Seismic data used in this study are a portion of a large multichannel 3D seismic survey conducted by Schlumberger WesternGeco in the northern Gulf of Mexico for oil and gas exploration between 2002 and 2003. Our study region (Fig. ) is at the continental slope of the northern Gulf of Mexico, outside the Mississippi River delta, where the water dynamics are mainly dominated by the Loop Current and the Mississippi River outflows . Although our seismic region is north of the Loop Current, it is still a high eddy-kinetic energy region due to the presence eddies and jets over the continental slope . Therefore, we expect this site to experience a range of temporal and spatial oceanic variability.

Seismic surveying lines run from northwest to southeast (inline azimuth, 330∘) over a gentle continental slope (average inclination, 1.5∘) where the seabed depth ranges from 800 to 1300 m. Seismic data were collected using two airguns of 5085 cubic inches, each in flip-flop configuration, and eight streamers, each accommodating 640 hydrophones. The streamer spacing was 100 m and the hydrophone spacing was 12.5 m. We received the raw shot gathers sampled at 2 ms and truncated at the seafloor with no processing applied apart from analog to digital anti-aliasing filtering (frequency range, 3–200 Hz). We received multiple CTD and XBT casts, including three CTD and two XBT casts inside the seismic area (see Fig. ).

Imaging the ocean water column using 3D seismic data from the oil industry is challenging. Acquisition geometry is not optimized for this target and ocean internal reflections are inherently weak and masked by noise of a different nature. However, oil industry contractors employ cutting-edge technology and therefore the obtained seismic data are generally of high quality. The seismic data processing began with reading the raw data that include the recorded outputs of all measuring instruments. The output is a time series that contains reflected signals and noise. The next step is to remove channels that show bad or null trace. To reduce the effect of long-range noise and keep computational costs low, we use the portion of the dataset that has the greatest signal-to-noise ratio (SNR), i.e., data within 4 km of the acoustic source. The strong seafloor signals were also excluded from processing to achieve a relative balance of amplitude levels. The data preprocessing used here employed standard techniques for sub-seafloor seismic imaging, as described in and adapted to work with the water-column specific issues. Typical marine noise was present in the water-column data, e.g., acquisition-related noise and unwanted non-reflected waves or reflection energy returns from previous field records. Environmental noises include wind, shipping activity, and inherent ocean noise. We designed a filtering strategy specifically tailored to remove or minimize as much noise as possible while preserving the weak signals of interest. Consequently, we used low-cut 5 Hz and high-cut 150 Hz Butterworth filtering after each processing step to attenuate noise. The unwanted direct waves interfered with the subtle internal reflections of the water column and complicated the imaging. We applied frequency–wavenumber (FK) filtering and radon filtering to fruitfully attenuate unwanted linear noise, previous shot effects, and seafloor contributions.

We processed the 3D data in a 3D fashion, which uses both inline and crossline data, rather than sequential 2D processing. The attenuation of seismic waves due to the spherical divergence was compensated using the conventional exponential gain function (with a power of 2). For velocity model building and offset effect removal, i.e., the normal move-out (NMO) correction, the data sort was converted to the common midpoints (CMP) between sources and receivers. Velocity model building was performed using a curve-fitting process to the reflections in each midpoint gather within a window of 1410 to 1590 m/s. An objective measure of coherence is needed to determine the best-fitting curve. Among the existing measures, we used the semblance norm , which is widely used in seismic data processing. This process, which is technically called semblance analysis, provides the time domain root mean square (RMS) velocity model that is a processing parameter required for further steps of seismic imaging. Semblance maxima (i.e., coherence) were picked manually in a distance interval of every 250 m in both the inline and crossline directions. Other useful information such as sound velocities derived from CTD measurements was also used to calibrate the seismic velocity model when there is a mismatch between model and measurement. The CTD casts are high-resolution data that are used as guidance for picking the semblance maxima during velocity analysis. Consequently, the stacking and migration processes were performed as imaging steps using the RMS velocities. For stacking, all traces within a midpoint gather are summed to produce a zero-offset single trace with higher SNR. Sorting the stacked traces along inline or crossline sections provides the first interpretable seismic images. The last correction was to correct the effect of the dipping reflectors via seismic migration that automatically moves reflectors to their true position in the subsurface and improves the continuity of the images.

We have also tested and performed a powerful multi-parameter imaging technique known as the common-reflection-surface (CRS) method for stacking and enhancing the data and also to avoid signal stretch after normal move-out correction . The size of the CRS stacking surface was, however, chosen conservatively small, particularly in the crossline direction (50 m), to avoid stacking traces from different swaths. The CRS showed improved results compared to the conventional CMP-based imaging especially along the inline direction . Afterward, post-stack time migration was performed using the estimated RMS velocities to correct the dipping events. Finally, the data were time-to-depth converted using the estimated velocity function integrated with in situ sound velocity casts.

The obtained 3D seismic volume of the ocean interior is illustrated in Fig. , showing the “slicing” configuration of 3D inline, crossline, and depth-line seismic images. A 3D seismic volume is generally a 3D matrix in which seismic reflections are color-coded according to the reflection amplitude and polarity. Each seismic reflection in the seismic images represents an “interface” between two water layers with different acoustic impedances, stemming from different temperatures and salinities in the water column . Because a seismic survey can be acquired in any planar direction, the notation in geographic coordinates (i.e., latitude and longitude) is not adopted. Instead, the planar position is organized in inline (the northwest–southeast direction, which is down the continental slope) and crossline (perpendicular to inline; the northeast–southwest direction, which is along the continental slope on a broad scale); see Figs.  and . The volume has a spatial resolution of 6.25 m in the inline direction, 25 m in the crossline direction, and 6–7 m in the vertical direction (assuming a sound speed of 1500 m/s and considering the dominant frequency of the airgun signal in the water column is between 50 and 60 Hz). Our seismic volume shows only the portion of the water column below 200 m because the geometry of sources and receivers, optimized for deep subsurface targets (several kilometers below the seafloor), poorly image the shallowest portion of the water column . The signal-to-noise ratio (SNR) of our inline seismic images ranges from 8–10, using the formula of .

For a 3D seismic oceanography study, inline sections are often acquired during short temporal intervals (typically a few hours considering the normal vessel velocity of 2.5 m/s). However, crossline sections, which are obtained by interpolating the inline sections perpendicularly, may embrace a wide temporal interval (from a few hours up to several months; the average interval is about 8 h within the linear acquisition period). We assume that water-column reflectors do not move during the shot and signal recording, i.e., the stick–slip model assumption . Considering our inline collection time (a few hours) is smaller than the timescale of the mesoscale ocean dynamics (days), we assume that each inline represents a seismic snapshot of the water column. The entire 3D seismic volume extends for 480 km3 of the water column in the Gulf of Mexico. It consists of 821 inline sections, 3463 crossline sections, and 256 depth slices (5 m intervals vertically). Temporally, it covers over 6 months, from August 2002 to February 2003, with most collected between September and October.

To test if the 3D volume accurately images vertical temperature gradients of water-column structures, we compared the seismic reflections with the acoustic reflection coefficient derived by inverting the temperature–salinity curves collected along with the CTD profiles . The acoustic reflection coefficient for normal incident sound waves is given by 1Rs=ρ1c1-ρ0c0ρ1c1+ρ0c0, where ρ0, ρ1 and c0, c1 are density and sound speed; the subscripts 0 and 1 specify adjacent layers. It has been widely demonstrated that the amplitude of seismic reflections is largely related to temperature gradients in the water column , but sometimes salinity may play a role up to 40 % in regions prone to diffusive convection . Figure a compares the reflection coefficient derived from concurrent CTD measurements (collected on 17 October 2002, location shown in Fig. ) with our seismic imaging (Inline #2130; seismic data collected on 17 October 2002). The overall agreement between the derived reflection coefficients with the seismic reflections suggests that our 3D seismic data successfully represents water-column thermohaline structures. Also, the derived reflection coefficient correlates highly (R=0.9970) with the temperature gradient (Fig. b), suggesting that the seismic amplitude in our seismic images can be interpreted as the vertical temperature gradient of the water column. In addition, the depth of the 3D volume seafloor reflection matches (within 5–10 m error) the seafloor depth from multibeam bathymetry, which is derived using highly accurate sound speed measurements collected by the NOAA ship Okeanos Explorer, confirming that our migration is correct and our seismic images accurately represent the water-column structure.

Now that we have established the oceanic nature of these seismic reflections, we can describe the distribution of the seismic volume in space and time. We emphasize that inline sections are generally acquired during a short temporal interval (typically a few hours considering the normal vessel velocity of 2.5 m/s), while crossline sections, which are obtained by interpolating the inline sections perpendicularly, embrace a wide temporal interval, in our case up to several months. It has been demonstrated that water-column reflectors do not move during the very short time between the shot and the recording of the reflected signal . In 3D surveys each shot is recorded simultaneously along with parallel streamers, generally referred to as swaths, which for our dataset count eight streamers. To better understand the temporal distribution of the inline or swath, we carefully investigate the acquisition time for each survey line by referring to the field acquisition records.

Figure  shows the seismic line number, which is equivalent to the center inline number of a seismic swath, as a function of the recording date and time. Considering that 3D surveys are ideally carried out by acquiring adjacent swaths consecutively, we should observe a linear relationship between sailing line number and survey time. However, it is common that owing to operational problems (e.g., weather, maritime traffic, etc.) and/or bad data acquisition, some portion of the area may be covered at a different time or some lines may be reacquired. Figure  shows a general linear trend, in particular for the portion including lines 1900–2140. In this interval, spatially contiguous seismic lines were collected consecutively in time, yielding a crossline surveying speed of 1.5 swaths per day (equivalent to 0.6 km crossline imaging per day). We refer to this portion as linear acquisition period and we will use it in the correlation analysis discussed below.

In this section, we present multi-directional images from our 3D seismic volume to improve the understanding of water-column 3D seismic images and the fundamentals of 3D seismic oceanography and, hopefully, shed light on ocean dynamics that can be resolved from our seismic images.

Figure shows three inline seismic images, each representing a snapshot along an inline transect in the linear acquisition period. We observe layers of strong seismic reflection in Fig. a, moderate reflection in Fig. b, and weak reflection in Fig. c, respectively, indicating vertical temperature gradients of the water column (see Sect. 2.4). Considering the spatial (2.5 km) and temporal (about 4 d) separations in these images, we observe that the water-column structures varied not only spatially (over 5 km) but, more importantly, temporally (over 8 d), suggesting a strong water-column mixing process may have occurred in this region. In other words, during the interval between 20–28 October 2002, over this portion (about 4 km) of the continental slope, the strong and thick thermoclines gradually weakened and thinned and eventually almost disappeared. We interpret the water-column variation as complex and diverse internal wave variation over the continental slope of the northern Gulf of Mexico, caused by a time-varying mesoscale oceanographic process (length > 10 km, depth > 800 m, time > 10 d), which transformed the water column from highly stratified to well mixed. The possible mesoscale processes causing this transformation will be further discussed in Sect. 4.

Figure shows one of the widest crossline seismic images in our seismic area. We observe significant discontinuities and a strong pattern of seismic swaths (e.g., vertical stripes). These discontinuities stem from abrupt changes of isopycnal depth (e.g., “heaving” of isopycnals) across seismic swaths, indicating the occurrence of temporal variation in the water column. In this region, the ocean dynamics that may change isopycnal depths include eddies, internal waves, internal tides, and Mississippi River outflow. Swaths collected over the linear acquisition period (marked by the red line) display some continuity, while swaths collected with longer time shifts can yield significant discontinuities. For example, seismic imaging of the contrast acquisition period (marked by the blue line) looks drastically different from its neighboring parts due to a 20 d time shift (cf. Fig. ). We also notice that the discontinuity varies with depth. The discontinuity is more significant at 200–350 m (Fig. b) than at 600–750 m (Fig. c), suggesting that temporal variation is more significant at shallower depths. Another proof for these discontinuities indicating temporal variation is the continuous seafloor, which has no temporal variation. Our analysis suggests that temporal variation of the water column, appearing as imaging discontinuity, is significant along the crossline direction, which lowers the quality of crossline images and may further hinder their interpretation.

Figure shows depth-slice seismic images at three different depths (250, 500, and 750 m) to represent the shallow, middle, and deep layers of our seismic volume. Water-column seismic depth slices are the most complex representation to conceptualize since they offer a new view of the ocean interior. We observe little discontinuity along the inline direction, suggesting that temporal variation is weak along the inline direction. However, we observe a significant swath pattern and strong discontinuity appearing along the crossline direction, and the discontinuity decreases from shallow to deep layers. This decreased discontinuity agrees with the fact that temporal variability of the top ocean is more significant than that of the deep ocean, as the ocean is mainly stirred from the top due to the wind-driven surface mixing . Due to the presence of temporal variation, interpreting seismic depth-slice images is extremely challenging, but we may get a hint of the timescale of the ocean dynamics in Fig.  based on the number of discontinuities and our 3D seismic setup (e.g., time separation between nearby seismic swaths). We suggest that in this region the ocean dynamics have a timescale less than 8 h at a depth of 250 m but longer than 8 h at 750 m. The analysis of depth-slice images suggests that in our seismic volume, temporal variation of the water column is non-negligible in the crossline direction but negligible in the inline direction. In other words, temporal variability is dominant at timescales greater than a few hours.

This qualitative analysis of 3D seismic images (Figs. –) reveals that temporal and spatial variations are both embedded in the 3D seismic volume and temporal variation is more significant along the crossline direction than the inline direction. Next, we will conduct two analyses to quantitatively assess temporal and spatial variations within a 3D ocean seismic volume. We conduct a simple (i.e., single frequency) theoretical and complex (i.e., multifrequency) data analysis to quantitatively assess temporal and spatial variations within a 3D ocean seismic volume.

Here we describe a theoretical example to gain insight into why temporal variation appears differently in inline and crossline directions in a typical seismic surveying setup. We assume a water column in which temporal variation occurs at one cycle per day (1 cpd, about 10-5 Hz) and spatial variation at one cycle per kilometer (10-3 cpm). The values here are arbitrary but can represent mesoscale oceanographic processes such as internal waves or internal tides for which seismic imaging is particularly useful.

Let us consider that this water column is imaged with a typical 3D seismic survey setup, having the same field geometry of our survey and carried out swath by swath like the linear acquisition period. The vessel sails at a speed of 2 m/s, the seismic swath is 400 m wide, and the vessel can collect two swaths per day. Accounting for the two-way travel time, surveying a 1 km inline section takes 21000[m]/2[m/s]×2[ways]=103[s]=0.012[d], which is negligible considering the concerned water-column dynamics have a temporal variation of 1 cpd. However, surveying a 1 km crossline section takes 31000[m]400[m/swath]×2[swath/d]×2[ways]=2.5[d], which fully covers 2.5 cycles of temporal variation of the concerned dynamics here. These numbers quantitatively illustrate how the temporal variation is not significant in inline imaging yet should be accounted for in crossline imaging when using the 3D seismic data to image mesoscale oceanographic processes.

The temporal and spatial variations are illustrated for inline and crossline directions in Fig. a and b, respectively. The illustration is for a survey over 1 km. The spatial variation (blue curves) is given by sin⁡(2πkx), where k is the wavenumber in the unit of cpm (recall here that k = 10-3 cpm). Now, to illustrate the temporal variation along the 1 km survey distance, consider how many temporal cycles have been covered over the time duration of the survey. In the inline direction, it takes 0.012 d for a 1 km survey (see Eq. ), corresponding to 0.012 cycles of temporal variation (recall that the temporal variation period is 1 cpd). That is 1.2×10-5 cycles of temporal variation per 1 m survey distance; effectively, it is 4k=1.2×10-5[cpm]. However, in the crossline direction, the corresponding numbers are 2.5 cycles of temporal variation over the temporal duration of a 1 km survey (see Eq. ); effectively, the wavenumber is 5k=2.5×10-3[cpm].

Based on above the settings, the temporal (red curve), spatial (blue curve), and total (sum of both; black curve) variations along the inline and crossline directions are illustrated in Fig. a and b, respectively. The results show that in the inline direction (Fig. a), temporal variation is subtle and the total variation mostly follows spatial variation. In the crossline direction (Fig. b), however, temporal variation is prominent and the total variation is a fair combination of both temporal and spatial variations, with temporal variation being more noticeable. This theoretical analysis confirms that with typical 3D seismic survey setups, temporal variation of mesoscale ocean dynamics is more significant in the crossline direction. In other words, inline images mostly represent spatial variation of the water column, while the crossline images represent a combination of temporal and spatial variations of the water column.

Finally, Fig. c and d show the spectral distribution of temporal (red curve) and spatial (blue curve) components in the wavenumber domain. Here we focused on wavenumber domains instead of frequency domains because seismic images are originally intended to resolve spatial variation rather than temporal variation. In both inline and crossline directions, the spatial component appears at the same location, e.g., k=10-3 cpm, while the temporal component appears at two different locations, e.g., at k=1.2×10-5 cpm in the inline direction (see Eq. ) and k=2.5×10-3 cpm in the crossline direction (see Eq. ). In other words, in inline images, temporal variation will appear as a low-frequency component far apart (about 2 orders of magnitude, 100 times) from the spatial component, while in crossline images, temporal variation appears as a high-frequency component very close (about 2.5 times, still on the same order of magnitude) to the spatial component. The red line in the inline spectrum (Fig. c) illustrates the theoretical position of the temporal component, which may be too weak to resolve when inline sections are too short (i.e., covering far less than one temporal cycle). These spectral analyses suggest that water-column temporal variation will appear differently in inline and crossline images, posing strong high-frequency aliasing in the crossline imaging. Importantly, separating temporal and spatial variations in crossline imaging could be challenging because they are spectrally close to each other.

Our theoretical analysis provides an idea for quantitatively analyzing temporal and spatial variations from the variations along the inline and crossline directions. However, real 3D ocean seismic volumes are more complicated than our single-frequency example. Here we developed a method to analyze temporal and spatial variations within our 3D seismic volume. The results will provide values of inline and crossline variations in our seismic volume, which can also be useful to other seismic oceanography studies with similar setups.

For a real 3D seismic volume, the variation along a direction can be quantified with the correlation, which measures the similarity between seismic images. The correlation for two seismic images, whose seismic amplitude stored in the 2D matrices x and y, is defined as 6r=∑i,j(xij-x‾)(yij-y‾)σxσy, where i and j form the 2D index, x‾ and y‾ are the means, and σx and σy are standard deviations of x and y. The correlation results are between -1 and 1, representing the image similarity with respect to the reference, with +1 being exactly similar, -1 exactly reversed, and 0 completely different. In practice, the variation along the inline direction can be simply derived from the correlation between crossline images (Fig. a), while the variation in the crossline direction can be derived from the correlation between inline images (Fig. b) due to the perpendicular relationship between inline and crossline directions.

We correlate the inline and crossline image series with corresponding reference images (IL #2140, or XL #18000), respectively. The correlations are plotted as functions of inline or crossline number. The inline axis can be further converted to lateral distances (spatial domain), while the crossline axis can be converted to both lateral distances (spatial domain) and acquisition time (temporal domain). When the water-column structure varies, either temporally or spatially, the similarity between seismic images varies as well. Therefore, the change of the image similarity summarizes the variation of the water column with little change of correlation value, indicating that there has been no spatial nor temporal variations of the oceanic processes between the compared images and a big change of correlation value indicating big temporal or spatial variations.

The correlation results allow quantitative analysis of the variation of a particular image series or at a particular depth range. We only use the linear period (see Fig. ) for a more meaningful analysis. To best represent the variation of the water column, we only use imaging of 250–750 m, excluding the seafloor and low-quality shallow portion data. Furthermore, we divide the water column into two parts, the upper part (250–500 m) and the lower part (500–750 m), to explore the depth dependency. Furthermore, the correlation function allows the examination of the correlation length, which is defined as the length to the first zero crossing, measuring the length required for two images to be completely different. The shorter the correlation length is, the faster the signal de-correlated. Finally, for periodical signals, the spectra of the correlation function can be used to study the inherent cycles within the signal.

Figure shows the results of the variation in the inline direction (namely, inline variation) and that in the crossline direction (namely, crossline variation) derived from our 3D seismic volume. Here, inline variation (Fig. c) is plotted as a function of space (i.e., lateral distance), while crossline variation (Fig. d) is plotted as a function of space or time (based on the acquisition time). Overall, both inline and crossline variations quickly decrease from unity and then fluctuate around zero. The upper and lower water columns vary differently and the variation of the whole water column is mostly dominated by the upper part. Comparing these two directions, three observations can be made: (a) the crossline variation fluctuates more than the inline variation, (b) the crossline variation has a shorter correlation length (about 0.4 km) than the inline variation (about 2.5 km), and (c) the crossline variation has more high-frequency fluctuations than the inline variation. These observations can only be explained with the presence of temporal variation in crossline variation, leading to more and faster fluctuation and shorter correlation lengths, agreeing with our theoretical analysis (see Fig. a and b). Thus, our analysis confirms that in our 3D seismic volume, the inline variation (Fig. c) is mostly associated with spatial variation, while the crossline variation (Fig. d) is associated with both temporal and spatial variations.

Figure shows the spectra calculated from the total variations (black curves in Fig. ) along inline and crossline directions. Note that the inline is only associated with space, while the crossline is associated with both time and space. Overall, we observe a similar descending trend in both spectra, providing us information about the spatial features in wavenumber domains. Comparing these two spectra, one can see that the many peaks seen in the crossline spectrum are absent in the inline spectrum. These peaks show the dominant frequencies of temporal variations. Particularly, we observe the semi-diurnal tidal cycle (marked as M2 in Fig. ) in the crossline spectrum, which is strong evidence that the crossline variation includes temporal variations, as continental marginal waters are often heavily influenced by tidal-forcing dynamics. A significant number of high-frequency components seen in the crossline spectrum are temporal variation, agreeing with our theory (Fig. d), which suggests that temporal variation appears as a high-frequency aliasing. Our spectral analysis, again, confirms that the crossline imaging captures temporal features of the ocean, while the inline imaging only captures the spatial features.