# Enhanced User Grouping and Pairing Scheme for CoMP–NOMA-based Cellular
Networks

Akhileswar Chowdary 1, Garima Chopra 1, Abhinav Kumar1, and Linga Reddy
Cenkeramaddi2  1 Department of Electrical Engineering, Indian Institute of
Technology Hyderabad, Telangana, 502285 India.
Email<EMAIL_ADDRESS>{garima.chopra<EMAIL_ADDRESS>2Department of Information and Communication Technology, University of Agder,
Grimstad, 4879 Norway.
Email<EMAIL_ADDRESS>
###### Abstract

Non-orthogonal multiple access (NOMA) has been identified as one of the
promising technologies to enhance the spectral efficiency and throughput for
the fifth generation (5G) and beyond 5G cellular networks. Alternatively,
Coordinated multi-point transmission and reception (CoMP) improves the cell
edge users’ coverage. Thus, CoMP and NOMA can be used together to improve the
overall coverage and throughput of the users. However, user grouping and
pairing for CoMP–NOMA-based cellular networks have not been suitably studied
in the existing literature. Motivated by this, we propose a user grouping and
pairing scheme for a CoMP–NOMA-based system. Detailed numerical results are
presented comparing the proposed scheme with the purely OMA-based benchmark
system, NOMA only, and CoMP only systems. We show through simulation results
that the proposed scheme offers a trade-off between throughput and coverage as
compared to the existing NOMA or CoMP based system.

###### Index Terms:

Coordinated multi-point transmission and reception (CoMP), Non-orthogonal
multiple access (NOMA), User grouping, User pairing schemes, fifth generation
(5G) and beyond 5G cellular networks.

##  I Introduction

Non-orthogonal multiple access (NOMA) has emerged as the promising multiple
access scheme to enhance the spectral efficiency (in turn the network
throughput) for the 5G and beyond 5G cellular networks. The primary idea of
NOMA is to serve multiple users utilizing the same resource (e.g. spectrum,
time, etc.) at the cost of increased complexity due to successive interference
cancellation (SIC) [1], [2]. In a typical downlink power-domain NOMA, the Base
Station (BS) sends the superpositioned signals of the users which differ in
their power. The user decodes its intended message either by treating
information of another user sharing the same resource as noise or through SIC.

Coordinated multi-point with joint transmission and reception which hereafter
is referred to as CoMP has been extensively researched in today’s wireless
networks [3]. It has been shown in [4], that joint transmission CoMP improves
the network coverage at the cost of reduced network throughput. This
throughput reduction in CoMP can be mitigated by utilizing NOMA with CoMP
networks [5]. Using joint transmission NOMA (JT-NOMA) for the CoMP system, in
this paper, we analyze NOMA for CoMP as well as non-CoMP users to improve the
network coverage and throughput.

Figure 1: System Model

The existing works on CoMP–NOMA system in [6, 7, 8, 9, 10, 11] have not
studied the performance of the network for different types of user pairs
possible, namely, CoMP–CoMP, (non-CoMP)–CoMP, (non-CoMP)–(non-CoMP), for a
generalized scenario (considering multiple cells, randomly distributed users,
and BSs). In a cluster of BSs, after differentiating CoMP and non-CoMP users,
grouping the users for performing NOMA pairing is not obvious. As shown in
[12], a CoMP user cannot act as both strong and weak user when paired with
multiple non-CoMP users. Thus, considering such conditions, the grouping of
CoMP and non-CoMP users before NOMA pairing is a non-trivial task.
Furthermore, after forming the aforementioned hybrid pairs of users,
scheduling them is also not straightforward. Motivated by this, we perform a
detailed study of throughput and coverage of CoMP–NOMA-based systems for a
grouping and pairing scheme. The main contributions of the paper are as
follows:

  1. 1.

We propose a user grouping and pairing scheme to study the performance of
CoMP–CoMP and (non-CoMP)–(non-CoMP) NOMA pairs considering network scaling.

  2. 2.

After pairing users from the groups formed, we perform scheduling for a
CoMP–NOMA-based system.

  3. 3.

The performance of the proposed scheme is compared with the purely OMA-based
benchmark system, CoMP only, and NOMA only systems. We show that the proposed
scheme results in superior throughput than the CoMP only system and superior
coverage than the NOMA only system.

The organization of the paper is as follows. Section II describes the system
model in detail. Section III presents the proposed user grouping and pairing
scheme for CoMP–NOMA-based system. The simulation results are discussed in
Section IV. Finally, the work is concluded in Section V.

##  II System Model

We consider a downlink cellular network where BSs and users are randomly
deployed according to a homogeneous Poisson point process (PPP) with densities
$\lambda_{b}$ and $\lambda_{u}$, respectively [13] as shown in Fig. 1. Let
$\mathcal{B}=\\{1,2,...,B\\}$ be the set of BSs deployed in an area
$\mathcal{A}$. Let $\mathcal{M}=\\{1,2,...,\text{M}\\}$ be the set of
subchannels for a given BS $b$. The users are associated with the BSs based on
the maximum received power [14].

###  II-A Channel Model

Assuming Time Division Duplexing (TDD), the signal-to-interference-plus-noise
ratio (SINR) of user $i$ from BS $b$ on subchannel $m$ is given as follows.

$\gamma_{i}^{b,m}=\frac{P^{b,m}g_{i}^{b,m}}{\sum\limits_{\begin{subarray}{c}\hat{b}\neq
b\\\
\hat{b}\in\pazocal{B}\end{subarray}}P^{\hat{b},m}g_{i}^{\hat{b},m}+\sigma^{2}}\,,$
(1)

where $P^{b,m}=\frac{P^{b}}{\text{M}}$ is the power allocated by the BS $b$ to
the subchannel $m\in\pazocal{M}$, $P^{b}$ is the maximum transmit power of BS
$b$, M is the total number of subchannels, $g_{i}^{b,m}=\|h_{i}^{b,m}\|^{2}$
is the channel gain between user $i$ and BS $b$ on subchannel $m$,
${\sum\limits_{\begin{subarray}{c}\hat{b}\neq b\\\
\hat{b}\in\pazocal{B}\end{subarray}}P^{\hat{b},m}g_{i}^{\hat{b},m}}$ is the
interference on the subchannel $m$, and $\sigma^{2}$ is the noise power. The
channel gain can be expressed as

$g_{i}^{b,m}=10^{\frac{-pl(d_{i}^{b})+g_{t}+g_{r}-f_{s}-v}{10}},$ (2)

where $pl(d_{i}^{b})$ is the path loss between user $i$ and BS $b$ at a
distance $d_{i}^{b}$, $g_{t}$ is the transmitter gain, $g_{r}$ is the receiver
gain, $v$ is the penetration loss, and $f_{s}$ is the loss due to shadowing
and fading. The corresponding link rate $r_{i}^{b,m}$ for a given SINR
$\gamma_{i}^{b,m}$ in (1) between user $i$ and BS $b$, can be expressed as

$r_{i}^{b}=\frac{\eta(\gamma_{i}^{b,m})sc_{o}sy_{o}}{t_{sc}}\text{M},$ (3)

where $\eta(\gamma_{i}^{b,m})$ can be obtained using the Adaptive Modulation
and Coding Scheme (MCS) as given in [14]. Further, $sc_{o}$, $sy_{o}$, and
$t_{sc}$ represent the number of subcarriers per subchannel, the number of
symbols per subchannel, and subframe duration (in seconds), respectively.

###  II-B CoMP

We consider a set of CoMP clusters denoted by $\mathcal{C}=\\{1,2,...,C\\}$ in
a given area $\pazocal{A}$. The BSs are clustered using K-means clustering
approach. Let $\mathcal{I}_{c}$ and $\mathcal{I}_{nc}$ be the set of CoMP
users and non-CoMP users, respectively, in cluster $c$ as shown in Fig. 1. If
$\gamma_{i}^{m}<\gamma_{th}$, then $i\in\pazocal{I}_{c}$, else the user
belongs to $\pazocal{I}_{nc}$. Let the set of BSs in the CoMP cluster $c$ be
denoted by $\mathcal{B}_{c}$, $\mathcal{B}_{c}=\\{1,2,...,B_{c}\\}$. All the
CoMP users in a cluster $c$ jointly receive information from the BSs for a
duration of $\theta_{c}$, whereas, for the remaining
$\left(1-\theta_{c}\right)$ time fraction, individual BSs in a cluster $c$
serve their respective non-CoMP users separately. Assuming each cluster $c$
has its typical value of $\theta_{c}$ [14], the SINR for the CoMP user $i$ in
cluster $c$ is given as follows.

$\gamma_{i}^{m}=\dfrac{\sum\limits_{\begin{subarray}{c}l\in\pazocal{B}_{c}\end{subarray}}P^{l,m}g_{i}^{l,m}}{\sum\limits_{\begin{subarray}{c}\hat{l}\in\pazocal{B}\\\
\hat{l}\not\in\pazocal{B}_{c}\end{subarray}}P^{\hat{l},m}g_{i}^{\hat{l},m}+\sigma^{2}},\forall
i\in\pazocal{I}_{c}\,$ (4)

where
$\sum\limits_{\begin{subarray}{c}l\in\pazocal{B}_{c}\end{subarray}}P^{l,m}g_{i}^{l,m}$
is the power received by the user $i$ from all the BSs present in the cluster
$c$, ${\sum\limits_{\begin{subarray}{c}\hat{l}\in\pazocal{B}\\\
\hat{l}\not\in\pazocal{B}_{c}\end{subarray}}P^{\hat{l},m}g_{i}^{\hat{l},m}}$
is the interference from other BSs in the system that do not belong to the
CoMP cluster $c$, and $\sigma^{2}$ is the noise power. Similarly, the SINR of
the non-CoMP users in cluster $c$ is computed using (1).

###  II-C NOMA

We now consider NOMA pairing for the CoMP based system. Let $\gamma_{s}$ and
$\gamma_{w}$ be the OMA SINRs of strong and weak users in a NOMA pair,
respectively, computed using (1). We consider the minimum SINR difference
(MSD) criteria given in [15] to pair two users and calculate the optimal power
fraction of the strong user. If the two users do not satisfy the MSD, then
they are served as OMA users. The SINRs of paired NOMA users with perfect SIC
are expressed as follows.

$\hat{\gamma}_{s}^{b,m}=\frac{\zeta_{s}P^{b,m}g_{s}^{b,m}}{\sum\limits_{\begin{subarray}{c}\hat{b}\in\pazocal{B}\backslash
b\end{subarray}}P^{\hat{b},m}g_{s}^{\hat{b},m}+\sigma^{2}},$ (5)
$\hat{\gamma}_{w}^{b,m}=\frac{(1-\zeta_{s})P^{b,m}g_{w}^{b,m}}{\zeta_{s}P^{b,m}g_{w}^{b,m}+\sum\limits_{\begin{subarray}{c}\hat{b}\in\pazocal{B}\backslash
b\end{subarray}}P^{\hat{b},m}g_{w}^{\hat{b},m}+\sigma^{2}},$ (6)

where $\zeta_{s}$ is the optimal power fraction allocated to the strong user
in the pair computed as in [15], $\hat{\gamma}_{s}^{b,m}$ is the SINR of the
strong user with perfect SIC after NOMA pairing, $P^{b,m}$ is the total power
assigned to the pair, $g_{s}^{b,m}$ is the channel gain of the strong user as
in (2), $\sum\limits_{\begin{subarray}{c}\hat{b}\in\pazocal{B}\backslash
b\end{subarray}}P^{\hat{b},m}g_{s}^{\hat{b},m}$ is the aggregate interference
received from other BSs, $\hat{\gamma}_{w}^{b,m}$ is the SINR of the weak user
after NOMA pairing, $(1-\zeta_{s})$ is the power fraction allocated to the
weak user, and $g_{w}^{b,m}$ is the channel gain of the weak user as in (2).
Throughout the work, we consider the adaptive user pairing algorithm (AUP)
proposed in [15] which uses MSD criteria for pairing users in two groups.
However, the formation of the strong and weak user groups for a CoMP–NOMA-
based system is not obvious. To address this issue, we next propose the
grouping and pairing scheme.

##  III Proposed CoMP–NOMA user pairing scheme

There are three types of NOMA pairs possible based on the users present in a
cluster: CoMP–CoMP, (non-CoMP)–CoMP, and (non-CoMP)–(non-CoMP). In this work,
due to the given space constraints, we study the performance of the CoMP–NOMA
system by proposing a user grouping and pairing scheme considering only
CoMP–CoMP and (non-CoMP)–(non-CoMP) NOMA pairs.

Figure 2: Illustration of the proposed grouping and pairing of users.

The group of all CoMP users is designated as group G1 and the group of non-
CoMP users associated with individual base stations is denoted as group
$\textbf{G2}_{b}$. After obtaining the groups, NOMA pairing is done as per the
AUP mentioned in [15]. However, in this scheme NOMA pairing is done within the
group as shown in Fig. 2. Therefore, no cross-group pairing occurs. The SINR
of a weak CoMP user in the CoMP–CoMP pair according to this scheme is as
follows.

$\bar{\gamma}_{i_{w}}^{m}=\frac{\sum\limits_{\begin{subarray}{c}t\in\pazocal{B}_{c}\end{subarray}}(1-\zeta_{t})P^{b,m}g_{i_{w}}^{t,m}}{\sum\limits_{\begin{subarray}{c}t\in\pazocal{B}_{c}\end{subarray}}\zeta_{k}P^{b,m}g_{i_{w}}^{t,m}+\sum\limits_{\begin{subarray}{c}\hat{l}\in\pazocal{B}\\\
\hat{l}\not\in\pazocal{B}_{c}\end{subarray}}P^{\hat{l},m}g_{i_{w}}^{\hat{l},m}+\sigma^{2}},$
(7)

where $\bar{\gamma}_{i_{w}}^{m}$ is the SINR of the weak user in the CoMP–CoMP
pair in cluster $c$ and $(1-\zeta_{t})$ is the power fraction allocated by
each BS in cluster $c$ to the weak CoMP user. The SINR of a strong user with
perfect SIC in the CoMP–CoMP pair in cluster $c$ is given as follows.

$\bar{\gamma}_{i_{s}}^{m}=\frac{\sum\limits_{\begin{subarray}{c}t\in\pazocal{B}_{c}\end{subarray}}\zeta_{t}P^{b,m}g_{i_{s}}^{t,m}}{\sum\limits_{\begin{subarray}{c}\hat{l}\in\pazocal{B}\\\
\hat{l}\not\in\pazocal{B}_{c}\end{subarray}}P^{\hat{l},m}g_{i_{s}}^{\hat{l},m}+\sigma^{2}},$
(8)

where $\bar{\gamma}_{i_{s}}^{m}$ is the SINR of the strong user with perfect
SIC in the CoMP–CoMP pair and $\zeta_{t}$ is the power fraction allocated to
the strong CoMP user by each BS in the cluster $c$. The SINRs of strong and
weak users of (non-CoMP)–(non-CoMP) pair are computed as given in (5) and (6),
respectively. The CoMP pairs are served collectively by all the BSs in the
cluster during the duration of $\bar{\theta}_{c}$. Using an $\alpha$-fair
scheduler ($\alpha$ is the fairness parameter [14]), each CoMP–CoMP NOMA pair
or OMA CoMP user (if any) is given a time fraction of $\bar{\beta}_{i}$ which
is given as follows (for $\alpha=1$)[14].

$\bar{\beta}_{i}=\frac{1}{|\hat{\pazocal{I}}_{c}|+|\tilde{\pazocal{I}}_{c}|},$
(9)

where $|X|$ denotes the cardinality of set $X$, $\hat{\pazocal{I}}_{c}$
represents the set of CoMP–CoMP NOMA pairs formed, and
$\tilde{\pazocal{I}}_{c}$ is the set of OMA CoMP users which could not be
paired (if any). The scheduling time fraction for (non-CoMP)–(non-CoMP) NOMA
pairs that are served by their respective BSs in the duration of
$(1-\bar{\theta}_{c})$ with an $\alpha$-fair scheduler is given as follows
(for $\alpha=1$)[14].

$\bar{\beta}_{i}^{b}=\frac{1}{|\hat{\pazocal{I}}_{nc}^{b}|+|\tilde{\pazocal{I}}_{nc}^{b}|},$
(10)

where $\hat{\pazocal{I}}_{nc}^{b}$ represents the set of (non-CoMP)–(non-CoMP)
NOMA pairs formed for a BS $b$ in the cluster $c$,
$\tilde{\pazocal{I}}_{nc}^{b}$ is the set of OMA non-CoMP users for a BS $b$
in the cluster $c$ which could not be paired (if any). We consider the
expressions of optimal $\bar{\theta}_{c}$, $\bar{\beta}_{i}$, and
$\bar{\beta}_{i}^{b}$ computed for a purely CoMP based system in [14] to study
the performance of this scheme. However, they may be sub-optimal for the
current CoMP–NOMA-based system.

TABLE I: Simulation Setup Parameter |  Value
---|---
Area, $\pazocal{A}$ ($\text{km}^{2}$) |  25
AWGN Power spectral density (dBm) |  $-174$.
Base Station density, $\lambda_{b}$ ($/\text{km}^{2}$) |  16, 30
Fairness parameter, $\alpha$ |  1
Number of subcarriers per subchannel, $sc_{o}$ |  $12$
Number of symbols per subcarrier, $sy_{o}$ |  $14$
Number of clusters, $K$ |  $10$
Number of iterations |  $10^{3}$
Path Loss ($d$ is in kms) |  $133.6+35\log_{10}(d)+\chi$
Standard deviation of shadowing random variable (dB) |  8
Subchannel Bandwidth (kHz) |  $180$
Total number of subchannels, M |  $100$
Transmission power, $P^{b}$ (dBm) |  $46$
User density, $\lambda_{u}$ ($/\text{km}^{2}$) |  $40,50,60,70,80,90,140,\newline 150$

##  IV Results and Discussions

In this section, we present the numerical results to evaluate the performance
of the proposed scheme through Monte Carlo simulations performed in MATLAB for
different combinations of $\lambda_{b}$, $\lambda_{u}$, and $\gamma_{th}$. The
details of the simulation parameters are summarized in Table I.

Figure 3: Illustration of variation of throughput with user density for
$\gamma_{th}=-6.5$ dB and $\lambda_{b}=16/\text{km}^{2}$ and
$\lambda_{b}=30/\text{km}^{2}$ .

Fig. 3 shows the variation of average throughput for a $\gamma_{th}$ of -6.5
dB with respect to $\lambda_{u}$ for $\lambda_{b}=16/\text{km}^{2}$ and
$30/\text{km}^{2}$, respectively. It is observed from Fig. 3 that with an
increase in user density there is a decrease in average throughput of the
proposed scheme. The proposed scheme is offering significantly better
throughput than that of the CoMP only system for lower values of
$\lambda_{u}$. However, the performance of this scheme is almost equal to that
of CoMP at higher $\lambda_{u}$. Similar trends are observed for higher values
of $\lambda_{b}$, but for all values of $\lambda_{u}$. The good performance of
the proposed scheme for lower $\lambda_{u}$ in the case of
$\lambda_{b}=16/\text{km}^{2}$ can be attributed to the availability of more
time fraction to the users at lower $\lambda_{u}$. As $\lambda_{u}$ increases,
$\bar{\theta}_{c}$ increases due to the increase in the number of CoMP–CoMP
NOMA pairs and this degrades the performance of the proposed scheme. At
significantly higher $\lambda_{b}$, a few NOMA pairs are formed and hence the
performance of the proposed scheme is similar to that of the CoMP only system.

Figure 4: Illustration of variation of throughput with $\gamma_{th}$ for
$\lambda_{b}=16/\text{km}^{2}$ and $\lambda_{u}=50/\text{km}^{2}$.

The variation of average throughput with respect to $\gamma_{th}$ for
Benchmark, CoMP only, NOMA only, and the proposed scheme is shown in Fig. 4.
It can be observed from the plot that the proposed scheme’s performance is
better than that of the CoMP only system for lower values of $\gamma_{th}$.
After a certain $\gamma_{th}$, the CoMP only system is performing better than
the proposed scheme due to the increase in CoMP users in a cluster because of
which the time fraction available for non-CoMP NOMA pairs reduces.

Similarly, the variation of coverage with respect to $\gamma_{th}$ is shown in
Fig. 5. It is observed that coverage increases with an increase in
$\gamma_{th}$. The coverage of the CoMP only system is the highest in
comparison to the benchmark followed by the proposed scheme, and NOMA only
systems. CoMP only system performs the best in terms of Coverage at the cost
of reduced throughput as dedicated CoMP resources are available for users with
lower channel gains. Thus, CoMP–NOMA system with the proposed scheme performs
better than the traditional CoMP in terms of throughput and traditional NOMA
in terms of coverage.

##  V Conclusion

In this work, we have studied the performance of the CoMP–NOMA system by
proposing a user pairing scheme considering only two types of pairs. The
proposed scheme results in enhanced coverage as compared to a NOMA-based
system. However, this enhancement comes at a cost of reduced network
throughput although, superior throughput trends are observed when compared to
CoMP only system for certain $\gamma_{th}$, $\lambda_{u}$, and $\lambda_{b}$.
Thus, the proposed scheme offers a trade-off between a CoMP system at one
extreme with high coverage-low network throughput and a NOMA system at the
other extreme with low coverage-high throughput. The presented scheme can be
used by cellular network planners to appropriately select the coverage-network
throughput trade-off by dynamically tuning the threshold and other parameters.
In the future, we plan to study this dynamic tuning using state-of-the-art
machine learning algorithms.

Figure 5: Illustration of variation of coverage of different schemes with
$\gamma_{th}$ for $\lambda_{b}=16/\text{km}^{2}$.

##  VI Acknowledgement

This work was supported in part by the Indo-Norway projects, no. 287918,
280835, and DST NMICPS through TiHAN Faculty fellowship of Dr. Abhinav Kumar.

## References

  * [1] Y. Tian et. al., “On the performance of opportunistic NOMA in downlink CoMP networks,” _IEEE Commun. Lett._ , vol. 20, no. 5, pp. 998–1001, May. 2016.
  * [2] L. Lei et. al., “Power and channel allocation for non-orthogonal multiple access in 5G systems: Tractability and Computation,” _IEEE Trans. Wireless Commun._ , vol. 15, no. 12, pp. 8580–8594, Dec. 2016.
  * [3] M. Sawahashi et. al., “Coordinated multipoint transmission/reception techniques for LTE-advanced coordinated and distributed MIMO,” _IEEE Wireless Commun._ , vol. 17, no. 3, pp. 26–34, Mar. 2010.
  * [4] V. Jungnickel et. al., “The role of small cells, coordinated multipoint, and massive MIMO in 5G,” _IEEE Commun. Mag._ , vol. 52, no. 5, pp. 44–51, May 2014.
  * [5] J. Choi, “Non-Orthogonal Multiple Access in Downlink Coordinated Two-Point Systems,” _IEEE Commun. Lett._ , vol. 18, no. 2, pp. 313–316, Feb. 2014.
  * [6] Z. Liu, et. al., “Power Allocation for Energy Efficiency Maximization in Downlink CoMP Systems with NOMA,” in _Proc. IEEE Wireless Commun. and Netw. Conf. (WCNC)_ , pp. 1–6, 2017.
  * [7] Q. Guo et. al., “Power control for coordinated NOMA downlink with cell-edge users,” in _Proc. IEEE Wireless Commun. and Netw. Conf. (WCNC)_ , pp. 1–6, 2018.
  * [8] Y. Tian et. al., “A Novel Opportunistic NOMA in Downlink Coordinated Multi-Point Networks,” in _Proc. IEEE 82nd Veh. Technol. Conf. (VTC2015-Fall)_ , 2015, pp. 1–5.
  * [9] Y. Tian, et. al., “On the Performance of a Multi-Tier NOMA Strategy in Coordinated Multi-Point Networks,” _IEEE Commun. Lett._ , vol. 21, no. 11, pp. 2448–2451, Nov. 2017.
  * [10] R. Singh, “Sub-channel assignment and resource scheduling for non-orthogonal multiple access (NOMA) in downlink coordinated multi-point systems,” in _Proc. 20th Conf. on Innova. in Clouds, Inter., and Netw. (ICIN)_ , 2017, pp. 17–22.
  * [11] A. Beylerian and T. Ohtsuki, “Coordinated Non-Orthogonal Multiple Access (CO-NOMA),” Glob. Commun. Conf. (GLOBECOM), pp. 1–5, Feb. 2016.
  * [12] M. S. Ali et. al., “Downlink Power Allocation for CoMP-NOMA in Multi-Cell Networks,” IEEE Trans. on Commun., vol. 66, no. 9, pp. 3982–3998, Sept. 2018,
  * [13] H. S. Dhillon et. al., “Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks,” IEEE Journal on Selected Areas in Commun., vol. 30, no. 3, pp. 550–560.
  * [14] Y. Ramamoorthi and A. Kumar, “Resource Allocation for CoMP in Cellular Networks With Base Station Sleeping,” _IEEE Access_ , vol. 6, pp. 12620–12633, Mar. 2018.
  * [15] N. S. Mouni et. al., “Adaptive User Pairing for NOMA Systems With Imperfect SIC,” _IEEE Wireless Commun. Lett._ , vol. 10, no. 7, pp. 1547–1551, Jul. 2021.


# WISE: full-Waveform variational Inference via Subsurface Extensions

Ziyi Yin1,*, Rafael Orozco1,*, Mathias Louboutin2, Felix J. Herrmann1
1Georgia Institute of Technology, 2Devito Codes Ltd * First two authors
contributed equally. Correspondence<EMAIL_ADDRESS>
###### Abstract

We introduce a probabilistic technique for full-waveform inversion, employing
variational inference and conditional normalizing flows to quantify
uncertainty in migration-velocity models and its impact on imaging. Our
approach integrates generative artificial intelligence with physics-informed
common-image gathers, reducing reliance on accurate initial velocity models.
Considered case studies demonstrate its efficacy producing realizations of
migration-velocity models conditioned by the data. These models are used to
quantify amplitude and positioning effects during subsequent imaging.

##  1 Introduction

Full-waveform inversion (FWI) plays a pivotal role in exploration, primarily
focusing on estimating Earth’s subsurface properties from observed seismic
data [1]. The inherent complexity of FWI stems from its nonlinearity, further
complicated by ill-posedness and computational intensiveness of the wave
modeling. To address these challenges, we introduce a computationally cost-
effective probabilistic framework that generates multiple migration-velocity
models conditioned on observed seismic data. By combining deep learning [2]
with physics, our approach harnesses advancements in variational inference
[VI, 3] and generative artificial intelligence [AI, 4, 5]. We achieve this by
forming common-image gathers [CIGs, 6], followed by training conditional
normalizing flows [CNFs, 7] that quantify uncertainties in migration-velocity
models.

Outline: First, we delineate the FWI problem and its inherent challenges.
Subsequently, we explore VI to quantify FWI’s uncertainty. To reduce VI’s
computational costs, we introduce _physics-informed summary statistics_ and
justify the use of CIGs as these statistics. Our framework’s capabilities are
validated through two case studies, which include studying the effects of
uncertainty in the generated migration-velocity models on migration.

##  2 Methodology

We present a Bayesian inference approach to FWI by briefly introducing FWI and
VI used as a framework for uncertainty quantification (UQ).

###  2.1 Full-waveform inversion

Estimation of unknown migration-velocity models, $\mathbf{x}$, from noisy
seismic data, $\mathbf{y}$ involves inverting nonlinear forward modeling,
$\mathcal{F}$, which links $\mathbf{x}$ to $\mathbf{y}$ via
$\mathbf{y}=\mathcal{F}(\mathbf{x})+\boldsymbol{\epsilon}$ with
$\boldsymbol{\epsilon}$ measurement noise. Source/receiver signatures are
assumed known and absorbed in $\mathcal{F}$. Solving this nonlinear inverse
problem is challenging because of the noise and the non-trivial null-space of
the modeling [8]. As a result, multiple migration-velocity models fit the
data, necessitating a Bayesian framework for UQ.

###  2.2 Full-waveform inference

Rather than seeking a single migration-velocity model, our goal is to invert
for a range of models compatible with the data, termed “full-waveform
inference”. From a Bayesian perspective, this involves determining the
posterior distribution of migration-velocity models given the data,
$p(\mathbf{x}|\mathbf{y})$. Algorithms to calculate statistics from the
posterior can be divided into: (i) sampling based algorithms, such as MCMC [9,
10, 11, 12]; or (ii) optimization based algorithms, such as VI [3].

Due to the high-dimensionality and expensive wave-based modeling, MCMC becomes
impractical for FWI [13]. For this reason, we focus on low-cost VI, which
exchanges the computational cost of posterior sampling for neural network
training [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].
Specifically, we employ amortized VI, which incurs offline computational
training cost but enables cheap online posterior inference on many datasets
$\mathbf{y}$ [30]. Next, we discuss how to use CNFs for amortized VI.

###  2.3 Amortized variational inference with conditional normalizing flows

During VI, the posterior distribution $p(\mathbf{x}|\mathbf{y})$ is
approximated by the surrogate,
$p_{\boldsymbol{\theta}}(\mathbf{x}|\mathbf{y})$, with learnable parameters,
$\boldsymbol{\theta}$. Because of their low-cost training and rapid sampling
[31, 32], CNFs are suitable to act as surrogates for the posterior with
training that involves minimization of the Kullback-Leibler divergence between
the true and surrogate posterior distribution. In practice, this requires
access to $N$ training pairs of migration-velocity model and observed data to
minimize the following objective:

$\underset{\boldsymbol{\theta}}{\operatorname{minimize}}\quad\frac{1}{N}\sum_{i=1}^{N}\left(\frac{1}{2}\|f_{\boldsymbol{\theta}}\left(\mathbf{x}^{(i)};\mathbf{y}^{(i)}\right)\|_{2}^{2}-\log\left|\det\mathbf{J}_{f_{\boldsymbol{\theta}}}\right|\right).$
(1)

Here, $f_{\boldsymbol{\theta}}$ is the CNF with network parameters,
$\boldsymbol{\theta}$, and Jacobian, $\mathbf{J}_{f_{\boldsymbol{\theta}}}$.
It transforms each velocity model, $\mathbf{x}^{(i)}$, into white noise (as
indicated by the $\ell_{2}$-norm), conditioned on the observation,
$\mathbf{y}^{(i)}$. After training, the inverse of CNF turn random
realizations of the standard Gaussian distribution into posterior samples
(migration-velocity models) conditioned on any seismic observation that is in
the same statistical distribution as the training data.

###  2.4 Physics-informed summary statistics

While CNFs are capable of approximating the posterior distribution, training
the CNFs on pairs $\left(\mathbf{x},\,\mathbf{y}\right)$ presents challenges
when changes in the acquisition occur or when physical principles simplifying
the mapping between model and data are lacking, both of which lead to
increasing training costs. To tackle these challenges, Radev et al. [33]
introduced fixed reduced-size _summary statistics_ that encapsulate observed
data and inform the posterior distribution. Building on this concept, Orozco
et al. [21] uses the gradient as the _physics-informed summary statistics_ ,
partially reversing the forward map and therefore accelerating CNF training.
For linear inverse problems with Gaussian noise, these statistics are unbiased
— maintaining the same posterior distribution, whether conditioned on original
data or on the gradient. Based on this principle, Siahkoohi et al. [20] and
Siahkoohi et al. [22] used reverse-time migration [RTM, 34], given by the
action of the adjoint of the linearized Born modeling [35, 36, 37], to
summarize data and quantify imaging uncertainties for a fixed accurate
migration-velocity model.

We aim to extend this approach to the nonlinear FWI problems. While RTM
transfers information from the data to the image domain, its performance
diminishes for incorrect migration velocities. Hou and Symes [38] showed that
least-squares migration [39] can perfectly fit the data for correct migration-
velocity models, but this fit fails for wrong velocity models. This highlights
a fundamental limitation in cases where the velocity model is wrong and RTM
does not capture the information, which leads to a biased posterior. For a
wrong initial FWI-velocity model $\mathbf{x}_{0}$,
$p\left(\mathbf{x}\middle|\mathbf{y}\right)\neq
p\left(\mathbf{x}\middle|\nabla\mathcal{F}\left(\mathbf{x}_{0}\right)^{\top}\mathbf{y}\right)$
with $\nabla\mathcal{F}$ Born modeling and ⊤ the adjoint. To avoid this
problem, more robust _physics-informed summary statistics_ are needed to
preserve information.

###  2.5 Common-image gathers as summary statistics

Migration-velocity analysis has a rich history in the literature [40, 6].
Following Liu et al. [41];Hou and Symes [42];Hou and Symes [43], we employ
relatively artifact-free subsurface-offset extended Born modeling to calculate
summary statistics. Thanks to being closer to an isometry—i.e, the adjoint of
extended Born modeling is closer to its inverse [44, 45] and therefore
preserves information, its adjoint can nullify residuals even when the FWI-
velocity model is incorrect as shown by Hou and Symes [38]. Geng et al. [46]
further demonstrated that neural networks can be used to map CIGs to velocity
models. Both these findings shed important light on the role of CIGs during VI
because CIGs preserve more information, which leads to less biased physics-
informed summary statistics for a wrong initial FWI-velocity model. Formally,
this means $p\left(\mathbf{x}\middle|\mathbf{y}\right)\approx
p\left(\mathbf{x}\middle|\overline{\nabla\mathcal{F}}\left(\mathbf{x}_{0}\right)^{\top}\mathbf{y}\right)$,
where $\overline{\nabla\mathcal{F}}$ is extended Born modeling. Leveraging
this mathematical observation, we propose WISE, short for full-Waveform
variational Inference via Subsurface Extensions. The core of this technique is
to train CNFs with pairs of velocity models, $\mathbf{x}$, and CIGs,
$\overline{\nabla\mathcal{F}}\left(\mathbf{x}_{0}\right)^{\top}\mathbf{y}$,
guided by the objective of Equation 1. Our case studies will demonstrate that
even with wrong initial FWI-velocity models, CIGs encapsulate more
information, enabling the trained CNFs to generate accurate migration-velocity
models conditioned by the data.

##  3 Synthetic case studies

Our study evaluates the performance of WISE through synthetic case studies on
two datasets: the CurveFault-A dataset of Open FWI [47] and 2D slices of the
Compass dataset [48]. We aim to compare the quality of posterior samples
informed by RTM alone versus those informed by CIGs.

###  3.1 Open FWI

The CurveFault-A dataset comprises velocity models with significant
variability across samples, which poses challenges for deep learning methods
[49, 50]. This is further compounded by faults and dipping events while
observations contain only reflected energy. Testing on this dataset allows us
to test WISE’s velocity-model generation capabilities.

Dataset generation and network training. We select $3000$ velocity models of
$640$ m by $640$ m, each with $64$ equally spaced receivers at $10\mathrm{m}$
tow depth and $16$ randomly placed sources [51]. The surface is assumed
absorbing. Using a $15\mathrm{Hz}$ central frequency Ricker wavelet with
energy below $3\mathrm{Hz}$ removed for realism, acoustic data is simulated
with Devito [52, 53] and JUDI.jl [54, 55]. Uncorrelated band-limited Gaussian
noise is added (S/N $12\mathrm{dB}$) before migrating each dataset with a 1D
initial FWI-velocity model calculated by averaging the corresponding true
model horizontally. CIGs are computed for $101$ subsurface offsets ranging
from $-250\mathrm{m}$ to $+250\mathrm{m}$ [56, 57]. The dataset is split into
$2800$ training, $150$ validation, and $50$ test samples. Two CNFs are
trained: one with velocity-RTM pairs and another with velocity-CIGs pairs.

Results. Results on two untested samples by our CNFs are included in Figure 1
and reveal notable variation in the posterior samples for sharp boundaries and
smooth transitions in the velocity. While the conditional mean estimate does
not fully replicate the true velocity, the standard deviations meaningfully
correlate with the errors, indicating that the uncertainty represented by the
standard deviation is informative. Across $50$ test samples, the mean SSIM
score for CIGs-based statistics is $0.87$, surpassing the $0.85$ mean for RTM-
based statistics. Motivated by these results, we will examine a more realistic
example with intricate geological structures next.

Figure 1: Applying WISE for two unseen test samples in Open FWI CurveFault-A
dataset.

###  3.2 Compass model

To validate WISE in a more realistic setting and examine uncertainty in
imaging, we consider the Compass dataset, known for its “velocity kickback”
challenge for FWI algorithms. By comparing, for a poor initial FWI-velocity
model, the conditional mean of samples of the migration-velocity model
informed by CIGs and RTM, we verify the superior information content of CIGs.
We also illustrate how uncertainty in migration-velocity models can be
converted into uncertainties in amplitude and positioning of imaged reflectors
[58].

Dataset generation and network training. We take $1040$ 2D slices of the
Compass model of $6.4$ km by $3.2$ km, with $512$ equally spaced sources towed
at $12.5\mathrm{m}$ depth and $64$ ocean-bottom nodes at random locations.
Source and noise setup remains the same. The arithmetic mean over all velocity
models is used as the 1D initial FWI-velocity model (shown in Figure 2(b)).
$51$ subsurface offsets ranging from $-500\mathrm{m}$ to $+500\mathrm{m}$ are
used to compute CIGs (shown in Figure 3(a)). The dataset is divided into $800$
training, $190$ validation, and $50$ test samples, with CNFs trained over
$200$ epochs.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 2: (a) an unseen ground-truth velocity model; (b) 1D initial FWI-
velocity model; (c) conditional mean estimate for RTM as summary statistics
($\mathrm{SSIM}=0.48$); (d) conditional mean estimate from WISE
($\mathrm{SSIM}=0.56$).

Results. Our method’s performance is evaluated on an unseen 2D Compass slice
shown in Figure 2(a). When RTM is used to summarize the data, the conditional
mean estimate (Figure 2(c)) does not capture the shape of the unconformity.
Thanks to the CIGs, WISE captures more information and as a result produces a
more accurate conditional mean (Figure 2(d)). For the $50$ test samples, the
SSIM scores with CIGs yield a mean of $0.63$, outperforming RTM-based
statistics with a mean SSIM of $0.52$.

Quality control. To verify the inferred migration-velocity model, CIGs
calculated for the initial FWI-velocity model (Figure 2(b)), plotted in Figure
3(a), are juxtaposed against CIGs calculated for the inferred migration-
velocity model (Figure 2(d)), plotted in Figure 3(b). Significant improvement
in near-offset focused energy is observed in the CIGs for the inferred
migration-velocity model. A similar focusing behavior is noted for the
posterior samples themselves.

(a) (a)

(b) (b)

Figure 3: CIGs calculated by the initial FWI-velocity model given by (a) the
1D background model (shown in Figure 2(b)) or by (b) the conditional mean
estimate (shown in Figure 2(d)).

Downstream imaging. While access to the posterior represents an important step
towards grasping uncertainty, understanding its impact on imaging with
($30\mathrm{Hz}$) RTMs is more relevant because it concerns uncertainty in the
final product. For this purpose, comparisons are made between the RTM computed
for the conditional mean (Figure 4(a)) and RTMs computed for individual
posterior samples that vary significantly judged by the point-wise standard
deviations in Figure 4(b). These deviations increase with depth and correlate
with complex geology where the RTM-based inference struggled. To understand
how this uncertainty propagates to RTM images, forward uncertainty is assessed
by carrying out RTMs for different posterior samples with results for the
standard deviations plotted in Figure 4(c). These amplitude deviations are
different because mapping migration-velocities to RTMs is highly nonlinear,
leading to large areas of intense amplitude variation and dimming at the edges
caused by the Born modeling’s null-space. While these amplitude sensitivities
are useful, deviations in the migration velocities also leads to differences
in reflector positioning. Vertical shifts between the reference image (Figure
4(a)) and RTMs for different posterior samples are calculated with a local
cross-correlation technique [59] and included in Figure 4(d) where blue/red
areas correspond to up/down shifts. As expected, these shifts are most notable
in the deeper regions and at the edges where velocity variations are the
largest.

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 4: Using WISE in the downstream imaging task: (a) imaged reflectors
using the conditional mean estimate (shown in Figure 2(d)) as the migration
velocity; (b) point-wise standard deviation of the posterior velocity samples;
(c) point-wise deviation of the imaged reflectivities; (d) point-wise maximum
depth shift. Note: (a) and (c) are normalized by the same constant. The images
are all grided for visualization purpose.

##  4 Discussion

Once the offline costs of computing 800 CIGs and network training are covered,
WISE enables generation of velocity models for unseen seismic data at the low
computational cost of a single set of CIGs for a poor initial FWI-velocity
model. The Open FWI case study demonstrates WISE’s capability of producing
realistic posterior samples and conditional means for a broad range of unseen
velocity models. In the case of the Compass model, the initial FWI-velocity
model was poor. Still, CIGs obtained from a single 1D initial model capture
relevant information from the non-zero offsets. From this information, the
network learns to produce migration-velocity models that focus at inference.
WISE also produced two types of uncertainty, namely (i) inverse uncertainty in
migration-velocity model estimation from noisy data, and (ii) forward
uncertainty where uncertainty in migration-velocity models is propagated to
uncertainty in amplitude and positioning of imaged reflectors. The latter can
also be assessed for tasks like horizon tracking [60, 12] and seismic
interpretation [61].

Opportunities for future research remain. One area concerns dealing with the
“amortization gap” where CNFs tend to maximize performance across multiple
datasets rather than excelling at a single observation [62]. To improve
single-observation performance, particularly for out-of-distribution samples,
computationally more expensive latent space corrections [22] can be employed
that incorporate the physics. Moreover, velocity continuation methods [63, 44]
could be used including recent advances in neural operators [64]. These could
offset the cost of running RTMs for each posterior sample.

##  5 Conclusions

We present WISE, full-Waveform variational Inference via Subsurface
Extensions, for computationally efficient uncertainty quantification of FWI.
This framework underscores the potential of generative AI in addressing FWI
challenges, paving the way for a new seismic inversion and imaging paradigm
that is uncertainty-aware. By having common-image gathers act as information-
preserving summary statistics, a principled approach to UQ is achieved where
generative AI is successfully combined with wave physics. Because WISE
automatically produces distributions for migration-velocity models conditioned
by the data, it moves well beyond traditional velocity model building. It was
shown that this distributional information can be employed to quantify
uncertainties in the migration-velocity models that can be used to better
understand amplitude and positioning uncertainty in migration.

##  6 Acknowledgement

This research was carried out with the support of Georgia Research Alliance
and partners of the ML4Seismic Center. The authors would like to thank Charles
Jones (Osokey) for the constructive discussion. The authors gratefully
acknowledge the contribution of OpenAI’s ChatGPT for refining sentence
structure and enhancing the overall readability of this manuscript.

## References

  * Virieux and Operto [2009] Jean Virieux and Stéphane Operto.  An overview of full-waveform inversion in exploration geophysics.  _Geophysics_ , 74(6):WCC1–WCC26, 2009.
  * Yu and Ma [2021] Siwei Yu and Jianwei Ma.  Deep learning for geophysics: Current and future trends.  _Reviews of Geophysics_ , 59(3):e2021RG000742, 2021.
  * Jordan et al. [1999] Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul.  An introduction to variational methods for graphical models.  _Machine learning_ , 37:183–233, 1999.
  * Ramesh et al. [2022] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen.  Hierarchical text-conditional image generation with clip latents.  _arXiv preprint arXiv:2204.06125_ , 1(2):3, 2022.
  * Herrmann [2023] Felix J Herrmann.  President’s page: Digital twins in the era of generative AI.  _The Leading Edge_ , 42(11):730–732, 2023.
  * Symes [2008] William W Symes.  Migration velocity analysis and waveform inversion.  _Geophysical prospecting_ , 56(6):765–790, 2008.
  * Winkler et al. [2019] Christina Winkler, Daniel Worrall, Emiel Hoogeboom, and Max Welling.  Learning likelihoods with conditional normalizing flows.  _arXiv preprint arXiv:1912.00042_ , 2019.
  * Tarantola [1984] Albert Tarantola.  Inversion of seismic reflection data in the acoustic approximation.  _GEOPHYSICS_ , 49(8):1259–1266, 08 1984.  doi: 10.1190/1.1441754.  URL http://dx.doi.org/10.1190/1.1441754.
  * Welling and Teh [2011] Max Welling and Yee W Teh.  Bayesian learning via stochastic gradient langevin dynamics.  In _Proceedings of the 28th international conference on machine learning (ICML-11)_ , pages 681–688, 2011.
  * Ely et al. [2018] Gregory Ely, Alison Malcolm, and Oleg V Poliannikov.  Assessing uncertainties in velocity models and images with a fast nonlinear uncertainty quantification method.  _Geophysics_ , 83(2):R63–R75, 2018.
  * Kotsi et al. [2020] M Kotsi, A Malcolm, and G Ely.  Uncertainty quantification in time-lapse seismic imaging: A full-waveform approach.  _Geophysical Journal International_ , 222(2):1245–1263, 2020.
  * Siahkoohi et al. [2022a] Ali Siahkoohi, Gabrio Rizzuti, and Felix J. Herrmann.  Deep bayesian inference for seismic imaging with tasks.  _GEOPHYSICS_ , 87(5):S281–S302, 08 2022a.  doi: 10.1190/geo2021-0666.1.  URL http://dx.doi.org/10.1190/geo2021-0666.1.
  * Gelman et al. [2003] Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin.  _Bayesian Data Analysis_.  Chapman and Hall/CRC, 07 2003.  doi: 10.1201/9780429258480.  URL http://dx.doi.org/10.1201/9780429258480.
  * Rizzuti et al. [2020] Gabrio Rizzuti, Ali Siahkoohi, Philipp A. Witte, and Felix J. Herrmann.  Parameterizing uncertainty by deep invertible networks: An application to reservoir characterization.  _SEG Technical Program Expanded Abstracts 2020_ , 09 2020.  doi: 10.1190/segam2020-3428150.1.  URL http://dx.doi.org/10.1190/segam2020-3428150.1.
  * Ren et al. [2021] Yuxiao Ren, Philipp A Witte, Ali Siahkoohi, Mathias Louboutin, Ziyi Yin, and Felix J Herrmann.  Seismic velocity inversion and uncertainty quantification using conditional normalizing flows.  In _AGU Fall Meeting 2021_. AGU, 2021.
  * Kumar et al. [2021] Rajiv Kumar, Maria Kotsi, Ali Siahkoohi, and Alison Malcolm.  Enabling uncertainty quantification for seismic data preprocessing using normalizing flows (nf) – an interpolation example.  _First International Meeting for Applied Geoscience & Energy Expanded Abstracts_, 09 2021.  doi: 10.1190/segam2021-3583705.1.  URL http://dx.doi.org/10.1190/segam2021-3583705.1.
  * Siahkoohi and Herrmann [2021] Ali Siahkoohi and Felix J. Herrmann.  Learning by example: Fast reliability-aware seismic imaging with normalizing flows.  _First International Meeting for Applied Geoscience & Energy Expanded Abstracts_, 09 2021.  doi: 10.1190/segam2021-3581836.1.  URL http://dx.doi.org/10.1190/segam2021-3581836.1.
  * Siahkoohi et al. [2021] Ali Siahkoohi, Gabrio Rizzuti, Mathias Louboutin, Philipp A Witte, and Felix J Herrmann.  Preconditioned training of normalizing flows for variational inference in inverse problems.  _arXiv preprint arXiv:2101.03709_ , 2021.
  * Zhang et al. [2021] Xin Zhang, Muhammad Atif Nawaz, Xuebin Zhao, and Andrew Curtis.  An introduction to variational inference in geophysical inverse problems.  In _Advances in Geophysics_ , volume 62, pages 73–140. Elsevier, 2021.
  * Siahkoohi et al. [2022b] Ali Siahkoohi, Rafael Orozco, Gabrio Rizzuti, and Felix J Herrmann.  Wave-equation-based inversion with amortized variational bayesian inference.  _arXiv preprint arXiv:2203.15881_ , 2022b.
  * Orozco et al. [2023a] Rafael Orozco, Ali Siahkoohi, Gabrio Rizzuti, Tristan van Leeuwen, and Felix J Herrmann.  Adjoint operators enable fast and amortized machine learning based bayesian uncertainty quantification.  In _Medical Imaging 2023: Image Processing_ , volume 12464, pages 357–367. SPIE, 2023a.
  * Siahkoohi et al. [2023] Ali Siahkoohi, Gabrio Rizzuti, Rafael Orozco, and Felix J. Herrmann.  Reliable amortized variational inference with physics-based latent distribution correction.  _GEOPHYSICS_ , 88(3):R297–R322, 04 2023.  doi: 10.1190/geo2022-0472.1.  URL http://dx.doi.org/10.1190/geo2022-0472.1.
  * Wen et al. [2023] Jeffrey Wen, Rizwan Ahmad, and Philip Schniter.  A conditional normalizing flow for accelerated multi-coil mr imaging.  _arXiv preprint arXiv:2306.01630_ , 2023.
  * Orozco et al. [2023b] Rafael Orozco, Mathias Louboutin, Ali Siahkoohi, Gabrio Rizzuti, Tristan van Leeuwen, and Felix Herrmann.  Amortized normalizing flows for transcranial ultrasound with uncertainty quantification.  _arXiv preprint arXiv:2303.03478_ , 2023b.
  * Zhang et al. [2023] Xin Zhang, Angus Lomas, Muhong Zhou, York Zheng, and Andrew Curtis.  3-d bayesian variational full waveform inversion.  _Geophysical Journal International_ , 234(1):546–561, 2023.
  * Orozco et al. [2023c] Rafael Orozco, Ali Siahkoohi, Mathias Louboutin, and Felix J Herrmann.  Refining amortized posterior approximations using gradient-based summary statistics.  _arXiv preprint arXiv:2305.08733_ , 2023c.
  * Zhang and Curtis [2023] Xin Zhang and Andrew Curtis.  Bayesian variational time-lapse full-waveform inversion.  _arXiv preprint arXiv:2308.08805_ , 2023.
  * Strutz and Curtis [2023] Dominik Strutz and Andrew Curtis.  Variational bayesian experimental design for geophysical applications.  _arXiv preprint arXiv:2307.01039_ , 2023.
  * Gahlot et al. [2023] Abhinav Prakash Gahlot, Huseyin Tuna Erdinc, Rafael Orozco, Ziyi Yin, and Felix J Herrmann.  Inference of CO2 flow patterns–a feasibility study.  _arXiv preprint arXiv:2311.00290_ , 2023.
  * Kruse et al. [2021] Jakob Kruse, Gianluca Detommaso, Ullrich Köthe, and Robert Scheichl.  Hint: Hierarchical invertible neural transport for density estimation and bayesian inference.  In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 35, pages 8191–8199, 2021.
  * Rezende and Mohamed [2015] Danilo Rezende and Shakir Mohamed.  Variational inference with normalizing flows.  In _International conference on machine learning_ , pages 1530–1538. PMLR, 2015.
  * Louboutin et al. [2023a] Mathias Louboutin, Ziyi Yin, Rafael Orozco, Thomas J Grady, Ali Siahkoohi, Gabrio Rizzuti, Philipp A Witte, Olav Møyner, Gerard J Gorman, and Felix J Herrmann.  Learned multiphysics inversion with differentiable programming and machine learning.  _The Leading Edge_ , 42(7):474–486, 2023a.
  * Radev et al. [2022] Stefan T. Radev, Ulf K. Mertens, Andreas Voss, Lynton Ardizzone, and Ullrich Kothe.  Bayesflow: Learning complex stochastic models with invertible neural networks.  _IEEE Transactions on Neural Networks and Learning Systems_ , 33(4):1452–1466, 04 2022.  doi: 10.1109/tnnls.2020.3042395.  URL http://dx.doi.org/10.1109/TNNLS.2020.3042395.
  * Baysal et al. [1983] Edip Baysal, Dan D Kosloff, and John WC Sherwood.  Reverse time migration.  _Geophysics_ , 48(11):1514–1524, 1983.
  * Schuster [1993] Gerard T. Schuster.  Least-squares cross-well migration.  _SEG Technical Program Expanded Abstracts 1993_ , 01 1993.  doi: 10.1190/1.1822308.  URL http://dx.doi.org/10.1190/1.1822308.
  * Nemeth et al. [1999] Tamas Nemeth, Chengjun Wu, and Gerard T. Schuster.  Least-squares migration of incomplete reflection data.  _GEOPHYSICS_ , 64(1):208–221, 01 1999.  doi: 10.1190/1.1444517.  URL http://dx.doi.org/10.1190/1.1444517.
  * Alsing and Wandelt [2018] Justin Alsing and Benjamin Wandelt.  Generalized massive optimal data compression.  _Monthly Notices of the Royal Astronomical Society: Letters_ , 476(1):L60–L64, 2018.
  * Hou and Symes [2016] Jie Hou and William W. Symes.  Accelerating extended least-squares migration with weighted conjugate gradient iteration.  _GEOPHYSICS_ , 81(4):S165–S179, 07 2016.  doi: 10.1190/geo2015-0499.1.  URL http://dx.doi.org/10.1190/geo2015-0499.1.
  * Zeng et al. [2014] Chong Zeng, Shuqian Dong, and Bin Wang.  Least-squares reverse time migration: Inversion-based imaging toward true reflectivity.  _The Leading Edge_ , 33(9):962–968, 09 2014.  doi: 10.1190/tle33090962.1.  URL http://dx.doi.org/10.1190/tle33090962.1.
  * Sava and Biondi [2004] Paul Sava and Biondo Biondi.  Wave-equation migration velocity analysis. i. theory.  _Geophysical Prospecting_ , 52(6):593–606, 2004.
  * Liu et al. [2013] Yujin Liu, William W Symes, and Zhenchun Li.  Multisource least-squares extended reverse-time migration with preconditioning guided gradient method.  In _SEG International Exposition and Annual Meeting_ , pages SEG–2013. SEG, 2013.
  * Hou and Symes [2015] Jie Hou and William W Symes.  An approximate inverse to the extended born modeling operator.  _Geophysics_ , 80(6):R331–R349, 2015.
  * Hou and Symes [2018] Jie Hou and William W Symes.  Inversion velocity analysis in the subsurface-offset domain.  _Geophysics_ , 83(2):R189–R200, 2018.
  * Yang et al. [2021] Mengmeng Yang, Marie Graff, Rajiv Kumar, and Felix J. Herrmann.  Low-rank representation of omnidirectional subsurface extended image volumes.  _GEOPHYSICS_ , 86(3):S165–S183, 03 2021.  doi: 10.1190/geo2020-0152.1.  URL http://dx.doi.org/10.1190/geo2020-0152.1.
  * Kroode [2023] Fons ten Kroode.  An omnidirectional seismic image extension.  _Inverse Problems_ , 39(3):035003, 01 2023.  doi: 10.1088/1361-6420/acb285.  URL http://dx.doi.org/10.1088/1361-6420/acb285.
  * Geng et al. [2022] Zhicheng Geng, Zeyu Zhao, Yunzhi Shi, Xinming Wu, Sergey Fomel, and Mrinal Sen.  Deep learning for velocity model building with common-image gather volumes.  _Geophysical Journal International_ , 228(2):1054–1070, 2022.
  * Deng et al. [2022] Chengyuan Deng, Shihang Feng, Hanchen Wang, Xitong Zhang, Peng Jin, Yinan Feng, Qili Zeng, Yinpeng Chen, and Youzuo Lin.  Openfwi: Large-scale multi-structural benchmark datasets for full waveform inversion.  _Advances in Neural Information Processing Systems_ , 35:6007–6020, 2022.
  * E. Jones et al. [2012] C. E. Jones, J. A. Edgar, J. I. Selvage, and H. Crook.  Building complex synthetic models to evaluate acquisition geometries and velocity inversion technologies.  _Proceedings_ , 06 2012.  doi: 10.3997/2214-4609.20148575.  URL http://dx.doi.org/10.3997/2214-4609.20148575.
  * Jin et al. [2021] Peng Jin, Xitong Zhang, Yinpeng Chen, Sharon Xiaolei Huang, Zicheng Liu, and Youzuo Lin.  Unsupervised learning of full-waveform inversion: Connecting cnn and partial differential equation in a loop.  _arXiv preprint arXiv:2110.07584_ , 2021.
  * Jin et al. [2023] Peng Jin, Yinan Feng, Shihang Feng, Hanchen Wang, Yinpeng Chen, Benjamin Consolvo, Zicheng Liu, and Youzuo Lin.  Does full waveform inversion benefit from big data?  _arXiv preprint arXiv:2307.15388_ , 2023.
  * Herrmann [2010] Felix J Herrmann.  Randomized sampling and sparsity: Getting more information from fewer samples.  _Geophysics_ , 75(6):WB173–WB187, 2010.
  * Louboutin et al. [2019] M. Louboutin, M. Lange, F. Luporini, N. Kukreja, P. A. Witte, F. J. Herrmann, P. Velesko, and G. J. Gorman.  Devito (v3.1.0): an embedded domain-specific language for finite differences and geophysical exploration.  _Geoscientific Model Development_ , 12(3):1165–1187, 2019.  doi: 10.5194/gmd-12-1165-2019.  URL https://www.geosci-model-dev.net/12/1165/2019/.
  * Luporini et al. [2020] Fabio Luporini, Mathias Louboutin, Michael Lange, Navjot Kukreja, Philipp Witte, Jan Hückelheim, Charles Yount, Paul H. J. Kelly, Felix J. Herrmann, and Gerard J. Gorman.  Architecture and performance of devito, a system for automated stencil computation.  _ACM Trans. Math. Softw._ , 46(1), apr 2020.  ISSN 0098-3500.  doi: 10.1145/3374916.  URL https://doi.org/10.1145/3374916.
  * Witte et al. [2019] Philipp A. Witte, Mathias Louboutin, Navjot Kukreja, Fabio Luporini, Michael Lange, Gerard J. Gorman, and Felix J. Herrmann.  A large-scale framework for symbolic implementations of seismic inversion algorithms in julia.  _GEOPHYSICS_ , 84(3):F57–F71, 2019.  doi: 10.1190/geo2018-0174.1.  URL https://doi.org/10.1190/geo2018-0174.1.
  * Louboutin et al. [2023b] Mathias Louboutin, Philipp Witte, Ziyi Yin, Henryk Modzelewski, Kerim, Carlos da Costa, and Peterson Nogueira.  slimgroup/judi.jl: v3.2.3, March 2023b.  URL https://doi.org/10.5281/zenodo.7785440.
  * Louboutin [2023] Mathias Louboutin.  _slimgroup/ImageGather.jl: v0.2.6_.  Zenodo, 06 2023.  doi: 10.5281/ZENODO.7997993.  URL https://zenodo.org/record/7997993.
  * Louboutin and Herrmann [2023] Mathias Louboutin and Felix J. Herrmann.  Wave-based inversion at scale on graphical processing units with randomized trace estimation.  _Geophysical Prospecting_ , 07 2023.  doi: 10.1111/1365-2478.13405.  URL http://dx.doi.org/10.1111/1365-2478.13405.
  * Poliannikov and Malcolm [2016] Oleg V Poliannikov and Alison E Malcolm.  The effect of velocity uncertainty on migrated reflectors: Improvements from relative-depth imaging.  _Geophysics_ , 81(1):S21–S29, 2016.
  * Hale [2006] Dave Hale.  An efficient method for computing local cross-correlations of multi-dimensional signals.  _CWP Report_ , 656:282, 2006.
  * Wu and Fomel [2018] Xinming Wu and Sergey Fomel.  Least-squares horizons with local slopes and multigrid correlations.  _Geophysics_ , 83(4):IM29–IM40, 2018.
  * Kaur et al. [2023] Harpreet Kaur, Qie Zhang, Philipp Witte, Luming Liang, Long Wu, and Sergey Fomel.  Deep-learning-based 3d fault detection for carbon capture and storage.  _Geophysics_ , 88(4):IM101–IM112, 2023.
  * Marino et al. [2018] Joe Marino, Yisong Yue, and Stephan Mandt.  Iterative amortized inference.  In _International Conference on Machine Learning_ , pages 3403–3412. PMLR, 2018.
  * Fomel [2003] Sergey Fomel.  Time-migration velocity analysis by velocity continuation.  _Geophysics_ , 68(5):1662–1672, 2003.
  * Siahkoohi et al. [2022c] Ali Siahkoohi, Mathias Louboutin, and Felix J Herrmann.  Velocity continuation with fourier neural operators for accelerated uncertainty quantification.  In _SEG International Exposition and Annual Meeting_ , page D011S092R004. SEG, 2022c.