We have developed an underwater gravity measurement system that uses an autonomous underwater vehicle (AUV) for exploration of seafloor mineral deposits. An improved air/sea gravimeter mounted on a gimbal mechanism in a pressure capsule 50 cm in diameter was installed in the AUV Urashima. We carried out 11 AUV dives in Sagami Bay as well as in deep-sea mineral deposit areas offshore Japan. The AUV was navigated at a constant speed of 2 knots and at either constant depth or constant altitude above the sea bottom. We obtained high-resolution Bouguer anomaly data through processing of gravity data, water pressure data, and the AUV’s navigational data, including pitch and roll motion. In addition to a vertical acceleration correction using a precise pressure meter with an in situ conversion factor from pressure to depth, three additional corrections were made to the gravimeter data: corrections for the effects of a spatial separation and a time delay between the depth sensor and the gravimeter, and an adjustment to the conversion from pressure increments to depth increments. These new corrections allowed us to obtain high-resolution data in a constant-depth survey of the southern Izena Hole at a depth of approximately 1550 m. The data had a 0.1 mGal rms crossover difference after leveling. This survey revealed two high Bouguer anomaly areas with amplitudes of 1–2 mGal. A model calculation suggested that the anomalies result from the presence of two buried cylindrical high-density mineral deposits. Our data processing method allows the detection of mineral deposits located at or just beneath the seafloor using gravity data collected aboard AUVs. However, further improvements in resolution are desirable, particularly for surveys in areas with rugged topography.
Although gravity measurements by sea-surface vessels are efficient, they cannot detect the gravity signatures of small-scale geologic features below the seafloor more than 500 m deep because of the large distances from the sources (Kinsey et al., 2013). Although high-accuracy gravity measurements can be obtained by on-bottom surveys, these measurements are time consuming because fewer than 20 stations can be measured per dive (Fujimoto et al., 2009; Kinsey et al., 2013). Recent advances in remotely operated vehicles (ROVs) and autonomous underwater vehicles (AUVs) provide new avenues for overcoming the drawbacks of the existing survey methods (Kinsey et al., 2013): One ROV or AUV dive can cover a survey area of more than with detailed track lines. We have developed an underwater gravity measurement system using an AUV to detect gravity signatures associated with high-density seafloor deposits (Fujimoto et al., 2011; Kanazawa et al., 2011; Shinohara et al., 2013, 2015a, 2015b). Although Kinsey et al. (2013) adopt 5 m as a typical altitude for a near-bottom survey, to ensure safe navigation, the altitude of an AUV survey ought to be closer to 50 m, even if detailed sea-bottom topography data are available in advance. Shinohara et al. (2018) use a model calculation to show that a survey conducted 50 m above the seafloor should be able to detect a gravity anomaly of approximately 0.4 mGal (or 0.2 mGal) given a typical-size deposit (defined therein as a high-density disk with a contrast of , a diameter of 400 m, and a center thickness of 20 m) located at (or 50 m below) the seafloor. We have developed a system composed of a gravimeter and a gravity gradiometer mounted in an AUV. Our resolution targets were 0.1 mGal for the gravimeter and 10 E () for the gradiometer. We chose the AUV Urashima of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) for this study due to its stable navigation performance. The Urashima has a large body (1.3 m in width, 10 m in length, and 1.5 m in height) with enough space to install the gravimeter and gradiometer (Figure 1; further details concerning the AUV Urashima may be found in Tsukioka et al.  and JAMSTEC ).
Araya et al. (2015) estimate the self-noise of the gradiometer system, which is composed of two identical gravity sensors with a vertical separation of 44 cm, to be 6 E at 2–50 mHz, but the gradient resolution during the first sea experiment was estimated to be 300 E due to the disturbance of the gradiometer platform and the noise level of the recording system. Further improvements to the operation of the platform and the sensor-recording configuration will allow the gradiometer to collect high-resolution data with a signal-to-noise ratio large enough to detect the gravity gradient above a typical ore deposit.
In this paper, after briefly introducing the gravimeter system, we describe the method used to process the gravimeter data, including the calculation of several correction factors, and discuss a subset of the experimental results.
We used the Micro-g LaCoste S-174 gravity sensor with an output rate of approximately 100 Hz. To acquire high-accuracy gravity data, the sensor must be kept at a constant temperature (60.4°C) and shielded from the effect of earth’s magnetic field. The former is accomplished by electrical heating combined with protective thermal insulation, and the latter is accomplished with a sheet of permalloy. The gravimeter is mounted on a gimbal system that is controlled by an inertial navigation sensor (IXSEA PHINS unit), and it is kept vertical (within 0.0004° under static conditions) with an efficient proportional-integral-derivative controller. The sensor and the gimbal system are housed in a pressure-proof spherical titanium alloy vessel 50 cm in diameter. When the AUV tilts, the sensor is kept vertical by rotating the gimbal system freely inside the sphere up to and in pitch and roll, respectively. Figure 2 shows an example of low-pass-filtered gravimeter tilt angle data obtained from the gimbal system and the Urashima’s tilt angle data during an experiment in 2015. The amplitude of the gravimeter pitch is generally 0.001° or less, and that of the gravimeter roll is much less than 0.001°. The estimated decrease in the observed gravity value is less than 1 μGal; thus, we omit tilt correction from our data processing procedure. The most important correction to data collected from a moving platform is the correction for the effect of vertical acceleration. To correct for this effect, we use a Paroscientific depth sensor with a sampling rate of approximately 40 Hz. This sensor is a transducer that measures ambient water pressure using resonant quartz technology and achieves measurement resolution better than 1 part per million of full scale. Further details of the gravimeter system may be found in Shinohara et al. (2018).
DATA PROCESSING METHOD
The data processing consists of low-pass filtering and several corrections. The first data processing step consists of the vertical acceleration correction and the corrections traditionally used for marine and sea-bottom gravity measurements (i.e., Eötvös, free-water, and Bouguer corrections). The second step addresses the accurate conversion of the water pressure data obtained by a depth sensor near the gravimeter to the vertical acceleration acting on the gravimeter. This is accomplished by correcting for the effects of the spatial separation and time delay between the depth sensor and the gravimeter and the error in the conversion from pressure increments to depth increments. We also use the crosscorrelation method (LaCoste, 1973) to determine linear correction parameters in the second step.
First step of the Bouguer anomaly calculation
Vertical acceleration correction
Eötvös and free-water corrections
Navigational data are obtained at a sampling rate of 1 Hz by the Urashima’s inertial navigation system (INS), assisted by a Doppler velocity log. The Eötvös and free-water corrections are calculated using the 1 Hz geographic and bathymetric data from the navigation system. Minor corrections are applied to the latitude and longitude data to account for the spatial separation between the gravimeter and the INS. In the case of the Urashima’s survey, the two instruments are approximately 4 m apart, leading to an Eötvös effect of up to during the AUV’s course changes for a speed of 2 knots.
Second step of the Bouguer anomaly calculation
Although the vertical acceleration, Eötvös, free-water, and Bouguer corrections are the major corrections, additional corrections lead to more accurate determination of gravity anomalies, mainly by addressing high-frequency noise. We investigated corrections for the following sources of error: the conversion factor from the water pressure increment to the depth increment, the effect arising from the gravimeter and the depth sensor not being perfectly collocated, and the effects of timing differences between data collected with different instruments.
Effect of spatial separation of the depth sensor and the gravimeter
Effect of time delay
After a laboratory test and the AUV Urashima’s two successful dives during the first sea experiment in 2012, which was conducted to evaluate the performance of the underwater gravimeter (Shinohara et al., 2018), we carried out nine sea dives (Figure 3; Table 1). Here, we describe the results of two experiments conducted in 2015, and we report a simple model calculation for the Bouguer anomaly data obtained in the second experiment.
Experiment in Bayonnaise Knoll in August 2015
In August 2015, two dives were carried out in the southeastern part of Bayonnaise Knoll in the Izu-Ogasawara Arc (Figure 4). The rapidly changing topography of the knoll is characterized by a caldera, and seafloor deposits have been found in the caldera rim of the southeastern part of the knoll (Iizasa et al., 2004). Using in situ CTD measurements, the conversion factor can be approximated as 98.99–0.058(P-8) m/MPa in this area (Figure 5). Measurements were made repeatedly along eight northeast–southwest transects and two east–west transects (Figure 4). The Urashima moved at a speed of 2 knots and maintained either a constant depth of 550–700 m (lines A1–A8, B3–B6, and C1–C8) or a constant altitude of approximately 50 m above the sea bottom (lines D1–D8). In the Bouguer anomaly calculation, we adopted a rock density of , and we applied a Gaussian low-pass filter with a 240 s filter width to the data to suppress high-frequency noise. Figure 6 shows the gravity data collected on 7 August 2015. The amplitude of the original data was generally greater than 1 Gal, but it was reduced to less than 1 Gal following the application of the low-pass filter. The gravity data collected during this experiment were of good quality throughout, except at 05:09 UTC on 6 August, when the gimbal system could not maintain verticality due to the AUV having inclined more than 30° in the pitch direction (Figure 7). The amplitudes of data components with periods smaller than 240 s (i.e., with spatial wavelengths shorter than 240 m for the AUV’s speed of 2 knots) were reduced after the application of the low-pass filter. The vertical acceleration signal obtained from the water pressure data is in good agreement with the gravimeter signal (Figures 7a and 8a). These signals display large amplitude variations when the AUV’s depth varies, i.e., along the constant-altitude tracks (lines D1–D8) and at the ends of the constant-depth tracks. Although the amplitude of gravimeter data was significantly reduced by the vertical acceleration correction, it remained in the range of 20–50 mGal. The free-water and Eötvös corrections further reduced the variability in the gravimeter data proportional to the AUV’s depth and the variability proportional to the AUV’s eastward speed, which can be calculated from its heading (Figures 7b and 8b). The Bouguer correction, which showed smaller amplitude variability, was calculated using the AUV’s depth data combined with the bathymetry data collected simultaneously by the multibeam systems of the AUV and the mother ship Yokosuka. Following these three corrections, the amplitude of the processed data was reduced to several mGal. The linear scale parameters obtained using the crosscorrelation method with a Gaussian high-pass filter with a width of 300 s, for the effects of the spatial separation , the gravimeter time delay , and the small error in the vertical acceleration , were 0.40 m, 0.38 s, and 0.0020 on 6 August, and 0.44 m, 0.44 s, and 0.0020 on 7 August, respectively (Figures 7e and 8e). The application of these three corrections greatly reduced the noise in the gravity data.
Figure 9a shows the Bouguer anomalies grouped by transect. Unfortunately, there is a problem of bias in the data on 6 August, and noise remained even after the application of the corrections, particularly in data along lines B3–B7. We subtracted from values along line A5, from lines A6 to A8, and from lines B3 to B7 (Table 2). Lines B3–B7 are at nearly the same depths as their counterparts, lines C8–C4 (Figure 9b), and the average variations of the Bouguer anomalies of the lines are similar, although lines B3–B7 include noisy, smaller scale variations. The rms differences between corresponding lines (e.g., lines C8 and B3) fall in the range of 0.2–0.3 mGal (Table 2). Lines C1–C8 are deeper than the corresponding lines A1–A8 by approximately 50 m, and the Bouguer anomalies of the former lines (particularly C3 and C4) are slightly higher than those of the latter in the middle parts of lines falling above the caldera floor. This tendency is very clear in lines D1–D8, which are at a constant altitude of approximately 50 m above the sea bottom and thus deeper than the other lines: The Bouguer anomalies in the middle parts of these lines are higher than the other lines by 1–3 mGal.
Figure 10a shows the Bouguer anomaly map made using the data collected along the constant-depth lines C1–C8. During most of the passes, the AUV remained at altitudes greater than 100 m over the deepest parts of caldera, which likely led to a reduction in anomaly amplitudes due to probable local sources just below the caldera floor. Figure 10b shows the Bouguer anomaly map created using data collected along the constant-altitude lines D1–D8. Compared with Figure 10a, higher amplitude anomalies are observed over the caldera floor in this map, reflecting the fact that the observations were made near the sea bottom. Although we do not attempt a detailed interpretation of the maps, we suggest that Figure 10a mainly shows larger scale anomalies that tend to increase northwestward toward the center of the knoll, and we suggest that the anomaly differences between the two maps reflect shallow geologic structures below the caldera floor, potentially including the effects of high-density mineral deposits.
Experiment in Izena Hole in August 2015
We carried out one more dive in 2015 in the relatively flat Izena Hole, one of the calderas in the middle Okinawa Trough (Figure 11). Hydrothermal activity has been identified in a location known as the Hakurei site in the southern part of the caldera (Ishibashi et al., 2015). We conducted detailed measurements along 14 lines in the north–south direction and 21 lines in the east–west direction. The sea bottom is 1550–1640 m deep in this area. The AUV moved at a constant depth of 1550 m (i.e., 50–100 m above the sea bottom), except during (1) the first two lines, when it moved at constant depths of 1580 and 1520 m along lines 1 and 2, respectively, and (2) the first part of line 12, at which point the Urashima rose to a depth of 1500 m. The conversion factor calculated from in situ CTD measurements was approximately 98.6205–0.046(P-16) m/MPa in this area (Figure 12). We adopted a rock density of for the calculation of Bouguer anomalies following the results of bulk-density measurements of drilled core samples in this area (Masaki et al., 2017). To suppress high-frequency noise, we applied a Gaussian low-pass filter with a 180 s filter width. The amplitudes of data components with periods smaller than 180 s (i.e., with spatial wavelengths shorter than 180 m for the AUV’s speed of 2 knots) were reduced after the application of the low-pass filter, and the vertical acceleration data obtained from the water pressure data were in excellent agreement with the gravimeter data (Figure 13a). Although the amplitude of gravimeter data was significantly reduced by the vertical acceleration correction, it remained in the range of 20–30 mGal. However, the amplitude was reduced to a few mGal following the Eötvös and free-water corrections (Figure 13b). The variability in the Bouguer correction was very small because the AUV’s depth was nearly constant and the density contrast was small in this case. By using the crosscorrelation method with a Gaussian high-pass filter with a 300 s width, we obtained linear scale parameters of 0.38 m, 0.34 s, and 0.0012 for , , and , respectively. The noise in the data was greatly reduced by applying these three corrections (Figure 13e).
Figure 14a shows Bouguer anomaly profiles along the east–west lines. The anomalies along lines 14–18 closely match those along corresponding lines 31–35 (Table 2). We made a line-leveling correction to minimize the crossover errors. The corrections were within , except for the corrections to lines 1, 2, and 7, which were , , and , respectively (Table 3). The plus signs in Figure 14a show the values of north–south lines at the crossover points after the line-leveling correction. We excluded the parts of lines over which the AUV depth was changing and the parts near the line course changes, where the measurement resolution might have been deteriorated. The rms crossover difference of the processed gravity data after the line-leveling correction is 0.10 mGal (Table 3). The detailed Bouguer anomaly map of the southern Izena Hole shows two anomaly highs in the northeastern and southwestern parts of the area (Figure 14b).
Model calculation of Bouguer anomalies in the southern part of Izena Hole
There is an anomaly high in the northeastern part of the Bouguer anomaly map of the southern Izena Hole and a northeastward projection of an anomaly high in the southwestern part. We used a simple model calculation to approximate these two anomaly highs. We assumed that the source bodies are two cylindrical disks with a density contrast of (i.e., a density of under the assumption of a rock density of ) as the source bodies. Disk A has a diameter of 350 m and a thickness of 30 m, whereas disk B has a diameter of 300 m and a thickness of 50 m (Figure 15). Figure 15 shows the sections of the two disks and compares the observations and the model results along the three virtual track lines. The bathymetry of this area gradually deepens northwestward. The disk tops are located approximately 30 m below the sea bottom. Figure 16 shows Bouguer anomalies obtained by using the model results to remove the effect of the two high-density cylindrical disks. The anomaly high in and around circle A and the projection of the anomaly high in and around circle B have nearly disappeared in this figure. The Bouguer anomalies and sea-bottom topography of the survey area are shown in Figure 17a and 17b, respectively. Circles A and B show the locations of the two supposed disks, and various marks in the topographical map indicate the locations where hydrothermal activity has been identified (Ishibashi et al., 2015). These are distributed surrounding the disks, supporting the supposition that there are buried hydrothermal deposits in those locations.
The largest correction that needs to be made to the gravity data is for the effect of the vertical acceleration. Thus, accurate determination of this effect is important for the data processing of gravity measurement made aboard an AUV. After determining the pressure-to-depth conversion factor using in situ CTD data, we introduced three additional corrections. These corrections are essential for determining the precise vertical acceleration acting on the gravimeter, and they consist of a spatial correction to account for the spatial separation of the depth sensor and the gravimeter, a temporal correction to account for the time delay between the depth sensor data and the gravimeter data, and, finally, a small adjustment to the pressure-to-depth conversion factor. More precise anomaly data can be obtained by applying these corrections.
Application of the crosscorrelation method to the data collected during the YK15-13 cruise in 2015 yielded the result that the gravimeter was located 0.38–0.44 m ahead of the depth sensor. However, direct measurement revealed that the gravimeter was only 0.33 m ahead of the depth sensor during this cruise (Figure 1). One possible cause for this discrepancy is the strong correlation between the effect of the spatial separation and that of the gravimeter time delay, which is visible in Figures 7e, 8e, and 13e. The normalized crosscorrelations of both quantities were generally greater than 0.85, meaning that it is difficult to determine the amplitudes of these quantities accurately and independently. We suggest that the crosscorrelation method may lead to a positive bias in the determination of these quantities.
Large variations in the seawater density were observed depending on location and water depth. The seawater density differs by approximately (or 0.5%) between Bayonnaise Knoll and Izena Hole. This result indicates the importance of the determination of the pressure-to-depth conversion factor using the seawater density data obtained from in situ CTD data. The gravity acceleration values can be determined easily to within 100 mGal (or with a precision better than 0.01%), and the water density calculated using the formulas of Fofonoff and Millard (1983) has an accuracy better than . However, the results of the crosscorrelation method suggest that the conversion factor obtained through these calculations is likely to be 0.1%–0.2% larger than the true values. We suggest that another effect (e.g., the failure of the rigid body assumption for the AUV) led to the apparent result of small corrections to the conversion factor.
Although the gravimeter was kept vertical during AUV course changes, the quality of the gravity data was degraded during those periods. This is probably due to inaccurate estimate of the Eötvös correction due to poor-quality navigational data. To address this issue, we excluded data near line ends from the analysis. The Bouguer anomaly data in the southern Izena Hole show large leveling corrections for lines 1, 2, and 7. Line 7 is near the eastern end of the survey area, and the above effect may remain in the crossing east–west lines. Lines 1 and 2 are at depths of 1580 and 1520 m, respectively. That these depths differ from 1550 m of the crossing east–west lines is likely the cause of the large leveling corrections: a negative value for line 1 and a positive value for line 2 (Table 3). Bouguer anomalies at different depths generally do not coincide. This result suggests that Bouguer anomaly in this area of Izena Hole decreases when the AUV increases its depth. This is also true for the survey in Bayonnaise Knoll, but in the opposite sense: The Bouguer anomalies over the caldera floor tend to increase when the AUV is deeper or closer to the sea bottom.
Geologic signals are reduced if their wavelengths are smaller than the wavelength determined by the Gaussian filter (i.e., 240 m for the Bayonnaise Knoll survey and 180 m for the southern Izena Hole survey). It is difficult to detect a source body that has a characteristic horizontal scale smaller than this length. On the other hand, the signal amplitude decreases as the altitude of the AUV above the source body increases. This is also true for the case in which the altitude increment is smaller than the characteristic scale of the source body, although the decrease is relatively small in that case. Figure 18 shows an example for lines A6, C6, and D3. To allow comparison of the amplitudes of anomalies observed at different depths, we introduced a simple source model that succeeded in reproducing the anomalies expected at the depth of line A6 (620 m). The model source body is a cylinder with a top depth of 820 m, a density contrast of , a diameter of 1100 m, and a thickness of 110 m. The observed lines are assumed to pass over the center of the cylinder. The gravity anomalies predicted by the model (shown as the dashed lines in Figure 18a) are in good agreement with the observed anomalies, although the agreement is not perfect, particularly for line D3. Line D0 is a virtual constant-depth line with a depth approximately equal to the maximum depth of line D3. The expected anomaly maxima of lines D0 and D3 are nearly identical, and the main difference between the lines is that the anomaly curve for line D0 is broader than that for line D3. This calculation shows that such an observed decrease in the anomaly amplitude with depth decrease is very plausibly explained by the source body in the calculation, although we do not explore the details of the source model here. The amplitude of a potential field with a horizontal wavelength of decreases as with upward continuation of (e.g., Blakely, 1995). Line A6 is 50 m above line C6 and 145 m above line D0, whereas the predominant wavelength is approximately 2500 m in this case, taking into account also the wavelength in the direction across this line. The decreases in amplitude that would result from these differences are estimated to be 0.88 for the A6–C6 pair and 0.69 for the A6–D0 pair, which approximately coincide with the results of the model calculation. These results show the importance of collecting data as close as possible to the sea bottom, particularly for smaller scale and smaller amplitude anomalies such as those in the Izena Hole area. The predominant horizontal wavelength of anomalies observed there is probably approximately 1000 m, and the amplitude decreases are estimated to be 0.73 and 0.53 for 50 and 100 m upward continuations, respectively.
It is difficult to obtain high-resolution measurements along constant-altitude lines in areas with rugged topography. We removed data in the parts of lines in which the depth of the AUV was changing when estimating the resolution for the Izena Hole survey. We estimate 0.2–0.3 mGal as the data resolution along lines D1–D8 in the Bayonnaise Knoll survey, although we do not yet have definitive data to confirm this estimate (the SD data for the difference of these lines from lines C8 to C1 in Table 2 only show the maxima of the resolution). We can obtain the rms errors by making repeat measurements at the same depth along the same constant-altitude line. These repeated measurements can also improve the resolution through stacking of the data.
We obtained high-resolution Bouguer anomaly data in underwater gravimeter surveys aboard an AUV by precise determination of vertical acceleration acting on the gravimeter. This was accomplished by introducing three new corrections to the gravimeter data: a correction for the effect of the spatial separation between the gravimeter and the pressure sensor, a correction for the effect of the time delay of the gravimeter data, and an adjustment to the pressure-to-depth conversion factor. We came close to meeting our target measurement resolution, obtaining a 0.1 mGal rms crossover difference after leveling under favorable conditions during the southern Izena Hole survey, during which the AUV moved at a nearly constant depth of 1550 m. The high-resolution measurement allowed us to succeed in the reliable detection of gravity anomalies due to buried mineral deposits with amplitudes of approximately 1–2 mGal. This method will be useful for surveys of similar-sized high-density mineral deposits located either at the seafloor or shallowly buried. However, at present, practical applications of the method may be limited to relatively flat areas such as Izena Hole. Many areas that have seafloor deposits are also characterized by rugged topography (e.g., Bayonnaise Knoll), and thus, require constant-altitude lines for near-bottom surveys. Our experience shows that maneuvering the AUV at a constant altitude of 50 m above the sea bottom is feasible given prior access to detailed topographical information. However, the amplitude of vertical acceleration increases in general, and the resolution of the measured anomalies deteriorates to 0.2–0.3 mGal, even after applying a low-pass filter with a longer width. Therefore, further improvements to the data resolution are necessary for reliable detection of mineral deposits in areas with rugged topography.
We thank H. Shiobara and K. Mochizuki for discussions concerning the development of the measurement system. We also thank T. Yagi and T. Matsuo of Earthquake Research Institute, the University of Tokyo; O. Matsubara, Y. Toyoshima, Y. Kakinuma, E. Kamimura, and S. Shimono of Q.I. Inc.; S. Sakurai and S. Kamoshida of LINK Lab. Inc.; K. Nozaki of Oyo Corp.; and Y. Shirasaki of Marin Eco Tech Ltd. for supporting the development of the gravimeter system and data collection using the system. We are obliged to the captain and crew of the M/V Yokosuka, JAMSTEC, for the success of the experiments. We are grateful for the skillful piloting of the AUV by the Urashima team, JAMSTEC. This research was supported by the Ministry of Education, Culture, Sports, Science and Technology, Japan, under the Program for the Development of Fundamental Tools for the Utilization of Marine Resources. Most of the figures were created using GMT (Wessel et al., 2013). We thank editor K. Biegert, the two anonymous reviewers, and the associate editor X. Li for their constructive comments and suggestions.
DATA AND MATERIALS AVAILABILITY
Data associated with this research are available and can be obtained by contacting Earthquake Research Institute, the University of Tokyo (Contact person : M. SHINOHARA).
PSD AND FREQUENCY RESPONSE OF GAUSSIAN LOW-PASS FILTERS
Figure A-1 shows PSD distributions for the constant-depth data collected on 10 August 2015 (Figure A-1a) and for the constant-altitude data collected on 7 August 2015 (Figure A-1b). The PSDs of the gravimeter data (the solid black line) and those of the vertical acceleration data (the purple line, largely hidden by the black line) are nearly coincident at frequencies less than , and the PSD of gravimeter data is greatly reduced following vertical acceleration correction. The PSD of the free-water correction (the solid red line) and the Eötvös correction (the orange line) have large amplitudes at frequencies less than . The corrections for the effect of the spatial separation of the gravimeter and the depth sensor (the green line) and the effect of the time delay of the gravimeter data (the dashed black line) further decrease the PSD of the gravimeter data at frequencies of . The effect of the error in the water pressure increment to depth increment conversion factor (the dashed red line) is small, particularly for the data collected on 10 August 2015, which is smaller than . The PSD of the gravimeter data following all these corrections (the blue line) is greatly decreased, but it retains noise at higher frequencies (particularly at frequencies greater than ), which can only be reduced by a low-pass filter. The gravimeter and vertical acceleration data collected on 7 August 2015 during the constant-altitude experiment (shown in Figure A-1b) have larger PSD amplitudes than do those for the constant-depth data collected on 10 August (shown in Figure A-1a). This holds true even after all the corrections have been applied at frequencies from to , suggesting that the constant-altitude data include noise in this frequency range that can be reduced only by increasing the window width of the low-pass filter.
The Fourier transform of a Gaussian function is a Gaussian function, and the Fourier transform of the convolution of two Gaussian functions is the product of their Fourier transforms. Thus, the frequency response of the convolution of two Gaussian low-pass filters with widths of and is equal to one Gaussian filter with a width of at low frequencies.
Figure A-2 shows various filters with low-frequency responses similar to that of an ordinary Gaussian low-pass filter with a window width of 180 s. Owing to its finite length, the ordinary Gaussian filter has a filter response of approximately in the frequency cut range (the black line). The response in the high-frequency part decreases in the two-step filter with widths of 10 and 180 s (the green line). Finally, a filter response of approximately is obtained by a filter with a width of 180 s but an actual () window width of 360 s (the red line). This figure also shows the frequency response of a filter with the width of 240 s and window width of 480 s (the blue line). The range of approximately in this filter extends to lower frequencies.
Figure A-3 compares the results of three different low-pass filters applied to the data obtained on 10 August 2015. High-frequency noise is retained after applying a simple low-pass filter with a width of 180 s, whereas noise is reduced by a double-width filter with a width of 180 s and by a two-step filter with widths of 10 and 180 s.