Ambient Vibrations

Encyclopedia

Various types of vibration sources are always producing so called

s) propagating on the surface. Low frequency waves (below 1 Hz) are generally called microseism

s and high frequency waves (above 1 Hz) are called microtremor

s. These ambient vibrations are used in practice to derive the elastic properties of the ground and the low-strain dynamic properties of civil-engineering structures (bridges, buildings, dams...). This information is useful for different purposes : fundamental seismology, engineering seismology, Earthquake engineering

, Seismic microzonation

, Structural health monitoring

, but also Hydrology

, Geotechnical Engineering, etc.

the peak between 0.1 and 0.3 Hz is clearly associated with the interaction of water waves of nearly equal frequencies but opposite directions. At high frequency (above 1 Hz), the wavefield is mainly produced by human activities (road traffic, industrial work...) but there are also natural sources like rivers. Around 1 Hz, the local atmospheric conditions (wind...) are also a major source of ground vibrations. The amplitude of ground ambient vibrations is typically in the range of 1e-6 m, i.e. in the order of the tenth of micrometers to tens of micrometers. Peterson provided high and low noise models as a function of frequency.

The ambient wave field is made of a small amount of body waves

(P- and S-waves), and a most generally predominant part of surface wave

s, i.e. Love

and Rayleigh wave

s. Theses waves are dispersive

, i.e. their phase velocity

varies with frequency (most generally, it decreases with increasing frequency). The dispersion curve

(phase velocity or slowness as a function of frequency) is tightly related to the variations of the shear-wave velocity with depth in the different ground layers: it can thus be used as a non-invasive tool to investigate the underground structure.

s at the end of 19th century. However, at that time, the famous Japanese seismologist Fusakichi Omori

could already record ambient vibrations in buildings, where the amplitudes are magnified. He found their resonance frequencies

and studied their evolution as a function of damage . After the 1933 Long Beach earthquake

in California, a large experiment campaign led by Carder in 1935 allowed to record and analyze ambient vibrations in more than 200 buildings. These data were used in the design codes

to estimate resonance frequencies of buildings but the interest of the method went down until the 1950s. The interest on ambient vibrations in structures rose again thanks to famous earthquake engineers, especially in California and Japan (G. Housner

, D. Hudson, K. Kanai and T. Tanaka ...). Ambient vibrations were however supplanted - at least for some time - by forced vibration techniques that allow to increase the amplitudes and control the shaking source and their system identification methods. Even if Trifunac showed as early as 1972 that ambient and forced vibrations led to the same results, the interest in ambient vibration techniques rose again only in the late 1990s. The relatively low-cost and easiness of implementation, the improvement of the recording material and of the computation opportunities make these techniques very popular nowadays, especially as the low-strain dynamic characteristics they provide were shown to be close enough to the measured dynamic characteristics under strong shaking, at least as long as the buildings are not severely damages .

The use of noise recordings on the ground started in the 1950s with the enhancement of seismometers to monitor nuclear tests and the development of seismic arrays. The main contributions at that time for the analysis of these recordings came from the Japanese seismologist K. Aki

in 1957 who first proposed the methods used nowadays (Spatial Autocorrelation method -SPAC-, Frequency-wavenumber -FK- method, correlation method...). However, the practical implementation of these methods was not possible at that time because of the low precision of clocks in seismic stations. Again, the opportunities of computations and the enhancements in the recording material led to a rise of interest in the 1990s. The first widely implemented method, rediscovered by Nakamura in 1989 is the Horizontal to Vertical Spectral Ratio (H/V) method to derive the resonance frequency of sites. Assuming that the shear wave dominates the microtremor, Nakamura indicated that the H/V spectral ratio of ambient vibrations were roughly equals the S-wave transfer function between the ground surface and the bedrock at a site. This assumption is however now criticized in the literature (e.g. SESAME project).

In the late 1990s , the array methods on ambient vibration data started to allow deriving the ground properties in terms of shear waves velocity profiles. The European Research project SESAME http://sesame-fp5.obs.ujf-grenoble.fr/index.htm (2004–2006) was one of the first structured attempts to standardize the use of ambient vibrations to retrieve the properties of the ground, in the aim of estimating site amplifications in case of earthquake (site effects).

, H/V peak, dispersion curves

and autocorrelation functions

.

Using an array of seismic sensors recording simultaneously the ambient vibrations allow to understand more deeply the wavefield and therefore to derive more properties of the ground. Due to the limitation of the available number of sensors, several arrays of different sizes may be realized and the results merged.

The information of the Vertical components is only linked to the Rayleigh waves, and therefore easier to interpret, but method using the 3 space components are also developed, providing informations about Rayleigh and Love wavefield.

s, ambient vibrations force into vibrations the civil engineering structures like bridge

s, building

s or dam

s. This vibration source is supposed by the greatest part of the used methods to be a white noise

, i.e. with a flat noise spectrum so that the recorded system response is actually characteristic of the system itself. The vibrations are perceptible by humans only in rare cases (bridges, high buildings). Ambient vibrations of buildings are also caused by wind and internal sources (machines, pedestrians...) but these sources are generally not used to characterize structures.

The branch that studies the modal properties of systems under ambient vibrations is called Operational modal analysis

or Output-only modal analysis

and provides many useful methods for civil engineering

.

The observed vibration properties of structures integrate all the complexity of these structures including the load-bearing system, heavy and stiff non-structural elements (infill masonry panels...), light non-structural elements (windows...) and the interaction with the soil

(the building foundation may not be perfectly fixed on the ground and differential motions may happen). This is emphasized because it is difficult to produce models able to be comparable with these measurements.

The power spectrum

computation of ambient vibration recordings in a structure (e.g. at the top floor of a building for larger amplitudes) gives an estimation of its resonance frequencies and eventually its damping ratio

.

Assuming ground ambient vibrations is the excitation source of a structure, for instance a building, the Transfer Function

between the bottom and the top allows to remove the effects of a non-white input. This may particularly be useful for low signal-to-noise ratio

signals (small building/high level of ground vibrations). However this method is not able to remove the effect of soil-structure interaction.

They consist in the simultaneous recording in several points of a structure. The objective is to obtain the modal parameters of structures: resonance frequencies

, damping ratio

s and modal shapes for the whole structure. Notice than without knowing the input loading, the participation factors of these modes cannot a priori be retrieved. Using a common reference sensor, results for different arrays can be merged.

Several methods use the power spectral densityIn statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

matrices of simultaneous recordings, i.e. the cross-correlation

matrices of these recordings in the Fourier domain

. They allow to extract the operational modal parameters (Peak Picking method) that can be the results of modes coupling or the system modal parameters (Frequency Domain Decomposition method).

Numerous system identification methods exist in the literature to extract the system properties and can be applied to ambient vibrations in structures

s are needed to compute these products (dispersion curve, modal shapes...) that could be compared with the experimental data. Computing a lot of models to find which agree with the data is solving the Inverse problem

. The main issue of inversion is to well explore the parameter space with a limited number of computations of the model. However, the model fitting best the data is not the most interesting because parameter compensation, uncertainties on both models and data make many models with different input parameters as good compared to the data. The sensitivity of the parameters may also be very different depending on the model used. The inversion process is generally the weak point of these ambient vibration methods.

and a digitizer

. The number of seismic stations depends on the method, from single point (spectrum, HVSR) to arrays (3 sensors and more). Three components (3C) sensors are used except in particular applications. The sensor sensitivity and corner frequency depend also on the application. For ground measurements, velocimeters are necessary since the amplitudes are generally lower than the accelerometer

s sensitivity, especially at low frequency. Their corner frequency depends on the frequency range of interest but corner frequencies lower than 0.2 Hz are generally used. Geophone

s (generally 4.5 Hz corner frequency or greater) are generally not suited. For measurements in civil engineering structures, the amplitude is generally higher as well as the frequencies of interest, allowing the use of accelerometers or velocimeters with a higher corner frequency. However, since recording points on the ground may also be of interest in such experiments, sensitive instruments may be needed.

Except for single station measurements, a common time stamping is necessary for all the stations. This can be achieved by GPS clock, common start signal using a remote control or the use of a single digitizer allowing the recording of several sensors.

The relative location of the recording points is needed more or less precisely for the different techniques, requiring either manual distance measurements or differential GPS

location.

or earthquake recordings used in Seismic tomography

.

Limitations of these methods are linked to the noise wavefield but especially to common assumptions made in seismic:

**Ambient Vibrations**on the Earth ground (also called ambient noise). These vibrations are mostly surface waves (Rayleigh waves, Love waveLove wave

In elastodynamics, Love waves are horizontally polarized shear waves guided by an elastic layer, which is "welded" to an elastic half space on one side while bordering a vacuum on the other side...

s) propagating on the surface. Low frequency waves (below 1 Hz) are generally called microseism

Microseism

In seismology, a microseism is defined as a faint earth tremor caused by natural phenomena. The term is most commonly used to refer to the dominant background seismic noise signal on Earth, which are mostly composed of Rayleigh waves and caused by water waves in the oceans and lakes...

s and high frequency waves (above 1 Hz) are called microtremor

Microtremor

Microtremor is a low amplitude ambient vibration of the ground caused by man-made or atmospheric disturbances. The term Ambient Vibrations is now preferred to talk about this phenomenon. Observation of microtremors can give useful information on dynamic properties of the site such as predominant...

s. These ambient vibrations are used in practice to derive the elastic properties of the ground and the low-strain dynamic properties of civil-engineering structures (bridges, buildings, dams...). This information is useful for different purposes : fundamental seismology, engineering seismology, Earthquake engineering

Earthquake engineering

Earthquake engineering is the scientific field concerned with protecting society, the natural and the man-made environment from earthquakes by limiting the seismic risk to socio-economically acceptable levels...

, Seismic microzonation

Seismic microzonation

Seismic microzonation is defined as the process of subdividing a potential seismic or earthquake prone area into zones with respect to some geological and geophysical characteristics of the sites such as ground shaking, liquefaction susceptibility, landslide and rock fall hazard, earthquake-related...

, Structural health monitoring

Structural health monitoring

The process of implementing a damage detection and characterization strategy for engineering structures is referred to as Structural Health Monitoring . Here damage is defined as changes to the material and/or geometric properties of a structural system, including changes to the boundary conditions...

, but also Hydrology

Hydrology

Hydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...

, Geotechnical Engineering, etc.

## Physical origin of the ambient vibrations

Bonnefoy-Claudet et al. reviewed the scientific work studying the origin of the noise wavefield. At low frequency (below 1 Hz), the noise sources are natural and mostly due to ocean waves. In particularthe peak between 0.1 and 0.3 Hz is clearly associated with the interaction of water waves of nearly equal frequencies but opposite directions. At high frequency (above 1 Hz), the wavefield is mainly produced by human activities (road traffic, industrial work...) but there are also natural sources like rivers. Around 1 Hz, the local atmospheric conditions (wind...) are also a major source of ground vibrations. The amplitude of ground ambient vibrations is typically in the range of 1e-6 m, i.e. in the order of the tenth of micrometers to tens of micrometers. Peterson provided high and low noise models as a function of frequency.

The ambient wave field is made of a small amount of body waves

Seismic wave

Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano that imparts low-frequency acoustic energy. Many other natural and anthropogenic sources create low amplitude waves commonly referred to as ambient vibrations. Seismic waves...

(P- and S-waves), and a most generally predominant part of surface wave

Surface wave

In physics, a surface wave is a mechanical wave that propagates along the interface between differing media, usually two fluids with different densities. A surface wave can also be an electromagnetic wave guided by a refractive index gradient...

s, i.e. Love

Love wave

In elastodynamics, Love waves are horizontally polarized shear waves guided by an elastic layer, which is "welded" to an elastic half space on one side while bordering a vacuum on the other side...

and Rayleigh wave

Rayleigh wave

Rayleigh waves are a type of surface acoustic wave that travels on solids. They are produced on the Earth by earthquakes, in which case they are also known as "ground roll", or by other sources of seismic energy such as ocean waves an explosion or even a sledgehammer impact...

s. Theses waves are dispersive

Dispersion relation

In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

, i.e. their phase velocity

Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

varies with frequency (most generally, it decreases with increasing frequency). The dispersion curve

Dispersion relation

In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

(phase velocity or slowness as a function of frequency) is tightly related to the variations of the shear-wave velocity with depth in the different ground layers: it can thus be used as a non-invasive tool to investigate the underground structure.

## History of their use

Ground ambient vibrations have very low amplitudes and cannot be felt by humans. Their amplitude was also too low to be recorded by the first seismometerSeismometer

Seismometers are instruments that measure motions of the ground, including those of seismic waves generated by earthquakes, volcanic eruptions, and other seismic sources...

s at the end of 19th century. However, at that time, the famous Japanese seismologist Fusakichi Omori

Fusakichi Omori

was a pioneer Japanese seismologist, second chairman of seismology at the Imperial University of Tokyo and president of the Japanese Imperial Earthquake Investigation Committee.-Education:...

could already record ambient vibrations in buildings, where the amplitudes are magnified. He found their resonance frequencies

Resonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

and studied their evolution as a function of damage . After the 1933 Long Beach earthquake

1933 Long Beach earthquake

The Long Beach earthquake of 1933 took place on March 10, 1933 at 17:55 PST , with a magnitude of 6.4, causing widespread damage to buildings throughout Southern California. The epicenter was offshore, southeast of Long Beach on the Newport-Inglewood Fault. An estimated fifty million dollars worth...

in California, a large experiment campaign led by Carder in 1935 allowed to record and analyze ambient vibrations in more than 200 buildings. These data were used in the design codes

Seismic analysis

Seismic Analysis is a subset of structural analysis and is the calculation of the response of a building structure to earthquakes...

to estimate resonance frequencies of buildings but the interest of the method went down until the 1950s. The interest on ambient vibrations in structures rose again thanks to famous earthquake engineers, especially in California and Japan (G. Housner

George W. Housner

George W. Housner was an eminent authority on earthquake engineering and National Medal of Science laureate. Housner received his Bachelors degree in Civil Engineering from the University of Michigan where he was influenced by Stephen Timoshenko...

, D. Hudson, K. Kanai and T. Tanaka ...). Ambient vibrations were however supplanted - at least for some time - by forced vibration techniques that allow to increase the amplitudes and control the shaking source and their system identification methods. Even if Trifunac showed as early as 1972 that ambient and forced vibrations led to the same results, the interest in ambient vibration techniques rose again only in the late 1990s. The relatively low-cost and easiness of implementation, the improvement of the recording material and of the computation opportunities make these techniques very popular nowadays, especially as the low-strain dynamic characteristics they provide were shown to be close enough to the measured dynamic characteristics under strong shaking, at least as long as the buildings are not severely damages .

The use of noise recordings on the ground started in the 1950s with the enhancement of seismometers to monitor nuclear tests and the development of seismic arrays. The main contributions at that time for the analysis of these recordings came from the Japanese seismologist K. Aki

Keiiti Aki

was a professor of Geophysics at the Massachusetts Institute of Technology , seismologist, author and mentor. He co-authored with Paul G. Richards, "Quantitative Seismology: theory and methods".Aki was born in Yokohama, Japan...

in 1957 who first proposed the methods used nowadays (Spatial Autocorrelation method -SPAC-, Frequency-wavenumber -FK- method, correlation method...). However, the practical implementation of these methods was not possible at that time because of the low precision of clocks in seismic stations. Again, the opportunities of computations and the enhancements in the recording material led to a rise of interest in the 1990s. The first widely implemented method, rediscovered by Nakamura in 1989 is the Horizontal to Vertical Spectral Ratio (H/V) method to derive the resonance frequency of sites. Assuming that the shear wave dominates the microtremor, Nakamura indicated that the H/V spectral ratio of ambient vibrations were roughly equals the S-wave transfer function between the ground surface and the bedrock at a site. This assumption is however now criticized in the literature (e.g. SESAME project).

In the late 1990s , the array methods on ambient vibration data started to allow deriving the ground properties in terms of shear waves velocity profiles. The European Research project SESAME http://sesame-fp5.obs.ujf-grenoble.fr/index.htm (2004–2006) was one of the first structured attempts to standardize the use of ambient vibrations to retrieve the properties of the ground, in the aim of estimating site amplifications in case of earthquake (site effects).

### Characterization of the ground properties

The analysis of the ambient vibrations leads to different products used to characterize the ground properties. From the easiest to the most complicated, these products are: power spectraSpectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

, H/V peak, dispersion curves

Dispersion relation

In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

and autocorrelation functions

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

.

**Single-station methods:**- Computation of power spectraSpectral densityIn statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

, e.g. Passive seismicPassive seismicPassive seismic is the detection of natural low frequency earth movements, usually with the purpose of discerning geological structure and locate underground oil, gas, or other resources. Usually the data listening is done in multiple measurement points that are separated by several hundred meters,...

. - HVSR (H/V spectral ratio): The H/V technique is especially related to ambient vibration recordings. Bonnefoy-Claudet et al. showed that peaks in the horizontal to vertical spectral ratios can be linked to the RayleighRayleigh waveRayleigh waves are a type of surface acoustic wave that travels on solids. They are produced on the Earth by earthquakes, in which case they are also known as "ground roll", or by other sources of seismic energy such as ocean waves an explosion or even a sledgehammer impact...

ellipticity peak, the Airy phase of the Love waveLove waveIn elastodynamics, Love waves are horizontally polarized shear waves guided by an elastic layer, which is "welded" to an elastic half space on one side while bordering a vacuum on the other side...

s and/or the SHS-waveA type of seismic wave, the S-wave, secondary wave, or shear wave is one of the two main types of elastic body waves, so named because they move through the body of an object, unlike surface waves....

resonance frequencies depending on the proportion of these different types of waves in the ambient noise. By chance, all these values give however approximately the same value for a given ground so that H/V peak is a reliable method to estimate the resonance frequency of the sites. For 1 sediment layer on the bedrock, this value f_{0}is related to the velocity of S-waves V_{s}and the depth of the sediments H following: . It can therefore be used to map the bedrock depth knowing the S-wave velocity. This frequency peak allows to constrain the possible models obtain using other seismic methods but is not enough to derive a complete ground model. Moreover, it has been shown that the amplitude of the H/V peak was not related to the magnitude of the amplification.

**Array methods:**Using an array of seismic sensors recording simultaneously the ambient vibrations allow to understand more deeply the wavefield and therefore to derive more properties of the ground. Due to the limitation of the available number of sensors, several arrays of different sizes may be realized and the results merged.

The information of the Vertical components is only linked to the Rayleigh waves, and therefore easier to interpret, but method using the 3 space components are also developed, providing informations about Rayleigh and Love wavefield.

- FK, HRFK
- SPAC method
- Correlations methods
- Refraction microtremor ReMIRefraction microtremorRefraction microtremor is a surface-performed geophysical survey developed by Dr. John Louie based on previously existing principles of evaluating surface waves and in particular Rayleigh waves...

### Characterization of the vibration properties of civil engineering structures

Like earthquakeEarthquake

An earthquake is the result of a sudden release of energy in the Earth's crust that creates seismic waves. The seismicity, seismism or seismic activity of an area refers to the frequency, type and size of earthquakes experienced over a period of time...

s, ambient vibrations force into vibrations the civil engineering structures like bridge

Bridge

A bridge is a structure built to span physical obstacles such as a body of water, valley, or road, for the purpose of providing passage over the obstacle...

s, building

Building

In architecture, construction, engineering, real estate development and technology the word building may refer to one of the following:...

s or dam

Dam

A dam is a barrier that impounds water or underground streams. Dams generally serve the primary purpose of retaining water, while other structures such as floodgates or levees are used to manage or prevent water flow into specific land regions. Hydropower and pumped-storage hydroelectricity are...

s. This vibration source is supposed by the greatest part of the used methods to be a white noise

White noise

White noise is a random signal with a flat power spectral density. In other words, the signal contains equal power within a fixed bandwidth at any center frequency...

, i.e. with a flat noise spectrum so that the recorded system response is actually characteristic of the system itself. The vibrations are perceptible by humans only in rare cases (bridges, high buildings). Ambient vibrations of buildings are also caused by wind and internal sources (machines, pedestrians...) but these sources are generally not used to characterize structures.

The branch that studies the modal properties of systems under ambient vibrations is called Operational modal analysis

Modal analysis

Modal analysis is the study of the dynamic properties of structures under vibrational excitation.Modal analysis is the field of measuring and analysing the dynamic response of structures and or fluids when excited by an input...

or Output-only modal analysis

Modal analysis

Modal analysis is the study of the dynamic properties of structures under vibrational excitation.Modal analysis is the field of measuring and analysing the dynamic response of structures and or fluids when excited by an input...

and provides many useful methods for civil engineering

Civil engineering

Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including works like roads, bridges, canals, dams, and buildings...

.

The observed vibration properties of structures integrate all the complexity of these structures including the load-bearing system, heavy and stiff non-structural elements (infill masonry panels...), light non-structural elements (windows...) and the interaction with the soil

Soil structure interaction

Most of the civil engineering structures involve some type of structural element with direct contact with ground. When the external forces, such as earthquakes, act on these systems, neither the structural displacements nor the ground displacements, are independent of each other...

(the building foundation may not be perfectly fixed on the ground and differential motions may happen). This is emphasized because it is difficult to produce models able to be comparable with these measurements.

**Single-station methods:**The power spectrum

Spectral density

In statistical signal processing and physics, the spectral density, power spectral density , or energy spectral density , is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time, which has dimensions of power per hertz...

computation of ambient vibration recordings in a structure (e.g. at the top floor of a building for larger amplitudes) gives an estimation of its resonance frequencies and eventually its damping ratio

Damping ratio

[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

.

**Transfer function method:**Assuming ground ambient vibrations is the excitation source of a structure, for instance a building, the Transfer Function

Transfer function

A transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system. With optical imaging devices, for example, it is the Fourier transform of the point spread function i.e...

between the bottom and the top allows to remove the effects of a non-white input. This may particularly be useful for low signal-to-noise ratio

Signal-to-noise ratio

Signal-to-noise ratio is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. It is defined as the ratio of signal power to the noise power. A ratio higher than 1:1 indicates more signal than noise...

signals (small building/high level of ground vibrations). However this method is not able to remove the effect of soil-structure interaction.

**Arrays:**They consist in the simultaneous recording in several points of a structure. The objective is to obtain the modal parameters of structures: resonance frequencies

Resonance

In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

, damping ratio

Damping ratio

[[Image:Damped spring.gif|right|frame|Underdamped [[spring–mass system]] with ζ 1 , and is referred to as overdamped.*Underdamped:If s is a complex number, then the solution is a decaying exponential combined with an oscillatory portion that looks like \exp...

s and modal shapes for the whole structure. Notice than without knowing the input loading, the participation factors of these modes cannot a priori be retrieved. Using a common reference sensor, results for different arrays can be merged.

- Methods based on correlations

Several methods use the power spectral density

Spectral density

matrices of simultaneous recordings, i.e. the cross-correlation

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...

matrices of these recordings in the Fourier domain

Frequency domain

In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

. They allow to extract the operational modal parameters (Peak Picking method) that can be the results of modes coupling or the system modal parameters (Frequency Domain Decomposition method).

- System identificationSystem identificationIn control engineering, the field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data...

methods

Numerous system identification methods exist in the literature to extract the system properties and can be applied to ambient vibrations in structures

## Inversion/Model updating/multi-model approach

The obtained results cannot directly give information on the physical parameters (S-wave velocity, structural stiffness...) of the ground structures or civil engineering structures. Therefore modelComputer simulation

A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...

s are needed to compute these products (dispersion curve, modal shapes...) that could be compared with the experimental data. Computing a lot of models to find which agree with the data is solving the Inverse problem

Inverse problem

An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...

. The main issue of inversion is to well explore the parameter space with a limited number of computations of the model. However, the model fitting best the data is not the most interesting because parameter compensation, uncertainties on both models and data make many models with different input parameters as good compared to the data. The sensitivity of the parameters may also be very different depending on the model used. The inversion process is generally the weak point of these ambient vibration methods.

## Material needed

The acquisition chain is mainly made of a seismic sensorSeismometer

Seismometers are instruments that measure motions of the ground, including those of seismic waves generated by earthquakes, volcanic eruptions, and other seismic sources...

and a digitizer

Digitizing

Digitizing or digitization is the representation of an object, image, sound, document or a signal by a discrete set of its points or samples. The result is called digital representation or, more specifically, a digital image, for the object, and digital form, for the signal...

. The number of seismic stations depends on the method, from single point (spectrum, HVSR) to arrays (3 sensors and more). Three components (3C) sensors are used except in particular applications. The sensor sensitivity and corner frequency depend also on the application. For ground measurements, velocimeters are necessary since the amplitudes are generally lower than the accelerometer

Accelerometer

An accelerometer is a device that measures proper acceleration, also called the four-acceleration. This is not necessarily the same as the coordinate acceleration , but is rather the type of acceleration associated with the phenomenon of weight experienced by a test mass that resides in the frame...

s sensitivity, especially at low frequency. Their corner frequency depends on the frequency range of interest but corner frequencies lower than 0.2 Hz are generally used. Geophone

Geophone

The term geophone derives from the Greek word "geo" meaning "earth" and "phone" meaning "sound".A geophone is a device which converts ground movement into voltage, which may be recorded at a recording station...

s (generally 4.5 Hz corner frequency or greater) are generally not suited. For measurements in civil engineering structures, the amplitude is generally higher as well as the frequencies of interest, allowing the use of accelerometers or velocimeters with a higher corner frequency. However, since recording points on the ground may also be of interest in such experiments, sensitive instruments may be needed.

Except for single station measurements, a common time stamping is necessary for all the stations. This can be achieved by GPS clock, common start signal using a remote control or the use of a single digitizer allowing the recording of several sensors.

The relative location of the recording points is needed more or less precisely for the different techniques, requiring either manual distance measurements or differential GPS

Differential GPS

Differential Global Positioning System is an enhancement to Global Positioning System that provides improved location accuracy, from the 15-meter nominal GPS accuracy to about 10 cm in case of the best implementations....

location.

## Advantages and limitations

The advantages of ambient vibration techniques compared to active techniques commonly used in exploration geophysicsExploration geophysics

Exploration geophysics is the applied branch of geophysics which uses surface methods to measure the physical properties of the subsurface Earth, in order to detect or infer the presence and position of ore minerals, hydrocarbons, geothermal reservoirs, groundwater reservoirs, and other geological...

or earthquake recordings used in Seismic tomography

Seismic tomography

Seismic tomography is a methodology for estimating the Earth's properties. In the seismology community, seismic tomography is just a part of seismic imaging, and usually has a more specific purpose to estimate properties such as propagating velocities of compressional waves and shear waves . It...

.

- Relatively cheap, non-invasive and non-destructive method
- Applicable to urban environment
- Provide valuable information with little data (e.g. HVSR)
- Dispersion curve of Rayleigh wave relatively easy to retrieve
- Provide reliable estimates of Vs30

Limitations of these methods are linked to the noise wavefield but especially to common assumptions made in seismic:

- Penetration depth depends on the array size but also on the noise quality, resolution and aliasing limits depend on the array geometry
- Complexity of the wavefield (Rayleigh, Love waves, interpretation of higher modes...)
- Plane wave assumption for most of the array methods (problem of sources within the array)
- 1D assumption of the underground structure
- Inverse problem difficult to solve