# Hazard Calculators¶

## Classical PSHA Calculator¶

The Classical Probabilistic Seismic Hazard Analysis (cPSHA) approach allows calculation of hazard curves and hazard maps following the classical integration procedure (Cornell [1968], McGuire [1976]) as formulated by Field et al. [2003].

Sources:

• Cornell, C. A. (1968).
Engineering seismic risk analysis.
Bulletin of the Seismological Society of America, 58:1583–1606.
• Field, E. H., Jordan, T. H., and Cornell, C. A. (2003).
OpenSHA - A developing Community-Modeling
Environment for Seismic Hazard Analysis. Seism. Res. Lett., 74:406–419.
• McGuire, K. K. (1976).
Fortran computer program for seismic risk analysis. Open-File report 76-67,
United States Department of the Interior, Geological Survey. 102 pages.

### Outputs¶

#### Hazard Curves¶

Hazard Curves are discrete functions which describe probability of ground motion exceedance in a given time frame. Hazard curves are composed of several key elements:

• Intensity Measure Levels (IMLs) - IMLs define the x-axis values (or “ordinates”) of the curve. IMLs are defined with an Intensity Measure Type (IMT) unit. IMLs are a strictly monotonically increasing sequence.
• Probabilitites of Exceedance (PoEs) - PoEs define the y-axis values, (or “abscissae”) of the curve. For each node in the curve, the PoE denotes the probability that ground motion will exceedence a given level in a given time span.
• Intensity Measure Type (IMT) - The unit of measurement for the defined IMLs.
• Investigation time - The period of time (in years) for an earthquake hazard study. It is important to consider the investigation time when analyzing hazard curve results, because one can logically conclude that, the longer the time span, there is greater probability of ground motion exceeding the given values.
• Spectral Acceleration (SA) Period - Optional; used only if the IMT is SA.
• Spectral Acceleration (SA) Damping - Optional; used only if the IMT is SA.
• Source Model Logic Tree Path (SMLT Path) - The path taken through the calculation’s source model logic tree. Does not apply to statistical curves, since these aggregates are computed over multiple logic tree realizations.
• GSIM (Ground Shaking Intensity Model) Logic Tree Path (GSIMLT Path) - The path taken through the calculation’s GSIM logic tree. As with the SMLT Path, this does not apply to statistical curves.

For a given calculation, hazard curves are computed for each logic tree realization, each IMT/IML definition, and each geographical point of interest. (In other words: If a calculation specifies 4 logic tree samples, a geometry with 10 points of interest, and 3 IMT/IML definitions, 120 curves will be computed.)

Another way to put it is:

`T = R * P * I`

where

• `T` is the total number of curves
• `R` is the total number of logic tree realizations
• `P` is the number of geographical points of interest
• `I` is the number of IMT/IML definitions

Hazard curves are grouped by IMT and realization (1 group per IMT per realization). Each group includes 1 curve for each point of interest.

Additionally, for each realization a hazard curve container (with `output_type` equal to `hazard_curve_multi`) is created. This container output could be used in contexts where you need to identify a whole group of hazard curves sharing the same realization as when you run a risk calculation supporting structure dependent intensity measure types.

##### Statistical Curves¶

The classical hazard calculator is also capable of producing mean and quantile curves. These aggregates are computed from the curves for a given point and IMT over all logic tree realizations.

Similar to hazard curves for individual realizations, statistical hazard curves are grouped by IMT and statistic type. (For quantiles, groups are separated by quantile level.) Each group includes 1 curve for each point of interest.

###### Mean Curves¶

Mean hazard curves can be computed by specifying mean_hazard_curves = true in the job configuration.

When computing a mean hazard curve for a given point/IMT, there are two possible approaches:

1. mean, unweighted
2. mean, weighted

Technically, both approaches are “weighted”. In the first approach, however, the weights are implicit and are taken into account in the process of logic tree sampling. This approach is used in the case of random Monte-Carlo logic tree sampling. The total of number of logic tree samples is defined by the user with the number_of_logic_tree_samples configuration parameter.

In the second approach, the weights are explicit in the caluclation of the mean. This approach is used in the case of end-branch logic tree enumeration, whereby each possible logic tree path is traversed. (Each logic tree path in this case defines a weight.) The total number of logic tree samples in this case is determined by the total number of possible tree paths. (To perform end-branch enumeration, the user must specify number_of_logic_tree_samples = 0 in the job configuration.

The total number of mean curves calculated is

`T = P * I`

where

• `T` is the total number of curves
• `P` is the number of geographical points of interest
• `I` is the number of IMT/IML definitions

Furthermore, also in that case a hazard curve set grouping all the mean curves is produced (of type `hazard_curve_multi`).

###### Quantile Curves¶

Quantile hazard curves can be computed by specifying one or more quantile_hazard_curves values (for example, quantile_hazard_curves = 0.15, 0.85) in the job configuration.

Similar to mean curves, quantiles curves can be produced for a given point/IMT/quantile level (in the range [0.0, 1.0]), there are two possible approaches:

1. quantile, unweighted
2. quantile, weighted

As with mean curves, unweighted quantiles are calculated when Monte-Carlo logic tree sampling is used and weighted quantiles are calculated when logic tree end-branch enumeration is used.

The total number of quantile curves calculated is

`T = Q * P * I`

where

• `T` is the total number of curves
• `Q` is the number of quantile levels
• `P` is the number of geographical points of interest
• `I` is the number of IMT/IML definitions

Moreover, also in that case curves sharing the same quantile are grouped into a virtual output container of type `hazard_curve_multi`.

#### Hazard Maps¶

Hazard maps are geographical meshes of intensity values. Intensity values are extracted from hazard curve functions by interpolating at a given probability exceedance. To put it another way, hazard maps seek to answer the following question: “At the given level probability, what intensity level is likely to be exceeded at a given geographical points in the next X years?”

The resulting geographical mesh is often depicted graphically, with a color key defining which color to plot at the given location for a given value or range values.

Hazard maps bear the same metadata as hazard curves, with the addition of the probability at which the hazard maps were computed.

For a given calulcation, hazard maps are computed for each hazard curve. Maps can be computed for one or more probabilities of exceedance, so the total number of hazard maps is

`T = C * E`

where

• `T` is the total number of maps
• `C` is the total number of hazard curves (see the method for calculating the number of hazard curves)
• `E` is the total number of probabilities of exceedance

Note: This includes mean and quantile maps.

##### Statistical Maps¶

Hazard maps can be produced from any set of hazard curves, including mean and quantile aggregates. There are no special methods required for computing these maps; the process is the same for all hazard map computation.

#### Uniform Hazard Spectra¶

Uniform Hazard Spectra (UHS) are discrete functions which are essentially derived from hazard maps. Thus, hazard map computation is a prerequisite step in producing UHS. UHS derivation isn’t so much a computation, but rather a special arrangement or “view” of hazard map data.

UHS “curves” are composed of a few key elements:

• Spectra Acceleration Periods - These values make up the x-axis values (“ordinates”) of the curve.
• Intensity Measure Levels - These values make up the y-axis values (“abscissae”) of the curve.
• Probability of Exceedance - The hazard map probability value from which the UHS is derived. The “Uniform” in UHS indicates a uniform PoE over all periods.
• Location - A 2D geographical point, consisting of longitude and latitude.

To construct UHS from a set of hazard maps, one can conceptualize this process as simply extracting from multiple hazard maps all of the intensity measure levels for a given location and arranging values in order of SA period, beginning with the lowest period value. This is done for all locations.

Note: All maps with IMT = SA are considered, in addition to PGA. PGA is equivalent to SA(0.0). Hazard maps with other IMTs (such as PGV or PGD) are ignored.

The example below illustrates extracting the IML values for a given location (indicated by x) from three hazard maps:

```Hazard maps PoE: 0.1

/--------------/
/              /<-- PGA [equivalent to SA(0.0)]
/     x        /-/
/--------------/ /<-- SA(0.025)
/     x        /-/
/--------------/ /<-- SA(0.1)
/     x        /-/
/--------------/ /<-- SA(0.2)
/     x        /
/--------------/
```

Assuming that the IMLs from the PGA, SA(0.025), SA(0.1), and SA(0.2) maps are 0.3, 0.5, 0.2, and 0.1, respectively, the resulting UHS curve would look like this:

```[IML]
^
|
0.5           *
|
0.4
|
0.3   *
|
0.2                   *
|
0.1                           *
|
+----|-------|-------|-------|----> [SA Period]
0.0    0.025    0.1     0.2
```

Uniform Hazard Spectra are grouped into result sets where each result set corresponds to a probability of exceedance and either a logic tree realization or statistical aggregate of realizations. Each result set contains a curve for each geographical point of interest in the calculation.

The number of UHS results (each containing curves for all sites) is

`Tr = E * (Q + M + R)`

where

• `Tr` is the total number of result sets (and also the number of files, if the results are exported)
• `E` is the total number of probabilities of exceedance
• `Q` is the number of quantile levels
• `M` is 1 if the calculation computes mean results, else 0
• `R` is the total number of logic tree realizations

The total of UHS curves is

`T = Tr * P`

where

• `Tr` is the total number of result sets (see above)
• `P` is the number of geographical points of interest

### Hazard Curves Post-Processing¶

Post processing functionality for the classical PSHA hazard calculator. E.g. mean and quantile curves.

`openquake.engine.calculators.hazard.post_processing.``compute_hazard_maps`(curves, imls, poes)[source]

Given a set of hazard curve poes, interpolate a hazard map at the specified `poe`.

Parameters: curves – 2D array of floats. Each row represents a curve, where the values in the row are the PoEs (Probabilities of Exceedance) corresponding to `imls`. Each curve corresponds to a geographical location. imls – Intensity Measure Levels associated with these hazard `curves`. Type should be an array-like of floats. poes (float) – Value(s) on which to interpolate a hazard map from the input `curves`. Can be an array-like or scalar value (for a single PoE). A 2D numpy array of hazard map data. Each element/row in the resulting array represents the interpolated map for each `poes` value specified. If `poes` is just a single scalar value, the result array will have a length of 1. The results are structured this way so that it is easy to iterate over the hazard map results in a consistent way, no matter how many `poes` values are specified.
`openquake.engine.calculators.hazard.post_processing.``do_uhs_post_proc`(job)[source]

Compute and save (to the DB) Uniform Hazard Spectra for all hazard maps for the given `job`.

Parameters: job – Instance of `openquake.engine.db.models.OqJob`.
`openquake.engine.calculators.hazard.post_processing.``make_uhs`(maps)[source]

Make Uniform Hazard Spectra curves for each location.

It is assumed that the lons and lats for each of the `maps` are uniform.

Parameters: maps – A sequence of `openquake.engine.db.models.HazardMap` objects, or equivalent objects with the same fields attributes. A dict with two values:: periods: a list of the SA periods from all of the `maps`, sorted ascendingly uh_spectra: a list of triples (lon, lat, imls), where imls is a tuple of the IMLs from all maps for each of the periods
`openquake.engine.calculators.hazard.post_processing.``hazard_curves_to_hazard_map`()

## Event-Based PSHA Calculator¶

The Event-Based Probabilistic Seismic Hazard Analysis (ePSHA) approach allows calculation of ground-motion ﬁelds from stochastic event sets. Eventually, Classical PSHA results - such as hazard curves - can be obtained by post-processing the set of computed ground-motion ﬁelds.

### Outputs¶

#### SESs¶

Stochastic Event Sets are collections of ruptures, where each rupture is composed of:

• magnitude value
• tectonic region type
• source type (indicating whether the rupture originated from a point/area source or from a fault source)
• details of the rupture geometry (including lat/lon/depth coordinates, strike, dip, and rake)

SES results are structured into 3-level hierarchy, consisting of SES “Collections”, SESs, and Ruptures. For each end-branch of a logic tree, 1 SES Collection is produced. For each SES Collection, Stochastic Event Sets are computed in a quantity equal to the calculation parameter ses_per_logic_tree_path. Finally, each SES contains a number of ruptures, but the quantity is more or less random. There are a few factors which determine the production of ruptures:

• the Magnitude-Frequency Distribution (MFD) each seismic source considered
• investigation time
• random seed

From a scientific standpoint, the MFD for each seismic source defines the rupture “occurrence rate”. Combined with the investigation time (specified by the investigation_time parameter), these two factors determine the probability of rupture occurrence, and thus determine how many ruptures will occur in a given calculation scenario.

From a software implementation standpoint, the random seed also affects rupture generation. A “base seed” is specified by the user in the calculation configuration file (using the parameter random_seed). When a calculation runs, the total work is divided into small, independent asynchronous tasks. Given the base seed, additional “task seeds” are generated and passed to each task. Each task then uses this seed to control the random sampling which occurs during the SES calculation. Structuring the calculation in this way guarantees consistent, reproducible results regardless of the operating system, task execution order, or architecture (32-bit or 64-bit).

#### GMFs¶

Event-Based hazard calculations always produce SESs and ruptures. The user can choose to perform additional computation and produce Ground Motion Fields from each computed rupture. (In typical use cases, a user of the event-based hazard calculator will want to compute GMFs.)

GMF results are structured into a hierarchy very similar to SESs, consisting of GMF “Collections”, GMF “Sets”, and GMFs. Each GMF Collection is directly associated with an SES Collection, and thus with a logic tree realization. Each GMF Set is associated with an SES. For each rupture in an SES, 1 GMF is calculated for each IMT (Intensity Measure Type). (IMTs are defined by the config parameter intensity_measure_types.) Finally, each GMF consists of multiple “nodes”, where each node is composed of longitude, latitude, and ground motion values (GMV). The sites (lon/lat) of each GMF are defined by the calculation geometry, which is specified by the region or sites configuration parameters. (In other words, if the calculation geometry consists of 10 points/sites, each computed GMF will include 10 nodes, 1 for each location.)

#### Hazard Curves¶

It is possible to produce hazard curves from GMFs (as an alternative to hazard curve calculation method employed in the Classical hazard calculator. The user can activate this calculation option by specifying hazard_curves_from_gmfs = true in the configuration parameters. All hazard curve post-processing options are available as well: mean_hazard_curves, quantile_hazard_curves, and poes (for producing hazard maps).

Hazard curves are computed from GMFs as follows:

• For each logic tree realization, IMT, and location, there exist a number of hzrdr.gmf records exactly equal to the ses_per_logic_tree_path parameter. Each record contains an array with a number of ground motion values; this number is determined by the number of ruptures in a given stochastic event (which is random–see the section “SESs” above). All of these lists of GMVs are flattened into a single list of GMVs (the size of which is unknown, due the random element mentioned above).
• With this list of GMVs, a list of IMLs (Intensity Measure Levels) for the given IMT (defined in the configuration file as intensity_measure_types_and_levels), investigation_time, and “duration” (computed as investigation_time * ses_per_logic_tree_path), we compute the PoEs (Probabilities of Exceedance). See `openquake.engine.calculators.hazard.event_based.post_processing.gmvs_to_haz_curve()` for implementation details.
• The PoEs make up the “ordinates” (y-axis values) of the produced hazard curve. The IMLs define the “abscissae” (x-axis values).

As with the Classical calculator, it is possible to produce mean and quantile statistical aggregates of curve results.

#### Hazard Maps¶

The Event-Based Hazard calculator is capable of producing hazard maps for each logic tree realization, as well as mean and quantile aggregates. This method of extracting maps from hazard curves is identical to the Classical calculator.

## Scenario Calculator¶

The Scenario Siesmic Hazard Analysis (SSHA) approach allows calculation of ground motion ﬁelds from a single earthquake rupture scenario taking into account ground-motion aleatory variability.

## Disaggregation Calculator¶

The Disaggregation approach allows calculating relative contribution to a seismic hazard level. Contributions are defined in terms of latitude, longitude, magnitude, distance, epsilon, and tectonic region type.

Sources:

• Disaggregation of Seismic Hazard
by Paolo Bazzurro and C. Allin Cornell
Bulletin of the Seismological Society of America, 89, 2, pp. 501-520, April 1999

### Outputs¶

#### Hazard Curves¶

Hazard curve calculation is the first phase in a Disaggregation calculation. This phase computes the hazard for a given location by aggregation contributions from all relevant seismic sources in a given model. (The method for computing these curves is exactly the same as the Classical approach.)

Mean and quantile post-processing options for hazard curves are not enabled for the Disaggregation calculator.

#### Disaggregation Matrices¶

Once hazard curves are computed for all sites and logic tree realizations, the second phase (disaggregation) begins. While the hazard curve calculation phase is concerned with aggregating the hazard contributions from all sources, the disaggregation phase seeks to quantify the contributions from the various ruptures generated by the source model to the hazard level at a given probability of exceedance (for a given geographical point) in terms of:

• Longitude
• Latitude
• Magnitude (in Mw, or “Moment Magnitude”)
• Distance (in km)
• Epsilon (the difference in terms of standard deviations between IML to be disaggregated and the mean value predicted by the GMPE)
• Tectonic Region Type

This analysis, which operates on a single geographical point and all seismic sources for a given logic tree realization, results in a matrix of 6 dimensions. Each axis is divided into multiple bins, the size and quantity of which are determined the calculation inputs. Longitude and latitude bins are determined by the `coordinate_bin_width` calculation parameter, in units of decimal degrees. Magnitude bins are determined given the `mag_bin_width`. Distance bins are determined by the `distance_bin_width`, in units of kilometers. `num_epsilon_bins` defines the quantity of epsilon bins. `truncation_level` is taken into account when computing the width of each epsilon bin, and so this is a required parameter. The number of tectonic region type bins is simply determined by the variety of tectonic regions specified in a given seismic source model. (For instance, if a source model defines sources for “Active Shallow Crust” and “Volcanic”, this will result in two bins.)

The final results of a disaggregation calculation are various sub-matrices extracted from the 6-dimensional matrix. These sub-matrices include common combinations of terms, which are as follows:

• Magnitude
• Distance
• Tectonic Region Type
• Magnitude, Distance, and Epsilon
• Longitude and Latitude
• Magnitude, Longitude, and Latitude
• Longitude, Latitude, and Tectonic Region Type

Each disaggregation result produced by the calculator includes all of these.

The total number of disaggregation results produce by the calculator is

`T = E * R * I * P`

where

• `T` is the total number of disaggregation results
• `E` is the total number of probabilities of exceedance (defined by `poes_disagg`)
• `R` is the total number of logic tree realizations
• `I` is the number of IMT/IML definitions
• `P` is the number of points with non-zero hazard (see note below)

Note: In order to not waste computation time and storage, if the hazard curve used to a compute disaggregation for a given point and IMT contains all zero probabilities, we do not compute a disagg. matrix for that point.