# Stacking repeat cycles of 40-Hz AltiKa data resolves the geoid anomalies of very small seamounts

**CoAuthors**

**Event: **2015 Ocean Surface Topography Science Team Meeting

**Session: **The Geoid, Mean Sea Surfaces and Mean Dynamic Topography

**Presentation type: **Type Oral

Seamounts are volcanoes on the world’s ocean floor that may number in the hundreds of thousands. It is important to know how many there are, and where they are, in order to correctly model ocean circulation and mixing, tsunami propagation, fishing grounds and habitats, hazards to submarines, etc. Because the oceans are mostly unmapped by ships, most of the world’s seamounts have been found when their geoid anomalies were revealed by satellite altimetry.

Small seamounts are much more common than large ones. One fractal model for their size-frequency distribution suggests that, as seamount height goes down two-fold, the number of seamounts goes up seventeen-fold. If a two-fold reduction in height implies an eight-fold reduction in mass, then one may expect that the amplitude of a seamount’s geoid anomaly will decrease very rapidly as its size decreases, making small seamounts much harder to find with an altimeter. Searching marine gravity maps derived from satellite altimetry for seamount anomalies finds many thousands, and their size-frequency distribution fits expected models when seamount heights are greater than 2 km tall. This is interpreted as evidence that altimeters have already found the seamounts 2 km tall and taller, but are failing to find the smaller ones.

Smith [Marine Geodesy, 2015, doi 10.1080/01490419.2015.1014950] compared seamount resolution by AltiKa to resolution by Envisat. He found that sea surface height profiles over seamounts less than 2 km tall showed seamount geoid anomalies much more clearly in AltiKa than in Envisat data. He attributed this to AltiKa’s lower noise level and denser sampling (5 cm at 40 Hz versus 8 cm at 18 Hz). His study characterized the ability of these altimeters to detect a seamount in only one pass over a feature, as would be necessary if an altimeter were put into a “geodetic mission” orbit.

Our present study characterizes the improvement in seamount resolution that can be achieved by “stacking” the repeated sea surface height profiles that AltiKa obtains every 35 days. We take all data at 40 Hz along-track sampling rate, group data across repeat cycles by aligning points nearest to a common latitude, and then take the mean or median of all such aligned points. The mean or median profile (the “stack”) has reduced noise and clarifies seamount anomalies. The noise variance decreases in inverse proportion to the number of cycles stacked. Because the median is a less efficient estimator than the mean, a median stack should be noisier than a mean stack; however, we find little penalty for using medians and we prefer them because they are robust with respect to outliers, which occur occasionally. We find that the root-mean-square noise level in the median stack is around (5.8 cm)/sqrt[(2N+1)/3] at 40 Hz, where N is the number stacked, and is below 2 cm at 40 Hz when 12 or more repeat cycles are stacked (about 1 year of data).

Stacking always reduces the noise in the mean sea surface height profile. Translating a reduction in noise into an improvement in seamount resolution requires also a specification of the seamount geoid anomaly being sought. In areas where AltiKa repeat tracks cross previously mapped small (< 1.5 km tall) seamounts, we can evaluate the coherency between sea surface height profiles and verified in situ bathymetry. The previous study [loc. cit.] found coherence at wavelengths greater than 17 km if only a single altimeter profile was used. In our present study, if we stack 9 profiles, the stacked geoid anomaly is coherent with seamount bathymetry at wavelengths as short as 10 km. This suggests that a seamount detection process using a matched filter approach will reliably detect seamounts smaller than 2 km, with reliability increasing as the number of repeat cycles stacked increases.

Small seamounts are much more common than large ones. One fractal model for their size-frequency distribution suggests that, as seamount height goes down two-fold, the number of seamounts goes up seventeen-fold. If a two-fold reduction in height implies an eight-fold reduction in mass, then one may expect that the amplitude of a seamount’s geoid anomaly will decrease very rapidly as its size decreases, making small seamounts much harder to find with an altimeter. Searching marine gravity maps derived from satellite altimetry for seamount anomalies finds many thousands, and their size-frequency distribution fits expected models when seamount heights are greater than 2 km tall. This is interpreted as evidence that altimeters have already found the seamounts 2 km tall and taller, but are failing to find the smaller ones.

Smith [Marine Geodesy, 2015, doi 10.1080/01490419.2015.1014950] compared seamount resolution by AltiKa to resolution by Envisat. He found that sea surface height profiles over seamounts less than 2 km tall showed seamount geoid anomalies much more clearly in AltiKa than in Envisat data. He attributed this to AltiKa’s lower noise level and denser sampling (5 cm at 40 Hz versus 8 cm at 18 Hz). His study characterized the ability of these altimeters to detect a seamount in only one pass over a feature, as would be necessary if an altimeter were put into a “geodetic mission” orbit.

Our present study characterizes the improvement in seamount resolution that can be achieved by “stacking” the repeated sea surface height profiles that AltiKa obtains every 35 days. We take all data at 40 Hz along-track sampling rate, group data across repeat cycles by aligning points nearest to a common latitude, and then take the mean or median of all such aligned points. The mean or median profile (the “stack”) has reduced noise and clarifies seamount anomalies. The noise variance decreases in inverse proportion to the number of cycles stacked. Because the median is a less efficient estimator than the mean, a median stack should be noisier than a mean stack; however, we find little penalty for using medians and we prefer them because they are robust with respect to outliers, which occur occasionally. We find that the root-mean-square noise level in the median stack is around (5.8 cm)/sqrt[(2N+1)/3] at 40 Hz, where N is the number stacked, and is below 2 cm at 40 Hz when 12 or more repeat cycles are stacked (about 1 year of data).

Stacking always reduces the noise in the mean sea surface height profile. Translating a reduction in noise into an improvement in seamount resolution requires also a specification of the seamount geoid anomaly being sought. In areas where AltiKa repeat tracks cross previously mapped small (< 1.5 km tall) seamounts, we can evaluate the coherency between sea surface height profiles and verified in situ bathymetry. The previous study [loc. cit.] found coherence at wavelengths greater than 17 km if only a single altimeter profile was used. In our present study, if we stack 9 profiles, the stacked geoid anomaly is coherent with seamount bathymetry at wavelengths as short as 10 km. This suggests that a seamount detection process using a matched filter approach will reliably detect seamounts smaller than 2 km, with reliability increasing as the number of repeat cycles stacked increases.