Volume 126, Issue 4 e2020JB021082
Research Article
Free Access

Constraining Floating Ice Shelf Structures by Spectral Response of Teleseismic P-Wave Coda: Ross Ice Shelf, Antarctica

Thanh-Son Phạm

Corresponding Author

Thanh-Son Phạm

Research School of Earth Sciences, Australian National University, Canberra, ACT, Australia

Correspondence to:

T.-S. Phạm,

[email protected]

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Hrvoje Tkalčić

Hrvoje Tkalčić

Research School of Earth Sciences, Australian National University, Canberra, ACT, Australia

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First published: 15 March 2021
Citations: 1

Abstract

The recent deployment of a broadband seismic array on the floating ice shelf in the Antarctica's Ross sea presents a great opportunity to study the shelf structure using broadband seismic data. In this study, we develop a further improvement of the P-wave coda autocorrelation method, which proved capable of characterizing grounded ice-cap structures. Ice shelves are floating ice sheets connected to a landmass, and in order to decipher their structures, a water layer has to be added to the problem. We construct the power spectrum stacks of P-wave coda data, waveform records that immediately follow P arrivals, in the spectral domain, which are equivalent, via a Fourier transform, to widely used autocorrelograms in the time domain. At half of temporary seismic stations under consideration, we report prominent resonant peaks in the spectral autocorrelograms, associated with the ice-water configuration of the ice shelf. The lack of clear resonant pattern for the rest of the stations is suspected due to a high noise level in the icy environment and significant lateral heterogeneity at the local scale. Subsequently, we develop a formalism to explain the observed resonance and devise a grid-search scheme to estimate ice- and water-thicknesses underneath the stations. Our water-thickness estimates agree well with the previously documented measurements, but there is a discrepancy in the ice thickness results. Therefore, the method has a great potential to complement the existing ice-shelf model, to be used in future monitoring applications of ice shelves, or near-future space exploration to icy planets.

Key Points

  • Resonant peaks are observed in spectral responses computed from earthquake-based data for receivers deployed on Ross Ice Shelf, Antarctica

  • Analytical formulation is developed to explain the observed resonances in spectral responses

  • Our water thickness estimates agree with the Bedmap2 datasets, but considerable difference is obtained for the ice thickness estimates

Plain Language Summary

The use of passive seismic data—continuous ground motion records—to study structures of the ice covers in some parts of the Earth has been limited either due to the lack of over-ice seismic deployment or the inefficiency of existing seismic techniques in the icy environment. This study proposes a new technique to analyze data of individual receivers, also known as the autocorrelation analysis. The technique is applied to a recent deployment of multiple seismic sensors over the largest floating ice shelf on Earth—Ross Ice Shelf, Antarctica. The main results are thickness estimates for the ice shelf and the sea water body right beneath the recording sites. The method will potentially find important applications in structural studies in other polar ice shelfs or seismic missions to icy planets in near future.

1 Introduction

The most recent deployment of a broadband seismic array on the floating ice shelf in the Antarctica's Ross sea (e.g., Baker et al., 2019; Bromirski et al., 2015; Chaput et al., 2018) presents a great opportunity to study structures of the ice shelf. Knowledge of ice shelf structures plays a critical role in understanding the dynamics of the Antarctic ice cover and furthermore on its influence on global climate change in the long term. For the period of two years from late 2014 to late 2016, there were 34 broadband seismometers deployed over Ross Ice Shelf, the largest ice shelf in Antarctica covering over 500,000 km2 (similar to the surface area of Spain or Thailand). This valuable data set has enhanced the understanding of dynamics and structures of the firn layer, the top unconsolidated snow cover of ice column (Chaput et al., 2018; Diez et al., 2016), and the interaction of ocean gravity waves with the ice shelf (Bromirski et al., 2015). In this study, we exploit the passive seismic dataset in a novel manner to increase constraints on the first-order shelf structures, in particular, the ice and water layer thicknesses, beneath individual receivers.

In seismology, autocorrelation method is a type of waveform data analysis involving only single-component recordings to study the crustal and mantle structure immediately beneath the receiver. This class of methods is of special importance in places with minimal access, where a seismic network is hardly feasible, such as in space missions to the Moon (e.g., Nishitsuji et al., 2016), Mars (e.g., Deng & Levander, 2020), or for experiments to the interior of Antarctica (e.g., Phạm & Tkalčić, 2018; Yan et al., 2020). It also serves as an effective tool to use older data from achieves with only vertical recordings (e.g., Kennett et al., 2015). In the vast majority of studies to date, the autocorrelation stacks are computed and then interpreted in the time domain as reflectivity records from virtual sources on the Earth's surface. This approach originated from the pioneering theoretical work of Claerbout (1968), who showed analytically that the autocorrelation of surface records of a vertically incident plane is equivalent to the reflectivity record at a colocated station. The important result was then extended to a slightly oblique incidence by Frasier (1970).

Employing these theoretical ideas had not been practical in passive seismology until the last decade. A large amount of research has focused on developing efficient data-processing procedures of seismic data to construct robust autocorrelation stacks in the time domain (also known as temporal autocorrelograms) to retain local structural signatures. Choices of input data for processing range greatly from several months of continuous ambient noise records (e.g., Gorbatov et al., 2013; Oren & Nowack, 2017; Tibuleac & von Seggern, 2012) to short portions of earthquake-based data (e.g., Delph et al., 2019; Kim et al., 2019; Phạm & Tkalčić, 2017; Ruigrok & Wapenaar, 2012; Sun & Kennett, 2016). In these past works, processing procedures employs at least one, but often both, of normalization operations in time and frequency domains. In some cases, filtering options, including pre-filtering of the input data and post-filtering of stacked autocorrelograms, appear to play a crucial role in revealing reflection signals from the top of subsurface discontinuities in resultant autocorrelograms (e.g., Helffrich, 2019). Possible discontinuities are then associated with either their large amplitude features (e.g., Gorbatov et al., 2013) or visible changes in the reflectivity pattern (e.g., Becker & Knapmeyer-Endrun, 2018; Sun & Kennett, 2016) in the autocorrelograms. Most recently, Deng and Levander (2020) employed the autocorrelation approach to ambient noise records at Mars thanks to the InSight mission (e.g., Banerdt et al., 2020) and provided some possible insights into the internal structures of Mars.

We previously developed an autocorrelation method of P wave coda from teleseismic earthquakes (Phạm & Tkalčić, 2017). The use of seismograms immediately following the first arrival of P waves, known as the P-wave coda, provides near-vertical illumination of subsurface structures. Additionally, it retains relatively high-frequency content compared to other types of passive seismic data, and hence enables imaging of shallow seismic structures at a high resolution. The use of short and selective P-wave coda seismograms also greatly reduced computational labor compared to autocorrelating ambient noise. As a consequence, the method enabled us to generate high-quality measurements of the ice-thickness at various places in Antarctica where the ice cap is grounded over bedrock (Phạm & Tkalčić, 2018). The effectiveness of the method was later confirmed by Yan et al., (2020) through a comparison with the results from the horizontal to vertical spectral ratio (H/V) method (Yan et al., 20182020).

Here, we make a slight modification to the P-wave coda autocorrelation approach and apply it to the ice-shelf seismic data by constructing power spectrum stacks. This can be thought of as spectral autocorrelograms because linearly stacked autocorrelograms in time and frequency domains are equivalent via a Fourier transform. We find that it is not straightforward to interpret the autocorrelation stack of ice-shelf data in the time domain as is the case for grounded-ice stations. This is because of a water layer, which has significantly lower impedance contrast with the floating ice in comparison with the impendence contrast between the ice layer sitting on top of bedrock. There were few in situ attempts to use passive seismic-noise records from short temporary campaigns (Diez et al., 2016; Zhan et al., 2014). They both analyzed the spectral ratio between vertical and radial spectral seismograms and interpreted it as the resonance phenomenon of seismic energy in the water layer.

The remainder of this paper is structured as follows. Section 2 is devoted to describing data retrieval and processing practices. In Section 3, we derive an analytical formulation characterizing the spectral response, or the theoretical spectral autocorrelation, of the ice-water system. The formulation, which is verified through a comparison with simulated data, forms the basis for a grid-search scheme for ice and water layer thickness. Section 4 reports on the main results of the method, which is followed by discussions on its limitation and potential in future studies. Section 5 is dedicated to concluding remarks. In Appendix A, we report on temporal autocorrelation results for the grounded stations, which do not fit the paper's main aspects, but are still important in providing nearby ice thickness measurements for those stations.

2 Data Retrieval and Processing

During 2-year period, from late 2014 to late 2016, 34 temporary seismic stations were deployed over the ice shelf in the Ross sea, the largest ice shelf (approximately 500,000 km2) in Antarctica (Bromirski et al., 2015). These stations belonged to two coordinated projects, RIS (Mantle Structures and Dynamics of the Ross Sea from a Passive Seismic Deployment on the Ross Ice Shelf) and DRIS (Dynamic Response of the Ross Ice Shelf to Wave-Induced Vibrations), whose sites are named with RS and DR prefixes respectively (see location map in Figure 1). Three-component seismic waveform data have been made available via the IRIS Data Management Center in mid-2019 for both broadband channels (20 samples per second) and very-high broadband channels (100–200 samples per second depending on sites). In this study, as we are interested in signals contained in a bandwidth below about 4 Hz, we do not see the difference in the two channel sets. For more information on instrumentation and installation details, we refer readers to a recent publication by Baker et al. (2019).

Details are in the caption following the image

Network maps of the Ross Ice Shelf experiment. (a) Ice thickness map of Antarctica from Bedmap2. The red box shows the area of Ross Sea, which is enlarged in panel (c). (b) Enlargement of the box indicated in panel (c). (c) Network configuration of seismic stations in the Ross Ice Shelf experiments. Black squares show stations deployed over grounded ice cap. Gray diamonds denote stations deployed near to the waterfront. Red and cyan circles are seismic stations whose data are processed in this study. Cyan stations yielded spectral responses with clear resonant peaks, while red stations yield noisy spectral responses. White triangle marks the location of the most recent drill borehole though the shelf, HWD2, in 2017 (Stevens et al., 2020). The color bar in (c) is also applied for maps in panels (a and b). (d and e) Cross-sections of the ice shelf along two long profiles marked by orange lines in panel (c). All data of ice-surface elevation, ice thickness and bedrock elevation used in this plot are from Bedmap2 datasets (Fretwell et al., 2013).

We selected teleseismic-earthquake waveforms for the events of Mw 5.5 or above cataloged in the Global Centroid Moment Tensor (GlobaCMT, Ekström et al., 2012) in the epicentral distance range 30–90°. The map of events during the operation period is provided in Figure S1. The waveform windows are 60-s long, starting from 10 s ahead of the predicted P-wave arrivals. In the following step, all waveforms are corrected for instrumental responses and high-pass filtered to remove periods longer than 10 s that have no sensitivity to the ice-shelf structures. We do not apply any strict data-quality criteria apart from the threshold magnitude. The rationale behind less-conservative data selection is that even weak teleseismic events can still provide some weak illumination of the structures, which can be effectively enhanced by stacking repeating contributions over many events (Tauzin et al., 2019). As we do not employ any subjective threshold to data selection, the results of similar quality are expected to be easily repeatable by other researchers.

We make some changes to the processing procedure that was used to construct temporal autocorrelograms for the grounded ice stations in Phạm and Tkalčić (2018). First, windowed P-coda waveforms are transformed to the frequency domain using a forward Fourier transform. Then, spectral whitening normalization to the complex spectrum is applied (Bensen et al., 2007; Phạm & Tkalčić, 2017),
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0001(1)

In Equation 1, the nominator, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0002, is the complex spectrum at the nth discrete frequency of the kth input waveform, the denominator is a smoothed average of its amplitude spectrum, and the division gives a whited complex spectrum. A spectral whitening width urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0003 defines the number of spectral points involved in the smoothening average. The value of the whitening width affects the relative amplitudes of spectral peaks revealed by the operation and will be further discussed in Section 3.2 using a data example. We fix urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0004 Hz in the analysis presented in this study unless otherwise specified.

Subsequently, we construct a linear stack of the whitened power spectra over all earthquakes in the dataset,
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0005(2)

Because the Fourier Transform is a linear operator, the inverse transform of the spectral stack is equivalent to the linear stack of the individual autocorrelation in the time domain. However, the power spectrum stack, which is hereafter referred to as the spectral response, will then be used as a direct observable comparable to the spectral response analytically derived in Section 3.1.

3 Theoretical Spectral Response

3.1 Analytical Derivation of Spectral Response

To explain the observed spectral response pattern, we derive an analytical representation of the spectral response for the ice-water system from a steep energy response. To simplify the theoretical consideration, we assume that the spectral response is caused by the perfectly vertical incidence of compressional energy, whereas possible interference of converted shear energy is ignored. Therefore, the ground motion is only in the vertical direction. This is because we are considering steeply arriving P-wave energy from teleseismic earthquakes (epicentral distance in 30–90°) and the water layer blocks the transmission of S-wave energy upward. Additionally, converted S-wave energy is minimal in the solid ice-layer due to steep incidence of P-wave in the water layer. Our synthetic experiments in Section 3.3 will further justify our assumption through numerically simulated waveforms. Thus, we neglect S-wave propagation in the ice in the following considerations.

Subsequently, we will derive an analytical expression of the spectral response for the ice-water system induced by a vertically incident impulsive wavefront. The formulation is derived using the matrix propagation method in a stratified medium (e.g., Claerbout, 1968; Kennett, 1983). Similar derivation for the grounded ice system, which is approximated as a layer over a half-space bedrock, can be found in Appendix A of Phạm and Tkalčić (2018). Structure of a floating ice shelf is modeled by two horizontal layers, of ice on the top and water on the bottom, over a half space bedrock as sketched in Figure 2. Associated seismic parameters for each layer are listed in Table 1. We note that the expression intended for the vertical ground motion was given in Claerbout (1968). Hence, the derived expression is comparable with spectral response constructed from vertical seismograms of steep, teleseismic P-wave incidence.

Details are in the caption following the image

Representation of seismic energy reverberations in ice-water system in high-frequency approximation. Background velocity model consists of ice and water layers lying over a half-space bedrock of parameters described in Table 1. (a) Incident seismic plane-wave arrives at the receiver via different ray paths. Black ray represents the direct arrival, orange rays are reverberations within the water layer, and blue rays represent multiple reverberations in the ice layer. The arrivals have different delay times. (b) Simplified representation of the seismic energy transport through stratified media. Up and down arrows indicate simplified energy fluxes at each layer interface.

Table 1. Velocity Model of the Ice-Water System
Layer P-wave velocity (m/s) S-wave velocity (m/s) Density (kg/m3)
Ice 3,900 1,950 900
Water 1,470 5 1,000
Bedrock 5,000 2,550 2,800
  • P and S-wave speeds are taken from Podolskiy and Walter (2016), while densities are rounded values from Griggs and Bamber (2011). The water layer has minimal S-wave speed as required by the forward waveform simulation program (Randall, 1989) that only works for solid media.
Let us consider ρi and vi be the density and compressional-wave speed in the i-th layer of the chosen velocity model. Then, the reflection coefficient ri of a downward energy flux across the base of the layer, ri, is given (e.g., Stein & Wysession, 2003) by
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0006(3)
and the corresponding transmission coefficient is urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0007. In the frequency domain, Ui and Di are upward and downward energy fluxes in the layer i (see Figure 2). According to Claerbout (1968), the transfer function corresponding to energy fluxes in two neighboring layers, is written as
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0008(4)
In Equation 4, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0009 and urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0010 are terms representing the propagation effects in the layer and j is the complex unity (i.e., urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0011). In the expression of wi, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0012 is the two-way travel time, Hi is the layer thickness and Qi is the acoustic quality factor. The transfer function over the stack consisting of L layers is a multiplication of the transfer matrices in Equation 4, that is
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0013(5)
In the top layer, the upward and downward energy fluxes are equal, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0014, because the reflection from the surface is total and we use X to denote both quantities. Underneath the ice-water system, we assume the upcoming energy from a distant earthquake is a Dirac signal in the time domain that is a spectral unity, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0015. There is also a fraction of energy that escapes the system and travels downwards in the half space, D3, but its quantity is not of our interest here. Given these definitions, Equation 5 can be re-casted in the following form for the ice-water system,
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0016(6)
We are interested in solving for the surface energy flux, X, that represents the spectral response at a surface receiver. To solve for X, we invert the transfer matrix and get
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0017(7)
By replacing Equation 7 into Equation 6 and after some algebraic reduction, we obtain X as a function of the reflection coefficients r1, 2 and propagation terms z1, 2:
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0018(8)

The power spectrum urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0019 will be then referred to as the spectral response of the ice-water system. The reflection coefficients are determined by the material properties, which are well known in the ice-water system through laboratory measurements (Table 1). Thus, the spectral response is simply a function of propagation terms, the layer thicknesses in particular. We will further elaborate on this dependence in the subsequence subsection, which will eventually lead to a grid-search scheme for the layer thicknesses.

To gain insights into the ice-water system, we visualize its spectral response (Equation 8) as functions of the layer thickness parameters in Figure 3. The response is featured by peaks at certain frequencies, describing the resonance phenomenon due to reverberating energy trapped within the layers. Figures 3a and 3b demonstrate the sensitivity of the resonant frequencies to the variation of layer thicknesses. The pattern becomes clearer in Figures 3c and 3d, in which the variations of the first and second resonant frequencies are shown in the parameter space of the two thickness parameters. Both resonant peaks are increasing with a decrease of thickness. However, the detailed trend is not proportional. This differs from the single layer case, where the first and second resonant frequency are proportional (see Appendix A in Phạm & Tkalčić, 2018). Note that the variation of frequency is more sensitive to the change in the water-layer thickness.

Details are in the caption following the image

Visualization of analytical spectral response (Equation 6) as a function of ice and water thicknesses. Variation of Spectral responses are featured by resonance peaks whose frequency locations are functions of (a) ice and (b) water column thickness, while water and ice thicknesses are fixed to 0.2 km respectively. (c) First and (d) second resonant frequencies as functions of ice and water thickness. (e) Determination of ice and water thickness at the intersection of the contour lines of the first and second resonant peak frequencies.

It follows that one can use the measured resonant frequencies to search for the thickness parameters if unknown. As demonstrated in Figure 3e, the intersection of the contour lines of the first and second resonant frequency gives an estimate of the layer thicknesses. It is also clear that water thickness is better constrained as it has higher sensitivity to the location peaks. We will exploit the useful properties of the spectral response in a grid-search scheme in Section 3.3.

3.2 Synthetic Experiments

In this section, numerical experiments are set up to achieve a two-fold purpose. First, we would like to validate a simplification we made by ignoring the interference of shear waves in the spectral response of the system. Second, we want to understand effects of high-frequency noise on the depletion of high-order resonance peaks.

Synthetic waveforms are generated through a two-step method. We refer to Section 4 in Phạm and Tkalčić (2017) for greater details on the generation of synthetic waveforms with graphical demonstrations shown in that paper's supplementary material. In the first step, impulse structural responses of oblique incidence are generated by the reflectivity code (Randall, 1989) based on the matrix propagation method (Kennett, 1983). As the reflectivity code does not allow for the presence of a fluid layer, we set a very small shear wave velocity for the water layer in the velocity model. Incident ray parameters are computed from epicentral distances and depths of 500 earthquakes in the waveform dataset of the pilot station DR14 (see Section 4.1). Second, the impulsive responses are convolved with randomly generated source time functions. The convolved seismograms are finally summed with random white noise.

We process synthetic waveforms in the same way we process real waveforms. To confirm the validity of the analytical derivation, we construct synthetic spectral responses of a structural response obtained from the first step in the waveform simulation procedure described above. In particular, a purely structural response without convolving source time functions and numerical noise, is computed analytically in Figure 4a. An excellent agreement of both the position and the relative amplitudes of resonant peaks confirm the analytical formulation. Furthermore, because oblique incidence simulating real earthquake-source geometries is used in the numerically generated waveforms, this result suggests that our approximation of the oblique incidence with the vertical incidence and neglecting the conversion of P-to-S energy at the water-ice interface are reasonable assumption in this problem.

Details are in the caption following the image

Comparison of analytical (theoretical) and numerical spectral responses. We use elastic media and realistic incidence angles in the numerical case while assuming acoustic media and normal incidence for the analytical case. (a) The thick gray line shows the spectral response computed from raw spectral response for the ice-over-water model using the synthetic waveforms for 500 teleseismic earthquakes at the station DR14. The dashed black line is the theoretical spectral response of the same system given by Equation 6. Panel (b) similar to (a), but the spectral response is computed from the convolution of structural responses and randomly generated STFs. Panel (c) similar to (b) but white noise is added to the computed seismograms. (d) The grid-search result for synthetic spectral autocorrelation without STFs and white noises involved according to the set up in (a). The background image shows the correlation coefficient for each combination of ice and water thicknesses with the computed spectral response. Coordinates of the yellow circle and red diamonds are the input and recovered ice and water thickness of the synthetic experiments. Red contour indicates the 95% credibility interval for the optimal model. Panels (d and e) are similar to (c) but for the set up in (b and c), respectively.

Next, we consider a possible influence of source-time function and high-frequency noise on the emergence of resonance peaks in the spectral response. The spectral response in Figure 4b is computed from synthetic waveforms having randomly STF convolved. As in the previous case, the location of the resonance peaks agrees well with the theoretical responses, but the match of relative amplitudes is much poorer. In Figure 4c, the presence of high-order resonance deteriorates after the addition of white noise to the synthetic waveforms. This is due to the fact that high-frequency noise disturbs appearance of high-order resonance especially the third resonance in this case. Although the high-order resonances are impacted, the resonant frequencies remain unchanged in the synthesized spectral response. The weak expression of the third or higher resonance peaks is possibly due to the high-frequency local noise.

Finally, we can conclude that the assumption of perfectly vertical incidence is reasonably made in the derivation of the analytical spectral response. This enables rapid evaluation of the spectral response which is beneficial when posing an inverse problem to search for two unknowns in the systems, which are ice and water thickness in our case. Due to this efficiency, a grid-search approach for ice and water column thicknesses based on observed spectral response is feasible.

3.3 Grid-Search for Ice and Water Column Thicknesses

A grid-search scheme is deployed to look for the best match in the resonant frequencies in the observed spectral response and the analytical prediction. We have the observed spectral autocorrelation, ak, defined in Equation 2 from data and the theoretical prediction urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0020 with Xk defined in Equation 8 at the same set of discrete frequencies as for the observation. For each pair of ice thickness, Hi, and water column thickness, Hw, the goodness of fit is quantified by the correlation coefficient between the theoretical and the observed spectral responses from 0 to 4 Hz,
urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0021(9)

This correlation measure of fit, urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0022, represents the fit on the pattern of the spectral stacks and tolerates the mismatch in the amplitude of resonant peaks. As noted in Section 3.1, the positions of the two resonant peaks, which are observed in the spectral response, are required to constrain the input parameters. Therefore, we will only progress with the grid-search for observed spectral responses satisfying this condition to report the results in Section 4.

The grid-search scheme is demonstrated by utilizing synthetic data from the previous section. Figures 4d–4f demonstrate recovery of the input parameters using the grid-search scheme for the three studied cases. The recovered results correspond to the maximum correlation coefficient. In these cases, the input thickness values are within the 95% confidence interval of the recovered model (see supporting information S1 for the estimate of the confidence interval), proving the feasibility of the data processing and searching scheme. It is also that the uncertainty in the ice thickness estimates is more considerable than those for the water thickness estimates as seen in the contour plots. We also note that in the third case, where both source time functions and high-frequency noise are present, the recovery is also successful with two visible resonant peaks. Because the second and the third resonant peaks are often weakly visible, this experiment proves the potential of the application of the proposed grid-search scheme to field data observations.

4 Results and Discussion

4.1 Results for the Pilot Station DR14

Spectral and temporal autocorrelation results for a pilot ice-shelf station, DR14 (see Figure 1 for location map), are shown in Figure 5. The 2-year long dataset of P-wave coda at this station is divided into subsets of 2 months with about 50 waveforms in each bin (see Figure 5a for the histogram of the dataset). All 2-monthly stacks of spectral responses (Figure 5b) show strikingly similar features with the overall 2-year stack (Figure 5c). Given the similarity in the spectral domain, it is not unexpected to observe a significant similarity also in the temporal autocorrelation (Figures 5c and 5d). The self-consistency of the autocorrelation results over time suggests that features in the stacks must strongly correspond to local structures. In stark contrast with the temporal autocorrelograms from the grounded ice stations (for example, see Figure 2 in Phạm and Tkalčić (2018) and Figure A1 in this paper), the temporal autocorrelograms constructed from the ice shelf data (Figures 5d5e, and 7b) are more complex and not readily interpretable in the traditional manner as reflection record from a virtual source in the time domain. However, we realize that more can be learned when comparing the observed spectral autocorrelograms with the analytical spectral response (Equation 8).

Details are in the caption following the image

The variation in short-operation windows of temporal autocorrelation and spectral response stacks at the pilot station DR14. (a) Histogram of Mw 5.5+ earthquakes recorded at the station in 2-month intervals from January 2015 to December 2016. (b) Spectral responses constructed by stacking events in 2-month intervals. (c) Overall spectral stack over the 2 years of operation. (d) One-sided autocorrelation stacks in the time domain of all earthquake in 2-month intervals. (e) Overall autocorrelation stack over 2 years.

Details are in the caption following the image

Results for pilot station DR14. Detail description of each subpanels can be seen in the caption of Figure 3.

Details are in the caption following the image

Spectral and temporal autocorrelograms as functions of spectral whitening widths. (a) Stacked spectral autocorrelograms of P-wave coda seismograms from teleseismic events of Mw 5.5+ recorded by the pilot station DR14. The stacks are computed with different spectral whitening widths. (b) Stacked temporal autocorrelations for different spectral whitening widths computed for the earthquake dataset in (a).

In Figure 6, we apply the grid-search scheme presented in Section 3.3 to the observed spectral autocorrelogram. The highest correlation coefficient between the observed and analytical spectral responses corresponds to some 320-m-thick ice (with a confidence interval from 280 to 380 m) and 540-m-thick water (with a confidence interval from 520 to 550 m) layers beneath the station (Figure 6b). The spectral response corresponding to those values shows a robust fit to the observation in terms of resonant peak positions along the frequency axis (Figure 6a). The optimal model and its confidence interval vary insignificantly with other values of P wave speed in the basement rock (e.g., 5,500 or 6,000 m/s rather than 5,000 m/s as in Table 1), because this value has minimal influence in the shape of the analytical spectral response (Equation 8) via the reflection coefficient r2. As discussed in Section 3.2, the mismatch in relative amplitudes of the second and third resonant peaks is somewhat expected given that source time functions of contributing earthquakes are heterogeneous and high-frequency ambient noise is rich in the icy environment (e.g., Baker et al., 2019; Chaput et al., 2018; Minowa et al., 2019; Olinger et al., 2019 and more details in the discussion).

In the above example, the spectral whitening width, which is a relevant but subjectively chosen controlling parameter in our autocorrelation processing, is fixed to urn:x-wiley:21699313:media:jgrb54798:jgrb54798-math-0023 Hz. In Figure 7, we demonstrate the robustness of the spectral features over a wide possible range of the spectral width W. Autocorrelation stacks for the 2-year earthquake dataset are constructed for whitening widths ranging from 0.25 to 1.75 Hz with an incremental step of 0.25 Hz. In the spectral and temporal stacks, we observe a clear variation of relative amplitudes among resonance peaks, but not their position along the frequency and time axis. The amplitude variation is ignored when using the correlation coefficient as a measure of fit in the search for the ice and water thickness parameters. Therefore, thickness estimates can be reliably obtained by repeating the grid-search for the set of spectral width values.

4.2 Layer Thickness Estimates for Other Stations

Among 34 stations in the Ross Ice Shelf experiment, seven stations were deployed on the grounded ice. Because they are not the main focus of this paper, we show the temporal autocorrelation results for the stations RS13 and RS14 in Appendix A. According to Bedmap2 dataset, the ice cap beneath other five stations is less than 1,000 m thick (see Table S1), which is below the resolution power of the earthquake-based autocorrelation method, which performs well for ice thickness larger than around 1,000–1,500 m (Phạm & Tkalčić, 2018; Yan et al., 2020). This lower threshold allows necessary separation between the reflection pulse and the central peak that is typical in any autocorrelogram. Additionally, we also do not report results for the stations DR01, DR02, and DR03 as they are deployed near the edge of the Ross ice shelf (e.g., Baker et al., 2019), in which the assumption of the 1D stratified layers underlying these stations does not hold.

We process P-wave coda datasets to construct spectral autocorrelograms for the remaining 24 stations sitting on the floating ice shelf. Half of the stations show at least two resonant peaks clearly in their spectral response of similar quality as in the pilot station DR14. For those, we further apply the grid-search to estimate ice and water layer thickness. The other half's spectral responses either have only the first resonant peak clear or show noisy patterns. Ice and water thickness estimates are summarized in Table 2, in which ice-thickness estimates vary from 300 to 500 m in those 12 stations, while values for water layer vary from 500 to 700 m. In the Supplementary Material, we report the spectral matching and grid-search results for the “good” stations (Figures S3–S13), and spectral autocorrelation stacks of the other stations that we found to be noisy (Figure S14).

Table 2. Summary of Results for 12 Ice-Shelf Stations and Two Grounded Ice Stations That We Characterized as of “Good” Quality
Ice thickness (m) Water thickness
Station Latitude Longitude Bedmap2 This study Confidence Difference (%) Bedmap2 This study Confidence Difference(%)
DR05 −78.63 −179.09 301 370 330–410 23 459 460 450–480 0
DR06 −78.79 −179.71 319 430 400–450 35 468 500 490–510 7
DR07 −78.93 179.20 313 390 350–430 25 515 610 600–620 18
DR08 −78.95 179.66 315 350 310–380 11 509 540 530–560 6
DR09 −78.96 179.89 326 400 360–460 23 489 460 450–470 −6
DR10 −78.97 −179.88 328 390 360–430 19 476 510 500–520 7
DR12 −79.01 −179.92 330 370 340–400 12 475 530 520–540 12
DR13 −79.05 −179.97 330 420 390–450 27 469 500 480–510 7
DR14 −79.14 179.95 327 320 280–370 −2 462 540 520–550 17
DR15 −79.49 −179.92 340 460 420–490 35 434 460 450–480 6
RS01 −78.18 169.96 239 460 410–480 92 681 670 650–700 −2
RS02 −78.49 173.35 327 430 380–500 31 535 620 580–650 16
RS13 −78.75 −144.01 1369 1463 7 0
RS14 −78.47 −140.46 1630 2009 23 0
  • Thickness measurements are in meters, the difference percentages are with respect to Bedmap2 data. The uncertainty associated with Bedmap2 ice thickness varies from places but is ∼100 m (Fretwell et al., 2013; Griggs & Bamber, 2011), while the uncertainty of the water thickness is not reported explicitly but it is much larger than ∼100 m.

Figure 8 is a comparison of thickness estimates obtained in this study with those extracted from Bedmap2 (Fretwell et al., 2013), which consists of seamlessly gridded datasets of ice surface elevation, ice thickness and bed elevation over the entire continent of Antarctica. In Bedmap2, ice surface elevation is compiled from data of multiple existing digital elevation models for parts or the continent (e.g., Bamber et al., 2009; Cook et al., 2012). The majority of direct thickness measurements for the ice cover in land were acquired using the radio-echo sounding method (see Bingham & Siegert, 2007; Plewes & Hubbard, 2001 for comprehensive reviews). Floating ice-shelf thickness was, on the other hand, indirectly measured by deducting from satellite altimetry for ice height above the sea level using the principle of hydrostatic equilibrium (see Griggs & Bamber, 2011 for great details of this method). Most recently, Das et al. (2020) presented an independent new dataset of ice thickness of the Ross Ice Shelf through extensive sounding measurements obtained via the ROSETTA-Ice project. High-resolution bed elevation beneath the grounded ice is obtained by merely subtracting ice surface elevation by ice cover thickness. However, seabed elevation underneath ice shelf was poorly constrained from spare seismic sounding campaigns (of ∼50 km pointwise apart) in the past (Albert & Bentley, 1990; Robertson & Bentley, 1990; Timmermann et al., 2010). Water thickness underneath the shelf is implied from surface elevation, ice-shelf thickness and seabed-elevation, therefore it is subjected to high levels of uncertainty. Thickness measurements from borehole drill HWD2 (Stevens et al., 2020; see location in Figure 1c) are valuable, but it is just the second drill in Ross Ice Shelf after 40 years.

Details are in the caption following the image

Comparison of estimated ice thickness and water thickness in this study with those extracted from Bedmap2 dataset (Fretwell et al., 2013). Estimates are from “good” ice shelf stations only. Orange circles denote ice thickness measurements and green diamonds denote water thickness measurements. Bedmap2 uncertainty is ∼100 m (Griggs and Bamber, 2011), while the measurement uncertainties from this study are listed in Table 2.

Our comparison shows a good agreement in water thickness between our estimates and the Bedmap2 data, and a large discrepancy in ice thickness comparison. In Table 2, the difference in terms of the water-thickness estimates is at most 20% relative to the Bedmap2 gridded data. This, reasonably good agreement, is on par with the previous studies using passive data to constrain subice shelf water thickness (Diez et al., 2016; Zhan et al., 2014). From our point of view, this could be the most appealing aspect of the method as the resonant signals in the spectral autocorrelograms are sensitive to the water thickness (see Section 3.2), which is not yet constrained well due to the lack of good seabed-elevation models.

However, the discrepancy in ice-shelf thickness with Bedmap2 is significant in most stations (up to 35%), and 85% for station RS01. Additionally, our estimated ice thickness is systematically thicker than Bedmap2 dataset. Even though the shelf thickness's uncertainty is reported at ∼100 m (Griggs & Bamber, 2011), the good comparison of independent ice thickness estimate from ROSETTA-Ice project (Table S2) confirms the high confidence in the quality of the ice shelf thickness values reported in both datasets. Thus, there is likely a systematic error in the simplified assumptions of homogeneously stratified media used in our approach. First, this is likely because spectral autocorrelograms are less sensitive to the ice thickness than the water dimension, as noted in Section 3.2. Second, this could be because we assume a homogeneous ice layer with a P-wave speed of 3.9 km/s, a reasonable estimate for the consolidated ice at depth. However, this value does not account for the possible radial variations within the ice column, especially near the top where there is a significant drop in P wave speed in a layer of unconsolidated snow (i.e., the firn layer). A possible solution for this is to account for depth-dependent velocity model to account for the firn (e.g., Diez et al., 2016), which is beyond the scope of the present study.

It is also noted that reasonable resonant patterns are observed at two ice-shelf stations: RS04 and RS16 (see Figures S15 and S16), but we think they the grid-search outcomes for layer thicknesses beneath them are not reliable. This is because the first resonant peak appears to be composed of multiple subpeaks. The match to each of them gives widely different results given the same level of fit to the second peak. Thus, we are reluctant to report the inverted results for these two stations and do not include them in the “good” stations group.

4.3 Discussion

We find an interesting analogy of the method presented in this study with the normal mode method widely used in global seismology and studies of tidal effects (e.g., Dziewonski & Gilbert, 1972; Lau et al., 2015). Prolonged reverberation of seismic energy emanated from a large earthquake excites long-period normal modes of the Earth, a finite body of approximately concentric spherical shells. In our local case, due to a very sharp negative impedance contrast between the water layer and the bedrock, compressional energy entering the ice-water system from steeply incident P waves is efficiently trapped for an extended period of time. The trapped energy, in turn, vibrates the finite stratified system and excites free oscillations. Therefore, this case study can be seen as an example of free oscillation study, but at the short-period end of the spectrum.

The emergence of multiple resonant peaks in the spectral autocorrelation constructed with spectral whitening (Equation 8) reinforces the advantages of the operation against a similar approach that uses spectral ratio between radial and vertical (or vice versa) spectral ratios (e.g., Sánchez-Sesma et al., 2011; Yan et al., 2018; Zhan et al., 2014) to characterize local site effects. Similar to Phạm and Tkalčić (2018), we demonstrate that high-order resonant peaks can be retrieved through the division of the original spectrum by its smoothing average. This operation is applicable to single recording components (e.g., only vertical or only horizontal components). As demonstrated in this study, the identification of higher-order resonance peaks allows us to resolve more complicated structures of a floating ice shelf rather than characterizing site effects under the assumption of a prominent resonant layer, for example, a sedimentary basin, as in the classical spectral ratio methods.

On a less positive note, the fact that a significant portion of the potential stations—nine of them—do not present the predicted resonant pattern for the flat ice-water system could somewhat weaken the appealing potential of the method proposed here. As shown in Figure S14, most of their spectral autocorrelograms show a clear first resonant peak, but the second and third peaks are not detectable, while other show a noisy spectral pattern. We noted in Section 3.2 via synthetic experiments, the absence of the high-order peaks might be due to a high level of incoherent noise, which is most likely present in the ice shelf environment due a high level of local glacial seismicity (e.g., Minowa et al., 2019; Olinger et al., 2019) and atmospheric activity (e.g., Baker et al., 2019; Chaput et al., 2018). Another possibility to cause a destructive contribution to the second resonant peaks could be due to rough interfaces of ice-water, water-seabed or both.

There are exciting but challenging questions relating to modeling and subsequently constraining attenuation factor in ice and water layers via the observation of the resonant peaks. The use of resonant peaks could be compared with the deep Earth attenuation studies via the normal modes. However, we caution that our method at this stage is not suitable to put meaningful constraints on attenuation properties though either absolute or relative amplitudes of the resonant peaks. First, we use a nonlinear operation, the spectral whitening (Equation 1), to reveal and amplify resonant peaks. Even though the operation is necessary because the source time function signature is strong in P-wave coda signals, it discards the actual resonance amplitudes. It only retains their significance and coherency in their vicinity (Phạm & Tkalčić, 2017). Second, in the local case, the resonant amplitudes are strongly affected by the background noise level at the recording sites (Figure 4). Thus, it requires precise knowledge of the background noise and its effect on resonant amplitudes. For these reasons, we preventatively do not aim to consider the attenuation effect in this study's scope.

5 Concluding Remarks

This study presents a spectral autocorrelation technique development and its application to the datasets from the Ross Ice Shelf experiments. We advocate the use of the spectral autocorrelation constructed from teleseismic earthquakes, which is equivalent to the temporal autocorrelation already widely employed in seismology. The observed resonance peaks can be explained by the theoretical resonance that exist in a layered structure consisting of ice and water layers over a half-space bedrock. A grid-search to match the observed with theoretically predicted, analytical spectral responses results in the thickness estimates for those layers. There is a relatively good agreement in water thickness with the Bedmap2 datasets obtained from nonseismological natures. The current study in conjunction with the previously developed technique that works well for the grounded ice stations present a great potential for future studies of polar ice caps and future space missions.

In retrospect, we propose a new measure to treat the observed autocorrelation as a direct observable instead of revoking the concept of reflectivity of a virtual source. The direct treatment of the autocorrelation in the spectral domain is in line with recent findings in correlation studies that the correlograms (or autocorrelograms in this study) are not equivalent to the reconstructed Green's function structure (e.g., Fichtner et al., 2017; Phạm et al., 2018; Sager et al., 2020; Tkalčić et al., 2020). This recognition lays ground for an emerging concept of “correlation seismology,” in which the correlation function can be treated as a direct observable for seismological inferences. Furthermore, this concept will most likely play an essential role in future studies of highly complicated Earth structures or heterogeneous source distribution and future planetary missions (e.g., Panning et al., 2018; Stähler et al., 2018).

Acknowledgments

The authors thank two reviewers for constructive comments that significantly improve this manuscript's quality. The facilities of IRIS Data Services, and specifically the IRIS Data Management Center, were used for access to waveforms, related metadata, and/or derived products used in this study.

    Appendix A: Results for Grounded Ice Stations

    There are seven stations in the experiment deployed on the grounded ice cap (marked by black hexagons in Figure 1). We apply the processing procedure from Phạm and Tkalčić (2018) to vertical waveform data recorded at these stations. Figure A1 shows temporal autocorrelation results for stations ST13 and ST14, which reveal negative-polarity signals at ∼0.7  and ∼1.0 s in the stacked autocorrelograms. The delays times are roughly the two-way travel time of P waves in the ice layer and are equivalent to some 1,450 and 2,000 m of ice beneath their sites assuming a homogeneous P-wave speed in the ice of 3,900 m/s (Kohnen, 1974). The measurements are slightly thicker but still agree reasonably well with Bedmap2 (see Table 2). At the other five grounded ice stations, the ice caps beneath their sites are thinner than 1,000 m, according to Bedmap2. Such thin structures are below the resolving power of the method being used with passive seismic data.

    Details are in the caption following the image

    Temporal autocorrelation computed for vertical components of two grounded ice stations ST14 and ST13. (a) One-sided autocorrelation functions of individual earthquakes are plotted as a function of epicentral distance to station ST14. (b) Autocorrelation stack is computed from all individual autocorrelation shown in (a). Panels (c and d) similar to (a and b) but for station ST13.

    Data Availability Statement

    Waveforms data are collected and made available thanks to Collaborative Research: Dynamic Response of the Ross Ice Shelf to Wave-Induced Vibrations and Collaborative Research: Mantle Structure and Dynamics of the Ross Sea from a Passive Seismic Deployment on the Ross Ice Shelf (https://doi.org/10.7914/SN/XH_2014).