Natural variability vs Knowledge uncertainty

Uncertainty is the result of imperfect knowledge concerning the present or future state of a system, event, situation, or (sub) population under consideration.  Uncertainty leads to lack of confidence in predictions, inferences, or conclusions.  It is important to distinguish uncertainty that results from a lack of knowledge from the uncertainty that results from natural variability. In this section, we will focus on how the different types of uncertainty enter into the software. In short, a given frequency function reflects the natural variability in our risk estimates, and all other sources of uncertainty reflect model uncertainty. The truth is often somewhere in between, but in this reduced form approach required by our Monte Carlo simulation, sides must be chosen. 

Natural Variability

Some variables are random and unpredictable and their values differ event to event. The distribution or spread of values within a natural “population” or data set.  This array of possible values in a population is caused by the inherent randomness of natural or social systems and is formally labeled aleatory uncertainty.  The values in the statistical population have some probability distribution, and only limited knowledge of the entire statistical population and the probability distribution may exist.  Sometimes variability is classed as a type of uncertainty although generally it should not be confused or interchanged with uncertainty as defined above. 

In HEC-FDA, natural variability enters the model through an instance of a frequency function, be it a flow-frequency function or a stage-frequency function. the different magnitudes of the hazard against their probabilities of occurrence in any given year. In the below image, the computation of expected annual damage is illustrated without uncertainty. An instance of natural variability is used to define an instance of variability in damage, from which we can calculate an average, which we term expected annual damage.  

Knowledge Uncertainty

All other uncertainty that enters the HEC-FDA software is taken to represent knowledge uncertainty, and contributes to the spread of expected annual damage. That uncertainty includes uncertainty about:

1. The frequency function. A given flow or stage frequency parametrization reflects an instance of natural variability, but we're unsure about the true parameterization, so we have to involve many possibilities of what the natural variability could look like. The instance of natural variability among all of the possibilities of what natural variability could look like is illustrated in the top right-hand side of the image below, where natural variability is the green line and the orange distributions represent the uncertainty distribution about the natural variability. 
2. Engineering transformation functions. For a given flow, we are uncertain about the stage. This uncertainty is illustrated in the top left-hand side of the image below. We are also uncertain about the regulated flow for a given unregulated flow and classify this as knowledge uncertainty. 
3.  Levee failure. We are unsure about the levee failure stage. This typically looks like a probability distribution of failure stages. In the current methodology, this distribution of failure stages (a relationship between stage and probability of failure) is knowledge uncertainty but is used in the calculation of an instance of EAD so does not contribute to the spread of expected annual damage. 
4. Consequences. All of the economic uncertainty parameters, to include uncertainty in the first floor elevation, structure value, content value, content-to-structure value ratio, and percent damage for a given depth are all knowledge uncertainties. These uncertainties contribute to the spread in damage for a given stage, which too, reflects knowledge uncertainty, and is illustrated in the bottom left-hand side of the image below. 

An instance of expected annual damage is calculated by integrating an instance of a damage-frequency function, where an instance of a damage-frequency function reflects an instance of natural variability. All of the model uncertainties produce variations in the damage-frequency function which result in the spread of expected annual damage.