J. Eur. Opt. Society-Rapid Publ. 22, 6( 2026) 49
2.2 Moving and scattering
Fig. 2.( a) Global coordinate system of the simulation model, and( b) sampling the point from the Gaussian spot.
u 0 ¼ 2p n rnd: ð3Þ
The photon’ s initial position on the water surface then be obtained as
8 x ¼ R G cos u 0 þ x st
>< y ¼ R G sin u 0 þ y st; ð4Þ
>: z ¼ H
where H is the flight height. The lidar system positions x st and y st will be changed with the UAV moves. The initial incident direction of the photon is 8
U 0 ðxÞ ¼ sin h 0 cos u 0 ><
U 0 ðyÞ ¼ sin h 0 sin u 0; ð5Þ
>:
U 0 ðÞ¼cos z h 0
where the emitting angle can be calculated by h 0 tan �1( R G / H). When the photon passes through the air – sea interface, the instantaneous state of the waves must be taken into account. The normal vector of the surface wave is defined as
8 U w ðxÞ ¼ sin h w cosu w
>< U w ðyÞ ¼ sin h w sin u w; ð6Þ
>:
U w ðÞ¼cos z h w
where u w is also sampled by equation( 3). The normalized probability density function of h w is proposed by Cox and Munk [ 64 ]:
pðh w Þ ¼ 2
� tan2 ðh w Þ exp tan ðh u 2 w r 2 w Þsec 2 ðh w Þ; ð7Þ
where r is the root-mean-square of surface slope and calculated by r ¼ ð0:003 þ 0:00512mÞ 1 = 2. m is the wind speed. The propagation direction U of the photon after entering the water can be obtained from U 0 and U w by applying Snell’ s law.
Absorption and scattering events happen when the photon propagates underwater. According to the randomly sampled optical distance, the photon’ s moving direction, and the attenuation coefficient profile of water, the geometric distance of the photon’ s random movement can be calculated [ 65 ]. The geometric distance is used to update the photon’ s position. At the new position, the photon’ s weight will be reduced, and the moving direction will be changed. For the scattering direction, its scattering polar angle h can be sampled by the scattering phase function, while the scattering azimuth angle u is uniformly distributed over [ 0, 2p ]. The above process can be found in the reference [ 49 ]. However, considering that the concentration of scattered particles in the stratified water varies with depth, which will affect the probability of the photon’ s colliding with particles, this effect must be included in the simulation. In this study, the scattering phase function of water molecules mixed with particles is used for scattering polar angle h sampling
bh ~ ð; zÞ ¼ b w b b ~ w ðhÞþ b p b b ~ p ðh; zÞ; ð8Þ
where the b ~ w ðhÞ is the normalized phase function of water and b ~ p ðhÞ is the Petzold average-particle phase function [ 66 – 68 ]. b is the total scattering coefficient, which is the sum of water scattering coefficient b w and the particles scattering coefficient b p respectively. It should be noted that, the scattering polar angle h is sampled with respect to the current photon propagation direction U, which serves as the reference axis of the local coordinate system. Consequently, to obtain the updated photon propagation direction U new, the directional transformation from the local scattering coordinate system to the global coordinate system is given by
2
UðxÞUz ðÞ
�UðyÞ
3 2 3 pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi UðxÞ
U new ðxÞ
1�UðzÞ 2 1�UðÞ z
6 2
7
6 U new ðyÞ7 4 5 ¼
UðyÞUz ðÞ
pffiffiffiffiffiffiffiffiffiffiffiffi
6 1�UðÞ z 2
4 U new ðÞ z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
� 1 � UðÞ z 2
2.3 Interaction with the target
UðxÞ
pffiffiffiffiffiffiffiffiffiffiffiffi
1�UðÞ z 2
UðyÞ
7 5
0 UðÞ z
2 3 sin h cos u
4 sin h sin u 5: ð9Þ
cos h
The target underwater is set up as a cylinder with a longitudinal length L C and a cross-sectional radius R C. As shown in Figure 3, the photon’ spositionisP i, based on the sampled geometric distance L step and the incident direction U, and the new position P step can be calculated by P step = P i + L step U. If the position P step( z) H + d w � R C, and the moving direction U( z) > 0, this will indicate that the photon moves downwards and it may collide with the target. When U( z) < 0 and P step( z) H + d w + R C, itindicatesa