Global Distribution and Morphology of Small Seamounts

Seamounts are isolated elevations in the seafloor with circular or elliptical plans, comparatively steep slopes, and relatively small summit area (Menard, 1964). The vertical gravity gradient (VGG), which is the curvature of the ocean surface topography derived from satellite altimeter measurements, has been used to map the global distribution of seamounts (Kim & Wessel, 2011, https://doi.org/10.1111/j.1365-246x.2011.05076.x). We used the latest grid of VGG to update and refine the global seamount catalog; we identified 19,325 new seamounts, expanding a previously published catalog having 24,643 seamounts. Seven hundred thirty‐nine well‐surveyed seamounts, having heights ranging from 421 to 2,500 m, were used to estimate the typical radially symmetric seamount morphology. First, an Empirical Orthogonal Function (EOF) analysis was used to demonstrate that these small seamounts have a basal radius that is linearly related to their height—their shapes are scale invariant. Two methods were then used to compute this characteristic base to height ratio: an average Gaussian fit to the stack of all profiles and an individual Gaussian fit for each seamount in the sample. The first method combined the radial normalized height data from all 739 seamounts to form median and median‐absolute deviation. These data were fit by a 2‐parameter Gaussian model that explained 99.82% of the variance. The second method used the Gaussian function to individually model each seamount in the sample and further establish the Gaussian model. Using this characteristic Gaussian shape we show that VGG can be used to estimate the height of small seamounts to an accuracy of ∼270 m.

• We used the latest vertical gravity gradient maps to update and refine a global seamount catalog, finding 19,325 new seamounts • Smaller seamounts (<2,500 m tall) having good bathymetry coverage (739) were modeled with a radially symmetric Gaussian function • Two modeling approaches show that smaller seamounts have a sigma to height ratio of 2.4 which agrees with an earlier study by Smith (1988) Supporting Information: Supporting Information may be found in the online version of this article. Wessel, 1997Wessel, , 2001. This model suggests that the majority of seamounts are small and there could be 50 to 100 thousand seamounts with heights above 1 km (Kim & Wessel, 2011;Wessel, 2007). Therefore, there is a lithosphere age to seamount size relationship; smaller seamounts generally form on young, thin lithosphere, while larger seamounts generally form on older, thicker lithosphere (Vogt, 1974;Watts et al., 2006).
The knowledge of the global distribution of seamounts is still incomplete because only 20% of the seafloor has been mapped by ships (Mayer et al., 2018). However, seamounts are valuable characteristics of the ocean floor since they provide insight on many of the Earth's geological, oceanographical, and ecological cycles and processes (Wessel, 2007). (a) From a geological perspective, seamounts are particularly important because they are windows into the composition and temperature of the mantle (Koppers & Watts, 2010). Since seamounts reflect the production and ascension of magma, scientists study their minor changes in isotopes to keep track of the chemical composition of lava and further understand the eruption process (Anderson et al., 2021;Zindler et al., 1984). They can also be used to explain the planet's tectonic evolution since plume-generated seamount chains serve as a record of absolute plate motion (Morgan, 1971;Müller & Seton, 2014). (b) From an oceanographic perspective, ocean floor bathymetry has an important effect on ocean circulation: large seafloor features such as ridges and plateaus act as barriers that inhibit deep cold water to mix with the warm water of the ocean surface (Roden et al., 1982). Recent studies suggest that smaller features such as seamounts can also play an important role oceanographically and have a greater influence on circulation which can help scientists better understand the uptake of heat and carbon dioxide in the ocean (Jayne et al., 2004). (c) From an ecological perspective, seamounts are centers for diverse biological communities. The ocean upwelling due to the presence of seamounts brings valuable nutrients from the deep water to the surface. This allows them to become the ideal habitat for fish and a variety of oceanic flora and fauna (Price & Clague, 2002;Rogers, 1994). Due to the impact that seamounts have on the ocean and ecosystems, they are important features to study, map, and classify.

Mapping Seamounts
There are two approaches for mapping seamounts-topographic mapping by echo sounders or multibeam sonar on ships and gravity field mapping by satellite altimetry (Hillier & Watts, 2007). Multibeam sonar mapping by oceangoing research vessels provides high resolution topography (100-200 m) (Epp & Smoot, 1989), although a great amount of the ocean (∼80%) remains unmapped because of the large gaps between ship tracks (Mayer et al., 2018). Since the majority of research surveys have been on young lithosphere, small seamounts near mid-ocean ridges have drawn significant attention for detailed investigation (Wessel et al., 2010). Swath surveys in remote areas or along transit cruises do not always cross the summit of a seamount so its height is poorly known (Wessel et al., 2010). Complete multibeam coverage of the global seafloor is time-consuming and expensive (Vogt & Jung, 2000), so scientists have turned to satellite altimetry to obtain a low-resolution (∼6 km) but global mapping.
Previous studies have shown that gravitational anomalies, derived from satellite altimetry, can be used to find seamounts taller than 2 km (Craig & Sandwell, 1988;Lazarewicz & Schwank, 1982;Watts & Ribe, 1984;Wessel, 1997). Satellite altimeters measure the geoid height which, through Laplace's equation, can be converted to gravity anomalies, and vertical gravity gradient (VGG). There are four main error sources when detecting and mapping seamounts from satellite-derived anomalies: upward continuation, measurement noise, seafloor roughness, and sediment cover (Wessel et al., 2010). (a) Upward continuation causes seamounts with diameters less than the mean ocean depth (∼4 km) to be smoothed and attenuated. (b) Ocean waves and currents introduce noise in the satellite altimeter measurements so short wavelength gravity anomalies (<20 km) are oftentimes not recovered . (c) There are a number of features that contribute to small scale gravity anomalies, including abyssal hills and ridges, and their signals can be confused with those of seamounts. (d) Lastly, older small seamounts are oftentimes obscured by sediment on the seafloor. The gravity anomaly will still appear above the buried seamount even though it is not visible in the topography .
Detection and mapping of seamounts shorter than 2 km has relied on multibeam surveys. In a study conducted by Smith (1988), multibeam (SeaBeam) data of 85 seamounts from the Pacific Ocean were analyzed. Smith (1988) found that there is a relatively uniform base radius to height (h) of 0.21, although there are variations in shape and flatness ( Figure 1). Large seamounts, for example, had smaller flatness values defined by f = d t /d b , where d t is the summit diameter and d b is the basal diameter (Smith, 1988). It was also found that the slope angle, defined by ɸ = arctan( ) where = 2h/(d b − d t ), was equal to ∼15° (Smith, 1988 (Smith, 1988).

Detecting Seamounts in Satellite Altimetry
Satellite altimetry is a valuable tool for estimating global topography at relatively low spatial resolution (∼6 km) and helping scientists find medium to large seamounts. The first global seamount maps were created from Seasat altimeter profiles. Seasat was launched in 1978 and collected sea surface profiles for just 105 days, which resulted in diamond shaped data gaps with dimensions of ∼100 km (Marsh & Martin, 1982). The analysis of Seasat altimetry profiles was able to identify 8,556 seamounts using Gaussian-shaped modeling (Craig & Sandwell, 1988). They also found that satellite altimetry can be used to determine the along-track locations of seamount centers with an accuracy of better than 10 km, but the cross-track location was more poorly determined due to the wide track spacing. Another measurable characteristic is the diameter of the seamount, which is equal to the distance between the peak and trough of the along-track vertical deflection (i.e., sea surface slope) profile. This study was able to use the locations of the seamounts to draw conclusions on the global distribution of seamounts. It found that the density of seamounts in the Pacific is higher than the Atlantic or Indian oceans and seamounts preferentially occur on the younger side of large fracture zones (Craig & Sandwell, 1988).
Since the Seasat mission, there have been a number of altimeter missions that have greatly improved the accuracy and coverage of the gravity field. This has enabled the construction of the VGG, which is the spatial derivative of the gravity field (Rummel & Haagmans, 1990). This spatial derivative amplifies short wavelengths and suppresses long wavelengths so it is a valuable tool for locating smaller features on the ocean floor (Kim & Wessel, 2011;Wessel & Kim, 2015). However, the spatial derivative also amplifies short wavelength noise, which limits seamount detectability. Since 2011 there have been two significant improvements in the accuracy of the VGG derived from altimetry. The first improvement (∼40% noise reduction) was due to the improved spatial coverage provided by the CryoSat-2, Envisat, and Jason-1 geodetic missions . The CryoSat-2 mission is still operating in 2023. Moreover in 2016, a new altimeter SARAL/Altika has provided a factor of 2 reduction in radar noise. These additional data have provided a second ∼40% noise reduction. These reductions in VGG noise have enabled the expansion of the seamount catalog.

Update to the Kim-Wessel (2011) Seamount Catalog
To begin the investigation we constructed high resolution VGG images for Google Earth that allowed for a better visualization of seamounts as well as already-digitized tectonic features. The data sets used in Google Earth included the VGG ( (Smith, 1988), where h is the seamount height, d b is the basal diameter, and d t is the diameter of the flattened summit. Flatness, f, is defined by d t /d b while the height to basal radius ratio is defined by 2h/d b . , global bathymetry and topography through the use of SRTM15+V2.3 (Tozer et al., 2019), and previous seamounts picked by Kim and Wessel (2011). To update this Kim-Wessel catalog, we used the newly refined VGG (Version 30), which has a spatial resolution of 6 km, revealing many smaller seamounts as well as resolving individual seamounts along ridges. We identified new seamounts by eye, avoiding seafloor features such as fracture zones, transform faults, ridge axes, and see-saw propagators since these can give signals that may look like seamounts in the VGG. Additional details, and comparisons with previous studies (Hillier & Watts, 2007;Kim & Wessel, 2011;Yesson et al., 2011), are provided in Supporting Information S1. Through this method we were able to identify 19,325 new seamounts. The new VGG also helped us to find 514 seamounts that were misidentified in the Kim-Wessel catalog of 24,643 (2011). These included any seamount picks that no longer showed a gravity signal in the VGG. After removing these and finalizing the new picks, the updated catalog came to a total of 43,454 seamounts.
The next step was to recenter all of the seamount picks using Python and the "grdtrack" module from the generic mapping tool (GMT; Wessel et al., 2019). To do this, we searched for the maximum VGG in a 5 × 5 pixel (∼5 min) area around the initial seamount pick. Although the location of the maximum in the VGG is oftentimes not the exact geometric center of the seamount, it is a good reference to use for modeling ( Figure 2).

Data Preparation
After the central longitude and latitude were found, we filtered the catalog for well-charted seamounts (i.e., those having at least 50% multibeam and single beam sonar data coverage of the seamount and complete coverage of its summit). This search was accomplished by using the source identification grid associated with the SRTM15+V2.3 global bathymetry (Tozer et al., 2019). This process resulted in 739 well charted seamounts <2,500 m tall; 554 from the KW catalog and 185 from the new SIO catalog. An example of a seamount with good data coverage is shown in Figure 3.
For each well-mapped seamount, we calculated the base depth and maximum seamount height. The base depth was taken as the median depth on a 30 km by 30 km area surrounding the center of the seamount. Most seamounts are surrounded by relatively flat seafloor so this base depth is well defined by the median of the depth histogram ( Figure 3c). The maximum seamount height above the base depth was derived from the shallowest depth in the same area (i.e., summit depth -base depth) (Figure 3d). It is important to note that the maximum seamount height is the shallowest point on the seamount and not necessarily the height at the VGG centered location.
The radius of each data point was calculated using the relationship between the point's coordinates and the VGG center of the seamount. The height above the base depth for each seamount, as well as the radius, were normalized by the maximum height. The normalized height was then median-filtered at 0.5 normalized radius increments using the "filter1d" function in GMT.
We use the Empirical Orthogonal Function (EOF) analysis (Hannachi et al., 2007;Preisendorfer & Mobley, 1988) to seek the basic structure of the 739 well charted seamounts. We construct a × two-dimensional matrix of the normalized heights at fixed normalized radii, where is 739 (the number of seamounts) and is 26 (the number of radius points). The first mode of EOF analysis explains 90.8% of the total variance, thus we neglect all other modes. Its expansion coefficients, which represent the structures in the sampling dimension, resemble a Gaussian shape. Based on this result we assume that each seamount has a radial symmetrical Gaussian shape and a common base to height ratio (i.e., amplitude divided by the standard deviation in Gaussian function). We then use two methods to compute this base to height ratio: an average Gaussian fit to the sample of 739 seamounts and an individual Gaussian fit for each seamount in the sample.

Method 1: Average Gaussian Fit
To prepare for the average Gaussian fit, we combined the radially normalized height data from all seamounts to obtain the median normalized heights and median absolute deviation. This data was then fit to the following Gaussian equation where is the seamount normalized radius from 0 to 12.5 with a 0.5 spacing, is the height, is the characteristic width, and is adjusted base depth. This analysis used the median normalized heights for and median-absolute 6 of 13 deviation as the error associated to find the and through least square fitting. Since this analysis is done with a profile stack of all the seamounts where the normalized median height converges to zero at larger radii, is set to zero.
This two-parameter Gaussian model produces the best-fitting height and characteristic width of our collective seamounts (Table 1). Our model had a ∕ℎ equal to 2.4 with a maximum absolute slope of 0.25 and explained ∼99% of the variance ( Figure 4). As discussed below, the maximum absolute slope of the best-fit model is in good agreement with a previous study based on the analysis of 88 seamounts where the seamount height was one fifth of the basal radius (Smith, 1988). The final ratio between sigma and height, / , with a value of ∼2.4 is important in defining the final model and ultimately, the gravity field of the Gaussian seamount that is used to construct a new global synthetic bathymetry (SYNBATH) where this factor is used to sharpen the shapes of predicted seamounts (Sandwell et al., 2022).  To evaluate the model, we applied the average Gaussian model to each seamount and computed the difference between topography extracted from the SRTM15+V2.3 and the Gaussian model created using the "grdseamount" function in GMT. GMT "grdseamount" takes the central longitude, central latitude, model height, and radius (3 ) as input. We used the median height at the summit of the seamount for the model height. This value is determined by filtering the real data in 0.5 km median increments and finding the maximum. Using the median height at the summit instead of the maximum height allows for less error in the fitting of the model that might have occurred due to singular sharp peaks at the summit. We examined the model fits to all 739 seamounts but only 6 are plotted below ( Figure 5) to illustrate some good fits as well as cases where the fits are poor.

Method 2: Individual Gaussian Fit
Since the first method has several seamounts with poor fits, we re-did the analysis by fitting a Gaussian model to each seamount individually. To test which set of seamount height series data is best for this fitting, we compared three types of data: all available bathymetric data, GMT "filter1d"calculated median heights from radii 0-12.5 km, and GMT "filter1d" calculated robust median heights from radii 0-12.5 km. The results showed that the median and robust median had better fits than the model using all available bathymetric data, but produced very similar results. Because of this, we chose to use the robust median height data for the second analysis. The robust median height data and radius input data (unit of km) are fit to Equation 1. Each seamount then receives its own unique , , and 0 values, which are height, sigma, and adjusted base depth respectively. In Figure 6 we have modeled the same 6 seamounts with GMT "grdseamount" as before but with their individual Gaussian fitting.
The median of the / ratio for these 739 individually fitted seamounts had a value of 2.39 and a mean of 2.6 ( Figure 7). This matches well with the value we obtained from the first approach. This indicates that the average seamount fitting and ratio of ∼2.4 is a good representation of the morphology of the majority of seamounts.

Comparing Method 1 and Method 2
The relationships between maximum height and model height from Method 1 and Method 2 respectively have been plotted below. Figure 8a shows that the relation between the maximum and model height is linear and therefore, the model serves as a good representation of the seamount height used for the Gaussian analysis. The model height is always less than or equal to the maximum height because the data that we used is filtered with GMT "filter1d" from 0 to 12.5 radii in 0.5 intervals. Each radius would have the median height value, which would naturally decrease the height value from the maximum height. Figure 8b shows the values for height obtained through the individual Gaussian fit against the maximum height of the seamount. Although this graph shows more variability in the data, it generally still follows a linear trend.

Comparing RMS Misfit
The root mean square (RMS) error is calculated from the difference of the model and real topography data available within a 30 by 30 km area. When comparing the results of the average ( Figure 5) and individual ( Figure 6) fitting of these six example seamounts we can see interesting results. From this sample, five of the six seamounts showed a better fit through Method 2. As shown in Table 2 below, we can see that the RMS for all but seamount KW-16423 decreased in error. This is understandable since seamount KW-16423 is elliptical and would not perfectly fit a radially symmetric Gaussian model regardless of the method. For seamounts such as this case, additional parameters such as ellipticity would need to be added for more accurate modeling (Kim & Wessel, 2011).
The RMS misfits of all 739 seamounts (both methods) are shown in Figure 9.
In general, the RMS misfit increases as seamount height increases. The Pearson coefficient between the height and RMS is 0.34 for the average Gaussian fit method, and 0.52 for the individual Gaussian fit method (both p-values are less than 0.01). The best linear fits (excluding outliers) between the height and RMS are shown overlapping with data and it indicates that the height error is typically ∼10% of the seamount height. The average Gaussian fit method shows a slightly wider range in RMS misfit distribution with a 2 = 0.11. In contrast, the individual Gaussian fit method RMS has less variation as the height increases with a 2 = 0.27.
When comparing the values directly, 454 seamounts showed improvement in the misfit after Method 2 while the other 285 had more error. The RMS of the 454 seamounts improved with a median value of −38.65 m while the RMS of the 285 diminished with a median value of 34.03 m. This shows that Method 2 serves as a better tool for modeling seamounts than Method 1.

Comparing Gaussian Model to Smith (1988)
In Smith (1988), 85 seamounts were analyzed based on their height to base radius ratios. In that study, Smith used a flattened cone model as seen in Figure 1 rather than a Gaussian model. Through her analysis, it was found that the seamounts' summit height is about one fifth (0.21) of the basal radius.
In order to compare the height to base ratio of our own analysis to that of Smith's, we fit a flattened cone model to our average Gaussian model. First, we created a 1,000 m tall.
Gaussian seamount where ∕ℎ = 2.4 using GMT "grdseamount." We then made several flattened cone seamounts, all of which were 1,000 m tall, but varied in their sigma values by 0.1. GMT "grdseamount" allows the user to input flatness value, . In Smith's study, a 17 point sample mean of seamounts (1,000-1,800 m tall) had a flatness value of 0.22 ± 0.11. Because of this, we use = 0.22 for our flattened cone models. The smallest RMS error obtained from the difference between Gaussian model and flattened cone models allowed us to find the sigma  equivalent for Smith's analysis. Through this method we determine that ∕ described by Smith (1988) is approximately the same as ∕2.1 in our analysis (Table 3).
The height to base ratios of the 739 seamounts from our sample and the 85 seamounts described by Smith (1988) are shown in Figure 10. In this figure, the seamount heights from Smith's sample are direct measurements from bathymetric surveys while the heights for our sample of 739 are calculated through the individual Gaussian method. Smith's sample has seamounts which are generally smaller than the seamounts in our sample. Our smallest seamount is 421 m but most are taller than 700 m because of the limitations of the VGG method in detecting smaller features. Over the height range where the two analyses have significant overlap (700-2,500 m), the height to base ratios of the two analyses show general agreement. Smith (1988) had revealed that the height to base ratio decreases slightly at larger sizes. This observation is also true for the larger seamounts of this analysis.

Conclusion
Improvement in the VGG allowed us to find 19,325 new seamounts, expanding the Kim-Wessel (2011) catalog which had 24,643 seamounts. The addition of these new seamounts and refinement of previous picks updated the catalog to a total of 43,454 seamounts. Prospective improvements in the VGG can further expand our knowledge of seamounts while surveying done by multibeam sonar remains limited.
By modeling a sample of 739 seamounts as a Gaussian we can conclude the following: 1. Two modeling approaches show that medium sized seamounts have a characteristic sigma to height ratio of 2.4 and a maximum slope of 0.25. This is in good agreement with an earlier study by Smith (1988) who found that the summit height is around one fifth of the basal radius. 2. Although our models provide a good representation of seamount shape, the radially symmetric Gaussian model has significant deviations from actual seamount shape. The way in which the center of the seamount is chosen can also have an effect on the model. It is common that the highest point of the seamount does not correspond to either the largest VGG signal or its geometric center. Table 3 Height to Base Ratio Comparison of Smith (1988) and This Study 85 seamounts (Smith, 1988)  3. When comparing the RMS misfit of both Gaussian Model methods, the individual seamount modeling method shows less error. However, both indicate that the error in modeling increases as seamount heights increase. 4. Our Individual Gaussian model was based on three parameters: height, sigma, and basal depth. Including additional parameters such as ellipticity in future analyses can help account for the shape of some seamounts when modeling and provide a better fit.
The modeling of seamounts as a Gaussian can help improve our understanding of their shapes and distribution. Most importantly, the characteristic sigma to height ratio of 2.4 can allow for the modeling of the majority of the seamounts that are identified through satellite altimetry, but have not been surveyed by ships. The VGG and the methods of Gaussian modeling can allow for clarity in understanding the morphology of globally distributed seamounts.

Data Availability Statement
The Google Earth overlays and the characteristics of the seamounts, can be found on the ZENODO repository, https://doi.org/10.5281/zenodo.7718512 (Gevorgian et al., 2023