*Note – If you are unfamiliar with any of the terms in this article, please review the seismic glossary of terms.*

The Earth is buzzing with energy, from deep magnetotelluric currents to microtremors of the subsurface. Passive-seismic methods rely on the ambient ground motion recorded between pairs of geophones. The waveforms collected for every geophone pair can then be analyzed across different frequencies and the coherency between the sensors for each frequency can be analyzed. And how geophone pairs are laid out is important. Why? Survey geometry has a big impact on passive-seismic data quality. It all depends on the direction of propagation of the microtremors used for analysis.

When microtremors propagate parallel to the survey line, phase velocity can be directly calculated.

If microtremors propagate perpendicular to the survey line, phase velocity cannot be calculated.

For microtremors that

propagate at an angle

to the survey line, the

input angle must be

known to accurately

calculate phase velocity.

Microtremors vary in propagation direciton and intensity from sources of unknown locations, and there is no way to know the propagation angles prior to measurement.

Passive-seismic field methods originally used two geophones. Now multiple geophones can be used, and all the pairs formed between them can be analyzed. Since propagation angles are unknown, the Spatial Autocorrelation (SPAC) method of analysis is used. SPAC assumes that microtremors radiate from all directions, making it independent of

source location.

Therefore, survey geometries with increased dimensionality are more robust in analyzing coherencies from multiple directions between geophones. Survey arrays which are less dimensional (such as a line) fair poorly compared to 2D arrays (such as an L-shape).

## 1D Survey Geometry: Line

A straight line survey is the current industry standard when it comes to MASW surveys. The reason why is the equipment used. Most seismographs connect to geophones to the AD chip using a spread cable. This spread cable has a fixed spacing, typically 5 meters between geophone takeouts. Here is a 24-ch MASW survey with 5 meter spacings:

**Pros:**

- Easy to setup
- Easy to stay consistent with geometry across surveys/jobs.
- Easier to understand/teach.

**Cons:**

- Lacks dimensionality.
- Coherencies will be lower, greater error in the data.
- Fixed geophone spacings.
- Cannot be expanded beyond spread cable length.
- Can be difficult to deploy in urban environments.
- Equipment is heavy and bulky.

## 2D Survey Geometry: L-shape

L-shape surveys are also relatively common for geotechnical applications. It’s easy to take a linear spread cable and add a 90 degree bend to it. It’s not easy taking a spread cable and making it into a triangle, circle, or more creative survey geometry however.

It is important to keep in mind when discussing survey geometries that the equipment used will determine the survey geometries made possible. The two main types of seismographs are spread cable based or nodal. We will look the pros and cons of a spread cable L-shape array, and then compare it to the advantages and disadvantages of a nodal L-shape array. First the fixed-separation spread cable L-shape array.

**Pros:**

- L-shape arrays are 2 dimensional.
- Easy to setup, use the perfect triangle rule to create an accurate 90 degree angle.
- Fits well into urban settings.

**Cons:**

- Fixed geophone spacings.
- Cannot be expanded beyond spread cable length.
- It is difficult to reduce the geophone interval.
- Equipment is heavy and bulky.

Better than a spread cable L-shape array is a nodal L-shape array. With a nodal seismograph system, each geophone connects to a seismograph with a battery, GPS, A/D chip, and potentially other electronics. Since passive seismic doesn’t require a source and an accurate trigger timestamp, nodal arrays are superior as they allow for unique geometrics and spacings between geophones. Through the many surveys completed for the NVSP, I have found that an L-shape array with spacings following the Fibonacci sequence to result in consistently high quality data. Lets examine this survey setup now.

**Pros:**

- Nodal seismographs enjoy completely flexible survey design, L-arrays in particular.
- Less channels (AUs) needed for better results.
- More spacings can be analyzed.
- If the survey area is large, it is easy to reduce the small spacings, moving AUs further out to capture longer wavelengths.
- If the survey area on one side is smaller, AUs can be removed from that side and added to the other, or added as offshoots whose location is measured via GPS.

**Cons:**

- Less channels. Spread cable seismographs are cheaper per channel as there is only one A/D chip per 24-channels, not one per channel.

For this array, the Fibonacci sequence was followed (0, 1, 2, 3, 5, 8, 13, 21, 34, 55 m) for each side of the L. Compare the spacings created for the first L-shape, and the spacings created for the Fibonacci L-shape (which has 5 less geophones).

5 meter spaced L-array geophone spacings: 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 55.0 60.0

## 2D Survey Geometry – Triangle

Triangular survey geometries are another common survey geometry. It is well studied with some geophysicists. Triangular surveys offer a reliable testing parameter. If many surveys are done as triangles, then it is easy to compare the data to each other. Having consistent survey parameters is important for passive-seismic applications, and the issue with triangular surveys is they typically result in lower quality data compared to other survey geometries. Triangular survey geometries have good overlap in pairings, but the number of spacings created are typically 1/2 to 1/3 of a conventional L-survey.

The number of spacings created, that is the distance between geophones, and the degree that they are equally distributed from 0 to N has a major effect on overall data quality for passive-seismic. With poorly distributed spacings, and not many of them, then big gaps will exist in the data, and interpolations will need to be made. This causes distortion. And if you only run triangles, it’s consistent. Expect less precise peak shear-wave velocity values through 0-30 Hz, more higher-mode data, and significant background noise beyond 10 Hz.

**Pros:**

- Consistent results.
- High confidence in data quality for pairings gathered.
- Two dimensional, can analyze microtremors from all directions well.
- Looks cool.

**Cons:**

- Poor number of total spacings.
- Poor distribution of spacings.
- Difficult to setup, more so as channels are added.
- Collects more background noise.
- Rigid survey design, can be made flexible with GPS placed AUs.

## 2D Survey Geometry – Circle

Nodal seismographs can also be laid onto the ground in more complicated geometries, such as circles. Circular surveys suffer many of the same problems as triangular surveys. The way you scale circles with size is to increase the number of rings, or choose which size ring you want. Each circular survey ring should have at minimum 8 AUs.

Circular arrays are excellent for ensuring that many spacings are collected evenly across the area that is being surveyed, but it is my experience that data collected with a circular array geometry will gather much more higher frequency noise than a L-shape or spiral array. My speculation would be because circular arrays typically don’t have good coverage of spacings below 10 meters. In practicality the geoscientist is limited in their array geometry by how many receivers they have, and to create a 8 receiver circle with a radius of 5 meters hinders the larger depth of investigation greatly as those eight receivers can no longer be positioned further out.

There are promising results (see N60 – Esther Deaver Park) that indicate an array geometry consisting of a circle paired with a Fibonacci line eliminates the high frequency noise problem while also enjoying the benefits of a circular array.

**Pros:**

- Circular arrays ensure very even receiver coverage over the chosen survey area.
- Consistent results.
- 2-dimensional
- 1 or 2 receivers can be removed from the circles with a minimal effect on data quality

**Cons:**

- Circular arrays require a knowledge of trigonometry.
- Time intensive to setup, as specific angles need to calculated in the field for each circle radius.
- More receivers are required for the same data quality of other array geometries.

## 2D Survey Geometry – Fibonacci Spiral

A Fibonacci spiral is an infinite shape created based on the Fibonacci sequence, otherwise known as the golden ratio. The math is simple:

for *n* > 1

To determine Fn2, it would be 1+0, so Fn2 equals 1. And Fn2 + Fn1 = 2, and the infinite Fibonacci sequence begins there. The first digits 0 through 1000 are:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, and more

This efficient scaling of numbers creates paring overlap and equal distribution of spacings. For example in a fib 0-55 survey line (which for a passive-seismic survey consists of 10 receiver locations) you’ll create a pairing for 21 twice (Fn8). First from location 0 to 21, and again with 34 to 55. and then single pairings will be created back down to zero for Fn8 minus Fn6 through Fn0.

To illustrate this concept, lets use a Fib21 array geometry, which is a straight line 21 meters in length which receivers setup at 0, 1, 2, 3, 5, 8, 13, and 21. Remember, since every distance between a geophone creates a SPAC spacing, all the spacings collected for a Fib21 array are 21, 20, 19, 18, 16, 13, 12, 11, 10, 8, 7, 6, 5, 4, 3, 2, 1. Only 17, 15, 14, and 9 are missing.

0 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | |

0 | N/A | 1 | 2 | 3 | 5 | 8 | 13 | 21 |

1 | N/A | 1 | 2 | 4 | 7 | 12 | 20 | |

2 | N/A | 1 | 3 | 6 | 11 | 19 | ||

3 | N/A | 2 | 5 | 10 | 18 | |||

5 | N/A | 3 | 8 | 16 | ||||

8 | N/A | 5 | 13 | |||||

13 | N/A | 8 | ||||||

21 | N/A |

Spacings that are never measured | Spacings that are measured once | Spacings that are measured twice | Spacings that are measured 3 times |

9 | 4 | 13 | 8 |

14 | 6 | 5 | |

15 | 7 | 3 | |

17 | 10 | 2 | |

11 | 1 | ||

12 | |||

16 | |||

18 | |||

19 | |||

20 | |||

21 |

If a five meter spacing was used for a twenty meter line (which is roughly comparable), you’d only collect data at 5, 10, 15, and 20 meters. Four spacings versus seventeen.

Once you wrap your mind around the Fibonacci sequence, the mathematics of it become quite simple to use in the field. To setup a Fibonacci array geometry most efficiently, you’ll want ground stakes, two tape survey tape measures, and a double right angle survey prism. The double right angle survey prism quickly allows you to make 90 degree angles, and then you can use a tape measure between the two receiver locations (in this example 0, 0 and 34, 34) to find the midpoint angle and place the offshoot receiver location (~24, ~10 m).

**Pros:**

- Excellent distribution of spacings and pairings
- Provides the highest quality data.
- Scales easily for smaller or larger arrays.

**Cons:**

- Requires a knowledge of Fibonacci numbers.
- Difficult to setup in the field relative to linear or L-shape arrays.

## Array Geometries – Results

As part of the NVSP, all five of these array geometries were collected on the same day under identical environmental and survey conditions. The goal was to establish a basis of comparison free of confounding variables. Below to the left are the phase velocity diagrams for each, as well as their pairings, spacing, and coherencies. Below to the right are the dispersion curves for each after 5 inversions, and the RMSE values.

### Linear Array

### L-Shape Array

### Triangle Array

### Circle Array

### Fibonacci Spiral Array

As you can see with the last result, the Fibonacci array, it was the greatest number of spacings, good bands of positive coherencies, the cleanest looking phase velocity diagram, and the lowest RMSE value.

## Practical Takeaway

Fibonacci Spiral arrays result in the best data quality, but they can be tricky to setup in the field. For passive-seismic surveys which demand the highest quality data, using a Fibonacci spiral array geometry is best. For a more practical alternative that is easier to setup, can be used in more constricted environments, and still results in excellent data quality, a Fibonacci L-shape survey is recommended.