14.3 ESTIMATION OF OBJECT PARAMETERS
We consider the problem of estimating a set of parameters for a scattering object, such as size, shape, orientation, sound speed and density, from measurements of the scattered field. We denote by u a vector of such parameters in a model of the scattering scenario, and by p(u) = p(x, t, и) the corresponding modelpredicted scattered field, computable by the methods described above. Denoting the experimentally observed field by q = q(x, t), the identification problem is then to find и such that a fitness function ®(u) measuring the distance between p(u) and q is minimized.
The top frame of Figure 14.5 shows a section of experimentally observed signals at receivers 2 to 6. The section contains (i) the specular echo from the upper surface of the boxshaped scatterer, followed by (ii) a ca. 1mslong lowamplitude period of returns from a very soft top sediment layer and (iii) stronger returns from underlying seabed layers. The bottom frame shows simulated echoes computed using the Ray Kirchhoff (RK  see Part I) method (solid) together with the specular echo intervals of the experimental traces (dashed). The dashed crosstrace curves show the arrival times of the echo reflected at the upper surface of the box, as predicted by ray theory.
The arrival times of modelpredicted echoes were seen to agree well with the experimental data. However, their amplitudes as a function of receiver depth were nearly uniform, with a weakly pronounced maximum at receiver 4, the depth at which specular reflection occurs at the flat upper surface of the box. In contrast, the experimentally observed echo amplitude increased with depth and was maximal at the deepest receiver.
Motivated by the characteristics of the experimental data and the accuracy properties of the RKpredicted transient echoes the fitness function ®(u) for the parameter inversion was defined as
where Ti (u) and Ai (u) are the computed arrival time and the normalized amplitude at receiver i; ri and at are the measured arrival time and the normalized measured amplitude at receiver i.
Figure 14.6 shows two examples of the fitness function (14.1) as a function of two parameters of the boxshaped scatterer: the roll and the yaw angles (left) and the x and zcoordinates of the centre point (right).
In general Ф(и) is a complicated nonlinear function, with multiple local minima, and the optimal parameter vector и must be sought from methods for global minimization. In this study two such algorithms were considered: a differential evolution algorithm (DE  Storn and Price, 1995) and a hierarchical genetic algorithm (GA  CantuPas, 1998), both combined with a final search of local minima by the downhill simplex method (Nelder and Mead, 1965). The RK method described was used as the forward model for computing the scattered field p(u) = p(x, t, и).
As a first step towards estimating the parameters of the scatterer, the methods described above were applied to calculate the roll and yaw angles of rotation of the
(14.1)
Figure 14.5. Signals in the middle five hydrophones of the vertical array: (top) experimental data; (bottom) predicted by the RK method (solid) and experimental data (dashed). Scatterer parameters based on prior information.
scatterer, keeping other parameters fixed. Table 14.1 shows the results of the minimization, the number of iterations and the number of function evaluations needed to locate a local minimum.
The GA method actually finds the minimum after only 7 iterations in this run, but the standard deviation in the population is in that stage still too high to terminate the search. The DE method needs 17 iterations to find the minimum and is terminated after 27 iterations by the condition that the population has not changed in 10 consecutive iterations. Figure 14.7 shows the fitness function Ф(и), with the best individual in each generation marked by dots. Ф(и) has two local minima in the
a
Ш
тз
25 30
Distance from sender
Figure 14.6. Fitness function Ф in Equation (14.1): (top) as a function of roll and yaw angle; (bottom) as a function of the x and zcoordinates of the centre point.
Table 14.1. Results of global search for roll and yaw angles.
Roll 
Yaw 
Iterations 
Function evaluations 
GA 0.86 DE 0.47 
11.8 10.1 
40 25 
800 450 
Table 14.2. Inversion for interior density and sound speed. Inversion results are underlined. 


Run 1 
Run 2 
Run 3 Run 4 Run 5 
Density (kg/m3) 
1740 
1420 
1490 1630 1490 
Sound speed (m/s) 
2680 
2570 
2530 2380 2570 
region shown. The individuals of the GA method (top frame) are seen to cluster around the local minima. For the DE method (bottom frame) fewer than 25 values are shown, reflecting that the updating procedure of the DE method leaves an individual unchanged unless the trial individual has a better fitness.
It should be noted that this is a simple example where not all the benefits of the global search methods come into play; a sound local minimizer will probably converge faster with the global minimum.
As a second step, the interior density and sound velocity ofthe box were estimated in five similar runs, with results as shown in Table 14.2. The underlined values are those resulting from the parameter search, other entries show values kept fixed during the search.
Similarly, in a third and a fourth step, the depth and range of the centre of the box and the pitch angle of the box, respectively, were inverted, keeping other parameters fixed at their values from previous inversions. The results of all inversion runs are collected in Table 14.3.
Table 14.3. Parameters of the scatterer determined by inversion.

Initial 
Inverted 
Range (m) 
24 
23.95 
Depth (m) 
74.85 
74.85 
Roll (deg) 
0 
0.9 
Pitch (deg) 
0 
4.8 
Yaw (deg) 
19 
11.7 
Density (kg/m3) 
1630 
1490 
Sound speed (m/s) 
2680 
2570 
Figure 14.7. Convergence history of parameter search: (top) GA method; (bottom) DE method.
In Figure 14.8 the modelpredicted received echoes obtained by the scatterer parameters shown in Table 14.3 are shown together with the experimentally observed specular echoes.
By comparing Figure 14.8 with the bottom frame of Figure 14.5, parameter inversion is seen to improve the match between the modelled and the experimental data significantly. Most notably, the echo amplitude as a function of receiver depth in the experimental data is well reproduced by the models after inversion. This improvement reflects the sensitivity of the vertical distribution of the scattered energy to the rotation of the scatterer, which was not accurately known at the outset.
Figure 14.8. Received signals: (dotted) experimental data; (solid) modelled by parameters from the acoustic inversion shown in the right column of Table 14.3. Top: RK. Bottom: BIE.