SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We now make use of the fact that  $\nabla \log p_{t,y}^{D}$  is a square-integrable Martingale in the reverse-time direction by Lemma 4. We therefore get a sequence of continuous  $L^2$ -bounded Martingales converging to a stochastic process. Since the space of continuous  $L^2$ -bounded martingale is closed and pointwise convergence translates to uniform convergence, we get that  $S$  is a  $L^2$ -bounded martingale, with the convergence of  $C^D\nabla \log p_{t,y}^{D}$  to  $S$  being uniform in time.
$$
Z _ {t} ^ {D} - Z _ {0} ^ {D} - \frac {1}{2} \int_ {0} ^ {t} Z _ {s} d s - \int_ {0} ^ {t} C ^ {D} \nabla \log p _ {s, y} ^ {D} (Z _ {s}) = \sqrt {(C ^ {D})} W _ {t} ^ {D}.
$$
Since all the terms on the left-hand side converge in  $L^2$ , uniformly in  $t$ , so does the right-hand side. Using again the closedness of the spaces of Martingales and Levy's characterization of Wiener process, we find that  $\sqrt{(C^D)} W_t^D$  converges to  $\sqrt{C} W_t$ . Therefore
$$
Z _ {t} = Z _ {0} + \frac {1}{2} \int_ {0} ^ {t} Z _ {s} d s + \int_ {0} ^ {t} S (t, Z _ {t}, y) + \sqrt {C} W _ {t}.
$$
Therefore,  $Z_{t}$  is indeed a solution to (34) and  $Z_{T} \sim X_{0}|Y = y$ . Using uniqueness of the solution we then conclude that this holds for any solution  $Z_{t}$ .
C.4 Proof of (23)
$$
\begin{array}{l} \mathbb {E} [ \| S (t, X _ {t}, y) \| _ {H} ^ {2} | Y = y ] = (1 - e ^ {- t}) ^ {- 2} \mathbb {E} [ \| \mathbb {E} [ X _ {t} - e ^ {- t / 2} X _ {0} | Y = y, X _ {t} ] \| _ {H} ^ {2} | Y = y ] \\ = (1 - e ^ {- t}) ^ {- 2} \mathbb {E} [ \| \mathbb {E} [ \int_ {0} ^ {t} e ^ {- (t - s) / 2} \sqrt {C} d W _ {s} | Y = y, X _ {t} ] \| _ {H} ^ {2} | Y = y ] \\ \leq \left(1 - e ^ {- t}\right) ^ {- 2} \mathbb {E} [ \| \int_ {0} ^ {t} e ^ {- (t - s) / 2} \sqrt {C} d W _ {s} \| _ {H} ^ {2} | Y = y, X _ {t} ] | Y = y ] \\ = (1 - e ^ {- t}) ^ {- 2} \mathbb {E} [ \| \int_ {0} ^ {t} e ^ {- (t - s) / 2} \sqrt {C} d W _ {s} \| _ {H} ^ {2} | Y = y ] \\ = (1 - e ^ {- t}) ^ {- 2} \mathbb {E} [ \| \int_ {0} ^ {t} e ^ {- (t - s) / 2} \sqrt {C} d W _ {s} \| _ {H} ^ {2} ] \\ = (1 - e ^ {- t}) ^ {- 1} \operatorname {T r} (C). \\ \end{array}
$$
C.5 Proof of Proposition 3
Note that with the assumptions in Proposition 3 with  $C_{\mu}$  and  $C$  having the same basis of eigenfunctions and the separability assumption on the Radon-Nikodym derivative for the modes, the system for the modes again diagonalizes. However, in this case the (conditional) distribution for  $X_0^{(j)}$  is non-Gaussian in general and the change of measure with respect to the Gaussian measure characterized by  $\psi^{(j)}(x^{(j)},y)$ . We let the superscript  $g$  denote the Gaussian case with  $\psi \equiv 1$ , then
we have:
$$
\begin{array}{l} S ^ {(j)} (t, x, y) \\ = - (1 - e ^ {- t}) ^ {- 1} \left(x ^ {(j)} - e ^ {- t / 2} \mathbb {E} \left[ X _ {0} ^ {(j)} \mid X _ {t} = x, Y = y \right]\right) \\ = - (1 - e ^ {- t}) ^ {- 1} \left(x ^ {(j)} - e ^ {- t / 2} \int_ {\mathbb {R}} x _ {0} \left(\frac {\mu_ {x _ {0} , x _ {t} ^ {(j)} | y} ^ {(j , g)} (x _ {0} , x ^ {(j)}) \psi^ {(j)} (x _ {0} , y)}{\mu_ {x _ {t} ^ {(j)} | y} ^ {(j)} (x ^ {(j)})}\right) d x _ {0}\right) \\ = (1 - e ^ {- t}) ^ {- 1} \left(e ^ {- t / 2} \int_ {\mathbb {R}} x _ {0} \left(\frac {\mu_ {x _ {0} , x _ {t} ^ {(j)} | y} ^ {(j , g)} (x _ {0} + x ^ {(j)} e ^ {t / 2} , x ^ {(j)}) \psi^ {(j)} (x _ {0} + x ^ {(j)} e ^ {t / 2} , y)}{\mu_ {x _ {t} ^ {(j)} | y} ^ {(j)} (x ^ {(j)})}\right) d x _ {0}\right) \\ = S ^ {(j, g)} (t, x, y) T ^ {(j)} (t, x ^ {(j)}, y) + \tilde {R} ^ {(j)} (t, x ^ {(j)}, y), \\ \end{array}
$$
for  $S^{(j,g)}$  the mode score in the Gaussian case given in (31) and with
$$
\begin{array}{l} T ^ {(j)} (t, x ^ {(j)}, y) = \left(\frac {\mu_ {x _ {t} ^ {(j)} | y} ^ {(j , g)} (x ^ {(j)}) \psi^ {(j)} (x ^ {(j)} e ^ {t / 2} , y)}{\mu_ {x _ {t} ^ {(j)} | y} ^ {(j)} (x ^ {(j)})}\right), \\ \tilde {R} ^ {(j)} (t, x ^ {(j)}, y) = (1 - e ^ {- t}) ^ {- 1} e ^ {- t / 2} \\ \times \int_ {\mathbb {R}} x _ {0} \left(\frac {\mu_ {x _ {0} , x _ {t} ^ {(j)} | y} ^ {(j , g)} (x _ {0} + x ^ {(j)} e ^ {t / 2} , x ^ {(j)}) (\psi^ {(j)} (x _ {0} + x ^ {(j)} e ^ {t / 2} , y) - \psi^ {(j)} (x ^ {(j)} e ^ {t / 2} , y))}{\mu_ {x _ {t} ^ {(j)} | y} (x ^ {(j)})}\right) d x _ {0}. \\ \end{array}
$$
We have for  $\phi_{\lambda}$  the centered Gaussian density at second moment  $\lambda$
$$
T ^ {(j)} (t, x ^ {(j)}, y) = \frac {\int_ {\mathbb {R}} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - v e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (v - x _ {y} ^ {(j)}) d v}{\int_ {\mathbb {R}} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - v e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (v - x _ {y} ^ {(j)}) \psi^ {(j)} (v , y) / \psi^ {(j)} (x ^ {(j)} e ^ {t / 2} , y) d v},
$$
for  $\lambda_t^{(j)} = \lambda_j(1 - e^{-t}),\mu_y^{(j)} = \mu_j / (1 + q^{(j)})$  and  $x_{y}^{(j)} = y^{(j)}q^{(j)} / (1 + q^{(j)})$  for  $y^{(j)} = \langle y,v_j\rangle$  Here  $x_{y}^{(j)},\mu_{y}^{(j)}$  are respectively the mean and variance of  $X_0^{(j)}$  given  $y$  and where we used the parameterization set forth in Section B.1. We then have
$$
| T ^ {(j)} (t, x ^ {(j)}, y) | \leq K ^ {2}, \quad \lim  _ {t \downarrow 0} T ^ {(j)} (t, x ^ {(j)} y) = 1.
$$
We moreover have
$$
\begin{array}{l} | \tilde {R} ^ {(j)} (t, x ^ {(j)}, y) | \leq e ^ {- t / 2} (1 - e ^ {- t}) ^ {- 1} L \left(\frac {\int_ {\mathbb {R}} x _ {0} ^ {2} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - x _ {0} e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (x _ {0} - x _ {y} ^ {(j)}) d x _ {0}}{\int_ {\mathbb {R}} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - v e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (v - x _ {y} ^ {(j)}) \psi^ {(j)} (v , y) d v}\right) \\ \leq e ^ {- t / 2} (1 - e ^ {- t}) ^ {- 1} L K \left(\frac {\int_ {\mathbb {R}} x _ {0} ^ {2} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - x _ {0} e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (x _ {0} - x _ {y} ^ {(j)}) d x _ {0}}{\int_ {\mathbb {R}} \phi_ {\lambda_ {t} ^ {(j)}} (x ^ {(j)} - v e ^ {- t / 2}) \phi_ {\mu_ {y} ^ {(j)}} (v - x _ {y} ^ {(j)}) d v}\right). \\ \end{array}
$$
We then find
$$
\limsup_{t\downarrow 0}|\tilde{R}^{(j)}(t,x^{(j)},y)|\leq \lambda^{(j)}LK,
$$
and, moreover
$$
| \tilde {R} ^ {(j)} (t, x ^ {(j)}, y) | \leq
$$
$$
\lambda_ {j} L K e ^ {t / 2} \left(\frac {1}{1 + (e ^ {t} - 1) p ^ {(j)} (1 + q ^ {(j)})} + \lambda_ {j} \frac {(x _ {y} ^ {(j)} - x ^ {(j)} e ^ {t / 2}) ^ {2}}{(\mu_ {y} ^ {(j)}) ^ {2}} \frac {(e ^ {t} - 1)}{(1 + (e ^ {t} - 1) p ^ {(j)} (1 + q ^ {(j)})) ^ {2}}\right).
$$
Consider in the Gaussian case in (22) a mode so that  $p^{(j)}(1 + q^{(j)}) \uparrow \infty$  and  $\lambda_{j}$  fixed, then the contribution of this mode to the score norm blows up in the small time limit. The situation with
$p^{(j)}(1 + q^{(j)})\uparrow \infty$  would happen for instance in a limit of perfect mode observation so that  $\sigma_B\downarrow 0$  and thus  $q^{(j)}\uparrow \infty$ . Indeed in the limit of small (conditional) target mode variability relative to the diffusion noise parameter the score drift becomes large for small time to drive the mode to the conditional target distribution. We here thus assume  $p^{(j)}(1 + q^{(j)})$  is uniformly bounded with respect to mode  $(j$  index), moreover, that  $C$  is of trace class. We then find that Assumption 1 is satisfied with the following bound
$$
\begin{array}{l} \sup  _ {t \in [ 0, T ]} \mathbb {E} \left[ \| S (t, X _ {t}, y) \| _ {H} ^ {2} | Y = y \right] \\ \leq 2 \left(\sum_ {j} \lambda_ {j} e ^ {T} \left(K ^ {4} p ^ {(j)} (1 + q ^ {(j)}) + 2 \lambda_ {j} e ^ {T} (L K) ^ {2} \left(1 + 3 \left(p ^ {(j)} (1 + q ^ {(j)}) (e ^ {T} - 1)\right) ^ {4}\right)\right)\right). \\ \end{array}
$$
We remark that in the case that we do not have a uniform bound on the  $p^{(j)}(1 + q^{(j)})$ 's it follows from (23) that the rate of divergence of the expected square norm of the score is at most  $t^{-1}$  as  $t \downarrow 0$  with  $C$  of trace class.
C.6 Proof of Proposition 4
We start from (30):
$$
\begin{array}{l} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \| S (t, x _ {t}, y) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right] = \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \| S (t, x _ {t}, y) \| _ {H} ^ {2} \right] \\ + \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \| s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right] - 2 \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \langle S (t, x _ {t}, y), s _ {\theta} (t, x _ {t}, y) \rangle \right]. \\ \end{array}
$$
From Definition 1 we have
$$
\begin{array}{l} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \langle S (t, x _ {t}, y), s _ {\theta} (t, x _ {t}, y) \rangle \right] \\ = - \left(1 - e ^ {- t}\right) ^ {- 1} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \left\langle x _ {t} - e ^ {- t / 2} \mathbb {E} _ {x _ {0} \sim \mathcal {L} (X _ {0} | X _ {t} = x _ {t}, Y = y)} [ x _ {0} ], s _ {\theta} (t, x _ {t}, y) \right\rangle \right] \\ = - (1 - e ^ {- t}) ^ {- 1} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \mathbb {E} _ {x _ {0} \sim \mathcal {L} (X _ {0} | X _ {t} = x _ {t}, Y = y)} \left[ \left\langle x _ {t} - e ^ {- t / 2} x _ {0}, s _ {\theta} (t, x _ {t}, y) \right\rangle \right] \right] \\ = - \left(1 - e ^ {- t}\right) ^ {- 1} \mathbb {E} _ {\left(x _ {0}, x _ {t}, y\right) \sim \mathcal {L} \left(X _ {0}, X _ {t}, Y\right)} \left[ \left\langle x _ {t} - e ^ {- t / 2} x _ {0}, s _ {\theta} (t, x _ {t}, y) \right\rangle \right]. \\ \end{array}
$$
We obtain that
$$
\begin{array}{l} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} \left(X _ {t}, Y\right)} \left[ \| S (t, x _ {t}, y) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right] \\ = B + \mathbb {E} _ {\left(x _ {0}, x _ {t}, y\right) \sim \mathcal {L} \left(X _ {0}, X _ {t}, Y\right)} \left[ \| - (1 - e ^ {- t}) ^ {- 1} \left(x _ {t} - e ^ {- t / 2} x _ {0}\right) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right], \\ \end{array}
$$
$$
B = \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \| S (t, x _ {t}, y) \| _ {H} ^ {2} \right] - \mathbb {E} _ {(x _ {0}, x _ {t}) \sim \mathcal {L} (X _ {0}, X _ {t})} \left[ \| (1 - e ^ {- t}) ^ {- 1} (x _ {t} - e ^ {- t / 2} x _ {0}) \| _ {H} ^ {2} \right]
$$
that does not depend on  $\theta$ . Since  $\mathcal{L}(X_t|X_0 = x_0, Y = y) = \mathcal{L}(X_t|X_0 = x_0)$  we finally get that
$$
\begin{array}{l} \mathbb {E} _ {x _ {t}, y \sim \mathcal {L} (X _ {t}, Y)} \left[ \| S (t, x _ {t}, y) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right] \\ = B + \mathbb {E} _ {x _ {0}, y \sim \mathcal {L} (X _ {0}, Y), x _ {t} \sim \mathcal {L} (X _ {t} | X _ {0} = x _ {0})} \left[ \| - (1 - e ^ {- t}) ^ {- 1} (x _ {t} - e ^ {- t / 2} x _ {0}) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right]. \\ \end{array}
$$
D Numerical experiments: details and additional results
In this section, we provide additional details regarding our numerical experiments. In both experiments, we parameterize the conditional score  $s_{\theta}(t,x_t,y)$  using discretization-invariant Fourier neural operators [FNOs; 53]. This parameterization enables mapping input triplets  $(t,x_t,y)$  to the score conditioned on  $y$  at time  $t$ . Once trained—by minimizing the objective function in equation (28) with respect to  $\theta$ —we use the FNO as an approximation to the conditional score to sample new realizations of the conditional distribution by simulating the reverse-time SDE in equation (14).
D.1 Stylized example
In this example, the conditional distribution that we approximate is defined using the relation
(b)
Figure 3: The proposed method's discretization invariance. Predicted samples (functions) on a uniformly sampled grid with (a) 20, (b) 30, and (c) 40 grid points.
$$
x _ {0} = a y ^ {2} + \varepsilon , \tag {35}
$$
where  $\varepsilon \sim \Gamma(1,2)$  and  $a \sim \mathcal{U}\{-1,1\}$ . Here,  $\Gamma$  refers to the Gamma distribution, and  $\mathcal{U}\{-1,1\}$  denotes the uniform distribution over the set  $\{-1,1\}$ . Having an explicit expression characterizing the conditional distribution allows us to easily evaluate the obtained result through our method—as opposed to needing to use a baseline method, e.g., Markov chain Monte Carlo.
Training data The discretization invariance of our model enables us to use training data that live on varying discretization grids. We exploit this property and simulate training joint samples  $(x_0, y)$  by evaluating the expression in equation (35) over values of  $y$  that are selected as nonuniform grids over the domain  $[-3, 3]$  that contain 15-50 grid points.
Architecture In this example, the FNO comprises (i) a fully connected lifting layer that maps the three-dimensional vector—including the timestep  $t$ , the  $y$  value, and the corresponding  $x$  value—for each grid point to a 128-dimensional lifted space; (ii) five Fourier neural layers, as introduced in [53], which contain pointwise linear filters applied to the five lower Fourier modes; and (iii) two fully-connected layers, separated by a ReLU activation function, that map the 128-dimensional lifted space back to the conditional score (a scalar) for each grid point.
**Optimization details** To train the FNO, we minimized the objective function in equation (28) using  $2 \times 10^{4}$  training steps. At each step, we simulated a batch of 512 training pairs selected from a grid with varying numbers of discretization points. We utilized the Adam stochastic optimization method [56] with an initial learning rate of  $10^{-3}$ , which decayed to  $5 \times 10^{-4}$  during optimization, following a power-law rate of  $-1/3$ . Regarding the diffusion process, we followed the approach outlined in Ho et al. [29] and employed standard Gaussian noise with linearly increasing variance for the forward dynamics, which was discretized into 500 timesteps. The initial Gaussian noise variance was set to  $10^{-4}$  and linearly increased over the timesteps until it reached  $2 \times 10^{-2}$ . The training hyperparameters were chosen by monitoring the validation loss over 1024 samples. The training process took approximately 7 minutes on a Tesla V100 GPU device. For further details, please refer to our open-source implementation on GitHub.
Additional results Figure 3 illustrates the predicted samples associated with the remaining testing grid sizes, whose densities were shown in Figures 1d-1f.
D.2 Linearized seismic imaging example
The inverse problem we are addressing in this example involves the task of estimating the short-wavelength component of the Earth's unknown subsurface squared-slowness model using measurements taken at the surface. This particular problem, often referred to as seismic imaging, can be recast as a linear inverse problem when we linearize the nonlinear relationship between surface measurements and the squared-slowness model, as governed by the wave equation. In its simplest acoustic form, the linearization with respect to the slowness model—around a background smooth squared slowness model  $m_0$ —results in a linear inverse problem for the estimation of the true seismic image  $\delta m^*$  using the following forward model,
$$
d _ {i} = J \left(m _ {0}, q _ {i}\right) \delta m ^ {*} + \epsilon_ {i}, \quad \epsilon_ {i} \sim p (\epsilon), \tag {36}
$$
where  $d = \{d_i\}_{i=1}^{n_s}$  represents a collection of  $n_s$  linearized surface measurements, i.e., data wherein the zero-order term of Taylor's series has been subtracted, and  $J(m_0, q_i)$  denotes the linearized
Born scattering operator, which is defined in terms of the source signature  $q_{i}$  and the background squared-slowness model  $m_0$ . Due to noise and linearization errors, the above expression contains the term  $\epsilon_{i}$ . In addition to this noise term, the forward operator has a non-trivial nullspace due to the presence of shadow zones and finite-aperture data [57, 58]. To tackle this challenges, we set seismic imaging into a Bayesian framework and learn the associated posterior distribution via our proposed method.
Training data We generated synthetic data by applying the Born scattering operator to 4750 2D seismic images, each with dimensions of  $3075\mathrm{m}\times 5120\mathrm{m}$ . These images were extracted from the Kirchhoff migrated Parihaka-3D dataset, which contains seismic images obtained by imaging the data collected in New Zealand [59, 60]. We parameterize the Born scattering operator using a background squared-slowness model (recall Figure 2b). The data acquisition geometry involves 102 sources with a spacing of  $50\mathrm{m}$ , each recorded for two seconds via 204 receivers spaced at  $25\mathrm{m}$  located on top the image. The source wavelet used was a Ricker wavelet with a central frequency of  $30\mathrm{Hz}$ . To replicate a more realistic imaging scenario, we add band-limited noise to the data, obtained by filtering white noise with the source function. To create training pairs, we first simulated noisy seismic data for all the 2D seismic images based on the aforementioned acquisition design. Subsequently, we reduce the dimensionality of seismic data by applying the adjoint of the Born scattering operator to the data. We use Devito [61, 62] for the wave-equation based simulations.
Architecture In this example, the FNO is composed of (i) a fully connected lifting layer that maps the five-dimensional vector—including the timestep  $t$ , the two spatial coordinates, the data value  $y$ , obtained after dimensionality reduction by applying the adjoint of the forward operator), and the corresponding  $x$  value—for each grid point to a 32-dimensional lifted space; (ii) four Fourier neural layers, as introduced in [53], which contain pointwise linear filters applied to the 24 lower Fourier modes; and (iii) two fully-connected layers, separated by a ReLU activation function, that map the 128-dimensional lifted space back to the conditional score for each grid point.
**Optimization details** We train the FNO according to the objective function in equation (28) with the Adam [56] stochastic optimization method with batch size 128 for 300 epochs. We use an initial stepsize of  $2 \times 10^{-3}$ , decaying to  $5 \times 10^{-4}$  during optimization with a power-law rate of  $-1/3$ . We use a similar diffusion process as the previous example. The training hyperparameters are chosen by monitoring the validation loss over 530 samples. The training process takes approximately two hours and 15 minutes on a Tesla V100 GPU device. For further details refer to our open-source implementation on GitHub.
Additional results Figures 4-7 illustrate more results regarding using the SDM to sample from the posterior distribution of multiple seismic imaging problem instances where the ground truth images (see Figures 4a-7a) are obtained from the test dataset. In each plot, we use the dimensionality-reduced data as the conditioning input to the FNO (see Figures 4b-7b). Through the SDM, we obtain  $10^{3}$  posterior distribution samples and use them to estimate the conditional mean (see Figures 4c-7c) and the pointwise standard deviation among samples (see Figures 4d-7d), with the former serving as a measure of uncertainty. In all cases, the regions of significant uncertainty correspond well with challenging-to-image sections of the model, which qualitatively confirms the accuracy of our Bayesian inference method. This observation becomes more apparent in Figures 4g-7g, displaying two vertical profiles with  $99\%$  confidence intervals (depicted as orange-colored shading) for each experiment, which demonstrate the expected trend of increased uncertainty with depth. Furthermore, we notice that the ground truth (indicated by dashed black lines) mostly falls within the confidence intervals for most areas. We also consistently observe a strong correlation between the pointwise standard deviation and the error in the conditional mean estimate (see Figures 4e-7e), which further asserts the accuracy of our method in characterizing the posterior distribution in this large-scale Bayesian inference problem. To prevent bias from strong amplitudes in the estimated image, we present the normalized pointwise standard deviation divided by the envelope of the conditional mean in Figures 4f-7f. These visualizations provide an amplitude-independent assessment of uncertainty, highlighting regions of high uncertainty at the onset and offset of reflectors (both shallow and deeper sections). Additionally, the normalized pointwise standard deviation underscores uncertainty in areas of the image where there are discontinuities in the reflectors (indicated by black arrows), potentially indicating the presence of faults.
(g)
Figure 4: Seismic imaging and uncertainty quantification. (a) Ground-truth seismic image. (b) Data after applying the adjoint Born operator (known as the reverse-time migrated image). (c) Conditional (posterior) mean. (d) Pointwise standard deviation. (e) Absolute error between Figures 4a and 4c. (f) Normalized pointwise standard deviation by the envelope of the conditional mean. (g) Vertical profiles of the ground-truth image, conditional mean estimate, and the  $99\%$  confidence interval at two lateral positions in the image.
(g)
Figure 5: Seismic imaging and uncertainty quantification. (a) Ground-truth seismic image. (b) Data after applying the adjoint Born operator (known as the reverse-time migrated image). (c) Conditional (posterior) mean. (d) Pointwise standard deviation. (e) Absolute error between Figures 5a and 5c. (f) Normalized pointwise standard deviation by the envelope of the conditional mean. (g) Vertical profiles of the ground-truth image, conditional mean estimate, and the  $99\%$  confidence interval at two lateral positions in the image.
(e)
Figure 6: Seismic imaging and uncertainty quantification. (a) Ground-truth seismic image. (b) Data after applying the adjoint Born operator (known as the reverse-time migrated image). (c) Conditional (posterior) mean. (d) Pointwise standard deviation. (e) Absolute error between Figures 6a and 6c. (f) Normalized pointwise standard deviation by the envelope of the conditional mean. (g) Vertical profiles of the ground-truth image, conditional mean estimate, and the  $99\%$  confidence interval at two lateral positions in the image.
(g)
Figure 7: Seismic imaging and uncertainty quantification. (a) Ground-truth seismic image. (b) Data after applying the adjoint Born operator (known as the reverse-time migrated image). (c) Conditional (posterior) mean. (d) Pointwise standard deviation. (e) Absolute error between Figures 7a and 7c. (f) Normalized pointwise standard deviation by the envelope of the conditional mean. (g) Vertical profiles of the ground-truth image, conditional mean estimate, and the  $99\%$  confidence interval at two lateral positions in the image.