Background

A copula is a mathematical function that describes the dependence among two or more random variables independently of their individual probability distributions (marginal distributions). By separating the dependence structure from the marginals, copulas provide a flexible framework for modeling multivariate relationships.

Compound flooding occurs when multiple flood drivers, such as heavy rainfall, storm surge, and fluvial flooding, occur simultaneously. In these situations, the frequency relationship of a response variable that depends on two or more contributing variables must be quantified. Historically, USACE studies have relied on the total probability method, commonly referred to as coincident frequency analysis, to evaluate joint probability. This approach is documented in Engineer Manual (EM) 1110-2-1415, Hydrologic Frequency Analysis (1993), and EM 1110-2-1413, Hydrologic Analysis of Interior Areas.

Why Copulas Instead of Coincident Frequency?

A Brief Primer on Coincident Frequency Analysis

When performing a coincident frequency analysis, the analyst must answer two questions:

  • Correlation: Are the magnitudes of extreme events at the two variables A and B related or independent?
  • Coincidence: Do the extreme events happen for both variables A and B at the same time?

Typically, the answers to these questions can be placed into one of five categories, shown below:

  1. Extremes are never coincident
  2. Magnitudes are not correlated and extremes are somewhat coincident
  3. Magnitudes are not correlated and extremes are usually coincident
  4. Correlated magnitudes and extremes are usually coincident
  5. Magnitudes are perfectly correlation and extremes are always coincident

Case 1 describes a univariate analysis. Case 5 is easy to describe since the marginal probabilities describe the joint probability.

When extremes at variables A and B are somewhat to usually coincident but the magnitudes are not correlated, development of the marginal distributions is relatively straightforward. When the variable magnitudes are correlated (case 4), development of the joint or condition distribution of the two input variables, A and B, rather than marginal distributions, is required. This requires significantly more data than developing marginal frequency curves for variables A and B.

If the degree of correlation is weak, analysts typically assume that the magnitudes are independent to simplify the problem.

However, correlation between variables A and B means that large events happen together and small events happen together. As a result, the upper end of the response frequency curve is higher (red curve in image below) than a response curve produced from the assumption of independence (purple dotted curve in image below).

Advantages of Copulas

Copulas offer several advantages over coincident frequency analysis that are detailed in the sections below.

Flexibility
Copulas are flexible. They can be used to model the dependence structure between variables modeled with any marginal probability distribution. The marginal probability distributions do not need to be the same. 

There are numerous copula families, which can capture different types of dependence. Elliptical copulas are derived from elliptical distributions, such as the multivariate normal and multivariate Student's t distribution. Elliptical copulas can capture linear dependence. Archimedean copulas are a large family of copulas constructed from a "generator function." They can model various forms of tail dependence. For example, the Joe copula models upper tail dependence and the Clayton copula models lower tail dependence (see image below).

No assumption of degree of correlation

Copulas can be used to capture any level of correlation, from no correlation (independence) to strong correlation. An independence copula can be used when the variables are independent. It is a specific type of copula where the joint cumulative distribution function (CDF) of the random variables is simply the product of their individual marginal CDFs.

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No assumption of dominant variable
Coincident frequency analysis requires the selection of a dominant variable, which is often ambiguous in compound flooding applications. Copulas eliminate the need to designate a dominant variable and allows for a two-sided analysis, where both variables are treated symmetrically, rather than designating one variable as the "driver."

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Simulations to describe the response variable are targeted

Coincident frequency procedures relies on simulating numerous combinations of values from the marginal distributions, but those combinations are not generated to satisfy a specified joint exceedance probability and therefore may not represent the desired design event.

Copulas can be used to derive the conditional distribution of variable values associated with a specified joint exceedance probability, enabling simulations that target a specific annual exceedance probability (AEP). 

Framework for uncertainty
Coincident frequency procedures do not provide a systematic framework for quantifying uncertainty and instead produce only a single best-estimate frequency curve for the response variable. Copula-based models naturally support uncertainty analysis through Monte Carlo simulation. There are two distinct sources of uncertainty we want to characterize: sampling (Monte Carlo) uncertainty in the design point, and parametric uncertainty in the fitted copula(s). 

Sampling and Parametric Uncertainty in Copulas

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Support for common analysis workflows

Copulas support several common analysis workflows, including:

  • Monte Carlo simulation: Fit probability distributions to the marginal variables, estimate the dependence structure using a copula, generate correlated samples in the uniform probability space, transform those samples back to the original variable space using the fitted marginal distributions, and perform the desired analysis for each realization.

  • Design event analysis: Identify combinations of variables (A, B) whose joint probability corresponds to a specified design frequency, and characterize both the most likely combinations and their associated uncertainty.

  • Retrospective (event attribution) analysis: Compute the joint cumulative probability associated with an observed or specified combination of variable values (A, B).

The image below is from a two-sided copula used to model the dependence between river flow and precipitation on the interior of a levee. Note that the 6 November 1985 historic event had a flow of 50,900 cfs and a precipitation of 5.54 inches and this event is basically the same as the 1/500 (0.002) AEP design point of 50,500 cfs of flow and 5.5 inches of precipitation.

Neptune Implementation

HEC-Neptune's Copula Analysis enables users to fit six copula families: Gaussian, Student's t, Gumbel, Clayton, Joe, and Frank. The analysis computes the Akaike Information Criterion (AIC) for each fitted copula to support model comparison and selection.

Tutorials

Copula Analysis Tutorials