Page 22 - Acoustics Today Summer 2011

Fig. 1. Illustration of the reverse time migration process in a free space with a scat- terer at location C. The propagation times for each path are included for the read- er’s reference. (a) Forward propagation step. (b) Backward propagation step.
conjugation, and –t represents a time reversal. The magnitude of the image Ix,y is then typically displayed to locate scatterers. Anderson et al. found that this traditional imaging condition does not work well in a highly reverberant medium and instead used the following imaging condition with better results
(2)
The experiments conducted by Anderson et al. found
that scatterers of a high impedance relative to the sample
impedance showed up as minima in the Mx,y image. In anoth-
er experiment at the Los Alamos National Laboratory, the
ward propagation (after these last set of data have been reversed in time). To the degree that the energy broadcast during the backward propagation step retraces the forward propagation paths, RTM allows imaging of passive scatterers.
Figure 1a illustrates the forward propagation of a RTM
experiment conducted in free space with a source at A, a
reversible transducer at B and a scatterer at C. The forward
signal is emitted from A after 7 time units. This forward sig-
nal is then directly received at B at a time of 11 units and the
reflection off of C arrives at B at a time of 12 units. The for-
ward signal arrives at C at a time of 10 units. The signal
recorded at B is now flipped in time and used as the source
signal for the backward propagation depicted in Fig. 1b (we
color the two emission signals from B to aid visualization of
this step). The red signal from B directly travels to A and
arrives at a time of 6 units. The red signal from B also
reflects off C and arrives at A at a time of 7 units. The blue
signal from B also directly travels to A and arrives at a time
of 7 units (constructively interfering with the reflected red
arrival producing the purple recorded signal). The blue sig-
nal from B also reflects off of C and arrives at A at a time of
8 units. The signal at A is typical of a symmetric time rever-
14
(1) where I represents a Fourier transform, * represents phase
high density scatterers was investigated . This experiment utilized a nearly semicircular aluminum plate of dimensions 6.54x179x396 mm (pictured in Fig. 2a). The characters “LANL” are milled out of the plate at a depth of 3.23 mm and a width of 10 mm. The characters “EES-11” are cut out of a 2.64 mm thick steel plate with a width of 10 mm and glued onto the plate. The removal of plate material for the “LANL” characters should present an incident wave with a low impedance scatterer, while the addition of material for the “EES-11” characters should present a high impedance scat- terer. In this experiment a single transducer (labeled as S in Fig. 2a and is slightly not in view for the photograph) is used as the source with a single receiver transducer (labeled as R
question of whether Mx,y could distinguish between low and 15
The signal recorded at C consists of the red signal delayed by 2 time units and the blue signal delayed by 2 units. Now, if the signal recorded at C is reversed in time and compared to the signal recorded at C during the forward propagation step, one will notice that the green arrival and the red arrival are synced in time. The traditional RTM image, I, is found through computing the cross correlation of the forward signal, F, at position (x,y) with the backward signal, B, at (x,y), after this later signal has been reversed in
time,
sal focus signal.
Fig. 2. (a) Photograph of a semicircular aluminum plate which has the letters
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“LANL” milled out of it and steel letters “EES-11” glued onto it . S and R denote
the source and receiver transducers named according to the forward propagation usage. (b) RTM image of the other side of this plate (mirror image is displayed). Amplitude is in dB units with an arbitrary reference.
18 Acoustics Today, July 2011