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Global Positioning System/Global Navigational Satellite System (GPS/GNSS) technology represents the current state of the art in making geodetic measurements, but given the century or longer recurrence of great (M > 8) subduction earthquakes10, it will be many decades before the first GPS/GNSS measurements throughout a complete seismic cycle are obtained. Elastic rebound in subduction zones has been demonstrated by geodetic measurements through only portions of the earthquake cycle, and inferred from growth patterns in Sumatran corals11. Furthermore, variation of the rate of strain accumulation between great earthquakes remains poorly understood. Palaeoseismic records in the regions of the 1960 Chilean12, 1964 Alaskan13, 2004 Sumatran11 and 2011 Japanese14 earthquakes suggest that neither the return time nor the size of megathrust earthquakes is regular, emphasizing the need to document deformation over full seismic cycles.

The written record of earthquakes in Chile—and of European visits to Isla Santa María—extends back to the mid-1500s15,16. Bathymetric charts made by early surveyors provide a rare opportunity to document vertical crustal movements over a period exceeding one entire seismic cycle. Isla Santa María is located about 80 km landward of the Chile Trench (Fig. 1) and 20 km above the subduction zone megathrust17. Captain Robert FitzRoy and the HMS Beagle visited the island about six weeks after the 1835 earthquake. FitzRoy estimated that the island rose 2.4–3 m during the earthquake from measurements on uplifted inter- and sub-tidal sea life attached to rocks5. Regarding the adjacent bay, Rada Santa María, he reported ‘The earthquake of 1835 raised the bottom nearly 9 feet [2.7 m]; so that where there was a depth of 5 fathoms [9.1 m] in 1834, the Beagle found only 3 1/2 [6.4 m] in 1835’ (ref. 18). Although Charles Darwin was crossing the Andes and not present when the Beagle charted the island, these measurements provided motivation for Darwin’s tectonic theories linking earthquakes with the uplift of the Andes19.

Figure 1: Isla Santa María.
figure 1

a, Location of Isla Santa María within the Golfo de Arauco along the coast of central Chile. Yellow lines are contours of slip (m) inferred for the 2010 Maule earthquake7. b, Isla Santa María with 2-m contours of water depths (m) measured in Rada Santa María. Yellow squares mark locations of the campaign GPS site STDO, and the continuous GPS site STAM (installed after the 2010 earthquake).

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Since 1835 two great earthquakes have occurred in the region. No vertical change was reported at or near Isla Santa María at the time of the great 1960 earthquake that affected the region to the south20. Following the 2010 Maule earthquake, however, measurements of uplifted sea life, campaign GPS measurements, and the resurvey of a geodetic benchmark indicate that the island was uplifted 1.6 to 2.2 m (ref. 21). Campaign GPS measurements showed that the island moved seaward 4.9 ± 0.1 m (ref. 7; all error and uncertainty estimates in this paper represent one standard deviation) and a slip inversion indicated a total displacement during the earthquake of about 6.6 m along the megathrust directly beneath the island7.

We collected data on water depth spanning two centuries in Rada Santa María, a shallow and flat-bottomed bay southeast of the island (Fig. 1). Its bottom is mostly smooth and slopes gently from a depth of about 3 m at the northwest corner of the bay, to a depth of 15 m about 5 km to the south and 7 km to the east. We included nautical charts surveyed in 1804 (ref. 22), 1835 (ref. 23) and 1886 (ref. 24), and our surveys made six weeks before, and three weeks after, the 2010 earthquake, as well as surveys in 2011 and 2013 (Fig. 2). We combined all post-earthquake surveys to make a detailed bathymetric map. Variations in water depth through time are shown in Fig. 3.

Figure 2: Surveys of Rada Santa María.
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a–d, Rada Santa María shown at the same scale with echo sounder tracks in January 2010 (red), and March 2010, January 2011 and March 2013 (blue) (a), and on three historical charts with soundings, in 1804 (b), in 1835 (c) and in 1886 (d). Black circles indicate prominent geographic locations chosen for georeferencing. Others outside map area not shown. Red circles indicate locations projected back onto the historic chart. All circles have a diameter of 200 m. Base images for b and c ©The British Library Board, Maps D.DE.H.SEC.7.(96.) and Maps SEC.10.(1303.), respectively.

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Figure 3: Elevation changes at Isla Santa María inferred from changes in water depth.
figure 3

a, Schematic depiction of elevation changes, the green arrow indicates gradual subsidence. b, Estimates of changes in the elevation of Isla Santa María. Red arrows indicate the amount of uplift in 1835 and 2010 earthquakes inferred from changes in water depth. Diagonal dashed line indicates the change in elevation if the rate of subsidence were a constant 10 mm yr−1. c, Histograms showing distributions of differences in water depths determined from historical nautical charts and modern surveys. One-sigma error bars reflect estimates of total uncertainty. See Supplementary Information for details.

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The water depth in the bay decreased on average 1.5 ± 0.2 m at the time of the 2010 earthquake. Decreases in water depth on the western edge of the bay ranged from about 1.4 to 2.1 m, and average about 1.7 m, in reasonable agreement with the measurements of uplift of 1.6 ± 0.1 to 2.2 ± 0.2 m reported along the shore21 and 1.7 ± 0.1 m at the GPS site STDO located 5 km northwest of the bay (Figs 1 and 4). Rada Santa María is east of STDO and measurement sites on the island, and the uplift is expected to decrease with distance eastward from the trench. A plane fit to the best data suggests an average tilt of about 0.05 m km−1 down to the east. As discussed in the Methods, the uplift data from both the island and the bay—including the inferred tilt beneath the bay—agree well with other regional uplift data at a comparable distance from the trench, as well as predictions from the slip model for the 2010 earthquake (Supplementary Fig. 3). Thus, we believe that slip on the underlying megathrust, rather than any surficial or secondary crustal faulting25, is the dominant cause of the observed vertical motions. GPS measurements in the three years following the earthquake show a decaying post-earthquake subsidence, most probably caused by afterslip, of slightly less that 0.2 m that we could not resolve in the bathymetric surveys (Fig. 4b and Supplementary Fig. 9).

Figure 4: Modelling elevation changes.
figure 4

a, Profile perpendicular to the strike of the megathrust at Isla Santa María. In calculations the locked zone is assumed to extend between downdip and updip limits. b, Changes in elevation of Isla Santa María determined by measurements at STDO from 2004–2013. Main panel, pre-earthquake data at expanded scale. Inset, all data at condensed scale, dashed line shows time of earthquake uplift. Error bars are one standard error. c, Rates of subsidence at Isla Santa María assuming a locked zone extending upward along the megathrust from 40 km to an updip limit at depths ranging from 7 to 40 km, and for locking ranging from 25 to 100%. Grey box shows approximate range of likely rates.

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After adjustments and corrections discussed in the Methods and Supplementary Information, the mean water depth in the bay was, relative to the aggregate of our post-earthquake surveys, 2.7 ± 0.4 m deeper in 1804, 0.4 ± 0.5 m deeper immediately after the 1835 earthquake, and 0.1 ± 0.4 m deeper in 1886 (Fig. 3 and Supplementary Table 2). Thus, we infer a history of vertical movements in which the elevation of the bottom of the bay was low in 1804; was uplifted 2.3 m in 1835; unchanged (within error bounds) to 1886; subsided about 1.4 m between 1886 and January 2010, and was uplifted 1.5 ± 0.2 m in the 27 February 2010 earthquake (Fig. 3).

Campaign GPS measurements covering six years before the 2010 earthquake at the site STDO showed that the island was subsiding at the rate of 10.2 ± 3.4 mm yr−1 (Fig. 4b). Landward displacement occurred at the rate of 42.1 ± 2.1 mm yr−1 (ref. 26). Long-time residents of the island described before the earthquake a continuous loss of onshore land by persistent flooding, supporting the conclusion that the island was subsiding during the twentieth century. If one assumes that subsidence occurred at a constant rate between 1835 and 2010, then a best-fit line to the data, considering uncertainties, yields a subsidence rate of 7.9 ± 2.9 mm yr−1. In contrast, if one assumes no subsidence between 1835 and 1886, followed by a constant rate of subsidence until 2010, the rate is 11.3 ± 4 mm yr−1. Considering uncertainties, neither of these rates is significantly different from the rate determined from GPS measurements before 2010.

A kinematic finite-element model with well-constrained fault geometry and rheological properties shows that the interseismic elevation changes between 1835 and 2010 agree with the pattern of elastic strain accumulation expected for a location above the seaward edge of a locked seismogenic zone of the megathrust (Fig. 4). In essence, the South American Plate is dragged downward and pushed landward by its locking to the descending Nazca Plate. A subsidence rate of 10 mm yr−1 would be consistent with an updip locking depth of 10 km and 100% locking or an updip locking depth of 15 km and 75% locking (Fig. 4c).

Differences between the depths of the bay in 1835 and 1886 are not statistically significant, suggesting that there may possibly have been a period of little or no subsidence—possibly from low locking—after 1835. However, in the first two years following the 2010 earthquake, the island subsided—with a decaying rate—a total of less than 0.2 m (Fig. 4c and Supplementary Fig. 9), indicating limited afterslip. Measurements over the coming years will resolve the question of whether there will be a period of little subsidence following the 2010 earthquake as may have occurred following 1835.

Debate rages about the extent to which observations of the accumulation of elastic strain provide any useful information about the level of probability for future earthquakes27. Although the uplift at Rada Santa María in 2010 was approximately equal to the accumulated subsidence since 1835, hinting that the earthquake was slip-predictable, the question of whether a similar pattern prevailed before 1835 remains open.

Although the process is clearly dominated by elastic rebound, the inhomogeneous nature of the slip confounds a simple accounting of the accumulation and release of elastic strain. Observations in 2010 and other large earthquakes show that slip is strongly inhomogeneous across the rupture area and caution against a simplistic interpretation of the relation between coseismic and interseismic displacements. The interseismic displacement (subsidence near the island) is probably controlled by a locked portion of the megathrust that extends over a wide area. In contrast, the coseismic vertical displacement will reflect spatially variable coseismic slip on the megathrust resulting from the dynamics of the rupture propagation, inhomogeneity in the degree of locking, slip in previous earthquakes, and other factors.

In addition to 2010 and 1835, large earthquakes in 1570, 1657 and 1751 probably ruptured the megathrust in this region15,28. Before the 175-year interval between 1835 and 2010, earthquakes occurred about every 88 years. Given the characteristic inhomogeneity of slip, the earlier apparent constancy of the recurrence interval was probably coincidental. Indeed, the earthquake in 1751 is believed to have been comparable to 2010 and larger than that in 1835 (ref. 28). Our observations cannot answer important questions about how the spatial distribution of slip in 1835 may have differed from that in 2010 beyond Isla Santa María, and specifically whether the slip distributions were similar or in any way complementary.

Uplifted Holocene beach berms8 (corrected for a +2 m mid-Holocene highstand predicted by the ICE-5G model29) and latest-Pleistocene (MIS-3) near-shore sediments9 on Isla Santa María indicate net uplift of ∼1.5 mm yr−1. The mean recurrence interval of strandline-forming events of ∼180 yr (ref. 8) suggests a mean net gain in elevation of ∼0.26 m per event. If these events are taken as a record of great earthquakes, the observed uplifts in 1835 and 2010 are typical, and great earthquakes are responsible for the net uplift, then as much as ∼10–20% of the seismic uplift could be permanent and a typical interseismic subsidence would recover only ∼80–90% of the coseismic uplift.

The apparently constant rates of plate convergence and the observation that the 2010 earthquake (as well as the 2004 Sumatran and 2011 Japanese earthquakes) resulted in nearly complete stress drops30 support the notion that a more sophisticated approach to the tracking of accumulated slip deficit may provide a basis for improving and updating the probabilistic estimates of future great earthquakes.

Historical bathymetric charts, in combination with other historical, geologic and geodetic data can provide constraints on past coseismic and interseismic vertical motions in subduction zones. These data at Isla Santa María support elastic rebound theory with allowance for spatial inhomogeneity in slip, but suggest a component of permanent deformation. They also hint at a variable subsidence rate during the seismic cycle, which will be confirmed or refuted by GPS observations underway at present. Data integrated from a variety of sources, including historical data, offer new possibilities for constraining models of strain accumulation and release during the earthquake cycle in subduction zones, and, possibly, the regional tracking of slip deficit.

Methods

Modern bathymetric surveys.

We carried out echo sounder surveys in a 6 × 8 km area of Rada Santa María in January and March 2010 (before and after the Mw 8.8 earthquake on 27 February), and in the summers of 2011 and 2013. Our analyses focused on a portion of the bay that is characterized by a smooth, gently sloping bottom with an average depth near 8 m. Our surveys were conducted in open boats using echo sounders and consumer-grade GPS units with typical horizontal position errors of a few metres. We corrected our surveys to a datum of mean sea level using the TPXO 8 Atlas tidal model31,32. We compared predictions from this model with a month-long tidal survey carried out on the island at Puerto Sur (on Rada Santa María) in 2009, with measurements at a tide gauge at Coronel 30 km across the Golfo de Arauco, and with our own measurements made during surveys in 2011 and 2013 with a portable tide gauge. Our conclusion is that using the TPXO model we can refer our surveys to mean sea level with an accuracy of about ± 0.1 m.

We combined the results of all post-earthquake surveys to construct one aggregate map of water depth in the bay. Although we know from GPS measurements on Isla Santa María that the island subsided less than about 0.2 m between March 2010 and January 2013 (discussed in more detail below) we were not able to resolve this amount with our surveys. Most of this subsidence occurred in the first several months following the earthquake. Combining all post-earthquake surveys in one model gives the highest geographic density of measurements and therefore reduces the uncertainties associated with interpolating to estimate depth within the bay. The aggregate model consists of Delaunay triangulation33 with linear interpolation within our post-earthquake echo soundings to predict the depth at any point within the bay. We estimated the consistency and uncertainties within the aggregate model by comparing the depths at the crossing points of post-earthquake track lines (Supplementary Fig. 1).

For additional details of our surveys including analysis of uncertainties related to transducer positioning, sound speed in water, and interpolation in the model, see the Supplementary Information.

Determination of 2010 coseismic uplift.

Comparison of the depths determined by the echo sounder in January 2010, before the earthquake, with the aggregate post-earthquake model resulted in a set of 1,427 depth differences distributed along the profiles within Rada Santa María. The mean of these differences yields an estimate of the mean uplift of 1.5 ± 0.2 m during the 2010 earthquake. The set of depth differences from our measurements are compared with other measurements of regional uplift during the 2010 earthquake from campaign and continuous GPS (ref. 7), and sea life attached to rocks21 in Supplementary Fig. 3. In Supplementary Fig. 3b all the uplift data are projected along the local strike of the megathrust fault (N15° E). The figure clearly shows that the uplift data from Isla Santa María and our bathymetry data from the adjacent bay are consistent with the other uplift data at a comparable distance from the trench in this region, mostly along the western coast of the Golfo de Arauco, to the south of Santa María. Second, the data are all in good agreement with a slip model7 derived from the inversion of GPS and InSar data. Although Isla Santa María was affected by some surface faulting25, the surface ruptures were short and the displacements rapidly died out suggesting shallow faulting. In addition there are no discontinuities or jumps in the pattern of uplift data shown in Supplementary Fig. 3b that might suggest significant splay faulting. Thus, although splay faults have played an important role in the long-term geologic history of the island17, and although there was some minor surface faulting on the island in 2010, vertical motions observed on the island and in the adjacent bay were most probably dominated by slip on the underlying megathrust.

Historical surveys.

We obtained scanned copies of the charts made from the 1804 and 1835 surveys22,23 from the British Library and a hard copy of the chart with the 1886 survey24 from the US Library of Congress, and digitized the soundings from the digital images of each chart. The general practices used in nineteenth-century nautical surveys are well documented34,35,36,37, although we lack precise details of the historical surveys used here. Preparation of such nautical charts was an issue of strategic importance to the leading nations of the day, and employed the best available technologies and highly skilled personnel. For our purposes the most important characteristics are: reasonable cartographic accuracy; correction for tides from the initially observed water depth to a known or inferred datum; knowledge of the year when the survey was carried out.

Shallow water hydrographic surveys were typically carried out in an open boat with a sounding pole for depths up to about 9–10 m, and a weighted line for deeper waters. Positions were determined by a variety of methods, including using a station pointer to fix a position from the measured angles to three points with known positions, or sighting sounding positions using theodolites based on shore. Distances were determined from baselines measured with a chain, or estimated by timing the arrival of the sound following the flash of a musket. Communication among the parts of the survey party onshore and afloat was carried out with flags. In some cases soundings were located by steering a constant azimuth from the main ship at anchor, and the distance was estimated by measuring the angle from the sea surface (or a stripe painted on the ship’s hull) to the top of the ship’s mast. True north and latitude could be determined with some precision from solar measurements. The coastline and topographic features were sketched with varying degrees of accuracy and artistry. Tides were either measured simultaneously, or estimated with primitive tidal models38. The surveys were reduced to a datum typically described as ‘low water ordinary springs,’ a datum typically between the data of mean lower low water and lowest astronomical tide in use today.

Soundings in the 1835 and 1886 surveys were reported in fathoms and metres, respectively. Soundings in the 1804 survey were reported on the published chart in units of ‘brazas [Spanish fathoms] of seis pies de Burgos,’ a less familiar but well-established unit equal to 1.672 m (ref. 39). For each sounding in the historical surveys we calculated the difference in water depth between the reported sounding and the value interpolated from our aggregate post-earthquake map at the same location (Fig. 3). We then computed the mean difference and other statistics. We applied corrections for truncation bias and to refer inferred water depths to mean sea level, as well as corrections for sea level rise and El Niño/Southern Oscillation as described in the Supplementary Information.

Georeferencing.

The historical charts present challenges in georeferencing the positions of the soundings to a modern coordinated system. Fortunately, the smooth and relatively flat bottom of Rada Santa María (Fig. 1b) reduces the impact of uncertainties arising from georeferencing the historic charts and of the position errors of individual soundings. There are no cultural features shown on the 1804 and 1835 charts that can be located today, and only one on the 1886 chart, the site of a since-replaced lighthouse. Therefore, we selected six distinctive geographic features that we judged to be apparent to sailors to be used as control points. Three are shown in Fig. 2; the fourth and fifth are at the westernmost and northeastern most points of the island, respectively. The sixth point is on a small island (Isleta Farallon) off the northern tip of Isla Santa María. We determined the modern coordinates of these points from a 2004 nautical chart40. Five of the six points are headlands, and we chose the coordinates of points at the base of the beach cliffs. On Isleta Farallon we chose the centre of the island. We then selected the coordinates (in pixels) of the same points, as best we could determine, on each of the historic charts. In georeferencing the 1886 chart we also included the site of the old lighthouse. We reduced each of the data sets to coordinates relative to its respective centroid, and then using least squares, determined the two-parameter Helmert41 or similarity transform between the pixel and UTM coordinate systems that minimized the misfit between the modern and historical control points. For the 1804 and 1835 charts we fitted the six points using two parameters with root-mean-squared (r.m.s.) errors of 33.6 and 24.1 m, respectively. The r.m.s. error for the seven points of the 1886 map is 15.6 m. Although there are no points to the east or south of the bay that we can use for georeferencing the historic charts, the locations of areas of steeper bottom slope towards deeper water on the east and south margins of the bay are reasonably well fitted by the transforms obtained. These transforms were then used to convert the positions of the soundings from pixel coordinates to UTM.

We used simulation techniques to estimate the uncertainties in the mean depth difference resulting from systematic uncertainties in georeferencing, and from the random, independent errors in the positions of the individual soundings. Details can be found in the Supplementary Information.

In comparing the differences between our model and the historical soundings we observed a tendency for outliers in the extreme southwest and northeast corners of the area surveyed where changes in water depth seem to be more abrupt, probably resulting at least in part from position errors in areas of submerged topography developed during sustained periods of lower sea level. Therefore, we reduced the area of consideration slightly. Supplementary Fig. 7 shows the depth differences plotted on a map of the bay for the 1804, 1835, 1886 and 2010 surveys together with the boundary of the final area used for comparison.

Uncertainty analysis.

Estimating the uncertainties in the differences in water depth formed an essential portion of our study. We included uncertainties arising from both random and systematic sources in a formal uncertainty analysis42,43. Random uncertainties included data variability, position errors of individual historical soundings, truncation, and interpolation. Systematic uncertainties included uncertainties related to correction to the historical and modern datum levels, sea level rise and El Niño/Southern Oscillation, georeferencing, and transducer depth. This analysis is described in detail in the Supplementary Information along with discussion of the normality of the distributions of depth differences and other statistical issues. Supplementary Table 2 contains a summary of the mean differences in depth between the earlier surveys and the post-2010 earthquake period, along with estimates of all adjustments and associated uncertainties.

Modelling and GPS/GNSS measurements.

To draw conclusions about the vertical motions at Santa María over the entire seismic cycle we combined our estimates of the long-term changes from bathymetry with GPS measurements made before and after the 2010 earthquake. Campaign GPS measurements were first made on Isla Santa María in 2004 and repeated every year before the 2010 earthquake, from which we calculated the pre-earthquake rate of subsidence, and six sessions following the earthquake. Rates of subsidence determined from both GPS measurements and the comparison of the historical and modern bathymetry were determined by least squares in which the sum of the squares was weighted by the reciprocal of the variance of the data or the variance of their mean.

We also compared the post-earthquake campaign GPS measurements with measurements from a continuous GPS station installed on the island after the earthquake to assess the role of afterslip in the complete seismic cycle. As discussed in more detail in the Supplementary Information, afterslip in the two and a half years following the 2010 earthquake resulted in a subsidence of Isla Santa María of less that 0.2 m (ref. 44). Afterslip on the megathrust in the vicinity of the island was significantly less than in neighbouring areas of the earthquake rupture and only a small fraction of the coseismic slip. Data are shown in Supplementary Fig. 9 and additional discussion of afterslip can be found in the Supplementary Information along with further details of the GPS processing.

The Green’s functions of ref. 7, originally determined to allow the inversion of geodetic observations to estimate the distribution of fault slip, also provide a means to investigate the vertical displacements at Isla Santa María during the interseismic period (typically subsidence) as well as for analyses of sensitivity of observed uplift to the distribution of coseismic slip (typically uplift). In this context the Green’s functions can be thought of as a matrix that relates the unit slip at one set of points covering the fault surface to the vertical displacements at a second set of points on the Earth’s surface. The values in this matrix were calculated with a finite-element model and depend on the geometry of the megathrust fault and on the values and distribution of elastic parameters assumed for the surrounding volume. Then for an assumed distribution of slip on the fault the resulting vertical deformations can be calculated by multiplying the matrix of Green’s functions times a vector containing the assumed fault slip. We used this technique to carry out the slip modelling shown in Fig. 4.

Other.

For discussion of the effect of sand mobility on the bathymetrical surveys and analysis of the sensitivity of uplift to the 2010 slip distribution, see the Supplementary Information.

Code availability.

Principal computer codes in MATLAB and the modern bathymetric data are available at http://www.github.com/rwesson-usgs/historical-bathymetry