2.2. Fresnel Zone
The Fresnel zone is an ellipsoidal region between a transmitter and a receiver, which can be interpreted as a region where scattered waves interfere with the direct wave. Different from the raypath, the Fresnel zone has considered the scattering effect of the wave propagation. A simplified ring array transducer is shown in
Figure 2 to illustrate the Fresnel zone.
and
are two of the transducer elements on the ring.
is the transmitter,
is the receiver, and
is an arbitrary spatial point. The straight line that links
and
is the central raypath. Considering that the frequency in USCT is finite, for the transmitter-receiver pair
and
, the points that affect wave propagation are not only on the raypath but through a zone around the central ray, called the Fresnel zone [
18,
19,
20,
21]. In
Figure 1, the gray part represents the Fresnel zone between
and
.
The Fresnel zone can be calculated from the eikonal equation
where
is the traveltime,
,
are two dimensional coordinates, and
is the slowness. By solving the eikonal equation with the finite difference (FD) method [
27], the travel times among the spatial points in the imaging area can be obtained. The range of the Fresnel zone is determined under the condition [
18]
where
is the travel time delay between the detour path
and the direct path
,
is the travel time from transmitter
to point
,
is the travel time from receiver
to point
,
is the travel time from transmitter
to receiver
.
is the constraint of travel time delay between the detour path and the direct path to determine the Fresnel zone, and
is the center frequency of the signal. The Fresnel zone between
and
illustrated by the gray area in
Figure 2 is composed of the points that satisfy Equations (2)–(4).
2.3. Fresnel Zone Travel-Time Tomography (FZTT) and Zone-Shrinking FZTT (ZSFZTT)
Fresnel zone travel-time tomography (FZTT) is an iterative inversion algorithm. The flowchart of FZTT algorithm is showed in
Figure 3. Firstly the traveltime
is detected from the captured ultrasound signals by the Akaike information criterion (AIC) method [
28]. Then set an initial slowness
and start the iterations. Calculate the traveltime maps for all transmitters by the FD method [
27].
is calculated by Equation (2) for all the transmitter-receiver pairs and to determine the Fresnel zone.
For transmitter-receiver
and
in
Figure 2, in raypath travel-time tomography,
where
is the travel time from
A to
,
is the raypath on
,
is the slowness (inverse of sound speed) on
. In Fresnel zone travel-time tomography,
where
is the sensitivity kernel of travel time (SKT) [
18], which reflects the sensitivity of
to the propagation medium. The higher the value of
, the more energy travels through
. After the Fresnel zone is determined, the SKT is approximated by [
18]
where
is a weighting parameter,
is the SKT on point
, and
is the length of a grid cell.
To illustrate the SKT in the Fresnel zone, a phantom model is simulated.
Figure 4a shows the model: a circular phantom (red) with SS 1560 m/s is immersed in water (blue) with SS 1500 m/s. When the transmitter positioned at
is activated, the traveltime map originated from
obtained by the FD method [
27] is plotted in
Figure 4b. In the same way, when a receiver positioned at
is activated, the travel-time map originated from
can be obtained and plotted in
Figure 4c. Using Equation (7), we can obtain the SKT in the imaging area (
Figure 4d). The SKT appears as a “banana-doughnut” shape as indicated in Marquering’s [
29] and Jocker’s [
30], the values of SKT in the central area of Fresnel zone are smaller than those on the boundary of Fresnel zone. Here, the SKT appears slightly bent because of the refraction phenomenon.
After SKT for all transmitter–receiver pairs are calculated by Equation (7), FZTT is formulated as an optimization problem with an objective function
. Assuming the number of transmitter–receiver pairs is
, the imaging grids are of size
. The length of the square imaging area is
, which is the same as the diameter of the transducer. Then the grid size is
To prevent an underdetermined problem, we set
Submitting Equations (8) to (9),
where
is the SKT matrix of size
, the slowness
is of size
, and the traveltime
is of size
. Then the objective function is designed as
which can be solved by the Quasi-Newton methods. Here, a limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method [
31] from the family of Quasi-Newton methods is adopted. By solving Equation (11), the optimal value of
is obtained. Repeat the loop until the value of the objective function
is smaller than a predefined threshold
determined by experience (approximately 200~500) or the iteration number
arrives at a predefined maximum iteration number
. Finally, the SS is formed by
.
FZTT can reconstruct an SS image with a high CNR, but due to the wide Fresnel zone, the reconstructed SS image is usually inaccurate. Here, a zone-shrinking Fresnel zone travel-time tomography (ZSFZTT) is proposed to solve the problem. A weighting factor
is introduced to shrink the zone. Equation (12) shows how the weighting factor
is performed.
With the increase of
,
decreases, which means fewer spatial points satisfy
; thus, the zone shrinks. In the proposed ZSFZTT, the calculation of the SKT is described by Equation (13),
The values of SKT get smaller because decreases with the increase of . Compared to FZTT, the calculation of Fresnel zone uses Equation (12) instead of Equation (3), and the calculation of SKT uses Equation (13) instead of Equation (7).
We tested how the Fresnel zone shrinks when
increases. The Fresnel zone between transmitter
and receiver
when
;
;
;
;
;
is plotted in red in
Figure 5. We can find that the Fresnel zone shrinks when
, while the zone does not shrink significantly when
. Compared to the zone area when
, the zones’ areas when
shrink to 85.7%, 80.3%, 77.5%, 75.7%, 74.5%, 73.7%, and 73.0% respectively. When
, the decrease of the percentage with the increase of
is less than 2%. n this research,
is imposed with a constraint defined by Equation (14) during the inversion process,
where
is the iteration number.