# Theoretical Solution and Applications of Ocean Bottom Pressure Induced by Seismic Seafloor Motion

## Abstract

Seismic signals captured by ocean bottom pressure sensors, which are designed to record tsunami waves, are largely ignored. In this paper, we derive a simple theoretical solution of the ocean bottom pressure as a function of prescribed seafloor motion. All the assumptions are clearly stated and analyzed. The solution is checked by comparing the seafloor displacement and pressure from three *M*7+ earthquakes, recorded by ocean bottom seismometers and pressure gauges located off the Japanese coast. We then show two applications. First, using the seafloor displacement data recorded by an ocean bottom seismometer, the pressure amplitude recorded by the associated pressure gauge is corrected, and vice versa. Second, pressure recordings from Deep Ocean Assessment and Reporting of Tsunamis during the 2011 Tohoku earthquake are converted to seafloor displacements, which are then utilized to estimate the earthquake focal mechanism. Thus, we demonstrate that seismic signals recorded by pressure sensors have great potential for fast estimate of earthquake source parameters.

## Key Points

- We derive the theoretical equation of ocean bottom pressure induced by seismic waves propagating on seafloor
- The equation is checked using observations of three earthquakes recorded by ocean bottom seismometers and pressure gauges
- The equation has two applications: correction between seafloor displacements and pressure and estimate of focal mechanism using pressure data

## 1 Introduction

Pressure gauges have been deployed on the bottom of ocean to track the ocean bottom pressure changes for various purposes. For example, the Deep Ocean Assessment and Reporting of Tsunamis (DART) system (Gonzalez et al., 1998; Titov et al., 2005), designed and operated by the National Oceanic and Atmospheric Administration (NOAA), provides real-time recordings of tsunami waves for early warning purposes. The tsunami data are routinely adopted in finite-fault inversions to investigate earthquake source parameters (e.g., An et al., 2014; Fujii et al., 2011; Heidarzadeh et al., 2016; Lay et al., 2014; Melgar et al., 2016; Satake, 1987; Satake et al., 2013; Wei et al., 2008). Also, a pressure gauge, either a differential pressure gauge (DPG) or an absolute pressure gauge (APG), is often installed with an ocean bottom seismometer (OBS) as an alternative channel for the vertical seismic channel. These pressure gauges also capture the ocean surface waves during an earthquake, and the data have been used to study tsunami characteristics (Gusman et al., 2016; Sheehan et al., 2015). The above mentioned observations are ocean bottom pressure change as a result of water waves on the sea surface. On the other hand, seismic waves propagating on the seafloor also cause ocean bottom pressure change. Relevant pressure records have long been recognized to be associated with Rayleigh waves (An & Liu, 2014; Filloux, 1982; Rabinovich et al., 2011). However, with exception of estimating the propagation speed, there is limited quantitative analysis of the pressure signals, especially for the DART pressure gauges where OBSs are missing to measure the seafloor motion.

Filloux (1982) first observed clear separation of Rayleigh R1 and tsunami waves in a bottom pressure record. He also suggested that, if the period of the seafloor motion is long compared to the travel time of sound across the ocean column, the pressure *p* associated with Rayleigh waves simply accelerates the water layer, that is, *p* = *ρ**h**a* (*ρ* sea water density, *h* water depth and *a* seafloor acceleration). He did not, however, provide any rigorous derivation of the relation. Ritsema et al. (1995) derived analytical solutions of ocean bottom pressure as a function of point source earthquake rupture and successfully inverted tsunami waves for earthquake source mechanism. Other relevant analytical studies are mostly related to the tsunami generation process, that is, as the seafloor deforms suddenly during an earthquake and elevates water, the ocean bottom pressure in the source area reflects the details of the earthquake rupture and initial water elevation (Kajiura, 1963; Saito, 2013, 2017; Saito & Tsushima, 2016). Many other studies take water compressibility into consideration, and thus, sound wave is included in the solution (e.g., Kajiura, 1970; Maeda & Furumura, 2013; Nosov, 1999; Sells, 1965), because the time scale of the seafloor deformation during an earthquake is comparable to the travel time of sound across the water depth. In this study, we ignore water compressibility and focus on the ocean bottom pressure caused by seismic waves propagating on the seafloor, instead of seafloor deformation during the earthquake rupture process. Thus, the boundary conditions at the seafloor are seismic waves, which allow us to assume periodic seafloor motion, leading to explicit solution of the ocean bottom pressure. Essentially, the relation proposed by Filloux (1982), *p* = *ρ**h**a*, is recovered. The derivation provides all the information on the necessary approximations and their justifications. In this study we also check the accuracy of the simple relation with three recent earthquake events.

Pressure records from DPGs and APGs installed with OBSs are widely used to study the infragravity waves in the ocean and to remove noise from the OBS seismograms (e.g., Bell et al., 2015; Crawford et al., 1991; Webb & Cox, 1984). Some studies utilize the data to investigate the underground Earth properties (e.g., shear modulus and shear velocities; Crawford et al., 1998; Yamamoto & Torii, 1986). However, DPG records often suffer from amplitude errors. For example, by matching the measurements of leading tsunami waves with model simulations, Sheehan et al. (2015) pointed out that wave amplitudes recorded by DPGs are up to a factor of 2 smaller than the synthetics. In this study, the theoretical solution of the ocean bottom pressure is used to provide accurate and straightforward corrections to the DPG data.

Besides, seismic signals from DART sensors are largely unused, but in this study we show that they can potentially contribute to fast estimate of earthquake magnitude and tsunami genesis. Land-based broadband seismometers sometimes saturate during large earthquakes if they are located close to the earthquake source (e.g., the 2011 Tohoku earthquake; Hoshiba & Ozaki, 2014), so researchers have been looking for alternative ways for rapid and accurate estimate of earthquake moment and focal mechanism (e.g., Melgar et al., 2015). DART data are not only free of saturation but also offer data coverage from the ocean side for subduction earthquakes. The fourth generation of DART (DART4G, http://nctr.pmel.noaa.gov/Pdf/brochures/dart4G_Brochure.pdf) will install pressure gauges near trenches and sample at a relatively high frequency (possibly 1 Hz; Y. Wei, personal communication, 2017),so the real-time recordings of ocean bottom pressure are of high potential for fast estimate of earthquake focal mechanism and evaluation of tsunami generation.

## 2 Theory

We consider a two-dimensional problem with horizontal and vertical coordinates being denoted by *x* and *z*, respectively (Figure 1). For simplicity the ocean is assumed to have a uniform still water depth of *h*. During an earthquake, seismic waves propagate on the seafloor and generate vertical seafloor displacement *D*(*x*,*t*) or vertical velocity *ζ*(*x*,*t*) = *∂**D*(*x*,*t*)/*∂**t*. The seafloor motion causes change of the sea surface elevation *η*(*x*,*t*), the velocity field **v**(*x*,*z*,*t*), and fluid pressure *p*(*x*,*z*,*t*). In this section, the relationship between the dynamic pressure at the seafloor and the seafloor displacement is sought after.

*ϕ*(

*x*,

*z*,

*t*) can be introduced such that the velocity field is . The governing equations and boundary conditions of the flow motions are written as (e.g., Mei, 1989) where

*ρ*is the constant fluid density and

*g*is the gravity acceleration. Note that the boundary conditions have been linearized to be applicable at

*z*= 0 and

*z*=−

*h*, which requires that

*η*≪

*h*and

*D*≪

*h*. The problem with a prescribed transient seafloor motion can by analyzed based on Laplace and Fourier transform method (e.g., Mei, 1989), and the details are given in the supporting information. Here we seek for special solutions by assuming a periodic seismic wave propagating on the seafloor.

*k*

_{1}, angular frequency

*ω*

_{1}, seafloor vertical velocity, and displacement can be written as

*ω*

_{1}/

*k*

_{1}. From the boundary condition at the seafloor (1d), the velocity potential

*ϕ*(

*x*,

*z*,

*t*) has the same phase as the seafloor motion, that is,

*k*

_{1}

*h*≪ 1 and is approximately 1. Thus, using (2b), 5 can be simplified and written in terms of seafloor displacement

*D*(

*x*,

*t*) or acceleration

*a*(

*x*,

*t*) as

Intuitively, this result is not surprising: the dynamic pressure on the seafloor is the result of accelerating the entire water column above the unit seafloor area. However, the present analysis provides a clear evidence that this result is valid only if (1) the water is assumed to be incompressible, (2) the seafloor displacement and water elevation are much smaller than the water depth, (3) the speed of seismic waves is much larger than that of the surface water waves, and (4) the water depth is much smaller than the wavelength of the seismic waves.

## 3 Validation of the Solution

In the past, seafloor displacements and dynamic pressure have been compared in the frequency domain to show high consistency in certain frequency ranges (Bolshakova et al., 2011; Matsumoto et al., 2012). In a most recent study, Saito (2017) numerically simulated the tsunami generation process in a compressible fluid. He showed that, for a slow rupture in which water compressibility can be ignored, the simplified solution 6 compares well with the fully physical simulation if the wavelength of the seafloor motion is much larger than the water depth (i.e., *k*_{1}*h* ≪ 1). In this section, we use field observations of seafloor displacement and ocean bottom pressure to check the accuracy of our solution 6. The data include displacement and pressure recordings of three earthquakes from a station operated by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) (Iwase et al., 2003; Kaneda, 2010; Kawaguchi et al., 2008) (https://www.jamstec.go.jp/donet/e/). There are three sites with deployment of sensors—Kushiro, Hatsushima, and Muroto. Hatsushima is the only one where the OBS and APG are installed at the same location. At the other two sites, the two sensors are apart by a few kilometers so that the recordings are not suitable for direct comparison. Therefore, we only examine the recordings at Hatsushima. Since the online data are available to the public after 1 November 2015 (Japan standard time), we select three large earthquakes after that time. They are the 2 March 2016, *M*7.8, Southwest of Sumatra, Indonesia earthquake; the 15 April 2016, *M*7.0, Kumamoto, Japan earthquake; and the 29 July 2016, *M*7.7, Northern Mariana Islands earthquake. The location of the three earthquakes and the site of the OBS and pressure gauge at Hatsushima are shown in Figure 2.

The raw data are first converted to physical quantities (displacement in meters and pressure in Pascals) by removing instrumental response. The static pressure of the still water column is then subtracted from the total pressure to give the dynamic pressure induced by the seafloor motion. Both the seafloor displacement and the pressure are then band-pass filtered between 0.02 and 0.2 Hz. Comparison of the pressure recordings and theoretical calculations from seafloor displacement is shown in Figure 2.

Compared with the Kumamoto earthquake, the other two earthquakes have either a larger epicentral distance (the Sumatra earthquake) or a larger focal depth (the Mariana earthquake), so the pressure disturbance induced by those two earthquakes is smaller than that by the Kumamoto earthquake. But it is clearly seen that, for all three earthquakes, the calculations using equation 6 match the observations very well, in terms of both amplitude and phase. In the supporting information, we also compare the pressure records with the theoretical calculations at different frequency bands, that is, 0.02–0.05 Hz, 0.05–0.1 Hz, and 0.1–0.2 Hz (Figures S2 to S4, middle three plots). Excellent matching is found for all frequency bands. For the Sumatra earthquake, although Figure 2 only presents the surface waves at late arrivals, shown in Figure S4 for a relatively higher-frequency range of 0.1–0.2 Hz, the *P* waves at early arrivals also perfectly match our solution. It is found that equation 6 works well for pressure as low as approximately 50 Pa, which corresponds to 5 mm water head.

## 4 Applications

### 4.1 Correction Between Displacement and Pressure for OBSs

We download the OBS data during the 2012 *M*7.8 Haida Gwaii earthquake from the IRIS Data Management Center (IRIS DMC). The OBSs are managed by the United States National Ocean Bottom Seismograph Instrument Pool, which has three institutional instrument contributors (Sheehan et al., 2015): Scripps Institution of Oceanography (SIO), Woods Hole Oceanographic Institution (WHOI), and Lamont Doherty Earth Observatory (LDEO). The pressure sensor is DPG for SIO and WHOI and APG for LDEO. As previously pointed out, DPGs installed by SIO and WHOI often suffer from amplitude errors. APG data from LDEO are correct, but the corresponding OBSs provide inaccurate amplitudes of seafloor displacements due to errors of instrumental response. In this section, we illustrate that equation 6 can be used to correct either the displacement or the pressure amplitude provided that one of them is correct. All the data are filtered between 0.02 and 0.2 Hz. Figure 3 (left) shows the focal mechanism of the Haida Gwaii earthquake and the locations of OBSs. Here we demonstrate the correction by using only two observation sites as an example for SIO, WHOI, and LDEO, respectively. A complete list of stations and the correction can be found in the supporting information (Figures S5 to S23).

For SIO and WHOI sensors, the amplitude of DPG data is incorrect (see, e.g., Sheehan et al., 2015) and the correction is shown in the first to fourth panels of Figure 3 (right). Since the seafloor displacement recorded by SIO and WHOI OBSs is correct, the corresponding pressure can be calculated from equation 6, which is plotted in blue thick lines. The incorrect pressure records are plotted in black thin lines. A constant coefficient is used to reduce the magnitude of the pressure so as to match the theoretical calculations, plotted in red thin lines after the correction. The constant is obtained using a least squares method to minimize the residual between pressure records and theoretical calculations. It is seen that the pressure after correction matches very well with calculations. We remark that the constant for all the SIO stations is negative, that is, the polarity of the SIO DPG records is reversed in addition to amplitude error. This is also observed by Sheehan et al. (2015) at the same SIO stations by comparing the pressure records of tsunami waves with numerical simulation. For the LDEO data, the seafloor displacement records are significantly smaller than the nearby SIO or WHOI records, so it is inferred that the LDEO OBSs provide incorrect displacements due to errors of instrumental response. The correction of LDEO displacement is shown in the fifth and sixth panels in Figure 3 (right). The pressure recorded by LDEO APGs is correct, which is plotted in blue thick lines. The theoretical pressure calculations based on OBS data are plotted in black thin lines, which do not match the APG observations. The calculations are corrected by multiplying a constant, plotted in red thin lines, to match the APG observations.

The correction for different frequency bands at all stations is given in the supporting information (Figures S6 to S23). The correction coefficients for different frequency bands are compared in Figure S24. Figure S24 also includes the correction coefficients for SIO and WHOI DPG records obtained from tsunami waves, which are obtained by comparing the DPG records with numerical simulations of tsunami waves (Sheehan et al., 2015). It is found that the coefficients vary slightly with frequency for SIO and WHOI DPGs, and it is almost constant for LDEO OBSs.

### 4.2 Estimating Earthquake Parameters Using DART Data

Another important application of equation 6 is to utilize DART pressure records to estimate earthquake source mechanism. We take the 2011 Tohoku earthquake as an example to illustrate that reasonable earthquake magnitude and rupture mechanism can be derived from DART records of seismic waves (Figure 4). The adopted DART data are the ocean bottom pressure associated with the seismic waves preceding the long-period tsunami waves (see Figure S25). The original data are collected at 39 DART stations; we select 12 of them based on data quality and perform an inversion for the earthquake focal mechanism. The original records of pressure are plotted in Figure S26 in the supporting information. Since the sampling rate is very low, which is about one data sample per 15 s, we filter the data between 0.01 and 0.02 Hz (50–100 s) to only include long-period components. We note that for large earthquakes, to derive a point source solution, longer wave period (>100 s) is necessary to accurately determine the earthquake magnitude (e.g., Stein & Okal, 2005). For the 2011 Tohoku earthquake of magnitude ∼9, the source size could be a bit large for wave period 50–100 s, possibly leading to notable estimate error of the earthquake moment. Nevertheless, considering that the wavelength of such a wave period band is close to the rupture dimension, we think that it is acceptable to use 50–100 s for rapid source characterization. In addition, we point out that waves of period >100 s in the DART data are contaminated by the ocean surface water waves, which is another reason why these components are excluded from the inversion. The pressure is first converted to seafloor acceleration using equation 6 and then integrated to vertical displacement. In previous sections, all the data have been analyzed between 0.02 and 0.2 Hz; however, equation 6 is also valid for lower frequencies, if the pressure records are resulted from seismic seafloor motion rather than ocean surface water waves. Details are given in section 5.

The Cut and Paste method (Zheng et al., 2009; Zhu & Helmberger, 1996) is adopted to seek the optimized earthquake focal mechanism in a least squares sense. Since the epicentral distances are far for the 12 DART stations and the period of the signals is long, we can apply the global frequency wave number method (F-K method; Zhu & Rivera, 2002) to calculate Green's functions. We fix the epicenter location and initial rupture time according to the Japan Meteorological Agency solution and vary the focal depth, earthquake magnitude, strike, and dip and slip angles to search for the best solution. Our results yield an earthquake focal depth of 20 km, magnitude of *M*_{w}9.36 (earthquake moment 1.23 × 10^{30} Nm) and fault plane of strike = 191°, dip = 20°, and rake = 68°. For comparison, the global centroid moment tensor (gCMT) solution estimates the earthquake focal depth to be 20 km, magnitude of *M*_{w} 9.11, and fault plane of strike = 203°, dip = 10°, and rake = 88°. Owing to the bad azimuthal coverage of data, very low frequency, and only vertical components, our solution does not exactly match the gCMT solution. The earthquake magnitude is estimated to be larger than the gCMT solution, and the rake angle is smaller, indicating less thrust rupture component. Nevertheless, the two parameters are still important for determining the probability of tsunami genesis and for fast tsunami warnings. The next generation of DART will record data at a relatively high sampling rate, and thus, more high-frequency signals can be included in the inversion, leading to more accurate estimate of earthquake focal mechanism. Additionally, our illustrating example uses DART stations far from the earthquake source, which requires a relatively long time to record seismic waves. More DART sensors near trenches will surely fasten the evaluation of earthquake source and tsunami genesis.

## 5 Discussion

In the above analysis, we have shown that equation 6 works well if the seismic frequency is between 0.02 and 0.2 Hz. If frequency is lower than 0.02 Hz, the pressure records are highly influenced by ocean surface waves or “infragravity waves” (e.g., Dolenc et al., 2007; Webb, 1998; Webb et al., 1991). These are surface water waves that have relatively long wavelength and can thus penetrate the ocean column to create detectable pressure at the seafloor. However, the corresponding seafloor deformation is very small and it is not easily detectable. Because of the surface water waves, the pressure records appear very noisy at frequencies lower than 0.02 Hz (e.g., 0.01–0.02 Hz, top panels of Figures S2, S3, and S4 and Figures S7, S13, and S19), and the seismic signals are buried in the background noise. For example, in Figure 5a, we show the pressure recordings and theoretical calculations at LDEO station J33B, filtered between 0.01 and 0.02 Hz. The pressure recorded by APG is noisy due to influences of surface water waves; the theoretical calculations from vertical seafloor displacements, after correction, do not match the pressure recordings. Webb et al. (1991) find that the period of such surface water waves is generally greater than 30 s (0.03 Hz), which is very close to the lower bound of our frequency range (0.02 Hz, 50 s). If the noise level is low and the seismic signals are conspicuous in the pressure records, equation 6 is still valid, as shown in Figure 5c at LDEO station FS13B (also see Figures S7, S13, and S19).

At higher frequencies (e.g., 0.2–0.4 Hz, bottom panels of Figures S2, S3, and S4 and Figures S11, S17, and S23), we observe that pressure records are still closely associated with seismic signals, but there exist phase lag as well as amplitude mismatch between the two. This is likely to be attributed to the manifestation of water compressibility at high frequencies. In such a high-frequency range, the period of the seafloor motion (2–5 s) is comparable to the travel time of sound in water through the water column (2 s), by assuming sound speed of 1.5 km/s in water and water depth of 3 km. Therefore, water cannot be treated as incompressible any more and sound waves must be incorporated in the formulation. Saito (2017) reported that the pressure is proportional to the seafloor velocity when accounting for water compressibility but only for the first motion. In Figure 5b, it is seen that in relatively shallow waters (station J33B, ∼350 m), where the travel time of sound in the water column is small compared to the period of seafloor motion, equation 6 is valid for the frequency range 0.2–0.4 Hz. On the other hand, in relatively deep waters in Figure 5d (station FS13B, 2,332 m), equation 6 does not apply for frequency range 0.2–0.4 Hz. By comparison of LDEO stations J33B and FS13B, it is found that, for low frequencies, equation 6 is more likely to be valid in deep waters, where it is more difficult for surface waves to penetrate the water column; for high frequencies, equation 6 is more likely to be valid in shallow waters, where water can be safely assumed to be incompressible.

In the theoretical derivation of equation 6 we have ignored the effect of finite wavelength, that is, term
, on the basis of long-wave assumption. According to the above discussion, to ignore water compressibility, it is required that the seismic period, *T* = 2*π*/*ω*_{1}=2*π*/(*k*_{1}*c*_{R}), is much larger than sound travel time across the water column, 2*h*/*c*_{s} (*c*_{R} Rayleigh wave speed and *c*_{s} sound speed in water). This leads to 2*π*/(*k*_{1}*c*_{R}) ≫ 2*h*/*c*_{s} or *k*_{1}*h* ≪ *π**c*_{s}/*c*_{R}. So the long-wave assumption, *k*_{1}*h* ≪ 1, is naturally satisfied if water compressibility can be ignored. Thus, the number of assumptions required for equation 6 to be valid can be reduced from 4 to 3, that is, (1) the water is assumed to be incompressible (period of seafloor motion is much larger than sound travel time in water layer), (2) the seafloor displacement and water elevation are much smaller than the water depth, and (3) the speed of seismic waves is much larger than that of the surface water waves. Note that in this study the forcing term is the seismic waves propagating on the seafloor, so the wavelength and period of the forcing term are closely associated by the seismic speed, that is, *ω*_{1}/*k*_{1}=*c*_{R}. The lower the frequency, the longer the wavelength, because the wave speed is nearly constant (e.g., 2–8 km/s). Saito (2017) investigated the tsunami generation process in which the forcing term is the seafloor deformation during the earthquake rupture. Under such circumstances, even if the deformation is slow to neglect water compressibility, it can still be small enough in length for term
to play an important role.

## 6 Conclusion

To conclude, in this paper, we derived the theoretical solution of ocean bottom pressure as a function of seafloor motion, which shows that the pressure is proportional to the seafloor acceleration with coefficient related to water density and depth. The solution is verified through comparison of realistic displacement and pressure records. Field data show that the solution matches the observations very well for frequency range 0.02–0.2 Hz. At lower frequencies, the solution is still valid, although pressure records are likely to be influenced by noises of infragravity waves; at higher frequencies, the solution does not apply and water compressibility must be considered. The solution can be used to convert the ocean bottom pressure recordings to seafloor displacement (or acceleration), which can be adopted in an inverse procedure to constrain earthquake source parameters. With the next generation of DART instruments installed near trenches, the solution will be useful to utilize ocean bottom pressure data for fast estimate of earthquake magnitude and tsunami genesis.

## Acknowledgments

This work made use of the GMT software. The Japanese data are acquired from JAMSTEC website (https://www.jamstec.go.jp/donet/e/); the SIO, WHOI, and LDEO OBS data are downloaded through the IRIS Data Management Center (IRIS DMC); the DART data are obtained from NOAA website (http://www.ndbc.noaa.gov/dart.shtml). The authors thank Tatsuhiko Saito, Takeshi Nakamura, Hiroshi Tsuruoka, Mohammad Heidarzadeh, Shiqing Xu, and Tomoki Watanabe for help on acquiring and processing the Japanese data. We also thank Aditya Gusman for sharing the correction coefficients of DPG data from tsunami waves. Thanks are extended to Han Yue, Yong Wei, Shiqing Xu, Tatsuhiko Saito, and Hua Liu for useful discussions. Two anonymous reviewers and the Editor provided valuable comments that improved this paper. This work is supported by the National Nature Science Foundation of China (NSFC) grant 11632012 (C. An), NSF grant 0841074 (C. Cai), NSFC grant 41574057 (Y. Zheng), and UCLA faculty research grant (L. Meng). Philip L. -F. Liu would like to acknowledge the support received from Cornell University (USA), from the National Research Foundation, Marine Research and Development Programme, Singapore (award MSRDP-05), and from Earth Observatory in Singapore (award RCA-16-162-NUS-EOS).