# Pulse Peak Delay-Total Focusing Method for Ultrasonic Tomography on Concrete Structure
## 2 Pulse Peak Delay Calibration
### Pulse Peak Delay-Total Focusing Method  
The visualization of the inside of a concrete structure requires a data set of signals from diverse wave paths, and the quality of the image increases as much as the diversity of the wave paths. The monostatic system, which was widely used in the past, shifts the location of a single transducer while transmitting and receiving an ultrasonic pulse simultaneously [24]. This method lacks diversity in the wave path and presents limitation in the quality of the resulting image. The ultrasonic array device A1040 MIRA used in this research is a bistatic system that measures wave signals from all possible wave paths except for the identical transducer pair [25], as shown in Figure 2a. The measuring method for the bistatic system is called full matrix capture (FMC) [26], and the measured signal data set is called full matrix data [27]. The MIRA device collects 66 time-domain signals with a single scan by using 12 sets of transducers.  
The collected full matrix data is almost impossible to be understood intuitively and visually as it is defocused. Therefore, an image reconstruction algorithm is required to convert the full matrix data into a focused image. An image reconstruction algorithm used by the MIRA device is the SAFT-C algorithm of which detailed information is undisclosed. But, it has been known that this algorithm is a delay-and-sum method. One popular image reconstruction algorithm using the delay-and-sum method for the FMC is the TFM, and the proposed method PPD-TFM (pulse peak delay-TFM) modifies the TFM.  
Figure 1: Frequency response spectrum and input/output signal of DPC (Dry Point Contact).  
Figure 2: (a) Ultrasonic array system and scanning scheme and (b) array geometry.  
The array geometry in the wave field is described in Figure 2b. The TFM uses full matrix data as input data, and the input data \(e\) has a 3-dimensional form as described in Equation (1):  
\[e(t,x_{T},x_{R})=\iint_{\infty}^{\infty}f(x,z)\delta(t-t_{d})\mathrm{d}x \mathrm{d}z, \tag{1}\]  
where \(t\) is the time; \(x_{T}\) and \(x_{R}\) are the transmitter and the receiver, respectively; \(f\) is the unknown wave field; \(x\) and \(z\) are the horizontal and the vertical coordinates at each pixel, respectively; \(\delta\) is the Dirac delta function; and \(t_{d}\) is the delayed time, as follows.  
\[t_{d}=\frac{\sqrt{(x-x_{T})^{2}+z^{2}}+\sqrt{(x-x_{R})^{2}+z^{2}}}{c}, \tag{2}\]  
where \(c\) is the wave speed.  
The TFM estimates the unknown wave field by superposing the signals from all transducer pairs at the delayed time to synthesize a focus at every point in the grid. The estimated wave field \(\hat{f}\) is given by  
\[\hat{f}(x,z)=\sum_{x_{T}=1}^{N_{e}-1}\sum_{x_{R}=x_{T}+1}^{N_{e}}e(t_{d},x_{T},x_{R}), \tag{3}\]  
where \(N_{e}\) is the number of array elements.  
The PPD-TFM has a procedure identical to the TFM except for the delayed time. The delayed time for the PPD-TFM, \(t_{d^{\prime}}=t_{d}+PPD\), includes an additional term, the pulse peak delay, as shown in Figure 1b. This value is calculated by the simulated band-pass filter and the input signal. Although little is known regarding the SAFT-C algorithm, it is known that the SAFT-C algorithm crops a resulting image as much as a delay predetermined by the speed and the frequency of an ultrasonic pulse to compensate the sagging phenomenon by the PPD. While the SAFT-C algorithm provides enough accuracy with a simple method to interpret results, the result of the SAFT-C is not the most accurate result as much as the MIRA device can provide. The PPD-TFM algorithm considers all possible information to simulate the output signal and focuses images at the strongest point in the signal to increase the performance.