Short-term retreat statistics of a slowly eroding coastal cliff

. The frequency, spatial distribution, and dimensions of coastal cliff retreats, a basic statistic underlying cliff top hazard assessment, are presented for 7.1 km of unprotected and slowly retreating coastal cliffs near Point Loma in San Diego, California, US. Using 8 airborne light detection and ranging (lidar) surveys collected over 5.5 years, 130 individual cliff edge failures (primarily rockfalls, block falls, and topples) were detected. Footprint areas varied from 3 to 268 m 2 , maximum landward retreats from 0.8 to 10 m, and alongshore lengths from 2 to 68 m. The failures with the largest landward retreats were also relatively long, and 13% of the slides accounted for 50% of the lost cliff area over the study period. On this short (5.5 years) time scale, “no change” was the most common observation (84% of the cliff edge). Probability distributions of non-zero cliff retreat during each time interval usually had a single peak between 1 and 2.5 m. Intervals with high mean retreat had elevated numbers of failure in all class sizes, and also contained the largest individual retreats. Small and medium slides tended to reoccur preferentially (relative to randomly) near previous small and medium slides, forming short-term hot spots, while large slides were less likely


Introduction
Seacliff retreat threatens cliff top infrastructure, and estimates of future cliff positions are needed for coastal planning. Multiple marine and subaerial erosive processes Correspondence to: A. P. Young (adyoung@ucsd.edu) interact with complex site-specific local geology to produce temporally and spatially variable retreat (Trenhaile, 1987;Sunamura, 1992). The limited available historical observations provide mean (time-and-alongshore averaged) retreat rates, but extrapolation of these rates to estimate sitespecific future risk is problematic because the magnitude variability (in space and time) of cliff retreat is not considered (Cambers, 1976;Lee et al., 2001;Moore and Griggs, 2002;Quinn et al., 2009). Retreat magnitude variability and episodicity are included in recent probabilistic characterizations of cliff retreat (e.g. Lee et al., 2001Lee et al., , 2002Hall et al., 2002;Williams et al., 2004;Teixeira, 2006;Milheiro-Oliveira, 2007;Furlan, 2008). These models highlight both the need for, and the scarcity of, observations of even basic statistics such as the dependence of cliff retreat frequency on retreat magnitude.
The statistics of inland mass movements have been studied extensively (e.g. Hovius et al., 1997;Pelletier et al., 1997;Dai and Lee, 2001;Dussauge-Peisser et al., 2002;Guzzetti et al., 2002;Hergarten, 2003;Brardinoni and Church, 2004;Brunetti et al., 2009;ten Brink et al., 2009). Historical records of coastal cliff failure statistics are relatively sparse. Dong and Guzzetti (2005) investigated the frequency statistics of soft cliff retreat for two data sets. The first (obtained from Hall et al., 2002) consisted of 32 (26 non-zero) measurements made along eight transects over four time intervals ranging from 7 to 29 years at East End in East Sussex, England. The second data set included 198 (168 non-zero) yearly measurements from 1953 to 1990 along six transects on the Holderness coast, England. For both datasets, the frequency of cliff retreat decreases with event size, and medium to large size events follow an inverse power-law distribution. Teixeira (2006) investigated the frequency statistics and geometric relationships of coastal mass movements in Miocene aged rock cliffs of the Algarve, Portugal. These datasets consisted of 140 field measurements obtained over Published by Copernicus Publications on behalf of the European Geosciences Union. 206 A. P. Young et al.: Cliff retreat statistics nine years, and 177 events obtained from aerial photographs over 44 years. The cumulative frequency of cliff retreat for both datasets fit a power-law distribution above a critical value, and deviated for relatively small events. Teixeira (2006) also found power-law relationships between volume, horizontal area, and retreat. Marques (2008) further investigated the magnitude-frequency statistics of strong, slowly retreating coastal cliffs (< 10 cm yr −1 ) in terms of volume, horizontal area, and cliff retreat. The database included about 600 events from regional scale observations obtained from aerial photographs spanning up to 53 years in Portugal and Morocco, and smaller scale insitu field observations (from Teixeira, 2006) in Portugal. Rosser et al. (2007) used high resolution monthly terrestrial laser scans collected over 32 months in the North York Moors National Park, UK to compile a large database of over 500 000 coastal rockfalls. Rockfall volume and scar area (parallel to the rock face) followed a power law, and the cumulative distribution followed a linear trend for log volume versus normalized frequency. Their results suggested that rock type influenced scar geometry, and that smaller rockfalls often preceded larger failures. Lim et al. (2010) recently compiled a database over 100 000 coastal rockfalls using monthly terrestrial laser scanning and orthophotography over 20 months at Staithes, North Yorkshire, UK. They found a linear trend in the non-cumulative distribution of log volumes, including the small changes that were resolved with their uniquely highresolution observations. These results suggest undersampling of small cliff failures caused the deviation of the magnitude-frequency relationships found previously in coastal (Dong and Guzzetti, 2005;Teixeira, 2006;Marques, 2008) and inland (e.g. Hovius et al., 1997;Stark and Hovius 2001;Dussauge-Peisser et al., 2002;Malamud et al., 2004) observations.
The different methods, measured parameters, and failure types make comparisons of failures from different studies difficult. For example Teixeira (2006) and Marques (2008) report horizontal (planimetric) area while Rosser et al. (2007) and Lim et al. (2010) consider area parallel to the cliff face. Some studies focus on magnitude-frequency relationships of cliff retreat (Dong and Guzzetti, 2005;Teixeira, 2006) while others on failure volume (Rosser et al., 2007;Lim et al., 2010). Cliff edge retreat failures, a subset of the cliff failures, are generally used in models to establish structural setbacks, while volumetric statistics are more appropriate for processed-based modeling. Many authors note that additional cliff failure inventories are required to assess the generality and predictive utility of observationbased, statistical characterizations of cliff retreat. The present study uses 8 airborne lidar surveys collected over 5.5 years to describe basic two dimensional (planimetric), short-term (5.5 yr) statistics of cliff edge failures over 7.  (Shepard and Grant IV, 1947), and resulted in building condemnation (Flick, 2005). Riprap and seawalls protect about 28% (2.8 km) of the cliffs, mostly in the northern section ( Fig. 1

Geology
Point Loma is a rock promontory uplifted by the Rose Canyon section of the Rose Canyon-Newport-Inglewood Fault. The seacliffs exposed on the seaward side of the promontory are generally 20 to 30 m high (range of 5 to 40 m), and are composed of two geologic units (Fig. 2). The lower unit is the Point Loma Formation, a lithified Cretaceous interbedded fine grain sandstone and shale, unconformably overlain by the Bay Point Formation, a poorly consolidated Pleistocene sandstone terrace deposit (Kennedy, 1975 (1) nearly entirely vertical and composed primarily of the Point Loma Formation, and (2) a sloped upper cliff composed of the Bay Point Formation (Fig. 3). According to Emery and Kuhn (1982), both profiles indicate that marine erosion dominates subaerial erosion. However, Kuhn and Shepard (1984) and Flick (1994) document accelerated local erosion caused by concentrated surface runoff, and the relative importance of marine versus subaerial processes is unknown. Although the statistical failure properties described are for unprotected cliffs, they may be affected by adjacent armored cliffs and cliff top development. Joints, fractures, and a series of northeasterly intersecting faults create weak areas in the lower cliff, resulting in alongshore variable cliff resistance, and a highly irregular shoreline containing sea caves, wave cut notches, surge channels, small headlands, a few pocket beaches, and an occasional sea stack, arch, or blow-hole. The shore platform,  composed of the Point Loma Formation, is often submerged at the cliff base and lacking beach sand (with the exception of a few narrow pocket beaches), with wave attack during average or higher tide stages. At some locations, the shore platform elevation allows wave attack at all tide stages. Cliff undercutting at the cliff base from wave action is often present and can result in cliff instability, as at other cliffs (Hampton, 2002;Kogure et al., 2006;Young and Ashford, 2008;Budetta, 2010). Cliff failures in the Point Loma Formation generally occur as rock falls, block falls, and topples (Varnes, 1978).

Historical erosion
Historical photographs of Sunset Cliffs-Point Loma (Shepard and Grant IV, 1947;Shepard and Wanless, 1971;Kuhn and Shepard, 1984) show the collapse of sea arches and sea caves, the erosion of sea stacks, and an earthquaketriggered cliff failure. Kennedy (1973) used old photographs and observations to estimate an average cliff retreat rate of 1.2 cm yr −1 over 75 years, and noted that some sea caves extend landward as much as 18 m from the cliff face. Average cliff retreat of 8 cm yr −1 between 1952 and 1994 was estimated using orthorectified aerial photographs Moore et al., 1999;Benumof et al., 2000). Spaulding and Crampton (2001) estimated retreat rates of 2 to 6 cm yr −1 for a small portion of Sunset Cliffs over a 70-year period beginning in 1928 from aerial photographs. Hapke and Reid (2007a) used 1934 T-sheets and a 1998 lidar survey to estimate cliff retreat rates along cross-shore transects that average to 17 cm yr −1 . Several studies (Shepard and Grant IV, 1947;Kennedy, 1973;Spaulding and Crampton, 2001) indicated no observable erosion at some locations over considerable time periods, while others (Kuhn and Shepard, 1984;Hapke and Reid, 2007b) found local cliff retreat rates as large as 100 cm yr −1 . The large range of average long-term cliff retreat estimates results from differences in data quality, time period evaluated, cliff armoring, and study section. There is general agreement that cliff retreat is episodic in both space and time. Shepard and Wanless (1971) state "the coast is retreating, although very unevenly", while Kennedy (1973) noted cliff retreat is "regionally negligible and locally high".

Climate
San Diego's semi-arid Mediterranean climate is characterized by dry summers and occasionally wet winters, with 85% of rainfall from November through March. Annual precipitation amounts vary from about 10 to 60 cm, and average 25 cm. Rainfall during the study period was average or less, except for winter 2004 to 2005 (Interval 2), when rainfall exceeded twice the annual average (Fig. 4).

Waves
The seacliffs are exposed to swell from distant storms in both hemispheres and waves generated by local winds. During winter, swell from the North Pacific and the Gulf of Alaska are most energetic, whereas swell from the South Pacific dominates in summer. The seasonal cycle produces maximum wave energy in winter (Fig. 4). Wave conditions vary alongshore, and were estimated from a wave buoy network (CDIP, http://cdip.ucsd.edu) and a spectral refraction wave model that accounts for the effects of complex bathymetry in the southern California Bight, varying beach orientation, and wave exposure (O'Reilly and Guza, 1991Guza, , 1998. Mean daily significant wave height (H s ) at a virtual buoy or "Monitoring and Prediction point" (MOP) located in 10 m water depth seaward of the cliff section ( Fig. 1), shows that wave heights were typical during the study period (Fig. 4). Although wave conditions vary alongshore, the patterns of seasonal and interannual variation are similar at all the study cliffs. Complex small scale cliff-base geometry precludes detailed estimates of location-specific marine erosional forces.

Seismicity
Minor ground motions are relatively common in the tectonically active San Diego region. However, no earthquakes with magnitude > 5.0 occurred within 100 km of the study site, during the study period (www.data.scec.org).

Lidar data collection
Eight airborne lidar surveys (Table 1) were conducted between October 2003 and March 2009 with an Optech Inc. Airborne Laser Terrain Mapper (ALTM) 1225. Each airborne survey consisted of 2 to 4 passes with a 40 • swath at an altitude ranging from 300 to 1150 m resulting in a point density of approximately 2 to 4 points m −2 . The alongshore aircraft trajectory was located approximately 0 to 100 m seaward of the seacliffs. ALTM elevation points are computed using three sets of data: laser ranges and their associated scan angles, platform position and orientation information, and calibration data and mounting parameters (Wehr and Lohr, 1999). GPS receivers in the aircraft and on the ground at control points provide platform positioning. The GPS receivers record pseudorange and phase information for post-processing. Platform orientation information comes from an Inertial Measurement Unit (IMU) containing three orthogonal accelerometers and gyroscopes. An aided-Inertial Navigation System (INS) solution for the aircraft's attitude is estimated from the IMU output and the GPS information.

Digital elevation model
Lidar data was occasionally obtained in over-hanging locations such as seacaves and notches, however the partial coverage did not permit construction of fully 3-D terrain models, and volume statistics could not be estimated. Therefore, the point data were processed into 0.5 m resolution 2.5-D digital elevation models using the second of two returns (the most representative of the ground surface) and a modified "natural neighbors" interpolation that removes over-vertical features and maintains vertical cliff edges and complex topography (see Appendix A, LEM-DEM).

Change detection and cliff retreat
Vertical topographic change for the seven time intervals (Table 1), obtained by differencing successive digital elevations models to create digital change grids (DCG), shows erosion (negative changes) at failure locations, and accretion (positive changes) at cliff base talus deposits (Fig. 3). Sources of errors in the DCGs include the basic lidar observations, spatial interpolation, and vegetation. The vertical root mean square difference between two surveys (RMS Z , Federal Geographic Data Committee, 1998), a measure of the total error, was estimated using three control sections of stabilized slopes and seawalls that represent the range of slopes and vegetative conditions of the studied seacliffs. The average RMS Z of all control sections and intervals was 18 cm, and ranged from 10 to 24 cm.
The DCGs were filtered and edited to remove noise and identify the individual failures causing seaward cliff edge retreat. First, all grid cells with a vertical change of less than 2 m ( RMS Z ) were removed. Next, a minimum topographic footprint was imposed, requiring at least 12 connected cells of negative change, thus enforcing a minimum change area of 3 m 2 . The vertical and footprint thresholds, selected through trial and error and tested on the control surfaces, significantly reduced noise. Finally, the digital data footprints were visually inspected, yielding the topographic footprints of individual seaward cliff-edge failures during each time interval. The seaward cliff edge was used as the shoreline reference feature because cliff retreat is readily identifiable for landslides occurring in both profile types (Fig. 3). The seaward edge was coincident with the cliff top edge when the upper cliff slope was vertical (Fig. 3c). These interval changes could represent combined individual, adjacent landslides that occurred during a particular time interval.

Cliff failure parameters
Cliff failure footprints were sampled on local cross-shore oriented transects spaced 1.0 m alongshore (Fig. 5). The DCGs usually were not aligned with the local shoreline orientation, resulting in a range of retreat measurements (including values less than the grid size) rather than increments of the 0.5 m grid size. The alongshore failure length was defined as the number of transects intersecting the failure footprint. Mean failure retreat was estimated as the average of intersecting transects for each slide, while the maximum failure retreat represents the longest intersecting transect. The cliff footprint area was measured as the planimetric area on the DCGs. Note this method can result in small differences between the measured planimetric area, and area calculated as the product of alongshore failure length and mean failure retreat, but provides a simple automated way to estimate cliff failure attributes with locally changing shoreline orientation.  between a rock revetment at the cliff base and an upper cliff retaining wall. Erosion (primarily of the terrace deposits) at location P2 occurred during Intervals 1 through 4 and 7, and involved a seawall failure with loss of residential property. These events (and others at protected cliffs) are excluded from further consideration because cliff protection significantly alters cliff failures.

Footprint transect retreat measurements
The 130 individual cliff edge failures from all time intervals were sampled on transects spaced 1 m alongshore, resulting in approximately 1300 non-zero transects. Transect retreats ranged from 0.01 to 10 m, with a mean and median of 2.3 m and 2.0 m, respectively. The maximum cumulative cliff retreat at single location during the study period was 10 m (1.8 m yr −1 ). Approximately 84% of the cliff edge experienced no detectable retreat over the study period based on the described methods.
The shapes of the retreat probability density functions for each time interval were similar, with a single peak ranging between 1 and 2.5 m (Fig. 9a), except for the bimodal Interval 5. The retreat events of combined large mean retreat intervals (1 and 2), small mean retreat intervals (3 through 7), all-intervals, and cumulative retreat are all positive skewed with a peak around 2 m (Fig. 9b). However, the retreat distribution during large and small mean retreat intervals are significantly different (k-s test, p = 0.000031, compare bold dashed with bold solid curve in Fig. 9b). During intervals of large mean section retreat, larger retreat events occur more often than expected compared to intervals with low mean section retreat. The converse is also true suggesting the distributions of cliff retreat may differ for time periods with varying amounts of overall failure activity.
The distributions of cumulative (e.g. total on transects) and average-all-interval retreat differ (Fig. 9b) because locations that retreat more than once are amalgamated into a single value in the cumulative distribution. The cumulative distribution of all individual retreat measurements (Fig. 10) fits better to an exponential distribution rather than a powerlaw, particularly for medium size events where the powerlaw fit deviates for values > 4 m. A single failure affects multiple adjacent transect retreat measurements, and the nonindependence of transects probably contributed to power law deviation, along with amalgamation and under sampling of small areas.

Mean interval cliff retreat
The mean cliff retreat (averaged over the unprotected cliff section, including transects with no change) during each interval ranged from 2 to 16 cm. The average retreat rate of 7.8 cm yr −1 is consistent with rates measured from 1952 to 1994 of about 8 cm yr −1 Moore et al., 1999;Benumof et al., 2000). Maximum mean cliff retreat, number of failures, and the extent of alongshore change (Table 1)  If the size distribution of retreats were constant from interval to interval, then the mean retreat and number of failures in the interval would be linearly related, as was only approximately observed (Fig. 11). Mean retreat values in Intervals 1 and 2 were elevated above the best fit line because large slides occur disproportionally more often in the intervals with the most slides (Fig. 11).

Spatial distribution of cliff retreat
Retreat occurred along less than 3% of the cliff length in all intervals except Interval 2, when 7% of the section retreated. In total, about 1150 m of the unprotected cliffs experienced measurable cliff retreat during the study period. Forty eight of the failure footprints were spatially adjacent to at least one other failure resulting in 20 amalgamated footprints and 135 m of cliff retreating more than once.
The potential presence of short-term cliff retreat hot spots (Young et al., 2009a) was investigated with a simple Monte Carlo simulation of cliff retreat for the study period. The simulated cliff retreat for each time interval used the observed interval failure events and random spatial placement along the 7.1 km (unprotected) cliff section.
The random spatial placement of the Monte Carlo simulation over-predicted the maximum cumulative cliff retreat by 6 m (Fig. 12), and the observed alongshore extent of cliff retreat was about 50 m smaller than the Monte Carlo mean (Fig. 13). The differences between the Monte Carlo simulation and observations are consistant with cliff retreat history at a particular location influencing subsequent cliff retreat. For example, the simulated alongshore extent of failures can be made to match the observations by forcing some small to moderate failures to coincide with previous events, creating non-random, short-term hot spots. Small failure events could redistribute stress to nearby locations causing strain accumulation and concentrating future failures in both space and time. That is, the observed failures may be part of a progressive failure (Bjerrum, 1967;Bishop, 1971;Petley et al., 2005), or sequence of successive coastal failures (e.g. Hutchison, 1969;Brunsden and Jones, 1976;Alveirinho Dias and Neal, 1992;Collins and Sitar, 2008;Young et al., 2009a) where an intial failure may result in adjacent unstable rock masses (Chowdhury et al., 2010). Rosser et al. (2007) suggested that small failures preceded larger events at the same and nearby locations. Lim et al. (2010) emphasized that sequential failures are a "connected process", and stressed the importance of including non-random events in coastal cliff erosion models. The random simulation over-prediction of maximum cumulative retreat suggests that a large failure is relatively unlikely to be followed by more large failures at the same location over short time periods. All locations that retreated at least 6.9 m during one time interval did not retreat in any other interval. Longer data sets are required to test these preliminary suggestions.

Temporal distribution of cliff retreat
Relationships between erosional processes or failure triggering events (waves, rainfall, and seismicity) and cliff retreat could not be established statistically. Although the most cliff retreat and rainfall both were in Interval 2, the second largest amount of retreat occurred during time Interval 1, when rainfall was relatively light (Table 1, Fig. 4). The retreat in Interval 1 might have resulted from wave action, however Intervals 4, 6, and 7, had similar wave conditions with relatively little cliff retreat. Heavy rainfall in Interval 2 might have triggered failures that would have otherwise occurred from wave action in subsequent time intervals, resulting in abnormally large cliff retreat during Interval 2 and low retreat afterwards. Longer data sets are required to confirm the suggestion that, in these slowly retreating cliffs, nonrandom failure sequences serve to decouple failures from local environmental forcing. In the comparatively softer, more erosive cliffs in northern San Diego County, cliff erosion and rainfall in the study period were well correlated (Young et al., 2009b).

Summary
This cliff failure inventory provides the basic statistics of cliff edge retreat caused by medium and large size planimetric cliff changes along 7.1 km of unprotected rocky coastal cliffs over 5.5 years. Footprint areas of the 130 individual cliff edge failures ranged between 3 and 268 m 2 . On this short (5.5 yearr) time scale, "no change" was the most common observation (84% of the cliff edge). Probability distributions of non-zero cliff retreat during each time interval usually had a single peak between 1 and 2.5 m. Intervals with high mean retreat had elevated numbers of failure in all class sizes, and also contained the largest individual retreats. Small and medium failures tended to reoccur preferentially (relative to randomly) near previous small and medium failures, forming short-term hot spots. Large failures were less likely than random to reoccur near previous large failures, but longer records are needed to confirm this statistically tentative result. Cumulative distributions of failure parameters (area, retreat, and length) follow an inverse power-law for medium to large size events and roll off for (poorly sampled) small events, similar to previously reported distributions of coastal and inland mass movements. between each DEM. Lastly, the final LEM-DEM (Fig. A1d) was produced by selecting the maximum grid cell elevation from all the DEM layers.