|NDT.net - November 1999, Vol. 4 No. 11|
|TABLE OF CONTENTS|
Where I[x,y,z] is the pressure of sound which reflected or scattered by point (x,y,z), N is the number of transducers, ri is the distance from transmitter to point (x,y,z) and from this point to ith receiver and c is the velocity of ultrasound in media.
|Fig 1: Pitch catch method|
In any method conventional or multi SAFT the measured signal by different receiver must be shifted and added to each other. The result is good if the source produce a sharp wave, which is broadband in frequency domain and as a result is very narrow in time domain. It is an ideal situation and cannot be achieved in real world. For example ringing time for very high frequency ultrasonic transducer, about 80 MHz with 160 MHz bandwidth is, at least, 300ns. Where as for very low frequency probes, about 100KHz, the best transducer which is built has, at least, 100 µs ringing time.
In this paper, a correlation method of the SAFT algorithm is introduced. In addition an innovation calculating method is discussed which reduce the calculation to less than 0.73% of using the introduced method directly. In this method we introduce a correlation matrix which has three dimension, i.e. x n x d, where is the maximum delay time, n is the number of receiving places in each scanning line and d is equal to the number of scanning line. This matrix is calculated before starting to construct image by using SAFT. This is the key, which make the calculation reduce. This method can be used for any testing system, which used SAFT algorithm for constructing the image in 2 or 3 dimensions.
And auto-correlation function is:
The signal x (t) is maximally correlated (i.e. similar) with itself when x(n+t)=x(n) . By comparing equation 2 and 3 we can say, the signal y(t) is maximally correlated (i.e. similar) with x (n+t) when y(n)=x(n+t).
Because of transducer ringing, the equation one must be write as follow:
(Hint: For more convenience we show Ix,y,z(n) by I(n)).
Assume the time series which shows the ringing of ultrasonic transmitter is y(n), then cross-correlation between y(n) and I(n) at time t=0 can show a measurable amount of reflected and scattered ultrasonic wave by point (x,y,z). The t=0 is considered because the flaw may be in any places. The formula can be written as follow:
|(6 - a)|
|(6 - b)|
By comparing right hand side 6-b with 2 the following equation is the result:
Correlation Matrix: Equation 7 shows for measuring ultrasonic reflected and scattered pressure in any point (x,y,z) the cross correlation between the transmitted signal and received signal at any receiving point at i must be calculated and then summing all of them.
The furthest place between transmitter point (x,y,z) and from this point to the receiving point can be find. It is obvious it can not be more than the length of time series Xi(n) when the length convert to number of sampled data.
A three dimensional matrix in which any i,j,k element shows the cross-correlation between y(n) and the received signal by jth receiver in kth scan line at time equal i is calculated. For showing the strength of this method in reducing the calculation, let consider an example. For calculating a 200x150 images when the time series of measured data has length of 650 sample there is 650x200x150x6x9=1.053x109 add operation. The numbers in this formula are, the length of time series, x, y dimension of image, number of scan line and number of receiving point in any line, respectively.
If the correlation matrix is calculated in advance 650x650x6x9=22815000 add operations is needed and after that only 200x150=90000 add operation is necessary for calculating a 2D image. Consequently all necessary add operations are 22905000. By comparing this two number we can see the last operation is less than 2.2% of previous one.
|Fig 1: Experimental system|
In this system following devices are used:
|2-||A pre amplifier|
|7-||A thank of water|
The data is collected from one surface. Choosing sampling frequency and number of sampling are important factors in collecting data in proper manner. Shannon's law states the lower band of sampling frequency. In this experiment because the nominal frequency of probes are 95 kHz, the sampling frequency must be more than 190KHz. Sampling frequency influences the lateral and depth resolution. If fs be sampling frequency and c be the ultrasound wave velocity then l=c/fs shows the distance that ultrasound travels between to adjacent point. In this experiment it is 2MHz. Number of samples depending on the system which is under test. For avoiding the effect of back wall echo the time duration, in which sampling is in process must be less than the time, which is necessary for travelling through the media, and reflected from back wall. This time is 360µs in here. When fs=2MHz is chosen the sampling period is 0.5µs which means 720 samples can be collected before back wall echo become appear. For using correlation method an exact pattern of transducer ringing is needed. Fig 2 shows such pattern for the system, which is used in this experiment.
|Fig 2: Transducer's ringing pattern|
In fig. 3 one recorded data is shown. There are two areas in data. Area one shows the area in which reflection can be exist from any difference in impedance in media. The starting point of area two shows the starting point of signal, which is reflected, from back wall. The data in area two can not be used for reconstructing image because they are redundancy data or they are from back wall.
|Fig 3: Recorded data in one point|
Fig 4: Reconstructing flaw by.
Fig 5: Reconstructing two flaws.
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