We calculated rates of shoreline change in two different ways to compare their respective effects on risk over time.

To represent exposure along the US Atlantic coast, we used county-level census data for the total value (adjusted to 2018 US dollars) of owner-occupied housing units in 51 coastal (ocean-facing) counties for each decade since 1970 (Table S3) (Minnesota Population Center, 2011). Because property value data are sparse for the 2010 census community survey (16 Atlantic coastal counties are missing), we instead used the 2009–2013 census 5-year survey. Several 5-year census surveys incorporate 2010, but we chose the 2009–2013 survey because it provides full coverage of all the Atlantic coastal counties, and its mean of total values is closest to the 2010 census community survey (for those Atlantic coastal counties surveyed in 2010). We adjusted the county-total values of owner-occupied housing units to 2018 US dollars and divided by the number of transects in each county to yield a proxy for property value per alongshore kilometre. Because of the range of values along the coast, we took a log-transform and normalized the results to fall between 0 and 1 (Figs. 2b, f, j and 3b).

We represented vulnerability (V) with a two-part relationship based on beach width (Vbw) and beach nourishment (Vbn) over time: 2V=05Vbn+05Vbw. Because the value of exposed property is not included in Vbw or Vbn, this formulation disentangles vulnerability from exposure – a subtle but important conceptual departure from the definition used by the National Research Council (NRC, 2014; Samuels and Gouldby, 2005), which includes property values in vulnerability.

We made the beach-width component (Vbw) inversely related to beach width, such that vulnerability increases as beach width decreases. We express the normalized beach-width component as 3Vbw=x0-xx0-xmin, where x0 is maximum beach width, xmin is minimum beach width, and x is beach width. Because real measurements are unavailable, we assumed that in 1970 all counties had the same initial beach width. (In the results presented here, xmin=10 m and x0=50 m; see also Table S4.) From this baseline, the county-scale shoreline erodes or accretes according to the linear rate determined by the hazard condition (historical, recent, long term, or derived from sea level). Because we used counties as the smallest spatial unit of comparison, our assumption implies that each county is fronted by beach. The physical geography of the real coastline is, of course, more spatially heterogeneous. Our analysis is too coarse to capture, for example, change at isolated pocket beaches in a predominantly rocky coastline, but counties with rocky coastlines will reflect very low or null rates of shoreline change. We consider only oceanfront shoreline, and we do not account for back-bay or estuarine shoreline.

For the beach-nourishment factor (Vbn), we collated beach-nourishment projects since 1970 by county from the beach-nourishment database maintained by the Program for the Study of Developed Shorelines (PSDS, 2017). We took Vbn as the running total number of nourishment projects per county (nc) over time (t, summed annually), and we normalized Vbn by the maximum number of projects among counties as of 2016 (nc*), such that the county that nourished the most has Vbn=1 in 2016. Each county starts with Vbn=0 in 1970, and Vbn increases incrementally with every nourishment project within the county boundary: 4Vbn=∑19701970+tncmaxnc*. We initiated Vbn in 1970 to match the census data for exposure (E). Because 80 % of beach-nourishment projects on the US Atlantic coast have occurred since 1970, we excluded a relatively small number of events. To test the sensitivity of our vulnerability and risk results to the 1970 start date, we examined the relative effects of (1) initiating Vbn from the first nourishment project in our record (in 1930) and (2) excluding the Vbn term altogether (Fig. S1 in the Supplement). Although the risk patterns resulting from these sensitivity tests changed in detail, their general characteristics did not.

In our routine, until a county nourishes for the first time, beach width (x) changes according to the county median linear erosion rate (γ): 5x(t)=xt-1+γt. The linear erosion rate (γ) applied to each county is either the (pre-normalized) historical, recent, or long-term shoreline-change rate or the rate derived from sea-level change, depending on the hazard scenario. The sign convention for γ is negative for erosion and positive for accretion.

Once a county has nourished – as determined by the empirical dataset of nourishment projects (PSDS, 2017) – beach width becomes a function of a linear erosion rate (γ), as in Eq. (5), and a nonlinear erosion rate (θ), which is applied to the nourished fraction of the total beach width (μ) to capture cross-shore and alongshore diffusion of nourishment deposition across and along the shoreface (Dean and Dalrymple, 2001; Lazarus et al., 2011; Smith et al., 2009): 6x(t)=(1-μ)x0+μe-θtx0+∑1tγt, where x0 is maximum beach width, θ is nonlinear erosion rate, μ is the fraction of the total beach width that the nonlinear rate applies to, γ is linear erosion rate, and t is the number of years since the last nourishment project. If a county nourishes at least once in a given year, its beach is restored to a maximum width in that year before it begins to erode. (Our minimum temporal increment was 1 year, and we assumed that nourishment always occurs at the end of a given year.) Maximum beach width (x0), nonlinear erosion rate (θ), and the fraction of beach width affected by the nonlinear rate (μ) are variables applied to the full spatial domain. Beach width (at the county scale) thus changes at a linear rate (γ), where a negative value is erosion and a positive value is accretion, with an additional nonlinear erosion rate (θ) over a fraction of the beach (μ) when nourishment occurs, until the beach is restored to maximum width by a subsequent nourishment project or reaches a specified minimum width (here 10 m). The Vbn term is ultimately normalized by the maximum and minimum beach widths.

Because vulnerability is normalized, the minimum beach width that we specify (xmin=10 m) affects the length of time it takes to reach maximum Vbw but does not affect the overall magnitude of V. A wider minimum threshold means that Vbw reaches a maximum faster, and vice versa. We used a minimum width of 10 m to avoid the numerical instabilities in Vbw that arise with a minimum width equal to or less than 0 m. The minimum width threshold does not affect the cumulative beach-nourishment factor.

We test the effect of altering x0, θ, and μ on both vulnerability and risk, under historical hazard and linear erosion rates (Fig. S1; Table S4). Sensitivity testing shows that vulnerability over time is highest in the case of a narrow beach (x0=25 m) with a high nonlinear erosion rate (θ=0.75) affecting a large fraction of the beach (μ=0.75). Vulnerability over time is lowest in the opposite case (x0=100 m, θ=0.05, μ=0.25) (Fig. S1). In calculating our results, we used a case in the middle of these extremes (x0=50 m, θ=0.5, μ=0.33), applying a value of μ similar to the value (μ=0.35) used by Smith et al. (2009) and Lazarus et al. (2011).

Like a ratchet, the cumulative beach-nourishment factor (Vbn) increases each time a county nourishes. This assumption represents the fact that nourishment projects for shoreline protection (as opposed to reactionary projects for emergency storm response) are cyclical within multi-decadal programmes (NRC, 1995, 2014). Nourishment at a given site rarely occurs only once. A community that initiates a nourishment programme will likely depend on periodic nourishment into the future. By comparison, the beach-width factor (Vbw) is more dynamic, reflecting the oscillatory behaviour of a nourishment cycle at multi-annual timescales by dropping to a minimum after a nourishment project (as the wide beach buffers property from hazard) and then increasing as the nourished beach erodes and coastal properties become more susceptible to hazard.

We examine relationships between the resulting spatial distributions of hazard, exposure, and vulnerability over time using a Kolmogorov–Smirnov test that quantifies, to 95 % confidence, relative differences between pairs of distributions. A Kolmogorov–Smirnov test does not require parametric distributions, and it evaluates the null hypothesis that a given pair of distributions are sampled from the same parent distribution. Rejection of the null hypothesis thus means the distributions are significantly different.

Our data-driven model generates a pattern of coastal risk that varies in space and time at the county scale along the US Atlantic coast (Fig. 1). From 1970, each county generates its own risk trajectory that represents the interaction of hazard, exposure, and vulnerability in that county (Fig. 1a). For visualization and analysis, we scaled each county by the number of 1 km transects they comprise (Fig. 1a). The result is a matrix of 2386 km over 47 years, in which each of the 2386 (1 km) rows is associated with a county. Alongshore mean values for the whole US Atlantic coast are taken from the full matrix so that they reflect the relative alongshore scale of each county (Fig. 1b).

We find that the collective trajectory of risk increases from 1970 to 2016 for all hazard scenarios – despite the occurrence of 998 beach-nourishment projects, ostensibly intended to reduce risk, during the same period (Figs. 2 and 3). The influence of beach-nourishment projects on vulnerability means that county-scale risk varies over time even if hazard forcing remains constant. Because hazard based on measured shoreline change (historical, recent, and long term) is spatially variable but temporally static (Figs. 2 and 3), changes in risk over time under this model condition are driven by either exposure or vulnerability.

The overall risk trajectory also increases with the spatio-temporally variable hazard condition derived from rates of sea-level rise (Fig. 4). The alongshore mean rate derived from sea-level rise shows close agreement with the mean recent shoreline-change rate, suggesting that our simplified bathtub representation of hazard is a reasonable proxy on a multi-decadal timescale (Fig. 5), even though bathtub models tend to underestimate shoreline erosion rates along barrier coastlines (Lorenzo-Trueba and Ashton, 2014; Wolinsky and Murray, 2009).

Individually, not all counties register rising risk trajectories over time. To compare how individual counties contribute to mean risk, we ranked each county by its risk index in 2016 (Table 1). We also examined in detail two examples of how individual counties responded to different hazards and beach-nourishment cycles (Fig. 6). Plymouth County, Massachusetts, demonstrates how vulnerability may respond to linear erosion rates (γ) that vary from eroding (negative, under the historical condition), to static (under the long-term and sea-level-derived conditions), to accreting (positive, under the recent condition) (Fig. 6a–d). Ocean County, New Jersey, demonstrates how the cumulative beach-nourishment factor (Vbn) can drive up risk (Fig. 6e–h). There, Vbn causes the local maxima and minima in vulnerability to increase over time (Fig. 6g), such that even when beaches are at full width, exposed property is still subject to vulnerability V>0. Ocean County highlights how the cumulative beach-nourishment factor functions as a ratchet that forces vulnerability to only increase over time. Because not every county practices beach nourishment, it is possible for a county to have V=0 if its shoreline is accreting (e.g. Camden and McIntosh counties, Georgia). A county that never nourishes will have a Vbn=0, and if a county nourishes only once or twice then their Vbn will remain negligible (but not negative). However, mean vulnerability is greater – and therefore mean risk is greater – when Vbn is left out (V=Vbw) (Fig. S1c and d) because its inclusion makes vulnerability less sensitive to changes in beach width. For example, a county that does not nourish could have a narrow beach but a low Vbn and therefore a lower vulnerability score than if its vulnerability were only a function of beach width.

Alongshore mean risk in our model also increases because of a well-documented national trend in exposure (NOAA, 2013). Exposure in an individual county may increase or decrease from one decade to the next, but mean exposure along the full span of the coast increases over time (NOAA, 2013; Union of Concerned Scientists, 2018). The 51 coastal counties in this analysis represent 1.6 % of all US counties, but since 1970 have constituted 6.9 %–9.25 % of the total value of all owner-occupied housing units in the country (Fig. S2). Thus, while our data-driven model includes simplifying assumptions, we suggest that the increasing risk trends in our findings represent a real phenomenon since exposure has risen at the coast decade for decade in real terms, and our cumulative beach-nourishment factor both dampens mean vulnerability and highlights the reality of long-term risk in counties that nourish continually.

Finally, we compared the statistical distributions of exposure in high- and low-hazard counties and in high- and low-intensity-nourishing counties (as an aspect of vulnerability), to examine whether the three components of risk, as we represent them, reflect temporal interrelationships. In keeping with the scaled stripes in Figs. 1, 2, and 4, we present these distributions (Figs. 7 and 8) at the transect scale rather than the county scale to better represent the contributions of counties by their coastal extents. For example, Queens County, NY, hosts a high density of exposure per alongshore kilometre – very high exposure and a short coastline – and contributes only four transects to the total (Fig. 2). Likewise, because of its size, Dare County, NC, has both high exposure and a longer shoreline, resulting in a lower value of exposure per alongshore kilometre that accounts for over 100 transects of the domain. Overall, Dare County is less densely developed than Queens County. However, our treatment of exposure does overlook concentrated areas of high-density development within otherwise low-density counties – hotspots at which hazard, exposure, and vulnerability (i.e. nourishment activity) may be closely related.

To explore potential relationships between exposure and hazard, we sorted the exposure time series (Fig. 2) into counties associated with “high hazard” (eroding shorelines) and “low hazard” (accreting shorelines) for historical and recent shoreline change (Figs. 7 and S3). We find that exposure increases each decade in zones of high and low hazard alike, for both historical and recent shoreline change. Under historical shoreline-change hazard, exposure of property value is greatest in zones of high hazard (Figs. 7a–h and S3a). Conversely, exposure to high hazard is relatively low for recent shoreline-change rates (Figs. 7i–p and S3d), in part because recent shoreline-change rates tend to be less erosional than their historical counterparts (Fig. 3a). The difference between relative distributions of exposure in high and low hazard zones for historical shoreline-change rates increases in significance decade for decade, with a decreasing Kolmogorov–Smirnov p value that reflects the significance of their divergence (Fig. S3c). There is no such temporal divergence of exposure in high and low hazard zones for recent shoreline-change rates (Fig. S3f).

To explore, in parallel, potential relationships between exposure and vulnerability, we sorted the exposure time series into nourishing and non-nourishing counties and then by the intensity of beach nourishment (high or low) according to whether counties fell above or below the 2016 median value of cumulative Vbn (Figs. 8 and S4). We find that although exposure increases each decade in nourishing and non-nourishing counties alike, more property is ultimately exposed in nourishing counties. Moreover, the mean value of that exposed property increases at a greater rate than in non-nourishing counties (Figs. 8a–h and S4a–c). Initially, all property is exposed in counties where nourishment intensity is present but low (their Vbn sits below the 2016 median) – which we expect because for counties to accrue enough nourishment events to match the 2016 median cumulative-nourishment factor requires time (Fig. 8i and m). Exposure in intensively nourished counties (counties that accrue enough nourishment projects to have Vbn above the 2016 median) shows a marked increase in the 1980s (Fig. S4d). Total exposure in intensively nourished counties overtakes total exposure in sparsely nourished counties by the 2010s (Fig. S4e), such that more property ends up exposed in counties where nourishment intensity is high (Figs. 8i–p and S4d–f).

Both of these temporal relationships in spatial patterns of exposure and hazard (Fig. 7) and exposure and vulnerability (Fig. 8) are likely two vantages of the same feedback, catalysed by beach nourishment. Higher property value is exposed where historical shoreline-change hazard was high (Fig. 7a–d) and recent shoreline-change hazard is low (Fig. 7m–p) because those places also practice relatively intensive use of beach nourishment (Fig. 9). The cumulative effect of beach nourishment may be sufficiently strong to mask “true” rates of shoreline change (Armstrong and Lazarus, 2019) – a defensive intervention that, by reducing apparent hazard, may spur further development (Fig. 8), increasing exposure and creating demand for additional protection (Armstrong et al., 2016).