Comparison of arrival time estimation methods

In order to accurately estimate the wall thickness of a sample, accurate arrival time estimation is key. Here techniques that are typically used for arrival time estimation are discussed. Their implementation is also described, and the repeatability of estimated wall thicknesses using each method is then compared in simulations.

3. Uncertainty in Arrival Time Determination (Signal Processing)

Time [µs]

Amplitude[V]

190 195 200 205 210 215

-0.1

Figure 3.2: The figure shows the Hilbert-envelope (green line) of a sample waveform (black line) and its estimated arrival times using P2P (green vertical lines).

Red horizontal lines show the calculated thresholds for each wavepacket. FA arrival times calculated based on the intersection of thresholds (6dB here) and the Hilbert envelope are also displayed (red vertical lines).

A commonly used arrival time estimation concept is based on calculating an envelope function for the waveform [39]. These methods are often referred to as Envelope Peak or Peak-to-Peak (P2P) methods. In this thesis the P2P notation is used throughout to avoid confusion. Some P2P methods use the raw signal, but most rely on computing an envelope function for the measured waveform. This ignores phase information and is believed to be more robust. One way to achieve this is via the Hilbert-transform. The Hilbert transform applies a 90^o phase shift to all frequency components of the signal [40], hence it can be used to calculate the envelope of the signal by:

E(t) = q

f (t)²+ H (f (t))² (3.1)

where f (t) is the function for which the envelope is calculated, H (f (t)) is the Hilbert transform of f (t) and E(t) is the computed envelope. A sample waveform along with its Hilbert-envelope is shown in Figure 3.2. Other methods exist that compute a similar envelope function, such as filtering using finite impulse response (FIR) filters [41, 42]. The result however is fundamentally the same as a

band-pass filter and the Hilbert-transform, namely an envelope function with a limited

3. Uncertainty in Arrival Time Determination (Signal Processing)

Time [µs]

Amplitude[V]

190 195 200 205 210 215

-0.1

Figure 3.3: The figure shows the cross-correlation function (blue line) of a sample waveform (black line) with a synthetised 180^o toneburst and the estimated arrival times of that waveform using XC (blue vertical lines).

frequency spectrum, where the phase information is ignored. Based on the computed envelope, the arrival times of the various wavepackets are estimated by determining the maximum peaks of the envelope function. P2P methods are simple to implement.

The calculation of the envelope is straightforward and since typically only a small number of peaks are present in it, their identification is easy and can be robustly automated.

First-Arrival (FA) is another method to estimate arrival times of wavepackets [11]. It also relies on calculating an envelope function and finding the peaks of that function.

FA then establishes a threshold as a function of the amplitude of each peak - e.g. the threshold for the each wavepacket is determined as -6 dB of its amplitude. Since for each wavepacket this threshold is determined independently, the threshold for each wavepacket will be different. The crossing of this threshold and the envelope function is then taken as the arrival time for a given wavepacket. This is shown in Figure 3.2 directly compared to a P2P arrival times. Since FA is based on an envelope as well, it is only marginally more complex to implement than P2P methods.

Cross-correlation (XC) is another popular arrival time estimation method [11, 39].

Cross-correlation is defined as the sliding dot product of a function with the complex conjugate of another. In the context of arrival time estimation, cross-correlation

3. Uncertainty in Arrival Time Determination (Signal Processing)

behaves as a form of similarity metric between two signals as a function of time-lag.

This can be used for arrival time estimation by cross-correlating the synthesised toneburst that is transmitted at the beginning of a measurement with the received signal. An example of a correlation function is shown in Figure 3.3. The peaks of the resulting cross-correlation function therefore represent the times where the two functions correlate well. The arrival times of the wavepackets in the waveform are then estimated by determining the time of the biggest peaks in the cross-correlation function.

A difficulty that may arise during this process is that in some cases the received wavepackets may be distorted compared to the sent toneburst. Phase shifts in the received signal are a typical form of distortion, caused by characteristics of the transducer itself, coupling to the sample and reflections from various interfaces in the sample. It is assumed here that such shifts do not occur after a given transducer has been coupled and so repeatability is assumed to be unaffected.

Beyond these three fundamental methods, there have been reports suggesting that combining some of them may result in more accurate arrival time estimation al-gorithms. Yu [39] proposes to estimate arrival times by calculating the envelope function of the received waveform and cross-correlating it with the envelope function of the sent toneburst. In the report by Yu [39], it was recognised that this approach was chosen since the standard P2P method appeared to be unstable because of what was described as dispersion effects. In this thesis it is assumed that signals are dispersion-free, in which case such an approach is not expected to offer any advantages.

In addition, other more complex methods also exist that can be used for arrival time estimation. The wavelet transform is one of the many possible methods that can be used for feature detection and arrival estimation [43]. Its potential has been recognised for defect detection, since it can be used to describe the temporal distribution of harmonic components [43]. However, such a method offers limited advantages in this thesis, since here the frequency spectrum of the reflection of ultrasonic waves is well defined and not expected to change. Since only the temporal shift of wavepackets is to be determined in this thesis, cross-correlation is expected to estimate arrival times with similar accuracy, hence the wavelet transform is not investigated in this

3. Uncertainty in Arrival Time Determination (Signal Processing)

thesis. The split-spectrum processing (SSP) method is another possible candidate for arrival time estimation [43]. SSP is also based on multi-frequency analysis, and therefore SSP is also omitted from this investigation for similar reasons.

Three arrival time estimation methods - P2P, FA and XC - were evaluated to find which method is capable of determining arrival time with the highest precision. The chosen methodology for this is as follows: a signal is simulated using the previously presented DPSM method by transmitting a 2 MHz 5 cycle Hann-windowed toneburst that has been reported to work well with the waveguide transducer [25] into a sample with a uniform wall thickness of 10 mm. This signal is simulated at 512 MHz sampling rate. No quantisation error is introduced (other than what is implicit in a double precision variable), therefore ideal sampling resolution is assumed. It is expected that the random noise level for a single raw experimentally measured waveform using a waveguide is approximately -52 dB, therefore -52 dB white Gaussian noise is added onto the simulated signals. Here, signal to noise ratio is interpreted as the ratio of the standard deviation of random noise and the maximum amplitude of the signal.

The noiseless simulated waveform and the signal with added noise are shown in Figure 3.4.

The simulated signal is then filtered and averaged 320 times following the outlined

Time [µs]

Amplitude[arb]

0 2 4 6 8 10 12

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

Figure 3.4: Simulated waveform using DPSM (green line). Signal with added

−52 dB white Gaussian noise (blue line).

3. Uncertainty in Arrival Time Determination (Signal Processing)

signal processing protocol. Since it is the repeatability of arrival times that is of interest, altogether 100 averaged waveforms are simulated using this protocol. These 100 waveforms are then evaluated using all three arrival time estimation methods. It should be noted that although waveforms are simulated with parameters that are characteristic of the waveguide transducer (i.e. characteristic SNR and transducer geometry), it is thought that the comparison of arrival time estimation methods is not specific to this transducer and the comparison is expected to apply to other transducers using a different geometry with signal to noise ratios of similar order of magnitude.

The distribution of calculated thicknesses for all three methods are shown in Figure 3.5 and their standard deviations are shown in Table 3.1. It is clear that the variability of XC is approximately an order of magnitude lower than any other method, which is as expected because of its better noise suppression compared to other considered methods. Because of this, XC is used as the default arrival time estimation method throughout this thesis.

In addition, it should be noted here that the absolute accuracy of all methods appears to be comparable. The mean thickness calculated in simulations for XC is 10.065 mm, for P2P it is 10.123 mm and for FA it is 10.175 mm. It is expected that the small differences come from small biases of each method, which may possibly be compensated. This is not considered here, since this performance evaluation is

Thickness distribution around the mean [nm]

Numberofmeasurements

Figure 3.5: Distribution of calculated thicknesses using all three signal processing methods for simulated signals. XC thicknesses are shown in blue, P2P results are shown in green and FA results are shown in red. All distributions are shown for each method around their corresponding mean calculated thickness.

3. Uncertainty in Arrival Time Determination (Signal Processing)

STD of arrival time [ps] STD of thickness [nm]

P2P FA XC P2P FA XC

53.9 51.6 3.63 88.1 84.4 5.95

Table 3.1: Standard deviations of time differences between backwall echo and surface wavepacket in the left three columns. In the right three columns, the standard deviations of thicknesses are shown. All values calculated for a simulated plate with a wall thickness of 10 mm and an ultrasonic velocity of 3250 m/s

aimed solely at improving repeatability, and since XC offers the best repeatability, no further effort is put into evaluating other methods.