# Physical Risk Input Description#

The main sources of input information required for a risk calculation
with the OpenQuake engine are an *exposure model* and a physical
*vulnerability model* or *fragility model* (in addition to the
calculation type and the region of interest). An *exposure model* for a
given category of asset (e.g. population, buildings, contents)
describes, at each location of interest within a given region, the
value of each *asset* of a given *taxonomy*. The physical *vulnerability
model* described the loss ratio distribution for a set of intensity
measure levels, while the *fragility model* provides the probability of
exceeding a set of damage states, given a set of intensity measure
levels.

## Exposure#

The OpenQuake engine requires an *exposure model* that needs to be
stored according to the respective Natural hazardsâ€™ Risk Markup
Language (NRML) schema (see
oq-nrmllib). More information
on the formats of this input model is provided in the OpenQuake
Engine User Manual. This file format can include several typologies
of asset such as population or buildings. The following parameters
are currently being used to describe each asset in the exposure
model:

Asset reference: A unique key used to identify the

*asset*instance;Location: Geographic coordinates of the

*asset*expressed in decimal degrees;Taxonomy: Reference to the classification scheme that describes the

*asset*;Number: A numerical value describing the number of units at the given location (e.g. building count).

Area: This parameter specifies the built-up area of the asset, and can be defined in the two following ways: the aggregated area (i.e. the total built-up area of all the units at a given location, with a certain

*taxonomy*); area per unit (i.e. the average built-up area for a single building);Structural cost: This parameter represents the structural replacement cost of the

*asset*. This value can be defined in three possible ways: the aggregated structural cost (i.e. the total economic value of all the units with a certain*taxonomy*at a given location); the cost per unit (i.e. the average value for a single building); the cost per unit of area. Further information about how the structural cost is handled within the OpenQuake engine can be found in the OpenQuake Engine User Manual.Non-structural cost: This parameter is used to define the cost of the non-structural components. This cost defined in the same way as the structural cost.

Contents cost: This parameter is used to define the contents cost, and it can be defined in the same way as the structural cost.

Retrofitting cost: This parameter is used to define the economic structural cost due to the implementation of a retrofitting/strengthening intervention. The retrofitting cost can be defined in the same way as the structural cost.

Occupants: This parameter defines the number of people that might exist inside of a given structure. Different values of occupants can be stored according to the time of the day (e.g day, night, transit).

Deductible: This parameter is used in the computation of the

*insured losses*, and it establishes the economic value that needs to be deducted from the*ground-up losses*. A*deductible*needs to be defined for each cost type (structural, non-structural and contents). This threshold can be defined in two ways:the direct (absolute) value that will be deducted;

the fraction (relative) of the total cost that will be deducted.

Limit: This parameter establishes the maximum economic amount that can be insured, and it also needs to be defined for each cost type. The limit can also be defined as an absolute or relative quantity.

## Physical Vulnerability#

Physical vulnerability is defined as the probabilistic distribution
of loss, given an intensity measure level. These *vulnerability
functions* can be derived directly, usually through empirical methods
where the losses from past events at given locations are related to
the levels of intensity of ground motion at those locations, or they
can be derived by combining *fragility functions* and *consequence
functions*. *Fragility functions* describe the probability of exceeding
a set of limit states, given an intensity measure level; limit states
describe the limits to performance levels, such as damage or injury
levels. *Fragility functions* can be derived by expert-opinion,
empirically (using observed data), or analytically, by explicitly
modeling the behavior of a given asset typology when subjected to
increasing levels of ground motion. *Consequence functions* describe
the probability distribution of loss, given a performance level and
are generally derived empirically.

Version 1.0 of the OpenQuake engine only supports physical
vulnerability through the aforementioned *vulnerability functions*. As
part of the plans for a risk modelling toolkit, calculators are
envisaged that will combine *fragility functions* and *consequence
functions* to produce *vulnerability functions* that can be input into
the engine.

### Vulnerability Functions#

#### Discrete Vulnerability Functions#

In the current version of the OpenQuake engine (v1.0) discrete
*vulnerability functions* are used to directly estimate fatalities and
economic losses due to physical damage. Discrete *vulnerability
functions* are described by a list of intensity measure levels and
corresponding mean loss ratios (the ratio of mean loss to exposed
value), associated coefficients of variation and probability
distributions. The uncertainty on the loss ratio can follow a
lognormal or Beta distribution. The figure below illustrates a discrete
*vulnerability function*.

#### Continuous Vulnerability Functions#

Continuous *vulnerability functions* may be implemented in future
versions of the OpenQuake Continuous engine. Continuous *vulnerability functions*
will probably be described by continuous distribu- tions of mean loss ratio and
other fractiles of loss ratio, with ground motion intensity. The figure below
illustrates this type of function, showing the distribution of mean loss and the
10 percent and 90 percent fractiles.

### Fragility Functions#

*Fragility functions* describe the probability of exceeding a set of
limit states, given an intensity measure level. When the asset
category concerns structures (e.g. buildings), the intensity
measure can either be structure-independent or structure-dependent.
The former can be calculated directly from recorded measurements of
ground shaking (e.g. peak ground accel- eration, peak ground
velocity, spectral acceleration at a given period of vibration, or
even macroseismic intensity). The latter requires information on the
characteristics of the struc- tures in order to be calculated, for
example spectral acceleration at the fundamental period of vibration,
or spectral displacement at the limit state period of vibration. The
calculation of these structural characteristics might be through a
simple formulae (e.g. a yield period- height equation, see e.g.
*Crowley and Pinho [2004]*) or through so-called non-linear static
methods, which are needed when the intensity measure is a non-linear
response quantity such as spectral displacement at the limit state
period of vibration (see e.g. *FEMA-440:ATC [2005]*). The oq-risklib
currently does not support non-linear static methods.

#### Discrete Fragility Functions#

*Fragility functions* can be defined in a discrete way by providing,
for each limit state, a list of intensity measure levels and
respective probabilities of exceedance. The first figure below presents a set of
discrete *fragility functions* using a macroseismic intensity measure.

#### Continuous Fragility Functions#

Continuous *fragility functions* are defined by the parameters of a
cumulative distribution function. In the second figure below an example of a set
of continuous fragility functions with a structure- dependent
intensity measure is presented.

#### Uncertainty in Fragility Functions#

The uncertainty in continuous *fragility functions* will be accounted
for in future versions of the engine. The third figure below shows a lognormal
distribution that has been fit to the data (i.e. the
fragility function), and the probabilistic distribution (i.e. mean
and standard deviation) to describe the uncertainty in both the
logarithmic mean and logarithmic standard deviation of the fragility
function. When a set of *fragility functions* for different limit
states are used, it is also necessary to provide information on the
correlation between the logarithmic means and logarithmic standard
deviations of each limit state.

### Consequence Functions#

*Consequence functions* describe the probability distribution of loss,
given a performance level. For example, if the asset category is buildings
and the performance level is significant damage, the *consequence function*
will describe the mean loss ratio, coefficient of variation and
probability distribution for that level of damage. The second figure below
presents the mean damage ratios for a set of performance levels
proposed by two different sources. Although these functions are not directly
supported, users can combine *consequence functions* with *fragility functions* to
produce *vulnerability functions* to be input into the engine.