Florian Lambauer, summer semester 2023


Coda wave interferometry (CWI) is an ultrasound technique for measuring state changes of a test object by the evaluation of so-called coda waves [1]. In this context, coda means "tail" in Latin and accordingly coda waves are the late scattered part of an ultrasound signal after the directly arriving phases [2,3].

a)

b)

Figure 1: (a) Measurement of coda waves in a cylindrical concrete test specimen using piezoelectric ultrasound transducers (transmitter on the left, receiver on the right)
(b) Dotted illustration of possible paths of a scattered ultrasound wave in the concrete cylinder shown in the half section [inspired by [18]]

Figure 2: Ultrasound signal over a millisecond measurement duration before and after a perturbation, recorded by the piezoelectric ultrasound sensor on the concrete cylinder from Figure 1 [inspired by [4,19,30,38,39]]

Introduction

Due to heterogeneities in the materials, an applied ultrasonic stress wave is scattered several times during propagation in a test object. The highly repeatable diffuse part of the recording of these scattered ultrasound waves is called the coda wave. These coda waves are sensitive to changes in the subsurface and can therefore provide information on state changes in a measured object using a suitable technique. Since this method measures the variation and interference of ultrasound waves in a test object, it is called coda wave interferometry (CWI). [1,4]
The CWI can be used to measure changes in temperature [5-7], relative humidity [8-11] and mechanical loading [4,12-16] in heterogeneous objects and to detect the formation of microcracks and fissures at an early stage [4,5,12-16]. Therefore, this technique is primarily used in geosciences [2,8,9] and in the non-destructive testing (NDT) of buildings [4,5,12-17].

Theoretical basics of CWI

The scattering of ultrasound waves in a heterogeneous test object generates coda waves [1]. As long as there are no changes in the test object or in the measurement setup, the coda waves will repeatedly follow the same travel paths in the object and scatter consistently [1]. Therefore, despite their chaotic appearance, coda waves were found to be highly reproducible and deterministic [18]. 
For spatially localized changes in propagation velocity or for changes in the source location, the travel-time perturbation may be different for different scattering paths. The coda waves that arrive within a certain time window are therefore subject to a distribution of travel-time perturbations. [1]
In order to quantify changes in the test object with CWI, the relative velocity change of the coda waves is estimated using several ultrasonic measurements before and after a perturbation of the test object. This relative velocity change can then be utilized to determine the extent of the change in state in the test object. [19]
However, different influencing factors also have different effects on CWI. For example, the variation of temperature has an approximately linear effect on the velocity variation of ultrasonic signals [5,7], while the relative humidity shows nonlinear effects [20]. Therefore, a trivial conclusion on an exact state of the test object based on a CWI is not possible.

Fundamentals: Path summation and change in the waves

The theory of CWI is based on path summation of the scattering paths P, where the wavefield u(t) at a given location is described as the sum of waves that propagate along all possible paths S_P(t) [1]

$u\left(t\right)=\sum_P{S_P\left(t\right)}.$

Several waveforms u_i(t) can be compared with each other within the same time window. Thereby, the selected time window restricts the evaluated paths of the ultrasound waves and should therefore be adapted to the time range of the coda waves. As long as the resulting coda waves are measured identically, variations between two waveforms indicate changes in the medium. If there are no changes in the scattering properties in the medium, the variations can be attributed to shifts in the scatterers, e.g., a change in the path length or the formation of new scatterers in the medium. Thus, the equation for the perturbed wavefield

$u_{\mathrm{perturbed}}\left(t\right)=\sum_P{S_P\left(t-{\tau }_P\right)}$

results from a travel-time perturbation {\tau }_P, which is caused by a change in phase. This theory is valid as long as the mean free path length of the scatterer is significantly larger than the wavelength. [1] 
Therefore, the greater the changes in the medium, the greater the change in the scattering of the ultrasound waves in the test medium [1,4]. Consequently, this differing scattering will result in a measurable change in the travel time of the coda waves.

Data processing procedures: Doublet techniques and stretching technique

The magnitude of the time lag between two different coda wave signals before and after a possible perturbation can be estimated by applying a time-shifted correlation coefficient

$CC\left(t_s\right)=\frac{\int^{t_2}_{t1}{u\left(t\right)\cdot u_{\mathrm{perturbed}}\left(t+t_s\right)\textrm{ⅆ}t}}{\sqrt{\int^{t_2}_{t_1}{u^2(t)\textrm{ⅆ}t}\cdot \int^{t_2}_{t_1}{u^2_{\mathrm{perturbed}}\left(t+t_s\right)\textrm{ⅆ}t}}},$

where t_1 and t_2 define a selected time window in the coda wave and t_s is the time-shift of the unperturbed and perturbed waves in the correlation [1,4].
The time-shifted correlation coefficient was developed considering doublet techniques which are able to measure temporal variations of crustal velocities and computing differences in arrival times [19,21,22]. In this processing technique the velocity variation is the slope of the mean travel time perturbation {\tau }_P [1,19]

$dv\ /v\ =\ -{\tau }_P/t.$

This assumes that the time-shift t_s is constant within the considered time window, which might be not the case [19].
Greater stability is achieved with the so-called stretching technique. This technique is a data processing procedure that interpolates the coda waves at times t(1-\varepsilon ) with various relative velocity changes \varepsilon = dv\ / v in the [t_1;t_2] time window. This corresponds to stretching the time axis and gives the stretching technique its name. The actual velocity change \varepsilon is obtained when the interpolated coda wave best fits the original data, and therefore, a maximum correlation

$CC\left(\varepsilon \right)=\frac{\int^{t_2}_{t1}{u\left(t\right)\cdot u_{\mathrm{perturbed}}(t\left(1-\varepsilon \right))\textrm{ⅆ}t}}{\sqrt{\int^{t_2}_{t_1}{u^2(t)\textrm{ⅆ}t}\cdot \int^{t_2}_{t_1}{u^2_{\mathrm{perturbed}}(t\left(1-\varepsilon \right))\textrm{ⅆ}t}}}$

is reached. Because there is no assumption of a constant time shift in the considered time window, the whole data can be processed at once, which results in a more stable and precise estimation of the relative velocity change dv\ /v\ (=\varepsilon ). One drawback is that this processing assumes a constant relative velocity change \varepsilon, which is not the case in some complex heterogeneous media. [19]
Nevertheless, due to its stability and sensitivity, this technique is applied in various non-destructive testing procedures [20,23].

Experimental setup and measurements

The measurement of coda waves is usually performed by the usage of piezoelectric ultrasonic transducers. Depending on the measurement configuration, the transmitter and the receiver are attached to the test specimen, while a firm and constant contact with the body must be ensured. The transmitter and receiver do not necessarily have to be on opposite sides and can sometimes consist of only a single transducer that both transmits and measures. When measuring seismological processes, usually a natural vibration is used instead of an actively controlled transmitter. After connecting the transducers to a measuring device and applying an ultrasound signal with a desired frequency and amplitude, the measuring device measures the decaying vibration response via the receiver. An exemplary practical setup for the measurement of coda waves in non-destructive testing is shown in Figure 1 and the measured ultrasonic response is illustrated in Figure 2.

Areas of application

Non-destructive testing

It can be challenging to perform conventional ultrasound test methods for the examination of concrete structures, because the variations in time of flight or in the propagation velocity are minute and difficult to measure [18,24]. This means that changes can often only be measured at mechanical loads above the cracking stress of concrete, which disqualifies these methods for many applications [4]. However, the later diffuse part of the ultrasonic waves can increase the measurement accuracy due to the longer travel times [4]. This makes the non-invasive CWI suitable for inspecting complex structures [4].
In non-destructive testing (NDT) the CWI can be used to assess the structural integrity of materials or to monitor changes in their properties over time. It is particularly useful for detecting and characterizing damage, such as cracks or corrosion, in various building structures. By analyzing the interferometric patterns in the coda waves, it is possible to identify the position and the extent of the defect. Additionally, CWI can provide information about the nature and severity of the damage, helping in the characterization of the structural integrity of the material. [4,17,25]
Thereby, CWI can be used for both static and dynamic monitoring of materials, allowing for continuous detection and assessment of changes over time. This enables, for example, the constant monitoring of the deterioration of structures such as bridges and can be integrated into early warning systems. [4,16-18,24,25,27-29]

Earthquake research and risk assessment

Plate displacement generates vibrations that can be measured and recorded at seismometer stations. With the help of these measured coda waves, the extent and distribution of the energy release during an earthquake can be quantified. This makes it possible to create detailed models of seismic activity and stress distribution in the underground. This information is important for understanding earthquake dynamics and predicting future earthquakes. [2,8,9,30]

Evaluation of soil conditions and exploration of oil deposits

Different geological materials cause different patterns of coda waves during an seismic measurement [31,32]. This makes it possible to analyze the nature of the subsurface using CWI [32]. For example, valuable soil information such as stiffness, bearing capacity, attenuation and creep behavior can be determined without altering the soil structure [31,33]. This information is important for the construction industry, as it is needed in the design of structures such as high-rise buildings or bridges [32]. In addition, due to its accuracy, CWI allows an accurate evaluation of potential oil fields and supports decision making in the placement of wells [31,34,35].

Exploration of geothermal resources

CWI is a high-resolution technique which can be used to locate deep geothermal reservoirs [36]. Thereby, the measurement of the coda wave velocity change can be utilized to provide a quantitative interpretation about the reservoir and its temperature [36]. Furthermore, geophysicists can obtain information about the properties of the rock layers, such as porosity and permeability, to determine appropriate locations for geothermal plants [29,31,36]. These evaluations enable an accurate characterization of potential deep geothermal reservoirs and a continuous seismic monitoring for the optimization of geothermal plants [29,31,36].

Measurement of volcanic activity

By analyzing coda waves, seismologists can obtain information about the rise of magma, the development of magma chambers, and other volcanic processes. This information is crucial for the prediction of volcanic eruptions and the monitoring of potentially endangered areas. [37]

Current research topics

An ongoing issue is to minimize the interfering signals and the measurement noise that limit the evaluation of low-amplitude coda waves. Besides this, one of the biggest challenges for CWI today is the simultaneous occurrence of condition changes that cause superimposed terms contributing to the velocity variation. [20,24]

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