Multivariate coastal flooding is characterized by multiple flooding pathways (i.e., high offshore water levels, streamflow, energetic waves, precipitation) acting concurrently. This study explores the joint risks caused by the co-occurrence of high marine water levels and precipitation in a highly urbanized semi-arid, tidally dominated region. A novel structural function developed from the multivariate analysis is proposed to consider the implications of flood control infrastructure in multivariate coastal flood risk assessments. Univariate statistics are analyzed for individual sites and events. Conditional and joint probabilities are developed using a range of copulas, sampling methods, and hazard scenarios. The Nelsen, BB1, BB5, and Roch–Alegre were selected based on a Cramér–von Mises test and generally produced robust results across a range of sampling methods. The impacts of sampling are considered using annual maximum, annual coinciding, wet-season monthly maximum, and wet-season monthly coinciding sampling. Although annual maximum sampling is commonly used for characterizing multivariate events, this work suggests annual maximum sampling may substantially underestimate marine water levels for extreme events. Water level and precipitation combinations from wet-season monthly coinciding sampling benefit from a dramatic increase in data pairs and provide a range of physically realistic pairs. Wet-season monthly coinciding sampling may provide a more accurate multivariate flooding risk characterization for long return periods in semi-arid regions. Univariate, conditional, and bivariate results emphasize the importance of proper event definition as this significantly influences the associated event risks.

Coastal flooding is a significant human hazard

Compound coastal flooding considers the combined impacts of marine and hydrologic forcings, typically within a physically relevant time window, and are considered multivariate events.
Typical events, such as precipitation or high water levels, occurring simultaneously may combine to generate extreme events

From a flood risk perspective there are multiple methods to characterize events. A univariate approach is often used where a single variable (e.g., water level) is considered. For example, the Federal Emergency Management Agency (FEMA) recommends characterizing multivariate events by developing univariate water level and discharge statistics and then adopting a smooth, blended result for transitional areas

Numerous studies have used a copula-based approach to study floods manifested by various combinations of variables (Table

Data sampling methods in multivariate studies influence distribution fitting. Two primary sampling methods exist: peaks over threshold

Coastal flooding studies primarily focus on locations defined by storm-surge-dominated oceanographic conditions with warm, humid

A non-exhaustive list of multivariate studies which utilized copulas to study the associated variables.

This study considers observed water level and precipitation influences for coastal multivariate events at Santa Monica (SM), Sunset Beach (S), and LA Jolla (SD) areas in Los Angeles, Huntington Beach, and San Diego, California (Fig.

Observed water levels from the Los Angeles (station ID: 9410660), La Jolla (station ID: 9410230), and Santa Monica (station ID: 9410840) tide gauges are available on NOAA's Tides and Currents for daily high–low, hourly, or 6 min intervals

The US Hourly Precipitation Data dataset provided by the NOAA's National Centers for Environmental Information

Multivariate flood probabilities are determined with combinations of sampling methods: annual maximum (AM), annual coinciding (AC), wet-season monthly maximum (WMM), and wet-season monthly coinciding (WMC). AM sampling pairs the single largest precipitation and OWL observations within a given year (without regard to co-occurrence), where AC sampling pairs the single largest precipitation observation within a given year to the largest OWL observation within its 24 h accumulation period. Each sampling method samples from a unique probability space and therefore will provide varying perspectives for a return period. A summary of each sites' associated gauges, observation windows, and number of pairs is provided in Table

Distributions are fit with existing precipitation observations greater than zero consistent with previous studies

Map displaying

Data pairs for each sampling method. Annual maximum (AM; blue x), annual coinciding (AC; green

Water level and precipitation observations at Santa Monica (SM), Sunset (S), and San Diego (SD) using annual maximum (AM), annual coinciding (AC), wet-season monthly maximum (WMM), and wet-season monthly coinciding (WMC) samplings.

Potential flooding events are determined with three different probability definitions: univariate, conditional, and bivariate. Assuming

Copulas are functions that associate random variables' univariate CDFs with their joint CDF (e.g.,

Notation and definitions from

“OR” scenario events have one or both random variables exceeding a specified threshold. That is, what is the probability of a water level or precipitation event exceeding a given value? Standard univariate CDFs make up the associated copula.

“AND” scenario events have both random variables exceeding a specified threshold. In this case the fundamental question is what the probability is of a particular water level and precipitation rate exceeding specified values. The survival copula (

The “Kendall” (K) scenario highlights an infinite set of OR events that separate the subcritical (i.e, “safe”) and supercritical (i.e., “dangerous”) statistical regions. In the OR scenario, events along an isoline (

The “survival Kendall” (SK) scenario highlights an infinite set of AND events which also separate safe and dangerous statistical spaces. AND events along a

The “structural” scenario considers the probability of an output from a structural function,

The Multivariate Copula Analysis Toolbox (MvCAT) developed by

Hydrologic events are commonly cast in the context of return periods

Multiple goodness-of-fit metrics and correlations serve to quantify the quality of distribution fits and dependencies between variables. Marginal fits are selected by the Bayesian information criterion (BIC; Eq.

Univariate, conditional, and bivariate probabilities were developed using four sampling methods (AM, AC, WMM, and WMC) and 17 different copulas. Two marginal distributions do not pass the chi-squared test at the standard 0.05 level of significance (San Diego AM OWL and Santa Monica WMM OWL). These distributions pass at reduced significance levels of 0.01. Four copulas almost always passed (Nelsen, BB1, BB5, and Roch–Alegre) the Cramér–von Mises test and are used for analysis. It is noted the Roch–Alegre (Roch.) did not pass at Sunset for WMM sampling, and the BB1 and BB5 did not pass at San Diego for WMC sampling. Additionally, Santa Monica's AM data are slightly negatively correlated (

Santa Monica, Sunset, and San Diego exceedance probabilities at the 10- and 100-year return periods for wet-season monthly maximum (WMM) and wet-season monthly coinciding (WMC) samplings.

The selected marginal distributions (Fig.

Historically, water levels have been described using a number of distributions including normal

Best-fitting univariate distributions for each location and sampling method (annual maximum, AM; annual coinciding, AC; wet-season monthly maximum, WMM; wet-season monthly coinciding, WMC).

BS – Birnbaum–Saunders, GP – generalized Pareto, E – exponential, R – Rayleigh, N – normal, L – log logistic, G – Gamma, W – Weibull, IG – inverse Gaussian, NA – Nakagami.

San Diego wet-season monthly coinciding conditional CDFs display individual copulas' effects (Fig.

San Diego wet-season monthly coinciding OWL (left column) and precipitation (right column)

Figures

San Diego wet-season monthly coinciding

San Diego wet-season monthly coinciding

San Diego 10-year marginal (M), conditional (C), and bivariate OWL (m) and precipitation (

San Diego 100-year marginal (M), conditional (C), and bivariate OWL (m) and precipitation (

San Diego conditional CDFs using the BB1 copula clearly present sampling effects (i.e., maximum versus coinciding and annual versus wet-season months).
It should be noted that each sampling method represents a unique probability space and accordingly results in alternative realizations of a given return period. Coinciding samplings exhibit similar OWL CDFs (green and red lines, Fig.

San Diego OWL (left column) and precipitation (right column)

Figures

San Diego

San Diego

Tables

San Diego 10-year marginal (M), conditional (C), and bivariate OWL (m) and precipitation (

San Diego 100-year marginal (M), conditional (C), and bivariate OWL (m) and precipitation (

A structural scenario is presented to consider flood severity along the Pacific Coast Highway (PCH) in Sunset Beach. PCH road elevation ranges from 1.7–2.4 m NAVD88 (Fig.

Areal precipitation flooding extent and depth can be estimated for water levels exceeding tide valve closure elevation. A water level equal to or greater than 1.68 m NAVD88 forces valve closures and frames the structural failure as a Conditional 1-type event. The local watershed is convex and drains an area of 94 897 m

Structural scenario precipitation and percent flooding (

Elevations within the Pacific Coast Highway boundary ranging from low (purple) to high (blue). Background imagery from

Structural scenario 5- (square), 10- (circle), and 100-year (diamond) return periods for annual maximum (AM; blue), annual coinciding (AC; green), wet-season monthly maximum (WMM; black), and wet-season monthly (WMC; red) data using the

Precipitation and percent flooding (

Previous multivariate studies typically use a small, popular group of copulas (e.g., Clayton, Frank, Gumbel, Student

The choice in sampling imparts a significant influence on event risk interpretation. When maximum versus coinciding sampling is considered, maximum samplings (AM and WMM) tend to provide the largest OWL at low return periods (Figs.

An important note is that each probability type appropriately describes a unique event, characterized by OWL and precipitation.

From a regulatory perspective, FEMA recommends individual (univariate) analysis to develop return periods for multivariate coastal flooding applications

Structural scenarios provide a quantitative context to frame flood vulnerability. In the structural failure context, annual coinciding sampling significantly underestimates flooding at all return periods, and annual maximum sampling underestimates severe (i.e., 100-year) events, echoing previous annual maximum and coinciding sampling issues. Similar values between most copulas support the suggestion that choosing a reasonable copula will provide robust results in these types of applications. Precipitation events in the structural scenario (Table

The maximum OWL and precipitation observations within the record are 2.33

Univariate and multivariate event risks from OWL and/or precipitation were explored at three sites in a tidally dominated, semi-arid region. Seventeen copulas were considered. Previous studies typically relied upon a small number of copulas (e.g., Clayton, Frank, Gumbel, Student

The annual maximum method is widely recognized for hazard assessments

Marginal OWL BIC values per fitted copula for Santa Monica (left column), Sunset (middle column), and San Diego (right column) using annual maximum

Marginal precipitation BIC values per fitted copula for Santa Monica (left column), Sunset (middle column), and San Diego (right column) using annual maximum

NOAA precipitation data are available for download at

JTDL conducted the primary analysis under the guidance and assistance of TG. Both authors wrote and edited the manuscript. TWG conceived of and funded the work.

The contact author has declared that neither they nor their co-author has any competing interests.

The content of the information provided in this publication does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. The authors’ acknowledge the USACE and USCRP’s support of their effort to strengthen coastal academic programs and address coastal community needs in the United States. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the agencies supporting the work. Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been supported by the US Coastal Research Program under contract W912HZ-20-200-004, California Department of Parks and Recreation under contact number C1670006, the National Science Foundation Graduate Research Fellowship Program under grant number DGE-1650604, the National GEM Consortium Fellowship, and the UCLA Cota-Robles Fellowship. The US Coastal Research Program (USCRP) is administered by the US Army Corps of Engineers® (USACE), Department of Defense. We would like to acknowledge the anonymous reviewers, Yeulwoo Kim, and Nikos Kalligeris for their constructive feedback, which strengthened this paper.

This research has been supported by the US Army Corps of Engineers (grant no. W912HZ-20-200-004), the California Department of Parks and Recreation (grant no. C1670006), and the Directorate for Engineering (grant no. DGE-1650604).

This paper was edited by Philip Ward and reviewed by three anonymous referees.