LifeSim

Uncertainty and Sampling Methods

Chapter Overview

The process of estimating life loss caused by a hazard is complex. Complexities include uncertainty in the spatial distribution of the initial PAR, the technological and sociological aspects of warning diffusion and mobilization, the human decision aspects of evacuation transportation, and determining fatality for those exposed to the hazard (Aboelata, Bowles, and Chen, 2004) [?]. By tracking the uncertain parameters in a simulation, a better understanding of the factors driving consequences is gained.

LifeSim employs uncertainty analysis using the Monte Carlo sampling method: the procedural steps that comprise LifeSim's methodology are repeated iteratively with a new set of sampled input parameters until a pre-defined number of iterations has been met. Each iteration provides a new set of sampled parameters from the uncertainty distributions (e.g., Warning Delay) resulting in a range of life loss estimates. However, LifeSim doesn't expressly consider the following:

• Effects of prolonged exposure to weather and other environmental factors.
• Spatial correlation of warnings.
• Indirect life loss.
• Water temperature.
• Debris or other hazards in the flood water.
• Rescue.
• Trip chaining or returning to the evacuation area.
• Initial traffic load prior to evacuation simulation.
• Hydrologic and hydraulic model uncertainties.

Types of Uncertainty

LifeSim accounts for two types of uncertainty: natural (aleatory) uncertainty and knowledge (epistemic) uncertainty.

Natural Uncertainty

Natural (or aleatory) uncertainty results from the natural randomness inherent in a process, or the natural process is so complex it seems random when viewed in isolation. It is often referred to as inherent uncertainty. Here, uncertainty can be estimated and quantified based on past observations, but it cannot be eliminated. LifeSim captures natural variability in the warning and evacuation process and estimating life loss when people are exposed to the hazard. Examples of parameters with natural uncertainty include annual maximum flood peaks and the willingness of a driver to enter a flooded road. These examples are a result of natural processes and human decision making that can never be completely understood or predicted.

Knowledge Uncertainty

Knowledge (or epistemic) uncertainty, often referred to as reducible uncertainty, results from limited knowledge about a system—either modeled or real. Here, uncertainty can be reduced as more information and data are obtained about a parameter providing a means of capturing knowledge uncertainty. In theory, this uncertainty can be eliminated if complete knowledge of a parameter is obtained. For example, percent damage vs. depth functions for an occupancy type are uncertain because each structure within that occupancy type is unique. However, one could reduce this uncertainty by improving the understanding of each building's foundation height, structure materials, and contents. Furthermore, if one had a perfect understanding of every building, the uncertainty could be nearly eliminated.

Application in Software

Table catalogs LifeSim parameters that are modeled with uncertainty. The table also states the type of uncertainty associated with each, where it is defined in the software, and the simulation level at which the parameter is sampled. Uncertainty on the parameters for which knowledge uncertainty applies can be reduced if more information and data are obtained about the parameter of interest. For the parameters with natural uncertainty, however, human reaction and decision making is inherent such that the uncertainty cannot be eliminated.

Many uncertain parameters are a combination of natural and knowledge uncertainty. For example, by performing a detailed assessment of a local community and the emergency management agency responsible for evacuating the community, uncertainty can be reduced about how quickly the warning will disseminate throughout the community. However, the order in which individual people will receive the warning is considered naturally variable. Another example of both types of uncertainty being applied at a single parameter is in vehicle stability thresholds. Uncertainty can be reduced on the thresholds required to make a vehicle lose stability, but even if all vehicles were the same make and model, other factors impact stability such as different tire tread, underlying surface (e.g., gravel, pavement), and debris in the water.

Table: LifeSim parameters and their uncertainty.

Uncertain Parameter	Uncertainty	Defined by…	Sampled by…
Structure Value	Knowledge	Occupancy Type	Structure/Iteration
Depth-Damage Functions (4 Types)	Knowledge	Occupancy Type	Occupancy Type/Iteration
Roof or Attic Access	Natural	Occupancy Type	Person / Iteration
Roof vs. Attic	Natural	Occupancy Type	Person / Iteration
Limited Mobility	Natural	Alternative / Age	Person / Iteration
Swimming Ability	Natural	Alternative	Person / Iteration
Foundation Height	Knowledge	Occupancy Type	Structure / Iteration
High Hazard Depth On Roof	Knowledge	Occupancy Type	Occupancy Type / Iteration
High Hazard Depth From Ceiling	Knowledge	Occupancy Type	Occupancy Type / Iteration
High Hazard Depth From Floor	Knowledge	Occupancy Type	Occupancy Type / Iteration
Structural Stability Threshold	Knowledge / Natural	User-defined	Structure / Iteration
Imminent Hazard ID Time	Knowledge	Alternative	EPZ / Iteration
Hazard Communication Delay	Knowledge	Alternative	EPZ / Iteration
Warning Issuance Delay	Knowledge	EPZ	EPZ / Iteration
First Alert Delay Function	Knowledge	EPZ	EPZ / Iteration
First Alert Received Time	Natural	EPZ	Structure or Evacuating Group / Iteration
Protective Action Initiation Delay Function	Knowledge	EPZ	EPZ / Iteration
Protective Action Initiation Time	Natural	EPZ	Structure or Evacuating Group / Iteration
Population that Evacuate in Vehicles vs. on Foot	Natural	Occupancy Type	Evacuating Group / Iteration
Population in High vs. Low Clearance Vehicle	Natural	Alternative	Evacuating Group / Iteration
Fraction who Reroute in Heavy Traffic	Natural	Alternative	Evacuating Group / Iteration
Willingness to Enter Flooded Road	Knowledge / Natural	Alternative / Vehicle Clearance	Evacuating Group / Iteration
Fatality Rate	Knowledge	Alternative / Hazard Zone	Evacuating Group / Iteration
Life Loss	Natural	Alternative / Hazard Zone	Person
Vehicle Stability Thresholds (3)	Knowledge / Natural	Alternative / Vehicle Clearance	Evacuating Group / Iteration
Human Stability Threshold (3)	Knowledge / Natural	Alternative	Person / Iteration

Uncertainty Sampling

Distribution Sampling

Many uncertain parameters, such as willingness to enter a flooded road, are defined as distributions in LifeSim. To estimate the parameter in a Monte Carlo simulation, the distribution is sampled with a random number between 0 and 1 using a random number generator. The distribution's inverse CDF function is used with the random number generator to provide the parameter. In other words, the random number represents a probability and the distribution returns the appropriate value for the given probability.

Curve Sampling

Curve sampling is the process of sampling a curve function from an array of distribution ordinates. In LifeSim, curve sampling is performed on depth-damage functions, building stability criteria, first alert dissemination, and PAI functions. With each of these functions, a distribution is defined conditional on some value such as depth or time. For example, in the depth-damage functions, each ordinate has a depth and the % damage at that depth is defined as a distribution.

To sample a curve function for the Monte Carlo simulation, LifeSim determines a random number between 0 and 1 using a random number generator. Then, for each ordinate in the curve function the distribution is sampled using the random number as described in Distribution Sampling. Monotonicity is ensured by using the same random number for each distribution.

Other Sampling Procedures

Uncertainty about the foundation height and structure values is defined using distributions at the occupancy type level. Sampling these parameters is unique because the uncertainty is defined as a % variation around the value defined at the structure level. For example, a structure with a value of 100 USD (Property Value) samples -50% for variation around value. The structure value used in the iteration is then 50 USD (Equation).

$$p_i = p_s + (p_s * v_i)$$

where:

p_i = Parameter value for iteration i.
p_s = Base parameter value used for variation such as structure value.
v_i = Variation of the value as a ratio sampled for iteration i.

Another sampling procedure in LifeSim is used when applying percentage ratios for population characteristics. For example, an occupancy type contains the fraction of population with limited mobility for both over and under 65 years old. Random sampling is used to determine which occupants have limited mobility and which ones don't. To sample limited mobility, for each person in a structure, a random number is generated between 0 and 1. If the sampled value is above the fraction with limited mobility for the appropriate age group, then the person does not have limited mobility. For example, if a person over 65 years old samples a 0.1 and the fraction with limited mobility over 65 for the person's occupancy type is 0.2, then the person will be assumed to have limited mobility.

First alert and PAI functions are defined in an iteration using curve sampling. These functions provide dissemination on a community level but don't provide the granularity required to know when specific people will receive a warning and take protective action. The procedure for evacuating groups is defined in Evacuating Groups and Their Attributes. The first alert and PAI function uncertainty can be reduced with more knowledge. Sampling the curve functions to determine when each group gets alerted and takes protective action is naturally variable.