# Array methods¶

This section briefly summarizes the available array methods for multi-station analysis.
The basics of the methods are described in a number
of scientific publications including Rost and Thomas [20], Schweitzer *et al.* [21],
Cansi [15] and Roessler *et al.* [19].

We divide the methods into

**Multi-station methods**applied to arrays with more than one station. Typically, the vertical components are used.**Single-station methods**which are applied to single stations with 1 (1C) or 3 components (3C). While multi-station array methods typically rely on stacking the data from several stations, analyzing single stations allow to add further observations from stations integrated into an array or network.

The methods can by applied in different modes

**Single time windows**which are are selected and the parameters represent the entire window.**Sliding time windows**where a defined time window is shifted across the selected data at defined increments. The parameters are computed for the defined time windows forming a time series with data points in intervals of the increments

## Inversion of phase picks¶

**Method:**multi-station**Mode:**single-time window**Provides:**arrival time

backazimuth

slowness

If at least 3 picks from the same phase are available at the array station
they can be inverted for the horizontal slowness.
Let a wave with a horizontal slowness s_{j} arrive at an array element
with index iand at location r_{i}.
Then the relative time delay t_{i} of the wave arriving at the i^{th} array
element with respect to the reference station is

(1)¶

where superscript *obs* indicates actual arrival times and subscript *0* refers
to the reference station.
The time of the pick made at the reference station can therefore be inverted for the
horizontal slowness

(2)¶

Similarly, the time differences between all station pairs can be used instead of referring to the reference station.

In reality, the phase arrivals are observed at time t_{i} ^{obs}.

(3)¶

where t_{0} is the absolute time of arrival at the reference station. When
t_{0} is available we simply invert t_{i} ^{rel} for s.

## Slowness-backazimuth analysis¶

**Method:**multi-station**Mode:**single-time window or sliding time window**Provides:**arrival time

backazimuth

slowness

The slowness-backazimuth analysis is implemented as a search over a defined range of horizontal slowness and backazimuth. Slowness and backazimuth define the relative time shifts applied to the data before stacking in order to compute the beam. As a measure of coherency semblance and beam power are computed. A detection is made when the semblance exceeds a configured threshold.

The beam computation is available as

Stacking in the frequency domain by the F-K method

Stacking in the time domain by beampacking

Re-picking is applied on the beam trace to determine the onset time. Only when re-picking is successful, a phase pick can be declared.

As a result the scalar horizontal slowness and the backazimuth of the detected wave is provided along with the time of the detection. The three values are integrated into a pick object.

## Beampacking¶

**Method:**multi-station**Mode:**single-time window or sliding time window

Beampacking is implemented in LAMBDA as a detection method in the time-domain equivalent to the F-K analysis.

However, as the differences in timeshifts between neighboring stations can be small, the sampling rate must must be adjusted to allow mapping these traveltime differences. The traveltime differences can be approximated from the inter-station distances and the considered slowness or wavenumber ranges.

## F-K analysis¶

**Method:**multi-station**Mode:**single-time window or sliding time window

F-K analysis is implemented in LAMBDA. The waveforms are shifted and stacked in
the frequency
domain for all grid points of a given grid of slowness or wavenumber and for the
selected frequency range. The considered slowness or wavenumber determines the
time shift of the waveform recoreded at a station with respect to the reference station.
For every grid point the beam trace is computed.
The method is described e.g. in Rost and Thomas [20] or Schweitzer *et al.* [21].

For each beam trace the semblance is calculated. A detection is declared and an pick is created if the parameters of the stack with the largest semblance within an individual location grids exceeds the defined thresholds. The pick is declared on the vertical channel of the array reference station. Before declaration, the pick time is refined by the AIC method applied to the beam trace.

In addition to picks from conventional phase pickers such as scautopick also the backazimuth and the slowness are provided. Both values are take from the grid point of the detection.

Data are used according to the the length of the configured time window. By shifting the time windows with the configured interval a time series of picks, maximium semblance, slowness and backazimuth is obtained.

## PMCC¶

**Method:**multi-station**Mode:**single-time window or sliding time window**Provides:**arrival time

backazimuth

slowness

The PMCC method has been implemented. It is based on the Progressive Multi-Channel Correlation methods described by Cansi (1995). PMCC creates families from single PMCC detections to summarize similar solutions and to reduce the amount or similar detections.

## Backprojection¶

**Method:**multi-station**Mode:**sliding time window**Provides:**source time

source location

Backprojection of waveforms to their potential sources is implemented in LAMBDA for single arrays as described in Roessler et al. (2010). By backprojection events are detected and located. The waveforms are shifted and stacked along traveltime curves for a given source - receiver pair and a considered phase. The time shifts are calculated by computing the traveltimes from all considered sources to all array elements. The traveltimes are read from the given traveltime interface and table. The sources are given by the a priori defined location grid.

The stacking is performed and the beam trace is computed

In frequency domain by the F-K method

In the time domain by beampacking

The absolute time shifts of traces for stacking are provided based on the source location, the location of the grid points and the selected traveltime interface and traveltime table.

For each stack, i.e. each grid point, the semblance is calculated. A detection is made when the semblance exceeds the defined thresholds. The stacking can be repeated for different source times providing the possibility of multiple detections.

For each detection re-picking based on AIC is applied on the beam trace to determine the onset time. Only when re-picking is successful, a phase pick and an origin are declared.

## N-th rooting¶

**Method:**multi-station**Mode:**single-time window or sliding time window

**no yet implemented**

## Vespagrams¶

**Method:**multi-station**Mode:**sliding time window

Vespagrams allow stacking of waveforms for a fixed backazimuth or slowness while varying slowness (slowness vespagram) or backazimuth (backazimuth vespagram), respectively. Currently slowness vespagrams are available in interactive analysis. The slowness range and and resolution can be adjusted interactively.

## 3C-polarization analysis¶

**Method:**single-station**Mode:**single-time window**Provides:**backazimuth

slowness

Single-station 3-component (3C) polarization analysis has been implemented. Within a single defined time window the polarization angles, backazimuth and incidence, are computed for each time sample. The values are selected at the maximum of their distribution.

## Conventional phase picking¶

**Method:**single-station**Mode:**single-time window**Provides:**arrival time

Conventional phase picking on single stations is available for any phase type which has a corresponding travel-time table. Backazimuth and slowness computed by other methods can be added to the picks.