The range-gated imaging systems are reliable underwater imaging system with the capability to minimize backscattering effect from turbid media. The tail-gating technique has been developed to fine tune the signal to backscattering ratio and hence improve the gated image quality. However, the tail-gating technique has limited image quality enhancement in high turbidity levels. In this paper, we developed a numerical model of range-gated underwater imaging system for near target in turbid medium. The simulation results matched the experimental work favorably. Further investigation using this numerical model shows that the multiple scattering components of the backscattering noise dominate for propagation length larger than 4.2 Attenuation Length (AL). This has limited the enhancement of tail-gating technique in high turbidity conditions.
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The range-gated imaging systems are well known for their capability to minimize backscattering effect from turbid media [1–3]. Tan et al.  and Witherspoon et al.  have confirmed that the tail-gating is superior than front gating. This is contrary to the results presented by McLean et al. . However, it is generally believed that the multiple scattering noises/photons could be dominant at higher level of turbidity. In this high turbidity level, the use of the tail-gating technique might not provide improvement to the image quality.
In the past, the research shows that the variation of backscattering noise due to fixed upper-limit convolution effect could improve the signal to backscattering noise ratio (assuming negligible multiple scattering effect from the target signals) towards the tail Reflected Image Temporal Profile (RITP) region up to C = 2.07/m, as experimented by Tan et al. . However, the experimental results have demonstrated the limited image quality enhancement in high turbidity level. In order to verify the experimental results with our assumption, we presents the estimation of the multiple scattering noises such that they become dominant components of the signal to noise ratio in a range gated imaging application. For complex geometrical conditions, Monte Carlo method simulation [6–10] of light propagation yields better results than an approximated analytical approach to estimate light propagation in turbid medium. Thus, this paper has proposed the use of Monte Carlo numerical approach to estimate the variation of backscattering noise and the multiple scattering noises in high turbidity medium.
The Monte Carlo model simulates the light propagation in short target distance (3 m target distance) in turbid seawater condition, which is the focus of many such range-gated system in recent years, i.e. He et al. , Grace et al. , and Hiroshi et al. . The simulation results match the experimental work favorably. Further investigation using this numerical model shows that the multiple scattering components of the backscattering noise dominate for propagation length larger than 4.2 Attenuation Length (AL).
2. Light components of reflected image temporal profile (RITP)
In range-gated underwater imaging system, the receiver is adjacent to the light source while the underwater target is behind the scattering medium. Thus, the Reflected Image Temporal Profile (RITP) is observed as super-position of a single target reflection and the backscattering noise in time domain. For an infinite length light pulse exerted into the medium, the backscattered energy at the receiver is modeled using a Temporal Point Spread Function  (TPSF), which represents the reflectance from the medium. The RITP is formed when the TPSF convolves with an approximately Gaussian  (actual measurement/pre-defined) pulse produced by the illuminator (light source).
Assuming the photons do not interact with each other, the received energy can be modeled as additive summation of individual photons based on their history. These summations of individual photons in any time instance will result in different signal to backscattering noise ratio [4,11]. Bissonnette [8,16] categorized the photons into different groups based on their propagation history, such as direct photons, single forward scattered photons, multiply scattered and backscattered photons. Those photon groups have been linearly summed up to obtain total reflected energy. Alternatively, Zege et al.  separated the returning energy from various layers of scattering media, such as water, atmosphere, water surface and underwater target. In this paper, we adopt Bissonnette’s approach to categorize the photons based on their individual history. The RITP energy, P(t), received at any time instance from scattering medium can be described as
PBSN(t) contains the single and multiple backscattering noise from turbid medium excluding target reflected photons within the receiver field of view (FOV). PS(t) contains the direct or forward scattered target reflected photons within the FOV (which is the desired signal in range-gated imaging system). PSMM(t) contains the target reflected photons within the FOV which have suffered multiple scattering such that the target information is lost.
The backscattering noise [PBSN(t)] is initiated at the entrance of the medium and dissipated to produce a “tail” due to its attenuated energy and optical ringing effects . The target signal [PS(t)] and signal scattered noise [PSMM(t)] contain photons from the target. On the way back to the receiver, they superimpose with the backscattering noise [PBSN(t)]. In order to quantify the signal to noise ratio, the photons return to the receiver are further classified into 18 groups based on their history . Equation (1) is further divided based on this classification, namely:
Each photon group in Eqs. (2)–(4) is indexed using the following notation: D – direct traveling with no scattering, B – backscattering from the medium, G – forward small angle scattering, T – target reflection, and M – multiple scattering. PMTG(t) represents the photons that reach the target surface after multiple scattering, reflect and return to the receiver with some forward scattering. Thus, PMTG(t) will be the later part of RITP signal compared to PDTD(t).
Generally, PBSN(t) and PSMM(t) increase proportionally, while PS(t) is reduced under increased scattering conditions. For near target turbid seawater and narrow FOV, Walker et al.  justified the light pulse propagation as a dispersion-less system, where the multi-path time is small and negligible compared with the pulse length. However, the multiply scattered photons reflected by the target (PSMM(t)) might be higher than PBSN(t) for higher attenuation length (higher turbidity level) and wider FOV condition when the multi path time is significant. PSMM(t) will be considered in higher attenuation length in section 4. In order to differentiate the interval of RITP which contains target signal (target irradiance pulse) from other temporal region without PS(t), we describe this specific interval with PS(t) as Actual RITP.
3. Backscattering noise within RITP analysis
In order to investigate the variation of total backscattering noise within the RITP, Eq. (2) is presented as
For a given semi-infinite medium, the backscattering elements in any given time slice is the convolution between the backscattering TPSF and the half-distance of the on-going laser pulse. Without considering the geometrical effect and forward scattering issues, Fritz et al.  defined the backscattering noise, as:
The laser emits a pulse of power profile Po(t) which starts at t = 0 and ends at t0. The kernel of evolution integral function Si(r) is the TPSF of each component (from i = DBD, direct-backscattering-direct, to i = MBM, multiple scattering-backscattering-multiple scattering), and v = speed of light in the medium. Equation (6) is adopted in this work for range-gated imaging system. For a laser pulse sent to an underwater target in turbid condition, the photons are assumed to have constant speed and traveling through virtual linear distance, r = vt/2. The returning pulse length will be stretched due to the longer photon transportation path in multiple scattering process, where <t> represent the mean multi-path time. If the emitted laser pulse with temporal profile, P0(t), hits an opaque target, and all the energy is either absorbed or reflected at the target, then the convolution process of backscattering effect will terminate at target distance, r0. We established here a formulation to explain this phenomenon, a fixed upper-boundary condition due to the opaque target is introduced. We rewrite Eq. (6) as
Assuming pulse stretching is negligible for short target distance (less than 10 attenuation length, AL [17,19]) in turbid seawater, <t> ≈0; then Pi(t) = 0 for t >(2r0/v + t0). This formulation describes the backscattering noise as a function of TPSF of the medium, outgoing pulse profile, pulse length t0, and time t. Direct interpretation of this function is proportional to the outgoing power P0(t) and Si(r). However, it is also affected by the fixed upper-limit at t >(2r0/v + t0), assuming that v(t-t0)/2 approaches (2r0/v + t0), PBSN(t) reduces. Based on Eq. (7), the backscattering noise is reduced at the tail end of RITP compared to the front of Actual RITP  region. For a coaxial system with the source at the receiver location, the backscattering results obtained from a semi-infinite scattering medium described by Eq. (5) and (7) can be modeled as
The multiple scattering photons from Eq. (4) can be modeled as
In higher turbidity level, the multiple scattering components, such as: PDBM,, PGBM, PMTM etc, are believed to have higher contributions to the total backscattering noise. In order to demonstrate the dominant effects of multiple scattering photon noises, we experiment the tail-gating condition as shown in Fig. 1 and 2 to higher level turbidity.
From Fig. 1, the gated camera and 532nm Nd:YAG 2nd harmonics Q-switched laser is used to capture images at 3 meter away in the water tank. The turbidity of the water condition is simulated by adding solutions of Aluminum Hydroxide / Magnesium Hydroxide in the water tank. Similar solutions are used by Witherspoon et al.  to simulate the turbid sea water in laboratory condition. The upper right picture is the photo-detector measurement (time domain) without a target placed in the water tank (4ns per division scale). Bottom right picture is the RITP with a target place at 3m from the camera. The RITP is displayed on the oscillograms (10ns per division scale).
A turbidity meter, AC9 from Wetlab, is used to measure the water turbidity. The control images (Fig. 2g–2i) are taken at lower turbidity with the tail-gating techniques for 31ns (2nd half of the RITP) and 37ns gate delays. As shown in Fig. 2, the gated system has been able to tune the gate opening time from 13 ns (the camera gate can be tuned to open before the arrival of RITP) to 37 ns relatively to the arrival of the Reflected Image Temporal Profile.
By using a common Modified Fidelity (MF)  index to evaluate the image quality quantitatively compared to the same image taken in clear water condition, the image quality shows little improvement at C = 1.82/m (2-way propagation length = 5.46 AL) and no change at C = 2.07/m (the propagation length = 6.21 AL). Figure 2a–2c are the target image at 3m distance that are captured at different gate opening time (fixed gated duration) at C = 1.82/m. At higher turbidity level (C = 2.07), the test results are depicted in Fig. 2d–2f. No quantitative (MF) or qualitative (observers eye) image enhancement is provided by tail gating for Fig. 2d to 2f. A set of control images are taken at lower turbidity level (C = 1.38/m) that has demonstrated the tail-gating concept. Note that Fig. 2c, 2f and 2i show higher backscattering effect, the uneven intensity level between right and left portion of the image are the resultant of the backscattering noise from the left side of the camera.
For control images taken at C = 1.38/m, the tail-gating technique shows significant image improvement at this level from MF = 0.81 (at 25ns, front gating) to MF = 0.84 (at 31ns, tail portion). At higher turbidity level, the same images are taken with tail-gating technique. However, the image enhancement is not very significant. This has shown that the multiple scattering noises are gradually getting dominant at higher turbidity level.
Though the signal to backscattering noise ratio is better for tail-gating of RITP as described by Eq. (8), it is believed that the multiple scattering noises have gained dominant effect at higher turbidity levels, as described by Eq. (9). Section 4 attempts to estimate the dominant level of the multiple scattering noises, when the signal to multiple scattering noise ratio is less than 1.
4. The Results of Multiple Scattering Noise versus Target Signals
In order to estimate the reflected signal of a seabed target, Walker et al.  established a simple analytical form to illustrate bottom return waveforms in shallow clear water when the transmitted pulse profile is known. The target signal can be modified from their analytical results asEq. (9) contains highly scattered reflected photons that have lost the original target reflection properties (such as location, orientation of the photon) .
One of the most important purposes of using Monte Carlo method is to investigate the limit of complex light scattering problem. The Monte Carlo code used has been modified from S. L. Jacques’  publicly available code. S. Jacque’s codes were written to address and verified the multi-layered light propagation issues with the time domain considerations. The details of the code are available on Tan et al. .
Figure 3 lists the parameter and coefficients used in the simulation. The code is verified and compared to Yaroslavsky et al.  within 20% deviation. Henyey Greentein approximation  is used as the phase function with g = 0.8. Theoretically, the Henyey-Greentein phase function is mostly adopted to approximate the actual phase functions for seawater conditions . 5 million photons are used to simulate the light propagation in the turbid water medium with Henyey-Greentein phase function . The field of View (FOV) and Field of Illumination (FOI) is set at 100 half angle from the camera position respectively.
Experimentally, it will be expensive to categorize photon based on the travelling history individually. Theoretically, it is complicated to solve the complex light propagation analytically. Thus, using the modified Monte Carlo based simulation, we managed to identify the photon groups described in Eq. (2)–(4).
In order to better differentiate the target photons with multiple scattering noises, the ballistics photons are recorded only if the said photons have less than or equal to 1 scattering events when it arrived at the FOV after it is reflected from the target (the photon should not deviated more than 100 from the optical axis). The records are separated at time domain (0.1 ns step) when the photons are directly reflected by the target. The multiple scattering photons are those photon noises that have reflected back from the target and have scattered more than 2 times after the photons return to the FOV. In this case, the differentiation of the ballistic photons and multiple scattering noises are clearly separated.
Figure 4 shows the ratio of multiple scattering to the ballistic photons of a 3m target range gated imaging system. The curves are least squared fitted to reduce the statistical noise error. By increasing the scattering coefficients, the ratio of the multiple scattering to the ballistic photons (signal) increases. The attenuation length can be calculated by taking propagation path (AL) = (attenuation co-efficient) x distance (3m), with absorption coefficient fixed at 0.2/m. Figure 4 provides the indication that the multiple scattering coefficient can be studied by varying the scattering coefficient from 0.6 to 1.5/m. It shows that the multiple scattering component become significant (more than 100% of the total signal return) once the propagation path reaches 4.2 AL (scattering coefficient is 1.2/m + absorption coefficient is 0.2/m). This will limit the tail-gating system from producing the highest signal to noise ratio for its auto-tuning gate imaging approach.
In this paper, we show the image enhancement limitation of the tail-gating technique in high turbidity level. This is due to the increased of multiple scattering noises, though backscattering noise components are reducing compared to the front part of the RITP. This is supported by our experimental results. A numerical model is developed to further investigate the multiple scattering phenomena. Further investigation using this numerical model shows that the multiple scattering components of the backscattering noise dominate for propagation length larger than 4.2 Attenuation Length (AL).
The authors gratefully acknowledge the support and funding from Nanyang Technological University and Universiti Tunku Abdul Rahman.
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