Ultrasonic Synthetic Aperture Focusing Using the Root-Mean-Square Velocity

When applying a single probe pulse-echo ultrasonic NDT to scan a point reflector in water, the B-scan, i(t,x), shows itself as a hyperbola in the image due to spreading of the ultrasound from the transducer (Fig. 1(a)). The idea of SAFT is to form a normal incident view from the probe and refocus the hyperbolic echoes into a high energy concentration image, such as i′(t ₀,x ₀) in Fig. 1(b).

Since the implementation of SAFT in the time domain is simple and direct, we will first illustrate the SAFT in the time domain for a point reflector in water. Figure 2 shows that a point reflector is scanned by immersion testing. The probe is positioned at x ₀, which is right above the flaw by a distance of d. The scan interval in the lateral direction is Δx. The two-way propagation time (t ₀) for the ultrasound to propagate from the probe to the point reflector can be expressed as where V is the velocity of the longitudinal wave of water. If the probe is not right on the top of the point reflector, the time-distance relationship of the ultrasound then becomes where nΔx is the horizontal distance measured from x ₀ to the current probe position; l is the distance between the current probe position and point reflector (Fig. 2 (a)). If we square both sides of Eq. (2) and incorporate it with Eq. (1), then Eq. (2) becomes The above equation shows a hyperbolic trajectory of the time-distance relation. Then, we can find the delay (Δt _n ) for the neighboring time histories with lateral distance nΔx, which is The SAFT image thus can be obtained by summing the delayed time histories of the scan, mΔx is the maximum lateral distance where the waves diffracted from the point reflector can be detected by the probe. Although the SAFT is based on the ultrasound diffracted from a point reflector, fortunately, an extended reflector can be thought of as a number of point sources and so SAFT work equally well in such cases. Assuming a scan environment where the vertical distance between the probe and point reflector is 1.5 cm and the velocity of the longitudinal wave of water is 1500 m/s (Fig. 2(a)), the time-distance relationship of the ultrasound diffracted from the point reflector calculated by Eq. (3) is shown in Fig. 2(b). Since the time-distance relationship is symmetrical with respect to the x ₀, only the positive nΔx is shown in Fig. 2(b).

Since the velocity and density of water are much lower than those of the solid specimens tested in ultrasonic NDT, the refraction effect will be stronger for the ultrasound propagating through the water-specimen interface. Therefore, the water-specimen medium is an extreme case of two-layer media for the refraction effect. This provides a good opportunity to evaluate the proposed method.

While scanning a point reflector inside a specimen during immersion testing, the ultrasound is refracted when propagating through the water-specimen interface. In this case, the vertical two-way propagation time should be expressed as where, d ₁ is the vertical distance between the probe and the specimen, d ₂ is the depth of the point reflector measured from the water-specimen interface, and V ₁ and V ₂ are the velocities of the longitudinal wave of water and specimen (Fig. 3(a)). When the probe is moved to a scan position at distance nΔx from x ₀, the two-way propagation time can be expressed as where l ₁ and l ₂ are the ray paths in water and the specimen, respectively (Fig. 3(a)). It may be cumbersome to determine the true root from the four roots by solving the Eq. (7) (a quartic equation) directly. Therefore, the commonly used method for obtaining t _n is the trial-and-error ray tracing method which shoots out dense rays from the probe and finds out which ray is closest to the target within the acceptable error as the ray of the target shot. Hereafter, in this study, the propagation time calculated by the trial-and-error ray tracing method is considered as the true propagation time which its accuracy is within 0.001 microsecond in order to fit the requirement of this study. Moreover, the true delay is taken as the difference between the true propagation time and vertical propagation time. However, the trial-and-error ray tracing method is complicated and time-consuming for the two-layer medium. For more than two layers cases, the difficulty and computation time increase dramatically.

To generalize to a continuous time-distance relationship, similar to Eq. (3), the squared reflection time of ultrasound is function of the squared lateral distance [20‐22] Taking x ² as the independent variable, the squared reflection time in the form of a Maclaurin power series (Taylor series expansion at x=0) becomes [20] Neglecting the third and higher order terms, Eq. (9) becomes However \(\frac{d}{dx^{2}}f(x^{2})|_{x = 0} = \frac{2^{2}}{V_{rms}^{2}}\), where V _rms is the root-mean-square velocity [20, 21].

Then, Eq. (10) becomes For an N-layer medium, V _rms is defined as Where V ₁, V ₂, V ₃,…, and V _N , are the velocities of the longitudinal wave of layers from 1 to N. t _I1, t _I2, t _I3,…, and t _IN are their one-way interval propagation time, for examples t _I1=d ₁/V ₁ and t _I2=d ₂/V ₂. For the discrete lateral distance, Eq. (11) becomes The best fitting time-distance relationship of the reflection ultrasound by a hyperbolic equation is [20, 21] where V _st is the stacking velocity computed by fitting the time-distance relationship of the reflection ultrasound. V _rms is generally smaller than V _st (V _rms ≤V _st ) [20] and it is a function of vertical propagation time. In ultrasonic NDT, the beam width of the sound field radiating from the transducer is usually small (smaller than 15 degrees), which follows the assumption (Maclaurin series expansion) of a very short lateral distance. Therefore, the approximated delays can be easily calculated by Eq. (12) and Eq. (13) and used in SAFT.

The error of using V _rms to calculate the propagation time of the ultrasound diffracted from a point reflector is estimated. Assuming the distance between the probe and specimen is 2 cm and the depth of the point reflector is 2 cm below the water-specimen interface, the velocities of the longitudinal wave of water and specimen are 1500 m/s and 5940 m/s (Fig. 3(a)), respectively. In Fig. 3b, the black line stands for the true propagation time of the ultrasound diffracted from the point reflector, and the blue and red lines are those calculated by the V _st and V _rms (Fig. 3(b)), respectively. The three lines are nearly coincident at a short lateral distance, i.e. nΔx<0.5 cm, where the incident angle of the ray is 2.8 degrees and the ratio of lateral distance to the flaw depth (d ₁+d ₂) is 1:8. As the lateral distance increases, the propagation time of the ultrasound computed by V _rms begins to deviate from the other two and the error of propagation time is 0.05 microsecond at lateral distance of 1.4 cm, where the incident angle of the ray is 7.2 degrees and the ratio of lateral distance to the flaw depth is 1.4:4. The dominant period (T _D ) of the ultrasound used in this study is 0.05 microsecond (or the frequency is 20 MHz). The pink line in Fig. 3(b) shows the ratio of the propagation time error computed by the V _rms to T _D , and the ratios at lateral distances 0.5 and 1.4 cm are 0.018 and 1, respectively. The beam width of the transducer used in this study is about 6 degrees; the half angle of the beam width is only 3 degrees. As we can see from these results, the error of the approximated propagation time is very small.

A steel specimen with three single-side-drilled holes located at different depths was fabricated (Fig. 5(a)). The thickness and width of the specimen are both 30 mm. Diameter of holes is 0.5 mm, and depths of centers of holes are 25 mm, 20 mm and 15 mm, respectively from left to right. These single-side-drilled holes are treated as the point flaws. Figure 5(b) shows a B-scan image of the steel specimen. The abscissa of the image is the lateral position of the scan and the ordinate is the depth measured from the top surface of the specimen. All the images in this study are displayed by a gray scale, black and white express the maximum and zero amplitudes of the echoes and the change from black to white is linear. Three curve events with apexes positioned at depths of 25 mm, 20 mm and 15 mm were imaged from the three single-side-drilled holes. Figure 5(c) is the SAFT image processed by the true delays. The three flaw images at different depths have been successfully synthesized with 1 mm horizontal size. Figure 5(d) is the SAFT image processed by the approximated delays. Since the delays used in SAFT are approximated values, the intensities of flaw images (Fig. 5(d)) are weaker than those in Fig. 5(c). Nonetheless, both SAFT images are alike.

The array performance indicator (API) [23] is a useful index to quantify the performances of the two methods, which measured the area (A _{−6 dB}) of the flaw image where the image intensity is within −6 dB of its peak intensity and normalized by the square of the bulk longitudinal wavelength. The A _{−6 dB} areas of flaw images of the three different depths flaws and their API values for the two methods are shown in Fig. 6. The average ratio of the API values of the two methods (\(\overline{\mathit{API}}_{\mathit{true}/\mathit{app}.}\)) is 0.94. Figure 7 shows the normalized amplitudes of SAFT images (Fig. 5(c) and (d)) along axial and lateral directions to the same scale. The red and blue amplitudes are the reconstructed images processed by the true delay SAFT and approximated delay SAFT. In Fig. 7(a), the normalized amplitudes of images run right through centers of the hole-flaws in axial direction. A higher noise level can be seen by using the proposed method. Furthermore, Fig. 7(b) shows the normalized amplitudes of images in lateral direction. Depths of normalized amplitudes of images are at 25 mm, 20 mm and 15 mm, from left to right, respectively. The lateral resolutions of images are almost the same for both SAFT methods. Therefore, based on Figs. 6 and 7, our simple and fast calculation method for approximated delays used in SAFT is feasible.

To test how the resolution is deteriorated by using the approximated delays for SAFT, a steel specimen was made with three sets of quad-side-drilled holes located at different depths. The dimension of holes is the same as those used in the previous specimen (Fig. 5(a)). For each quad-side-drilled set, there are four holes and the horizontal intervals between the holes are 3 mm, 2 mm and 1 mm, respectively, from left to right (Fig. 8(a)). Depths of the three sets are settled at 25 mm, 20 mm and 15 mm. The B-scan image of the specimen is shown in Fig. 8(b). As we can see from the image, the holes in the sets were not resolved. The image was then processed by the SAFT with the true and approximated delays, respectively; the results are demonstrated in Fig. 8(c) and 8(d). As seen Fig. 8(c) and 8(d), the holes with 3 mm and 2 mm in between them can be easily identified. However, this is not the case for those with 1 mm horizontal intervals in the different depth quad-side-drilled holes sets. Again, the reconstructed images of holes illustrated in Fig. 8(d) show weaker intensity than those demonstrated in Fig. 8(c), but both SAFT images are alike.

The A _{−6 dB} areas of flaw images of the three different depths quad-side-drilled holes sets and their API values for the two methods are shown in Fig. 9. Figures 9(a), 9(b) and 9(c) are the images of holes sets at depths 25 mm, 20 mm and 15 mm, respectively. The columns of figures in Fig. 9 from left to right are the images of left, left second and right two holes of quad-side-drilled holes. The \(\overline{\mathit{API}}_{\mathit{true}/\mathit{app}.}\) is 1.13. Figure 10 shows the normalized amplitudes of SAFT images (Fig. 8(c) and (d)) along axial and lateral directions. The red and blue amplitudes are the images processed by the true delay SAFT and approximated delay SAFT, respectively. In Fig. 10(a), the normalized amplitudes of images run through the lateral distances at 25 mm, 50 mm and 75 mm in axial direction. Again, in the figure, the noise levels are higher by using the approximated delay SAFT. Figure 10(b) shows the normalized amplitudes of images in lateral direction. Depths of normalized amplitudes of images are at 25 mm, 20 mm and 15 mm, from left to right, respectively. The lateral resolutions of both SAFT images are almost the same. Based on Figs. 9 and 10, the resolutions of both SAFT images are almost the same.