(2 天津市成像与感知微电子技术重点实验室, 天津 300072)
(3 天津市红外成像技术工程中心, 天津 300072)
(2 Tianjin Key Laboratory of Imaging and Sensing Microelectronic Technology, Tianjin 300072, China)
(3 Tianjin Infrared Imaging Technology Engineering Center, Tianjin 300072, China)
Airborne Lidar Bathymetry (ALB) is an accurate, cost-effective and efficient technique for shallow water measurements. As the laser pulse travels through the water column and encounters the bottom of sea, it undergoes absorption, scattering and refraction. These processes attenuate the laser return energy, ultimately limiting the depth at which sea bottom can be detected^{[1-3]}.
In general, the return signal reflected from the water column varies dramatically with water depth^{[4]}. Therefore, the common survey system prepares two bottom detecting receiving channels, such as the SHOALS system, which equips two receiving channels measuring the green laser return from shallow water depths, 1~12 m, and intermediate water depths, 7~40 m^{[5]}. Actually, the Photomultiplier Tube (PMT) of Hamamatsu, H1156-20-NN, exhibits typical dark current of 10 nA, while the max output current reaches 100 μA. The dynamic range of the PMT^{[6-7]} is calculated by DRPMT=20×lg(100 μA/10 nA)=80 dB.
However, the available maximum dynamic range of a typical high-speed 8 bit Analog to Digital Converter (ADC) device is 48.16 dB. Therefore, it is impossible to process the signal received aptly all the time even though the noise level is not considered.
For the HawkEye Ⅱ system, the received signal from the two green channel receivers is applied with a Time-Varied Gain (TVG) filter to enhance the bottom return ^{[8]}.However, it would cause the problem of waveform distortion. Moreover, on account of bottom radiance and water turbidity^{[9-12]}, the amplitude of bottom return is variable and unknown^{[13]}, so it is not impeccable to only depend on optical attenuation or gain adjustment.
In this paper, we aim to design a new processing structure to receive the signal with 80 dB dynamic range and verify the reliability of the design by waveform stitching method and fitting algorithm, along with the objective of making the entire predictive system more accurate.
1 Materials and methods 1.1 Waveform simulationThe waveform data set is generated by using the recently developed Wa-LID simulator^{[14]}, which is the summation of the laser pulse convolution with the impulse response functions of the surface, column and bottom, as well as additional noises. In our simulations, the partial values of system parameters acting in the Wa-LiD equations and partial environment parameters are listed in Table 1 and Table 2, respectively.
In order to compute accuracy statistics on the bathymetry estimates, the data set is generated. Data set contains 1000 simulated waveforms for each water depth.
1.2 Design and analysis of three-channel processing structureIn order to receive a signal with 80 dB dynamic range, a three-channel processing structure illustrated in Fig. 1 has been introduced into the design.
PMT with high sensitivity is used to record the return signal events and process the return signal in conjunction with the three-channel processing structure. Here, we use the Trans-impedance Amplifier (TIA) and the filter to pre-process the return. Afterwards, the strong signal (shallow water returns), the middle signal (intermediate water returns) and the weak signal (deeper returns) with low, middle and high gain in parallel processing mode are amplified, respectively, by utilizing a Variable-Gain Amplifier (VGA), a Fixed-Gain Amplifier (FGA) and the combination of the two. The D_{l}, D_{m} and D_{h} are the data sampled by the high-speed ADC in the low, middle and high gain channel respectively, finally captured by Field Programmable Gate Array(FPGA).
In this paper, we set low gain G_{l}=1, middle gain G_{m}=10 and high gain G_{h}=100, which could achieve dynamic range of 88.1 (20×lg (256×G_{h})) dB theoretically. The three-channel waveform at the depth of 18 m quantified by ADC is illustrated in Fig. 2. The ADC used in the structure is a 2GSPS, 8 bit device. The horizontal axis in Fig. 2 is the number of sample points and the vertical axis is the output code from the ADC.
In Fig. 2, it is appropriate that the bottom return is amplified by high gain channel obviously.
1.3 Bathymetry estimation from three-channel waveformFor the purpose of utilizing the waveforms recorded by the design preferably, the three-channel simultaneous recordings are stitched. The stitching method is given by
$ D\left\{ \begin{array}{l} {D_1} + {D_{\rm{m}}} + {D_{\rm{h}}}\;\;\;\;\left( {{D_{\rm{h}}} = {{2.}^\wedge}{Q_{\rm{b}}} - 1, {D_{\rm{m}}} = {{2.}^\wedge}{Q_{\rm{b}}} - 1} \right)\\ \;\;\;\;{D_{\rm{m}}} + {D_{\rm{h}}}\;\;\;\;\;\;\;\left( {{D_{\rm{h}}} = {{2.}^\wedge}{Q_{\rm{b}}} - 1, {D_{\rm{m}}} < {{2.}^\wedge}{Q_{\rm{b}}} - 1} \right)\\ \;\;\;\;\;\;\;\;{D_{\rm{h}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{D_{\rm{h}}} < {{2.}^\wedge}{Q_{\rm{b}}} - 1} \right) \end{array} \right. $ | (1) |
where Q_{b} is quantization bits, D is the data sets after stitching. Furthermore, there are advantages of keeping the correlation among the waveform data in this process and avoiding the problem of waveform distortion existing in the system which uses the TVG filter.
After stitching and smoothing, apeak detection procedure is used on the smoothed waveforms. It regards a peak as any local maxima in the lidar waveform that has an amplitude much higher than the noise level, 8 times in this paper. Thus, the dynamic range limited by peak detection is changed to 70.0 dB (88.1-20lg (8)).
The peak with the largest amplitude is attributed to the surface position (A_{s}, μ_{s}) while the peak with the largest time is attributed to the bottom position (A_{b}, μ_{b}). After peak detection, the fitting procedure is applied to the waveforms that have a detectable bottom. The pentagonal function is used to fit the water column contribution on account of the different slopes in column return caused by different gains. The surface and bottom returns are considered to be Gaussian functions. So the fitted lidar waveform can be expressed as
$ R\left( t \right) = G\left( {t;{A_{\rm{s}}}, {\mu _{\rm{s}}}, {\sigma _{\rm{s}}}} \right) + Q\left( {t;a, b, c, d, e, g, h, f} \right) + G\left( {t;{A_{\rm{b}}}, {\mu _{\rm{b}}}, {\sigma _{\rm{b}}}} \right) $ | (2) |
where G(t; A_{s}, μ_{s}, σ_{s}) is the Gaussian function defined as
$ G\left( {t;{A_{\rm{s}}}, {\mu _{\rm{s}}}, {\sigma _{\rm{s}}}} \right) = {A_{\rm{s}}}\exp \left( { - {{\left( {t - {\mu _{\rm{s}}}} \right)}^2}/2{\sigma _{\rm{s}}}^2} \right) $ | (3) |
where A_{s}, μ_{s}, and σ_{s} are the amplitude, the mean and the standard deviation of the Gaussian function, respectively.
The Q(t; a, b, c, d, e, g, h, f) is given by
$ Q\left( {t;a, b, c, d, e, g, h, f} \right) = \left\{ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {t \le a} \right)\\ \;\;\;\;\;\;\;e\left( {\left( {t - a} \right)/\left( {b - a} \right)} \right)\;\;\;\;\;\;\;\;\;\;\;\;\left( {a \le t \le b} \right)\\ \;\left[{ec-bg + t\left( {g-e} \right)} \right]/\left( {c - b} \right)\;\;\;\;\left( {b \le t \le c} \right)\\ \left[{gd-hc + t\left( {h-g} \right)} \right]/\left( {d - c} \right)\;\;\;\;\left( {c \le t \le d} \right)\\ \;\;\;\;\;\;h\left( {\left( {f - t} \right)/\left( {f - d} \right)} \right)\;\;\;\;\;\;\;\;\;\;\;\left( {d \le t \le f} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {t \ge f} \right) \end{array} \right. $ | (4) |
where a, b, c, d, and f are the x-axis points of five corners for the pentagonal; and e, g, and h are the ordinates for three vertices, respectively.
The G(t; A_{b}, μ_{b}, σ_{b}) is the Gaussian function defined as
$ G\left( {t;{A_{\rm{b}}}, {\mu _{\rm{b}}}, {\sigma _{\rm{b}}}} \right) = {A_{\rm{b}}}\exp \left( { - {{\left( {t - {\mu _{\rm{b}}}} \right)}^2}/2{\sigma _{\rm{b}}}^2} \right) $ | (5) |
where A_{b}, μ_{b}, and σ_{b} are the amplitude, the mean and the standard deviation of the Gaussian function, respectively.
A Nonlinear Least-Squares (NLS) approach using the Levenberg-Marquardt optimization algorithm is performed to fit the sum of three functions. The initial values of NLS fitting are listed in Table 3, where T_{0} is pulse width.
In order to verify that the three-channel processing structure could enhance the bottom return and make the entire predictive system more accurate, some data sets are generated. Of the overall waveforms, the dynamic range DR_{0}, i.e., the ratios of the surface peak and the bottom peak is calculated. After simulation, it proves that the DR_{0} exceeds 71.1 dB at water depths greater than 27 m, which is beyond the dynamic range limited by peak amplitude definition (70.0 dB). Thus, the dynamic range DR_{0} at water depths of 1 m to 27 m are shown in Fig. 3 (a), and the peak detection rates of the three-channel and the one-channel processing approach are shown in Fig. 3 (b).
At the water depths of 1-26 m, the DR_{0} reaches 68.9 dB, which means 86.9 dB (68.9+20lg (8)) for the received signal when the limit of peak detection taken into account. As a result, the advantage of the three-channel processing structure is evident. The bottom return can still be detected when water depth is greater than 8 m, at which the dynamic range reaches 29.6 dB, about 47.6 dB (29.6+20lg (8)) for the received signal, while the one-channel is out-of-range for the property.
The Signal to Noise Ratio (SNR) is calculated for each water depth, which is defined here by the ratio of the bottom peak amplitude in the waveforms to the noise amplitude. For the water depths of 1~26 m, the maximum SNRs of bottom return with a detectable bottom in three-channel are shown in Fig. 4.
In Fig. 4, The maximum SNR decreases with water depth, which reduces from 804.2 at 1m water depth to 1.4 at 26 m water depth in low gain channel, from 8 808.7 to 15.7 in middle and from 14 851.6 to 28.2 in high.
By using the fitted parameter values, the bias, i.e., the mean difference betweenthe estimated and simulated water depths as well as the standard deviation of the bathymetry estimates are computed. The fitting results and the accuracy of fitting algorithm at water depths of 1 to 26 m are shown in Fig. 5.
The simulation result shows that the bias ranges from 1.6 to 4.7 cm at water depths of 1 m to 26 m. Besides, the standard deviation is better than 1.1 cm. The overall bias and standard deviation for all the used water depths are shown in Table 4, as well as the results in Ref.[15] and Ref.[16].
The result of error statistics shows improvements of 2.7 cm in bias, 7.3 cm in standard deviation and 16 m for limiting depth compared with Ref.[15]. Additionally, it takes on superiority of 1.9 cm in standard deviation and 11 m for limiting depth compared with Ref.[16]. This proves that it is feasible to reach a greater dynamic range by using the three-channel processing structure which could achieve more accurate performance for bathymetry estimate and greater limiting depth.
3 ConclusionIn conclusion, we propose a novel processing structure and method to process the signal with 80 dB dynamic range and verify the reliability of bathymetry estimates through simulation and analysis. The result has shown that the processing structure could measure the echoes of water depth of 1 m to 26 m, with the dynamic range of 86.9 dB, better than 47.6 dB achieved by the one-channel. The bias of the bathymetry estimates is ranging from1.6 to 4.7 cm with the standard deviation better than 1.1 cm. This processing structure satisfies the requirement of receiving and processing the signal with a wide dynamic range in lidar bathymetry. We will continue to study various coastal areas to develop a practical and reliable method for bathymetry.
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