This paper presents a Bayesian uncertainty analysis method for underwater 3D reconstruction using Neural Radiance Fields (NeRFs). The proposed method introduces a spatial perturbation field $ D_{\omega} $ based on Bayes' rays in SeaThru-NeRF and performs Laplace approximation to estimate the uncertainty of the parameters $ \omega $. The diagonal elements of the covariance matrix $ \Sigma $ correspond to the uncertainty at each spatial location. A thresholding method is also employed to remove artifacts from the rendered results. Numerical experiments demonstrate the effectiveness of this approach in both synthetic and real-world underwater scenes. The paper discusses the challenges of applying NeRFs in underwater environments due to light absorption and scattering. SeaThru-NeRF is introduced as an extension of NeRFs for underwater scenes, capable of separating the clean appearance and geometry from the effects of the scattering medium. However, existing methods treat NeRFs as deterministic models, ignoring their inherent uncertainty. The proposed method addresses this by quantifying uncertainty through Bayesian inference, allowing for the analysis of model reliability and enhancement of robustness. Uncertainty quantification is crucial for improving the reliability of NeRFs in practical applications. Various methods such as deep ensemble, variational inference, and MC-dropout have been explored. The Laplace approximation method is used to approximate the posterior distribution with a Gaussian distribution around the mode of the posterior, reducing computational costs. The proposed method introduces a learnable spatial perturbation field and performs Laplace approximation to quantify the uncertainty of the pre-trained SeaThru-NeRF model. This approach avoids additional training, costly sampling, or accessing training images, and can be used for post-processing to remove artifacts. The paper also discusses the scientific background of SeaThru and NeRFs, explaining how they handle underwater scenes with light absorption and scattering. SeaThru-NeRF extends NeRFs to handle scattering media by introducing separate color and density parameters for the medium and objects. The method uses a reparameterization approach to estimate uncertainty, allowing for the explicit inference of spatial uncertainty in both synthetic and real-world underwater scenes. Numerical experiments are conducted to evaluate the performance of the proposed method on real and synthetic datasets. The results show that the method achieves high accuracy in uncertainty estimation, with minimal impact on reconstruction loss. The method is also tested on a smaller neural network model (SeaThru-NeRF-lite), which provides slightly lower image quality but still produces excellent results. The ablation study shows that the parameter M (grid size) significantly affects the accuracy of uncertainty estimation, with optimal performance achieved at a moderate grid size. The regularization parameter $ \lambda $ has minimal impact on the AUSE metric, indicating the robustness of the method. The number of iterations also affects the performance, with more iterations generally leading to better results but at the cost of increased computational time. The proposed method provides a reliable and efficient way to quantify uncertainty in underwater 3D reconstruction usingThis paper presents a Bayesian uncertainty analysis method for underwater 3D reconstruction using Neural Radiance Fields (NeRFs). The proposed method introduces a spatial perturbation field $ D_{\omega} $ based on Bayes' rays in SeaThru-NeRF and performs Laplace approximation to estimate the uncertainty of the parameters $ \omega $. The diagonal elements of the covariance matrix $ \Sigma $ correspond to the uncertainty at each spatial location. A thresholding method is also employed to remove artifacts from the rendered results. Numerical experiments demonstrate the effectiveness of this approach in both synthetic and real-world underwater scenes. The paper discusses the challenges of applying NeRFs in underwater environments due to light absorption and scattering. SeaThru-NeRF is introduced as an extension of NeRFs for underwater scenes, capable of separating the clean appearance and geometry from the effects of the scattering medium. However, existing methods treat NeRFs as deterministic models, ignoring their inherent uncertainty. The proposed method addresses this by quantifying uncertainty through Bayesian inference, allowing for the analysis of model reliability and enhancement of robustness. Uncertainty quantification is crucial for improving the reliability of NeRFs in practical applications. Various methods such as deep ensemble, variational inference, and MC-dropout have been explored. The Laplace approximation method is used to approximate the posterior distribution with a Gaussian distribution around the mode of the posterior, reducing computational costs. The proposed method introduces a learnable spatial perturbation field and performs Laplace approximation to quantify the uncertainty of the pre-trained SeaThru-NeRF model. This approach avoids additional training, costly sampling, or accessing training images, and can be used for post-processing to remove artifacts. The paper also discusses the scientific background of SeaThru and NeRFs, explaining how they handle underwater scenes with light absorption and scattering. SeaThru-NeRF extends NeRFs to handle scattering media by introducing separate color and density parameters for the medium and objects. The method uses a reparameterization approach to estimate uncertainty, allowing for the explicit inference of spatial uncertainty in both synthetic and real-world underwater scenes. Numerical experiments are conducted to evaluate the performance of the proposed method on real and synthetic datasets. The results show that the method achieves high accuracy in uncertainty estimation, with minimal impact on reconstruction loss. The method is also tested on a smaller neural network model (SeaThru-NeRF-lite), which provides slightly lower image quality but still produces excellent results. The ablation study shows that the parameter M (grid size) significantly affects the accuracy of uncertainty estimation, with optimal performance achieved at a moderate grid size. The regularization parameter $ \lambda $ has minimal impact on the AUSE metric, indicating the robustness of the method. The number of iterations also affects the performance, with more iterations generally leading to better results but at the cost of increased computational time. The proposed method provides a reliable and efficient way to quantify uncertainty in underwater 3D reconstruction using