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**Solid Earth**
An interactive open-access journal of the European Geosciences Union

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SE | Articles | Volume 10, issue 6

Solid Earth, 10, 1833–1855, 2019

https://doi.org/10.5194/se-10-1833-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/se-10-1833-2019

© Author(s) 2019. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Advances in seismic imaging across the scales

**Research article**
30 Oct 2019

**Research article** | 30 Oct 2019

Full-waveform inversion of short-offset, band-limited seismic data in the Alboran Basin (SE Iberia)

^{1}Barcelona Center for Subsurface Imaging, Institute of Marine Sciences (ICM), CSIC, 08003, Barcelona, Spain^{2}ICREA, Passeig de Lluís Companys, 23, 08010, Barcelona, Spain

^{1}Barcelona Center for Subsurface Imaging, Institute of Marine Sciences (ICM), CSIC, 08003, Barcelona, Spain^{2}ICREA, Passeig de Lluís Companys, 23, 08010, Barcelona, Spain

**Correspondence**: Clàudia Gras (gras@icm.csic.es)

**Correspondence**: Clàudia Gras (gras@icm.csic.es)

Abstract

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We present a high-resolution P-wave velocity model of the sedimentary cover and the uppermost basement to ∼3 km depth obtained by full-waveform inversion of multichannel seismic data acquired with a 6 km long streamer in the Alboran Sea (SE Iberia). The inherent non-linearity of the method, especially for short-offset, band-limited seismic data as this one, is circumvented by applying a data processing or modelling sequence consisting of three steps: (1) data re-datuming by back-propagation of the recorded seismograms to the seafloor; (2) joint refraction and reflection travel-time tomography combining the original and the re-datumed shot gathers; and (3) full-waveform inversion of the original shot gathers using the model obtained by travel-time tomography as initial reference.

The final velocity model shows a number of geological structures that cannot be identified in the travel-time tomography models or easily interpreted from seismic reflection images alone. A sharp strong velocity contrast accurately defines the geometry of the top of the basement. Several low-velocity zones that may correspond to the abrupt velocity change across steeply dipping normal faults are observed at the flanks of the basin. A 200–300 m thick, high-velocity layer embedded within lower-velocity sediment may correspond to evaporites deposited during the Messinian crisis. The results confirm that the combination of data re-datuming and joint refraction and reflection travel-time inversion provides reference models that are accurate enough to apply full-waveform inversion to relatively short offset streamer data in deep-water settings starting at a field-data standard low-frequency content of 6 Hz.

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How to cite.

Gras, C., Dagnino, D., Jiménez-Tejero, C. E., Meléndez, A., Sallarès, V., and Ranero, C. R.: Full-waveform inversion of short-offset, band-limited seismic data in the Alboran Basin (SE Iberia), Solid Earth, 10, 1833–1855, https://doi.org/10.5194/se-10-1833-2019, 2019.

1 Introduction

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Seismic methods are one of the most powerful existing geophysical tools to
extract information on the structure and properties of the Earth's
subsurface. These techniques have been, and currently continue to be, widely
used to obtain images of the sediments and crust and to map the variations in
physical properties, particularly P-wave velocity (*V*_{P}). The
*V*_{P} distribution provides information on the subsurface geology
and on the type of rocks constituting the different structures. The most
objective and widely used method to retrieve *V*_{P} models is
seismic tomography, using either travel-time information only, as in
travel-time tomography (TTT) (e.g. Zelt and Smith, 1992; Korenaga et al., 2000; Hobro et al., 2003), or a more complete set of waveform attributes including both
phase and amplitudes as in full-waveform inversion (FWI)
(e.g. Virieux and Operto, 2009). Whereas TTT is a robust, moderately non-linear
technique that provides coarse *V*_{P} models, FWI is
computationally demanding and strongly non-linear, but it has the potential
to provide higher-resolution *V*_{P} models. For their
characteristics, TTT and FWI are considered to be complementary, so that they
are often combined and applied together. In general, TTT is applied first to
get a moderate-resolution *V*_{P} model, which is then used as an initial model for a multi-scale FWI. The key of a successful combined
inversion is to obtain an initial FWI model that is kinematically correct, in
the sense that the simulated and recorded waveforms are not cycle-skipped at
the lowest frequency available to perform FWI (e.g. Shipp and Singh, 2002).
Long-offset refracted waves containing long-wavelength information are a key
ingredient to obtain a kinematically correct reference macro-velocity model
through TTT. Due to the intrinsic limited offset of near-vertical reflection
acquisitions, refracted waves are typically not observed in seismic
reflection recordings, so that they are barely used to perform seismic
tomography. These narrow-azimuth reflection acquisitions sample only small
values of scattering angles at a diffracting point if the background model is
smooth. In the framework of the single-scattering approximation, for small
values of scattering angles only the high-wavenumbers of the subsurface can
be recorded and, thus, reconstructed from the wave fields applying conventional
FWI (Brossier et al., 2014).

One example of near-vertical seismic acquisition is marine multichannel reflection seismic (MCS) systems. The set-up of MCS systems is designed to record near-vertical reflections with a high redundancy (large number of channels), but offset is limited to the streamer length. In deep-water settings, the critical distance for refracted waves is often beyond this distance. Near-vertical reflections hold indirect information on the velocity field and on the depth of the layer where they are reflected. Due to the inherent velocity–depth trade-off and to the possible errors in the identification of the reflector boundaries, we cannot expect a precise velocity model by travel-time inversion using only one or few reflectors with no lateral or vertical continuity.

Aside from the lack of long offsets or wide-aperture acquisition systems and, hence, of refracted phases and wide scattering angles needed for TTT, the ability to solve low-wavenumber information of the subsurface is also influenced by the low angular frequency content of the signal (Brossier et al., 2014). The most commonly used sources in MCS systems are air guns, which are typically not capable of generating energy below 4–5 Hz, so in those cases the signal-to-noise ratio is very low. The combination of (1) a lack of refracted waves that make difficult to apply TTT and (2) a lack of low-frequency components in the data set that aggravate the issue of cycle skipping makes it even more challenging to perform TTT–FWI with MCS data.

A common approach to mitigate the above-mentioned issues and apply TTT–FWI to
marine MCS data in deep-water settings is by re-datuming the collected data
(Arnulf et al., 2011, 2014; Henig et al., 2012) to a virtual recording surface
first (Qin and Singh, 2017), so that refracted waves can be identified as first
arrivals in the virtual recordings to perform TTT. Then first arrival TTT is
applied, and the resulting model is used as a reference for multi-scale FWI
starting at the lowest available frequency. The success of this approach
relies on obtaining a TTT *V*_{P} that allows overcoming cycle
skipping. This issue is outstanding and difficult to overcome when data lack
frequencies below 3–4 Hz (e.g. Shah et al., 2012; Jiménez Tejero et al., 2015), which is the case of most field MCS data.

Our goal here is designing and testing a data processing or modelling strategy
that allows applying FWI to limited-offset MCS data in deep-water settings
starting FWI at frequencies higher than 5 Hz. For this, we propose a
modified version of the joint TTT–FWI approach that combines wave equation
datuming with joint refraction and reflection TTT (instead of first arrival
TTT), prior to FWI. The performance of the proposed approach is demonstrated
by applying it to field MCS data acquired with a 6 km streamer in the
∼2 km deep Alboran Basin (W Mediterranean). The final FWI model
displays a number of details of geological interest that cannot be seen in the
initial TTT model, including a layer that may correspond to evaporites
embedded in the sedimentary basin, faults, and short-wavelength highs and lows
of the crystalline basement. Finally we show that this *V*_{P} model
allows us to significantly improve pre-stack depth migrated (PSDM) images.

2 Methodology

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Having a good initial model is essential to overcome the inherent
non-linearity of FWI for band-limited seismic data. A key ingredient to build
that model is using travel times of refracted waves, which for deep water and
short-offset settings are not visible as first arrivals. The objective of our
workflow is solving these challenges and applying FWI under these adverse
conditions. The proposed approach consists of the following steps: (1) data
re-datuming by downward continuation (DC) of the recorded data to the
seafloor, (2) joint refraction and reflection TTT of the original and
re-datumed shot gathers, (3) FWI of the original shot gathers using the model
obtained in (2) as initial reference, and (4) pre-stack depth migration of the
original shot gathers using the FWI *V*_{P} model. In the next
sections we describe the fundamental aspects of the different methodologies.
The general workflow is summarized in Fig. 1.

As stated in the introduction, refractions are essential to build a macro-velocity model with the correct low-wavenumber content on it. As MCS acquisition systems have a limited offset (Fig. 2a), the first arrivals in deep-water environments are typically dominated by the direct wave as well as reflections, whereas refractions are hidden behind these phases (Fig. 2b). Several methods have been proposed to recover refracted signals by changing the plane of the acquisition geometry (e.g. Schuster and Zhou, 2006; Vrolijk et al., 2012; Wapenaar et al., 2014; Cho et al., 2016). The goal of wave equation datuming and, particularly, of DC, is to extrapolate the recorded data to a virtual datum surface so that refractions can be retrieved as first arrivals. The technique used here to back-propagate the recorded data follows the steps proposed by Berryhill (1979, 1984) and the scheme of McMechan (1982, 1983). This procedure is easy to implement and can deal with spatial velocity variations within the water column and irregular datum surfaces, which are often located at the seafloor (Fig. 2c).

The idea is eliminating the effect of the water layer by simulating a sea bottom acquisition (both sources and receivers). Without the water layer, the direct-wave signal and the seafloor reflections disappear, and the critical distance is reduced to zero offset. Thus, the virtual recordings contain early refractions visible as first arrivals (e.g. Fig. 2d).

In 2-D, the recorded wave field can be expressed as

$$\begin{array}{}\text{(1)}& \mathbf{S}({x}_{\mathrm{s}},{z}_{\mathrm{s}};t)\cdot \mathbf{G}({x}_{\mathrm{s}},{z}_{\mathrm{s}};{x}_{\mathrm{r}},{z}_{\mathrm{r}};t)=\mathit{u}({x}_{\mathrm{r}},{z}_{\mathrm{r}};t),\end{array}$$

where *t* represents the time, *x* and *z* the horizontal and vertical
spatial coordinates, **S** the source signal, **G** the Green's function that
defines a forward wave field extrapolation process in a heterogeneous
subsurface observed at (*x*_{r},*z*_{r}) from a source at
(*x*_{s},*z*_{s}), and ** u** is the seismic trace. In the
streamer configuration (Fig. 2a), there are two contributions of the wave
propagation through the water column that can be removed from the data by
applying wave equation datuming. Following the matrix operator notation used
by Berkhout (1981, 1997a, b), the true two
dimensional (2-D) wave field propagation from sources to receivers is also
formulated here by means of matrices but divided into three equations, each
considering a different part of the total trajectory of the wave field, as

$$\begin{array}{}\text{(2)}& {\displaystyle}{\mathbf{S}}^{\mathrm{T}}({x}_{\mathrm{s}},{z}_{\mathrm{s}};t)\cdot {\mathbf{G}}_{\mathrm{d}}({x}_{\mathrm{s}},{z}_{\mathrm{s}};{x}_{\mathrm{d}},{z}_{\mathrm{bat}};t)={\mathit{u}}_{\mathrm{d}}^{\mathrm{T}}({x}_{\mathrm{d}},{z}_{\mathrm{bat}};t),\text{(3)}& {\displaystyle}{\mathit{u}}_{\mathrm{d}}^{\mathrm{T}}({x}_{\mathrm{d}},{z}_{\mathrm{bat}};t)\cdot {\mathbf{G}}_{\mathrm{e}}({x}_{\mathrm{d}},{z}_{\mathrm{bat}};{x}_{\mathrm{u}},{z}_{\mathrm{bat}};t)={\mathit{u}}_{\mathrm{u}}^{\mathrm{T}}({x}_{\mathrm{u}},{z}_{\mathrm{bat}};t),\text{(4)}& {\displaystyle}{\mathit{u}}_{\mathrm{u}}^{\mathrm{T}}({x}_{\mathrm{u}},{z}_{\mathrm{bat}};t)\cdot {\mathbf{G}}_{\mathrm{u}}({x}_{\mathrm{u}},{z}_{\mathrm{bat}};{x}_{\mathrm{r}},{z}_{\mathrm{r}};t)={\mathit{u}}^{\mathrm{T}}({x}_{\mathrm{r}},{z}_{\mathrm{r}};t),\end{array}$$

where Eq. (2) corresponds to the downward propagation
(**G**_{d}) from the source (**S**) surface
(*x*_{s},*z*_{s}) to the seafloor
(*x*_{d},*z*_{bat}) through the water column;
Eq. (3) describes the effects of the wave propagation through the
Earth's subsurface (**G**_{e}) for an acquisition system deployed at
the seafloor from an incoming wave field at
(*x*_{d},*z*_{bat}) to an outcoming wave field at
(*x*_{u},*z*_{bat}), and Eq. (4) represents the
upward propagation (**G**_{u}) from the seafloor
(*x*_{u},*z*_{bat}) to the recording surface
(*x*_{r},*z*_{r}) through the water media; ${\mathit{u}}_{\mathrm{d}}^{\mathrm{T}}$, ${\mathit{u}}_{\mathrm{u}}^{\mathrm{T}}$, and
*u*^{T} represent the pressure field expressed at its corresponding
datum surface as a row vector, “T” being the transpose symbol. The
**G**_{d} and **G**_{u} operators are equal only if sources and
receivers are located at the same horizontal coordinates and thus the
downward and upward vertical propagation follow the same path.

Here the inverse extrapolation follows the boundary value migration (BVM) scheme of McMechan (1982, 1983). In this case, the inverse extrapolation is described by the source, wave equation and boundary conditions used. The source function through time is formed by the observed wave field, which is introduced in the 2-D model plane sequentially and backwards in time as a boundary value. The wave field transferred from the seismogram to the model at each time step is taken as an equivalent source. The complete seismogram is used; thus no preselection and arrival identification are needed. Prior to re-datuming, we (1) deleted noisy channels, (2) applied Butterworth 1/3-40/60 filtering, and (3) muted direct arrival. We did not apply amplitude corrections. The proper source spacing and optimal recording time step are the ones that avoid aliasing issues and grid dispersion and reduce the effects caused by the discrete approximation of the wave field.

The inserted wave field is propagated back in time following the acoustic wave equation solved by the forward extrapolation scheme (Dagnino et al., 2016). The following 2-D acoustic differential equation,

$$\begin{array}{}\text{(5)}& \begin{array}{rl}{\displaystyle \frac{\mathrm{1}}{\mathit{\kappa}(x,z)}}{\displaystyle \frac{{\partial}^{\mathrm{2}}\mathit{u}(x,z;t)}{\partial {t}^{\mathrm{2}}}}& +\mathrm{\nabla}\cdot \left({\displaystyle \frac{\mathrm{1}}{\mathit{\rho}(x,z)}}\mathrm{\nabla}\mathit{u}(x,z;t)\right)\\ & =s(x,z;t),\end{array}\end{array}$$

where *x* is the horizontal space coordinate, *z* the depth or vertical space
coordinate, *t* the time, ** u** the pressure wave field,

Unlike re-datuming approaches that use the Kirchhoff implementation
(Berryhill, 1979, 1984; Shtivelman and Caning, 1988; Arnulf et al., 2011), the
computation of extrapolated data at intermediate depth levels is needed.
Despite the fact that it increases the computational complexity as compared
with the Kirchhoff method (Arnulf et al., 2014), the wave equation imaging
allows including laterally and vertically variable water velocity, in
contrast with other techniques in which the water velocity is considered
constant (Cho et al., 2016) or needs a specific ray tracing
(Shtivelman and Caning, 1988). Recursive extrapolation makes that spatial velocity
variations can be handled properly. To define the 2-D water velocity model we
use in situ oceanographic measurements of water properties obtained with
expendable bathy-thermographic (XBT) probes. We have also tested the effect
of the water velocity model used to DC the shot gathers versus the travel-time
picks. In our area of study the velocity changes in the water layer are small
(<20 km s^{−1}), so considering either a
realistic or a homogeneous model has a minor effect on the first arrival
travel-time picks, smaller than the travel-time picking uncertainty (±0.035 s). Moreover, the new virtual positions do not necessarily
have to be in a flat surface or at several metres above the seafloor surface as
in other studies (e.g. Harding et al., 2016); here the new virtual surface
is the seafloor relief, itself extracted from bathymetric data. In contrast
to Harding et al. (2016), who use the DC data to perform FWI, we only use
first arrival travel times to perform TTT, so surface-related effects do not
affect the inversion results here. The main drawback of this DC strategy is
the computational cost, worse wave field reconstruction in the far offsets
caused by the truncation of the recorded data, and potential artefacts due to
propagation effects.

In the inverse extrapolation process, complex-frequency-shifted perfectly matched layers (CFS-PML) (Zhang and Shen, 2010) are imposed as absorbing boundary conditions to avoid spurious reflections in all model boundaries that may cause interference and mask the correct seismic phases. The only exception is the time-dependent boundary values associated with the equivalent sources.

In summary, the main steps of the DC approach are the following: first, we back-propagate the shot gathers through the water layer from the receiver positions to a virtual surface located at the seafloor. Then, we sort the resulting virtual seismograms into common receiver gathers. After that, we back-propagate the receiver gathers through the water layer, this time from the virtual surface to the source positions, in the opposite direction to its true trajectory. Finally, we re-sort the resultant wave field in the original shot gather domain (Fig. 2d). The rearrangement of the data is possible because of the reciprocity principle, so that if source and receiver have identical directional characteristics, then interchanging the positions of sources and receivers yields the identical seismic trace (Berryhill, 1984).

The final virtually recorded wave field with sources and receivers located at the seafloor can be expressed as

$$\begin{array}{}\text{(6)}& {\mathbf{S}}^{\mathrm{T}}({x}_{\mathrm{s}},{z}_{\mathrm{s}};t)\cdot {\mathbf{G}}_{\mathrm{e}}({x}_{\mathrm{s}},{z}_{\mathrm{bat}};{x}_{\mathrm{r}},{z}_{\mathrm{bat}};t)={\mathit{u}}_{r}^{\mathrm{T}}({x}_{\mathrm{r}},{z}_{\mathrm{bat}};t).\end{array}$$

Because the re-datumed wave field (*u*_{r}) at
(*x*_{r},*z*_{bat}) is built by using finite, discrete, and
single-sided recordings, it is not the pressure wave field that would be
recorded by performing the experiment with the virtual geometry set-up.
Moreover, the energy coming from the direct and surface waves is not
included. In general, the amplitude of the resulting wave field is reduced due
to the energy loss through the absorbing model boundaries during the inverse
extrapolation process. Therefore, amplitude is not preserved as in other
re-datuming approaches (Schuster and Zhou, 2006) and its attenuation factors are
related to geometrical spreading and transmission losses. As amplitudes are
affected by the re-datuming process, it is not straightforward performing FWI
using DC data. However, arrival times of the different phases are correct
(which is our main objective) if the *V*_{P} model of the water
layer is accurate enough. A better approximation of the true wave field, and
thus a better result, would be achieved using denser and larger arrays. For
sparse data sets the re-datumed arrivals would be less focused due to a larger
amplitude loss and noise effects, but it would nevertheless be correctly located.

The accuracy of the resulting wave field also depends on the precision of the forward solver. The numerical algorithm (Dagnino et al., 2016) uses a Runge–Kutta method of fourth order in time (Lambert, 1991) and a sixth-order approximation in space discretized with a staggered grid (Virieux, 1986). It employs the extrapolated output as input for the next extrapolation step (recursively) plus four weighted average increments.

Another effect that can affect the result is the presence of artefacts such as diffraction tails located at the edges of the array and therefore where the observed wavefronts are truncated. These artefacts are due to the finite aperture of the recording system. Their effects are larger for small aperture set-ups, where they can interfere with the energy focusing in the new datum surface and mask the true arrivals. Moreover, the finite aperture of the acquisition array causes a non-uniform illumination of the wavefronts, which results in energy mitigation near the recording limits. Finally, the resulting extrapolated wave field is also influenced by numerical dispersion, which attenuates high frequencies as it travels backwards in time through the medium. Thus, the spectrum of the frequency bandwidth of the data is reduced due to the Earth effect that acts as a filter (Berkhout, 1997a, b).

The goal of this step is to obtain a reference *V*_{P} model by TTT
of some predefined seismic phases; in particular, we combine first arrivals
identified at the virtual DC recordings with reflections recorded for the
original MCS data. Therefore, the data input file contains both types of
travel-time picks, each with their corresponding source–receiver positions.
In the case of DC arrivals, the 2-D coordinates will be the ones in the
virtual surface, but for the reflections they will be at the actual location
of the air gun source and of the individual streamer channels. Further
details for our particular study area and experiment are presented in
Sect. 4.2. As a rule of thumb, it is considered that *V*_{P} models
obtained by TTT are correct to a depth that is about a half of the maximum
experimental offset.

We use a version of the TOMO2D code (Korenaga et al., 2000) modified by
Begovič et al. (2017) to enable locating both sources and receivers within the
water layer so as to invert streamer data. A two-dimensional *V*_{P}
model is parameterized in our study as a regular mesh hanging from the
seafloor, i.e. the model is limited to the subsurface medium and does not
include the water layer. Propagation velocity in the water layer is fixed to
1.5 km s^{−1}. Arrival time differences using a realistic water
*V*_{P} model are much smaller than the expected data misfit,
typically several tens of milliseconds. A 1-D floating reflector is defined
along the profile. This means that it is parameterized as a node array that is
independent from the velocity mesh, and both reflector array and velocity
mesh can be spatially variable. Travel times for given *V*_{P} and
reflector depth models are computed using a ray tracing strategy that
combines the graph method (Moser, 1991) with a subsequent ray bending
refinement (Moser et al., 1992).

The partial derivatives of travel-time residuals with respect to velocity
(for first arrivals and reflections) and reflector depth (only for
reflections) are introduced in the Fréchet matrix with additional
regularization constraints (smoothing, damping). Thus, this interface depth
model is inverted simultaneously with the *V*_{P} model. The inverse
problem to iteratively update the *V*_{P} and depth models is solved
using the sparse matrix solver of the LSQR (least square root) algorithm of Paige and Saunders (1982).
The difference between observed and synthetic travel times is iteratively
minimized as a least-squares problem until a stopping criteria is fulfilled.
Regularization constraints (smoothing and damping) are applied to stabilize
the minimization process avoiding singularity of the matrix. For further
details, see Korenaga et al. (2000) and Meléndez et al. (2015).

The last inversion step of the proposed approach is using the
*V*_{P} model obtained by TTT as initial one for adjoint-state FWI.
In this case, we use the original seismograms recorded by the MCS system as
input to refine the *V*_{P} model. As in the case of the DC, the FWI
code uses the acoustic domain FD solver of Dagnino et al. (2014, 2016)
for the forward- and back-propagation of the wave equation.

For the forward problem, we consider the 2-D acoustic wave equation in the time domain expressed in the form that appears at Eq. (5) to simulate the synthetic pressure wave field. The code includes CFS-PML (Zhang and Shen, 2010) as boundary conditions, and we use 25 layers on the left, bottom, and right boundaries to eliminate numerical reflections and a free surface at the top of the model that represents the water–air discontinuity. The optimal temporal step to solve the forward problem fulfils the usual Courant–Friedrich–Lewy stability condition to avoid instabilities in the PML (Dagnino et al., 2014). To estimate the source for the first iteration we use a Ricker wavelet with central frequency of 20 Hz. Then the source wavelet is inverted at each iteration as proposed by Pratt (1999) prior to velocity inversion, so the updated source signature is used in the subsequent iterations (Dagnino et al., 2016). The source signature is calibrated and inverted for all the individual shots up to 0.5 km offset. The initial (T/6) and final (6T) oscillations, where T is the period of the source signal, are removed to avoid the introduction of noise in the synthetic data.

The goal of the inversion is to find the parameters *m*, in our case
*V*_{P}, that minimize the discrepancy or misfit between some
predefined waveform attributes of the observed (*u*^{o}) and
synthetic (*u*^{s}) seismic traces at each iteration
(Jiménez Tejero et al., 2015). We use the classical L2-waveform objective
function (*χ*) that is built as the least-squares norm of the
corresponding misfit (Tarantola, 1987) for a source as follows:

$$\begin{array}{}\text{(7)}& \mathit{\chi}=\sum _{\mathrm{r},\mathrm{t}}{\displaystyle \frac{{\left({\mathit{u}}^{\mathrm{s}}({x}_{\mathrm{r}},{z}_{\mathrm{r}};t)-{c}_{\mathrm{r}}{\mathit{u}}^{\mathrm{o}}({x}_{\mathrm{r}},{z}_{\mathrm{r}};t)\right)}^{\mathrm{2}}}{\mathrm{2}}},\end{array}$$

where *x*_{r} is the receiver position, *t* the time, and
*c*_{r} the calibration term. The calibration term is calculated
independently for each source–receiver as in Dagnino et al. (2016), but fixing
the sea bottom reflection instead of the direct water wave. This term is
useful to correct the source signature changes due to an irregular bathymetry
and the inhomogeneous source directivity. The wave field of each shot gather is
resampled to fit on the finite difference grid. The synthetic data for the
misfit calculation is simulated solving the forward problem previously
explained in a reference model and its updates.

The acoustic 2-D forward modelling code compensates for the actual 3-D amplitude
decay of the field data multiplying the field data by $\sqrt{t}$
(Hicks and Pratt, 2001; Dagnino et al., 2016) before the calculation of the residuals.
Moreover, the energy of the shots is normalized dividing it by the total
source energy to overcome problems due to elastic effects. The density model
of the subsurface used for the inversion also affects the reflectivity part
of the data set, so it is important to interpret the reflectors not only as
velocity contrasts. In our case, it is updated after the inversion of each
frequency band using the *V*_{P} model obtained at the previous
iteration and the Nafe–Drake (Ludwig) experimental relationship
(Mavko et al., 1998).

The following step consists of updating the model in the direction where the misfit decreases. For this purpose, we calculate the gradient of the misfit following the adjoint-state method proposed by Lailly (1983) and Tarantola (1984). The adjoint field is obtained by back-propagating the residuals in time for each receiver position. The adjoint source is calculated as $\mathit{f}(x,z;t)=-{\mathrm{\nabla}}_{{u}^{s}}\mathit{\chi}$.

The gradient of the objective function with respect to the model,
∇_{κ}*χ*, is computed by convolution of the forward-propagated
wave field of the source term and the adjoint or back-propagated wave field of
the residuals from the receiver location (Tarantola, 1984).

$$\begin{array}{}\text{(8)}& {\mathrm{\nabla}}_{\mathit{\kappa}}\mathit{\chi}=\sum _{\mathrm{s}}{\displaystyle \frac{\mathrm{1}}{\mathit{\kappa}(x,z)}}\underset{t}{\int}{\displaystyle \frac{\partial \mathit{u}(x,z)}{\partial t}}{\displaystyle \frac{\partial \mathit{f}(x,z)}{\partial t}}\end{array}$$

Because of non-linearity, the gradient can point toward a local minimum. In the FWI code, a gradient-based preconditioning concentrates model updates in the regions where the gradient is more reliable. Moreover, it also adds a weighting to compensate for the amplitude decay of the signal away from the source. Furthermore, the gradient is balanced to reduce its sensitivity to the presence of “spikes” with a smoothing regularization based on a 2-D low-pass zero-phase Butterworth filter. Details on the gradient preconditioning techniques can be found in Dagnino et al. (2014, 2016).

We use the steepest descendent (SD) approach in the optimization problem
because it has particularly low sensitivity to noise. The step used to find
the direction in the model space where the misfit function locally decreases,
is just the opposite of the gradient in the SD approach, ${\mathit{p}}_{i}=-\mathrm{\nabla}{\mathit{\chi}}_{i}$. In the FWI algorithm, we use a normalized search direction. Then,
the optimal step length (*α*_{i}) is determined by imposing ${\mathit{\alpha}}_{i}={min}_{c>\mathrm{0}}\mathit{\chi}({\mathit{m}}_{i}+c{\mathit{p}}_{i})$. The final step is obtained after acquiring the
minimum of a polynomial approximation over three steps calculating the
search direction of the misfit function, *p*_{i}. Then, the model can be
updated as ${\mathit{m}}_{i+\mathrm{1}}={\mathit{m}}_{i}+{\mathit{\alpha}}_{i}{\mathit{p}}_{i}$.

A multi-scale approach where frequency bands are inverted sequentially, from low to high frequencies (Bunks et al., 1995), is applied to reduce the risk of falling into a local minimum and to add details to the model progressively. In particular a low-pass Butterworth filter is applied to the data, which allows us to solve different frequency bands of the same data at each iteration, restricting the inversion to a limited bandwidth. Implementing this strategy mitigates the non-linearity because the objective function is smoothed when the data are filtered. High wavenumbers are then incorporated sequentially into the model's successive iterations. The maximum number of iterations per frequency step is fixed to 10. An additional stopping criterion that we use in the inversion is the Arminjo rule (Nocedal and Stephen, 2006) when changes are <0.01.

We apply a 2-D Kirchhoff depth migration to image the subsurface structures
directly at depth by using the *V*_{P} models obtained and the high-frequency content of the data set. The migration module applied is part of
the Seismic Unix platform. We use the sukdmig2d command for the data
migration. The travel timetable required for the depth migration is obtained
with the rayt2d command, which calculates the 2-D ray tracing along the
*V*_{P} model.

3 Study area and data set

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The field data used to test the described approach correspond to an experiment conducted in the Alboran Sea, a complex basin located in the westernmost Mediterranean (Fig. 3) (Booth-Rea et al., 2007, 2018). The target of this survey was to image the structure and properties of the sediment units and the nature of the uppermost basement.

The seismic records used in this study were collected in 2011 on board
Spanish R/V *Sarmiento de Gamboa*, as part of the TOPOMED cruise. We
select an 80 km long section of the TM28 profile that crosses the
central and deepest part of the basin, across a volcanic arc
(Gómez de la Peña et al., 2018), to test the approach. The input data consist of
series of seismograms (traces) recorded with a 6 km long streamer
(Fig. 4). The streamer has 480 channels with a group interval of
12.5 m. The length of the trace record used is 8 s and the
time sampling is 2 ms. In total, we use 1517 air gun shots with an
average spacing of 50 m. The source power was 4600 cubic inches, producing a
central frequency of ∼20 Hz. The system was towed at a depth of
10 m, and the nearest-offset distance was 203.7 m. The early
arrivals in the streamer recordings are dominated by the shallow
near-vertical sediment reflections, as can be seen in Fig. 4. Hence, data
preconditioning is essential to identify first arrivals and make the data
set appropriate for TTT.

4 Results

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Here we show the results after applying each step of the modelling strategy
described in the previous sections (see Fig. 1). We work under the premise
that we have no a priori information on the structure and properties of the
subsurface, so the goal is to recover all the possible information on the
*V*_{P} model from the MCS data alone. Before starting our
processing and modelling workflow, we apply a 2-D band-pass minimum-phase
Butterworth filter to the data set for swell noise removal. The low cut and
high cut of the band-pass filter are 2 and 60 Hz respectively. The
DC result of the streamer recordings is first shown to make the early
refractions visible as first arrivals. Then, a macro-velocity model is
obtained by joint refraction and reflection TTT, which is then used as
initial model to perform FWI starting at ∼6 Hz, the lowest
signal frequency available in data set. Afterwards, we perform a PSDM of the
recorded data using the 2-D *V*_{P} models obtained by TTT and by
FWI.

We compute the data re-datuming by wave equation DC of the streamer
recordings to the seafloor surface. Figure 5 shows the result after the first
step of the DC for three shot gathers derived using the full wave field of each
shot and a heterogeneous water layer model built as explained in the
Methodology section. The grid size in both the *x* and *z* direction is
12.5 m. We image the wavefronts of all the seismic events recorded in
the streamer traces reconstructed by the solver after its back-propagation to
the virtual receiver positions in the correct reversed time (i.e. the
OBS (ocean bottom seismometer)-type acquisition setting). In this way, we recover several first arrivals
that were obscured in the original recordings. However, the main first
arrival along the recordings is still the seafloor reflection.

In the second step, we re-sort the virtual OBS-type wave field in receiver gathers, and then, the data from all shot gathers are combined to construct the final virtual shot gather. Figure 6 shows the final result for the three shot gathers also shown in Fig. 5. The DC has collapsed the seafloor reflection towards a single point, so refractions from shallow subsurface can now be identified and tracked as first arrivals from zero offset. First arrivals are more difficult to identify at long offsets because of amplitude attenuation and the appearance of diffraction tails. Lower panels (d–f) in Fig. 6 show the first arrivals used as input for the TTT (blue dots).

The inversion parameters used in the TTT are shown in Table 1. We do not use
the entire data set to reduce computational burden, but all receivers are used
to ensure data redundancy. The selected reflection corresponds to the top of
the basement (TOB), which is a clear event (e.g. see Fig. 9). Reflection
travel times are picked from MCS common midpoint gathers where it is a
bright continuous reflection, which is also interpreted at the same time in a
fully processed stacked image (Fig. S1 in the Supplement). The source and
receiver positions were projected to a straight line defined between first
and last shots, preserving their offset distance. Correlation lengths for the
velocity model are set at the top and bottom nodes of the grid and are
linearly interpolated for the intermediate nodes. The velocity gradient is
stronger in the vertical direction, so the vertical correlation lengths are
selected to be shorter than the ones defined in the horizontal direction. Our
inversion process follows the layer-stripping strategy as described in
Meléndez et al. (2015). In the first inversion step, the initial TOB depth
model is a flat boundary located at 2.25 km depth, and the initial
*V*_{P} model follows the linear function of depth from the seafloor
${V}_{\mathrm{P}}\left(z\right)\phantom{\rule{0.125em}{0ex}}\left[\mathrm{km}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}\right]=\mathrm{1.61}+\mathrm{0.72}\cdot z\phantom{\rule{0.125em}{0ex}}\left[\mathrm{km}\right]$,
increasing from 1.61 km s^{−1} at the seafloor to
4.13 km s^{−1} at a depth of 3.5 km below the seafloor
(Fig. 7a). In the second inversion step we use the inverted TOB from the
first step (Fig. 7b) as initial TOB depth model and the same data set and
inversion parameters. Regarding the *V*_{P} model, we build a new
initial model that is equal to the *V*_{P} model above the TOB
obtained in the first step, whereas below it, *V*_{P} is defined as
${V}_{\mathrm{P}}\left(z\right)={V}_{\mathrm{TOB}}+(z-{z}_{\mathrm{TOB}})\cdot \frac{({V}_{{z}_{\mathrm{f}}}-{V}_{\mathrm{TOB}})}{({z}_{\mathrm{f}}-{z}_{\mathrm{TOB}})}$, varying from 3.6 km s^{−1} at the TOB
interface (*V*_{TOB}) to 6 km s^{−1} (${V}_{{z}_{\mathrm{f}}}$)
at a depth of 8.6 km below the seafloor (*z*_{f}) (Fig. 7c).
The goal of the layer-stripping strategy is to recover the sharp velocity
contrast at the sediment–basement reflecting boundary, which might otherwise
appear as a smooth velocity gradient, contributing also to the improvement of
the deepest part of the model.

The final macro-velocity model is presented in Fig. 7d. The joint refraction
and reflection TTT allows recovering the long-wavelength geometry of the
sharp sediment–basement boundary. The ray coverage of the model inversion is
quantified by the derivative weight sum (DWS) (Toomey et al., 1994) (Fig. 7),
and it is influenced by the geometry of the experiment and the subsurface
velocity distribution. Thus, ray coverage decreases to the edges of the model
and with depth, and it is denser beneath the source locations. The model is
generally well constrained in the central part that is covered by both
refractions and reflections, whereas the lateral areas mapped only by
reflections are subject to a higher degree of velocity–depth ambiguity. In
the central area, the overall trend is an increase in velocity with depth,
from ∼1.6 km s^{−1} at the seafloor to ∼4.0 km s^{−1} at the bottom. The results indicate that a high-velocity anomaly (>3.5 km s^{−1}) is located at ∼115 km distance and ∼3 km depth. Slower velocities are
obtained at shallower depths below the TOB. The velocity contrast that
accurately follows the geometry of the TOB is delimiting steeply dipping
discontinuities. As an example, we show that on the left-hand side of the
profile in Fig. 7d the basement is located just below the seafloor (∼200 m with respect to the seafloor), whereas ∼15 km further
along the profile the basement position is at a depth of ∼3.5 km. On the other side, the geometry of the TOB is gently wavy
around an average depth of 2 km. The TOB boundary marks strong
velocity changes with a velocity jump of ∼0.5 km s^{−1}
between sediment and basement velocities (>2.7 km s^{−1}).

Figure 8 shows histograms of travel-time residuals for three data groups
after the first and last iterations of the two inversion steps of the
layer-stripping strategy. The distributions of data misfit with the initial
model (red lines) are asymmetric and wide for both refractions and
reflections. After the first inversion step (orange lines), the distributions
are symmetric and narrower with the highest pick around zero. At the initial
stage of the second inversion step of the layer stripping (green lines), an
increase in positive travel-time residuals is shown with respect to the
distribution after the first step (orange lines) due to the insertion of
higher velocities below the TOB to recover the sharp velocity contrast. Thus,
the distributions (green lines) are slightly asymmetric and wider than after
the first step of the layer stripping (orange lines). The final distributions
of travel-time residuals (blue lines) are narrower than the previous ones and
approach to a Gaussian centred at zero with the largest counts. This fact
evidences the improvement of the velocity model, and therefore the
corresponding travel-time fitting during the inversion. Lower panels (d–f)
in Fig. 6 show the first arrivals for three DC shot gathers used as input for the
TTT (blue dots) and the first arrivals recovered after TTT (red squares). The
final velocity model in Fig. 7d shows a good convergence with root mean
square (rms) residual travel times of ∼20 ms, which is of the
order of the picking uncertainty (*χ*^{2}≈1). In contrast, the
initial rms before the inversion (Fig. 7a) is ∼420 ms.

Moreover, to assess the reliability of the inversion results, we convert the
velocity and TOB models to two-way time and superimpose the results on top of
the time-migrated image (TMI) (Fig. 9). The figure shows that the TTT
*V*_{P} and TOB models and the MCS image are both consistent, since
both the velocity contrasts and the depth of the TOB are located in the same
two-way time where the TMI displays important reflectivity changes and the
TOB discontinuity. The good match validates our travel-time picks, given that
neither the interpolation of the reflected picks or the inclusion of the
far-offset first arrival DC travel times appear to introduce substantial
artefacts or errors in the model. However, the TTT model lacks resolution
compared to the time-migrated MCS image. It does not reproduce the sharp
geometry and changes in amplitude of the anomalies in detail, such as the highs
and lows of the TOB, the faults at the flanks, or the sediment layering seen
in the TMI.

After TTT, we perform FWI of the original streamer shot gathers starting from
a smoothed version of the TTT *V*_{P} model of Fig. 7d. A total of
1512 shot gathers (>720 000 seismograms) are used for the inversion. The
length of the traces is 4.6 s. In order to reduce computational
costs, we invert sequentially four different sets of 378 shot gathers for each
frequency band. Source and receiver positions are also projected to a
straight line defined between first and last shots, preserving their offset
distance. As in TTT, no decimation is applied to receiver sampling.

We apply a data preconditioning to emphasize reflections between the first
arrival, typically the seafloor reflection, and its first multiple, using data windowing. We define a time window centred at the seafloor reflection.
We identify this phase by using a maximum kurtosis and k statistics criterion
(Saragiotis et al., 2002). We calculate the travel time of the seafloor
reflection approximately using a water velocity of 1.5 km s^{−1},
the seafloor depth, and the source–receiver offset distance. When the first
arrival is not found, for example due to a noisy channel, or the observed and
simulated seismograms are cycle-skipped for the inverted frequency, then the
trace is eliminated. The trace value is set to zero before the first arrival
travel time, and after that it is balanced by a function defined as
$\sqrt{t}$ to compensate for the amplitude decrease at depth. Finally, the trace
is also set to zero after the travel time of the first multiple. Therefore,
the gradient calculation focuses on information of the data
set coming from the near-vertical reflections. Additionally, when the
absolute difference between the maximum amplitudes of a field and its
corresponding synthetic traces is higher than 3 times the maximum
amplitude of the synthetic trace, it is considered to be a noisy channel and
the trace is eliminated. Another issue to consider when comparing synthetic
and field data is having a high signal-to-noise ratio (SNR), so each trace is
stacked with its two neighbours at frequencies lower than 8 Hz.

We start the multi-scale FWI at 6 Hz. The maximum frequency inverted
is 16 Hz. A frequency step of 0.5 Hz is applied at the
beginning and 1 Hz at the final stages of the multi-scaling
inversion. A total of 14 different frequencies in the multi-scaling strategy
are inverted. The size of the space grid for the inversion ($\mathrm{d}x<\frac{{\mathit{\lambda}}_{\mathrm{inv}}}{\mathrm{5}}$) depends on the frequency step and the
minimum velocity of the model, being smaller at high frequencies. The minimum
and maximum velocities were constrained as gradient preconditioning and set
at 1.4 and 5.0 km s^{−1}.

Figure 10 displays the final FWI velocity model. This model has higher
resolution than the TTT one (Fig. 7d), showing velocity contrasts of
intermediate and short wavelength and a number of geologically meaningful
structures that cannot be identified in the TTT model. The improvement due to
the higher inverted frequencies is clearer in the shallow part of the model.
The velocity contour of ∼3.25 km s^{−1}, which corresponds to the
sediment–basement boundary, is better focused and detailed highs and lows are
recovered. Thus, in the right-hand side of the profile the velocity
variations reproduce the TOB geometry accurately. Dipping low-velocity
features are shown with high resolution. The velocity differences between
these anomalies and the surrounding media are 0.5–1.0 km s^{−1}
at ∼2.25 km of depth between 137 and 165 km along the
profile. The sedimentary package shows velocities from ∼1.7 km s^{−1} near the seafloor increasing to ∼3.0 km s^{−1} just above the basement. The blue colour found in the
deeper parts of the model corresponds to high velocities of more
consolidated rocks. However, there are areas where the high velocities are
shallower than in others, due to the action of normal faults, as in the left
edge of the model.

A basin from 95 to 137 km distance is displayed between two
sloping faults. In addition, a 200–300 m thick, ∼3 km s^{−1} high-velocity layer ∼25 km long
embedded within the sedimentary package is shown up to a volcano-like
structure at ∼2.6 km depth. The key point is that all these
features have been imaged with a streamer of only 6 km thanks to
the workflow followed here.

To show the model improvement in the data domain, in Fig. 11 we compare a recorded shot gather (Fig. 11c) and a synthetic one generated with the FD solver (Dagnino et al., 2014) using the initial TTT (Fig. 11a) and final FWI (Fig. 11b) velocity models. In contrast with the synthetic data generated with the TTT velocity model (Fig. 11a), the shot gather generated with the FWI velocity model (Fig. 11b) presents some near-vertical reflections. Thus, the seismogram simulated with the final model shows a larger number of seismic events compared to the initial one, which only recovered the first arrival phases and the TOB reflection from the TTT. The data-driven preconditioning of the FWI strategy targeted the energy corresponding to the near-vertical reflections. Therefore, in that region final residuals (Fig. 11e) are smaller than initial ones (Fig. 11d) but are larger around the seafloor reflection. Again, in Fig. 11f aside from wave amplitudes, the main difference between the data generated with the TTT model (orange line) and the target (black line) seismogram is the presence of reflected waves. We observe a better fit between the final (blue line) and real data set (black line), except for the effects that are not modelled, such as 3-D diffractions or the arrivals not included in the data-based preconditioning.

The misfit reduction for three steps of the multi-scale strategy is shown in Fig. 12. Normalized least-squared misfit is reduced after the whole inversion, and the final residual tends to zero typically after five iterations.

As in the case of the TTT model, we have converted the FWI *V*_{P}
model to two-way time and we have superimposed it to the TMI (Fig. 13).
Velocity contrasts present a remarkable match with the reflectors of high
amplitude of the TMI. The geometry of the TOB interface reflector, for
example, is recovered with great detail. Here, velocity differences are of
short wavelength, so many of the details that were previously not displayed
can now be properly identified (Fig. 15a).

The seismic data used for the PSDM are not the same as for FWI but data that have been processed to attenuate coherent and incoherent noise. Streamer field data were processed using a Wiener filter and a surface-consistent deconvolution to increase the vertical resolution attenuating the ringing of the source. Data were sorted in the CDP (common depth point) domain. An amplitude balance (quality factor of 100) was applied to recover the energy lost by geometrical spreading.

We interpolate the FWI result to a 3.125 m grid interval in the
horizontal and 3 m in the vertical for the ray tracing. The receivers
were spaced at 12.5 m, so the horizontal midpoint distance was
6.25 m for the migration. First, we perform a 2-D Kirchhoff depth
migration using the processed data set and the TTT velocity model (Fig. 14a).
The same PSDM image is also shown in Fig. 14b superimposed with the TTT
velocity model, providing additional information on the rock properties. The
resolution of the velocity models obtained through TTT is similar to the
typical velocity models built for PSDM, although the latter are based
fundamentally on reflections. Thus, the PSDM result shown in Fig. 14 should
be comparable to the one that would be obtained with conventional velocity
analysis and PSDM. Although the main features are imaged, i.e. the basin
geometry, some structures and interfaces are not as well defined as when the
PSDM is performed using a high-resolution *V*_{P} model, as is
shown below.

Finally, we repeat the PSDM but using the final FWI *V*_{P}
model instead, obtained after the processing–modelling sequence proposed in this work
(Fig. 15b and c). An accurate image of the real structure of the shallow
subsurface is displayed directly at depth in Fig. 15b. The sediment layer is
thicker on the left side of the profile (∼1.5 km), where we
find the basin, than on the right (∼0.5 km), where the basement
gets shallower. As can be seen in both the FWI velocity model as well as
the PSDM image, the basement is severely folded and cut by clear fault
structures along the whole profile. The strong lateral velocity changes
produced by the faults have caused some migration smiles at the TOB. The
deepest part and edges of the profile where FWI and TTT have the largest
uncertainties have likewise the largest uncertainty in the PSDM result. On
the whole, the vertical section in Fig. 15b shows the correct geometry of the
structures at depth with a high resolution, clearer than in Fig. 14 when a
TTT model was used.
As an example, the depth and location of the TOB boundary is now
comparatively more precise.

A good fit is obtained between the shape of the geological features and the isovelocity contours in Fig. 15c. The volcano-like structure and dipping faults, coincide with well-defined velocity anomalies. Furthermore, a high-velocity layer, which was not visible with TTT in Fig. 14b, is also clearly imaged within the sediment package. The combined interpretation helps us to better define the geological structures and to obtain a proper characterization of the nature of the features.

5 Discussion

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Information on the structure and properties of the subsurface in the Alboran
margin is retrieved by applying a combination of TTT and FWI, particularly
the *V*_{P} of the sedimentary basin and the geometry of the TOB
discontinuity. The area has a heterogeneous vertical and horizontal velocity
gradient, with a *V*_{P} distribution showing velocity contrasts
delineating boundaries of a geologically complex area.

Using the TTT information alone, a coarse *V*_{P} model is obtained
from no a priori information. No refractions are observed as first arrivals
in the original data, so the shot gather recordings are first re-datumed to
the seafloor by wave equation downward continuation. We simulate a virtual
acquisition system with both sources and receivers located at the seafloor
for the TTT, so that refractions can be tracked as first arrivals from zero
offset. These near-offset crustal refractions provide excellent structural
detail just in the upper portion of our model. But by improving the velocity
detail in the upper region of the model (Fig. 7b), we obtain better
resolution of the deeper structure (as shown in the second TTT inversion,
Fig. 7d). In our case, some deep regions of the model are fitted only by the
TOB reflection. Under these circumstances, the inversion result suffers from
velocity–depth ambiguity. Streamer TTT exploits the dense and even spatial
distribution of MCS data and is further improved upon by DC, which allows
for the inclusion of shallower refractions that were only previously
obtainable using either seafloor receivers and sources (Henig, 2013). It
is important to notice that these highlighted phases would correspond to
ray paths in the upper region of the subseafloor section. Moreover, Fig. 9
validates the TTT model obtained, and thereby the travel-time picks used because of the matching of TWT (two-way time)-converted velocity and TOB models with the
TMI.

While some studies use waveform modelling in downward continued data (Qin and Singh, 2018), here only the travel-time information is used for inversion. The wave field during the back-propagation is altered due to several factors. The wave field at the virtual positions obtained is affected by grid dispersion and clearly shows amplitude attenuation. Moreover, the field data used to reconstruct the virtual seismogram are a discrete single-sided recording, and, therefore, part of the signal is missing during the back-propagation, hindering the complete reproduction of the wave field. This justifies using travel-time information from the DC shot gathers instead of the whole waveform.

The data re-datuming allows recovering and identifying the refracted phases
as first arrivals, but not all the energy collapses at its corresponding
point, making the first arrival picking even more difficult (see Fig. 6). To
check the first arrival travel-time errors introduced by the DC procedure we
have reproduced the re-datuming processing but using synthetic shot gathers
simulated with the streamer acquisition geometry and the final FWI
*V*_{P} model (Fig. 10). As is shown in Fig. 16a–c, the first
arrivals identified in the real DC shot gathers (blue dots) coincide
remarkably well for all offsets with the first arrival travel times of the
final TTT *V*_{P} model (red squares) and also with the ones in the
DC wave fields that are obtained using the synthetic data. The similar results
for the first arrivals justify the inclusion of the DC travel times even for
large offsets.

The target of applying FWI is to obtain a high-resolution model of the
properties of the subsurface. However, due to the non-linear behaviour
typical FWI applications use data with low-frequency content or good
kinematical starting models from a priori information
(e.g. Sirgue and Pratt, 2004; Brossier et al., 2009, 2014; Morgan et al., 2013, 2016). Here, we have implemented and tested a methodological solution
(Fig. 1) to overcome this challenge. In our case, the TTT model (Fig. 7d) is
essential to perform FWI from data which lack of low frequencies
(<6 Hz). As stated in the Introduction, TTT is a very robust technique
and commonly used to build good background initial models to FWI
(e.g. Shipp and Singh, 2002; Dagnino et al., 2014; Qin and Singh, 2017, 2018). The TTT result is
improved here during the FWI by introducing smaller wavenumbers into the
*V*_{P} model especially from near-vertical reflections (Fig. 11).
Despite the higher level of detail, not all the observed signal is matched by
the acoustic approach of FWI (Fig. 11). Possible mismatches most likely
originate from the presence of noise, elastic (Warner et al., 2013; Marjanovic et al., 2019), anisotropy, or attenuation effects, which were not taken into
account. The seafloor reflection is one example of a discontinuity that is
hard to fit to the acoustic formulation of the wave propagation.

In both TTT and FWI *V*_{P} models, we clearly identify the TOB as an
abrupt velocity change. However, the shape of this discontinuity and its
velocity contrast differs between the two models in some areas. The FWI
result shows sharper boundaries with pronounced dipping contacts at ∼137–167 km along the profile, which are not displayed in the model
obtained by TTT. Those dipping contacts, which may represent fault planes in
some cases, are of short wavelength and separate zones of abrupt velocity
changes that might be associated with lithological variations. This difference
becomes more evident at ∼115 km along the profile, where a
volcano-like structure is imaged. This feature might be a volcano due to the
location of the profile, which crosses an area where there is supposed to be
a volcanic arc. Moreover, the geometry of the volcano-like structure and
irregular TOB boundary is only clearly imaged at depth when the PSDM is
performed with the high-resolution *V*_{P} model provided by FWI
(compare Figs. 15 and 14). In addition, the high-velocity layer within
sediments on top of this structure is only clearly defined in the FWI
*V*_{P} model (Fig. 10). This layer is likely to be an evaporite
layer deposited during the Messinian crisis.

To assess the match of the velocity model to the seismic boundaries obtained
with more traditional processing and imaging, the *V*_{P} model is
converted to two-way time and compared with the TMI. As state above, the
two-way time-transformed velocity model has an excellent match with MCS time
migration; velocity changes nicely follow major reflectivity contrasts and
fault locations (Fig. 13), which further support the quality of the inversion
result and workflow.

6 Geological structure

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The seismic line was collected to image the structure of the eastern Alboran Basin in a region where basement is interpreted to be made up of magmatic arc rocks (Booth-Rea et al., 2007, 2018; Gómez de la Peña et al., 2018). The time migration and PSDM show a sediment basin with basement highs on either end of the image. The large basement highs in the flank display some indications of extensional faulting and block tilting, particularly in the NE. These faults worked during the formation of the basin and may have some syntectonic sediment tilted in between rotated fault blocks, but the faults were not active during the deposition of most of the sediment cover.

The basement top under the basin and also along the southern ridge displays several small-scale highs with steep flanks and a triangular shape with fairly symmetric flanks that do not support tectonic rotation and may indicate small-scale volcanoes, so that the image supports the basement formation by the interplay between magmatic and extensional processes as expected in a magmatic arc.

The sediment cover displays several units bound by unconformities and cut by
small sub-vertical, currently active faults. Based on regional geology, we
infer that most of the sediment sequence is possibly Pliocene but the oldest
layers may be late Miocene, although no drill hole in the vicinity provides a
calibration. The seismic velocity distribution in the basin infill may
provide some further clues regarding the age and nature of the sediment. The
*V*_{P} model of the largest basin shows a gradual increase in
velocity with depth to *V*_{P} ∼3 km s^{−1} (change from reddish to orange colour in Fig. 15a, c) that changes abruptly by ∼0.5 km s^{−1} in a slower-velocity body (change from orange to reddish
in Fig. 15a, c). This high- to low-velocity change is marked by a high-amplitude reflection that marks an unconformity – possibly erosional – with
the underlying unit. The high velocity is anomalous because clastic material
– turbidities in most of the basin – typically increases in *V*_{P}
with a compaction-driven decrease in porosity. The high-velocity layer may
indicate the presence of evaporites from the end of the Messinian period characterized
across much of the Mediterranean by a relatively large drop-down of the water
level and the deposition of oversaturated deposits. In the deep basin further to the
east there are considerably thick salt deposits that can be easily identified in
seismic reflection images due to the mobility and formation of diapiric
structures (Booth-Rea et al., 2007). Here, in the region of this study there are no
known salt deposits, and the presence of Messinian deposits without drilling
information is unconstrained. A potential interpretation is that the high-velocity layer represents an anhydrite-rich layer that has relatively high
velocities (like salt layers), but it is not as mobile.

7 Conclusions

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We show that the proposed workflow, including DC of MCS data to the seafloor,
joint refraction, and reflection TTT and FWI, allows obtaining a detailed,
geologically meaningful *V*_{P} model from a non-optimal data set.
Although the streamer length is only 6 km, the water depth is fairly great (∼2 km), and seismic records lack low frequencies
(<6 Hz), the inversion converged, successfully providing a detailed
velocity model to a depth of 3–4 km. The excellent match between the
two-way time-transformed velocity model and the time-migrated MCS image
validates the results. The PSDM images together with the *V*_{P}
model reproduce the structures and properties of the sedimentary layers and
the uppermost part of the basement with high resolution directly in the depth
domain. Aside from the irregular TOB, a volcano-like structure and steeply
dipping anomalies at the flanks of the basin that may correspond to faults
are clearly imaged. Moreover, the model allows identifying the location,
geometry, thickness, and velocity of a high-velocity layer (possibly
evaporites on top of the Messinian unconformity) embedded within sediments.
We conclude that the proposed method is valid and can be applied under these
circumstances. Applying it to longer streamer data should improve
*V*_{P} models and allow extending the models to greater depths.
Here, we recover all the possible subsurface information from the MCS
recordings alone. Future research can be directed to apply this strategy to
longer-offset data from wide-angle acquisition geometries to increase the
coverage and to better constrain the model properties.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/se-10-1833-2019-supplement.

Author contributions

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Author contributions.

CG performed the DC, modelled the seismic data, and led the writing of the text. DD performed the FWI code and helped with the assessment of the inversion parameters, together with CEJT. AM helped with the assessment of the TTT inversion parameters. CEJT and AM also contributed to the writing and structure of the text. VS participated in the writing of the text and the guidelines of the research together with CRR, who also led the geological interpretation.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Advances in seismic imaging across the scales”. It is a result of the EGU General Assembly 2018, Vienna, Austria, 8–13 April 2018, and of the 18th International Symposium on Deep Seismic Profiling of the Continents and their Margins, Cracow, Poland, 17–22 June 2018.

Acknowledgements

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Acknowledgements.

We would like to thank everyone involved in the TOPOMED survey. Special thanks to the officers and crew of the R/V *Sarmiento de Gamboa* as well as the UTM technicians. This is a contribution of the Barcelona Center for Subsurface Imaging that is a Grup de Recerca 2017 SGR 1662 de la Generalitat de Catalunya. In particular, thanks to Alcinoe Calahorrano for her assistance in the data preprocessing. We also thank Laura Gómez de la Peña for providing the bathymetric map of the studied area included in Fig. 3, Milena Marjanovic and an anonymous reviewer for their constructive comments and questions. Generic Mapping Tools (Wessel and Smith , 1995) was used in the preparation of this paper.

Financial support

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Financial support.

The TOPOMED survey was funded by the Spanish Ministerio de Ciencia e Innovación (grant no. CGL2008-03474-E). The work has been partially financed by REPSOL (CODOS project).

Review statement

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Review statement.

This paper was edited by Michal Malinowski and reviewed by Milena Marjanovic and one anonymous referee.

References

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Special issue

Short summary

We present a workflow that combines different geophysical techniques, showing that a detailed seismic velocity model can be obtained even for non-optimal data sets, i.e. relatively short-offset, band-limited streamer data recorded in deep water. This fact has an important implication for the Marine seismic community, suggesting that many of the existing data sets should be revisited and analysed with new techniques to enhance our understanding of the subsurface, as in the Alboran Basin case.

We present a workflow that combines different geophysical techniques, showing that a detailed...

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