation rate is 1/70 and the number of repeated measurements is 160 times, Fig. 5(a) illustrates the time tag distribution when the detector was triggered, and Fig. 5(b) illustrates the averaged uncorrected ranges Runc, the corrected ranges Rcor, and the truth range Rtruth of 30 groups. When the number of repeated measurements is 1600 times, Figs. 6(a) and 6(b) show the similar bar and curves as those of Figs. 5(a) and 5(b). The corrected results are all shown in Table 2.
From Table 2, even 160 repetitive measurements are sufficient to correct most of the range walk errors and the results are very close to the residual errors shown in Table 1 with 10000 repetitive measurements. When 1600 repetitive measurements are aggregated, the residual errors are identical to the results with 10000 repetitive measurements in Table 1.
In the verified experiment, residual errors still existed after the correction, mainly because the used photon-counting detector is a Gm-APD, which has a long dead-time (50 ns), whereas the correction method is derived for a photon-counting lidar with short dead-time detectors. A long dead-time makes the triggered time tag distribution have a steeper leading edge and a tail, as illustrated in Fig. 6(a), resulting in underestimation of the range. This underestimation is the reason that the corrected offsets calculated based on Eq. (14) are larger, and residual errors are most positive values. Nevertheless, the new correction method performed well, even if a detector with a long dead-time was used. For the ICESat-2's lidar with 16 short dead-time PMT detectors, the derived correction method will be more suitable, and 160 or 1600 repetitive times mean only 10 or 100 successive along-track measurements, respectively.
In addition, if the laser pulse is received at the beginning or the end of the range gate, the Gm-APD or PMT might not detect the complete signal pulse, which might cause that the correction method may not work very well. Fortunately, it can be guaranteed that the ATLAS on ICESat-2 is able to receive the complete signal pulse in the overwhelming majority of measurements. The ICESat on-board search for a valid echo pulse uses a digital elevation model with surface type, and real-time ICESat position to estimate the expected time of flight . Moreover, the range gate of the airborne MABEL is up to approximately 1500 m in the vertical direction, which is sufficient enough to capture the complete echo pulse . As the most advanced satellite lidar at present, the ICESat-2 will also operate these steps to guarantee the quality of received data.
A theoretical ranging performance model for photon-counting lidars with multiple detectors was first derived, by which the range walk error and ranging precision were quantified when inputting different values of the number of detectors, the mean received photons, and the mean pulse width. The result indicates that multiple detectors are very useful to maintain a consistent ranging precision when the mean signal photons change from 0.1 to 10. For a typical target surface with a 0.3 m roughness and 1° slope, the ICESat-2 lidar can achieve approximately 7.5 cm ranging precision after discarding the background noise photons.
Moreover, a new theoretical correction method for range walk error was proposed that is convenient to operate, i.e., only the number of repetitive measurements, the time tag and the number of the photon-counting detector triggering are required. A Gm-APD lidar system was used to verify the theoretical ranging performance model and the correction for the range walk error. The result indicates that for a satellite photon-counting lidar system such as ICESat-2 (0.1-10 mean signal photons and 2 ns received width), this correction method may achieve a centimeter ranging accuracy with 100 successive measurements of ICESat-2, with a mean and standard deviation of the residual errors of 1.14 cm and 1.23 cm, respectively. The ICESat-2 is coming soon in the second half of 2018 and the proposed correction method is potentially suitable to offset the range walk error with only aggregate 100 successive measurements.
The National Natural Science Foundation of China (41506210); the National Science and Technology Major Project (11-Y20A12-9001-17/18, 42-Y20A11-9001-17/18); the Postdoctoral Science Foundation of China (2016M60061).
The comments from the two anonymous reviewers improved the manuscript and we thank the reviewers and the editor very much. This is publication number 51 of the Sino-Australian Research Centre for Coastal Management.
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