# A Statistical Approach to Computer-based Modeling of Storm Surge

According to the National Hurricane Center, storm surge poses the most severe threat to property and life in coastal towns and communities during a hurricane. It manifests as an abnormal, rapid rise of water forced towards shore by strong cyclonic winds and causes extreme flooding—and thus tremendous damage—in coastal areas. Many of the deaths that occur when a hurricane makes landfall are either a direct or indirect consequence of storm surge. When storm surge occurs during normal high tide, the resulting “storm tide” can easily exceed 20 feet and relentlessly erode beaches, batter and flood coastal structures, and damage marinas and estuaries.

Storm surge is a complex phenomenon, and its severity is dependent on factors including storm intensity, speed, size, central pressure, and angle of landfall, as well as shape of coastal land features. The width and slope of an affected shoreline’s continental shelf is also influential; a shallow slope (like the Louisiana coast) results in a higher surge, while a steep drop-off (like Miami Beach) limits surge height. As a result, accurate estimation of storm surge levels in coastal areas could better prepare communities for hurricanes and limit loss of life, ecological destruction, and socio-economic damage. During the 2018 SIAM Conference on Mathematics of Planet Earth, currently taking place in Philadelphia, Pa., Whitney Huang of the Statistical and Applied Mathematical Sciences Institute addressed the challenges of hurricane modeling and used computer simulation to predict consequential storm surge.

“As a statistician, the first logical thing to think about is observations of storm surge,” Huang said. However, the lack of available hurricane data makes observation-based approaches unfeasible. For example, tide gauges—devices that measure vertical change in sea level—provide beneficial estimates of storm severity, but only six gauge stations are present on the entire North Carolina coastline. Furthermore, hurricanes—as so-called “extreme events”—occur fairly infrequently. While their irregularity is of course beneficial to coastal populations, it does result in a dearth of relevant data.

In lieu of observational models, Huang uses computer-based modeling to study hurricane motion. “We simply don’t have enough observations to estimate storm surge,” he said. “So we put physical knowledge into a computer model to simulate storm surge for a given hurricane.” The primary driving force of rising water is a low-pressure system, and a planetary-boundary model depicts this nicely. Computational models must also account for wind and pressure fields, water levels, and waves/radiation stress.

Wind and pressure fields, water levels, and waves/radiation stress all contribute to computer-based models of storm surge activity. Figures courtesy of Gangai (Dewberry) and Danforth (FEMA).

Huang then delved further into the model’s components. “To create wind and pressure fields for input for the whole model, we need to somehow parametrize the low pressure system,” he said. He identifies key inputs and distinguishes variables for pressure, wind speed, and the distance of a storm’s trajectory from the coastal point of interest. This distance is perhaps the single most important factor, as it determines the severity of impact. Huang uses the input distribution to create a synthetic hurricane and prepare the computer simulation.

Unfortunately, the high spatial resolution necessary to capture local topography makes the resulting simulation particularly time-consuming. “Running this kind of simulation for a single hurricane can take a lot of computing time,” Huang said. “You can’t run it many, many times.” The limited number of feasible trials eliminates the possibility of a Monte Carlo approach and means that one must diligently choose input configurations and sample points that maximize extraction of the input/output relationship. “You really need to choose the input configurations and the synthetic hurricane wisely to learn as much as possible about extreme storm surge,” Huang said. The computational expense of a single model run also means that uncertainty quantification is not straightforward, resulting in input uncertainty, structural uncertainty, and uncertainty in the input-output relationship.

Key model inputs reflect hurricane statistics. Figure courtesy of Toro (2008).
Despite these aforementioned challenges, deriving the input-output relationship from a computer model can still yield valuable insights into extreme storm surge. Huang employs a statistical formulation of the joint probability model to calculate surge level, identify peak surge height, and estimate joint distribution of storm characteristics under certain conditions. He also recognizes a surge response function and an error term that accounts for unexpected tides or parameter inconsistencies. During this input modeling phase, one must identify criteria to classify “relevant” storms of historical significance along the line of distribution. For example, Huang considers relevant storms to be those that fell within a 100-kilometer circle around his point of reference. 52 hurricanes met this precedent, and he collected historical data from those storms only. “With that limited amount, it’s quite difficult to determine the right input distribution and even harder to assess the fit,” Huang said.

“Borrowing” information to improve input modeling offers a solution. Huang borrows strength across time and space to exploit the large-scale spatial structure of storm characteristics and explore their temporal dependence. He applies the stochastic-deterministic track method to enlarge the sample size and simulate a large number of synthetic storms, and uses a high-resolution fast hurricane model for dynamical downscaling; overall, the combination of these methods yields more complete data. Finally, Huang conducts tail modeling to estimate the $$r$$-year return level.

Ultimately, Huang combines physical and statistical modeling in the context of storm surge risk to simulate hurricanes’ projected impact on coastal areas. While input uncertainty—such as landfall location and the specific characteristics of a given storm—remains the biggest challenge, statisticians are well-trained to choose inputs widely, borrow information to improve input modeling, choose inputs wisely, sensibly quantify uncertainty, and estimate the output’s upper tail. A practical approach to hurricane modeling and an improved understanding of storm surge will ultimately help coastal communities better prepare for natural disasters.

 Lina Sorg is the associate editor of SIAM News.