# Comparing Kirchhoff-approximation and boundary-element models for computing gadoid target strengths

TL;DR: In computations of target strength as a function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of the same four frequencies.

Abstract: To establish the validity of the boundary-element method (BEM) for modeling scattering by swimbladder-bearing fish, the BEM is exercised in several ways. In a computation of backscattering by a 50-mm-diam spherical void in sea water at the four frequencies 38.1, 49.6, 68.4, and 120.4 kHz, agreement with the analytical solution is excellent. In computations of target strength as a function of tilt angle for each of 15 surface-adapted gadoids for which the swimbladders were earlier mapped, BEM results are in close agreement with Kirchhoff-approximation-model results at each of the same four frequencies. When averaged with respect to various tilt angle distributions and combined by regression analysis, the two models yield similar results. Comparisons with corresponding values derived from measured target strength functions of the same 15 gadoid specimens are fair, especially for the tilt angle distribution with the greatest standard deviation, namely 16°.

## Summary (2 min read)

### Introduction

- The 20 or so methods cited in a 1991 study3 have since been augmented significantly by a number of new te niques, including both empirical methods8,9 and theoretical models, especially those based on the deformed-cylin model10 and boundary-element method.11.
- These have been accompanied by novel applications, for example, to salm and trout,8,9 cod ~Gadus morhua!,12 orange roughy~.

### II. KIRCHHOFF-APPROXIMATION MODEL

- In the Kirchhoff approximation, the field on the scatte ing surface is assumed to be knowna priori.
- For a swimbladder-bearing fish at rather high frequencies, wavelengths which are rather small compared to the m mum length of the swimbladder, the fish is represented b pressure-release surface conforming to the inner wall of swimbladder.
- The integration in Eq.~1! is performed numerically us ing Gauss quadrature over curvilinear surface elements which the position vectorr is interpolated quadratically from nodal values.
- A similar relations exists for eight-node quadrilateral elements using 333 or more Gauss points.
- This translates to a co tion that the element side-to-wavelength ratio should be than 2/5.

### III. BOUNDARY-ELEMENT METHOD

- To develop the acoustic boundary-element meth ~BEM!, the wave equation for the pressurep is reduced to the Helmholtz form by assuming the harmonic time dep dence exp(ivt), wherev is the angular frequency in radian per second, hence¹2p1k2p50, wherek5v/c is the wave number.
- The use of the centroids, rather than the nodes, as calculation points for the normal-derivative form is found be sufficient to overcome the problem of the critical freque cies while not increasing the computational effort unduly.
- The accuracy of geometrical representa depends on the degree of undulation of the surface, bu should be noted that the quadratic interpolation allows elements to be curved.

### IV. SWIMBLADDER MORPHOMETRY

- First, contours of the swimbladder in planes perpendicular to major axis of the fish, and hence perpendicular to the mic tomed sections, are determined at intervals along the m axis, by finding the points of intersection of each plane w the original digitized sections.
- The n meshes have been used in computations with the Kirchh approximation model in parallel with the BEM.
- At 120.4 kHz the nodal spacing, to satisfy thel/6 condition for accuracy of the BEM and Kirchhoff approximation model, should be less than 2.03 mm.
- The backscattered pressure from a rigid sphere has been computed u similar conditions, again with excellent agreement, wh avoiding discrepancies at the critical frequencies.

### VI. COMPARISON OF MODEL COMPUTATIONS

- The target strength for an immersed void with the sha of the swimbladder shown in Fig. 1 has been computed function of tilt angle for both the dorsal and ventral aspe at each of four frequencies.
- Both the Kirchho approximation model and BEM have been examined.
- The target strength corresponding to each avera backscattering cross section, denotedTS, has been compute by substituting the value ofs̄ from Eq. ~14! in Eq. ~3!.
- The standard error of the regression has been computed each derived regression equation.

### A. Model validation computations

- To validate the BEM for application to the gadoid swim bladder, a 25-mm-radius spherical void in sea water has b chosen as a test case in order to have a shape for whi rather simple analytical solution exists and whose surf area is greater than that of the largest swimbladder in data set.
- Sensitivity to axial orientation is negligible long as the maximal nodal separation does not exceedl/6.
- The Kirchhoff-approximation model is exercised with th identical meshes but performs less well than the BEM; it inherently different, as is proved by the difference in resp tive exact and analytical solutions for the two models for t pecial shape.
- Differences in the two models are also evident in t logarithmic measures presented in Table IV.
- In contrast, Kirchhoff approximation could be exercised with far mo elements than used here and thus, in principle, could be m amenable to computation at higher frequencies.

### B. Swimbladder-shape-based computations

- The detailed computations of target strength as a fu tion of tilt angle are shown for a single specimen, No. 205 Fig. 3. Both the Kirchhoff-approximation model and BEM results are shown for the swimbladder as represented in 1.
- Sign cantly for this work, the Kirchhoff-approximation and BEM results are quite similar.
- The new mapping, for consistency with the BEM, conta fewer but curvilinear elements spanning the swimblad surface.
- Reveals a greates discrepancy of 0.2 dB, with median discrepancy of 0.1 d for the dorsal aspect.

### C. Summary of comparisons

- Earlier validation exercises with the BEM have be upplemented by a new example, that of a spherical void cibels ents for he as ho nts ier ers which a simple analytical solution is known.
- Nonetheless, in the c of the swimbladder-shape-based computations, the Kirch J. Acoust.
- Soc. Am., Vol. 111, No. 4, April 2002 K. G. Foote and e ff approximation, when exercised with the curvilinear eleme used in the BEM, yields results that agree well with earl computations carried out using meshes with larger numb 1651D.
- Differences in predictions, as pressed through the regression coefficient in Eq.~15!, are less than 1 dB in all cases except at 38.1 kHz where greatest difference is 1.3 dB.
- There is some expectation the discrepancy might be largest at the lowest frequency the Kirchhoff approximation assumes high frequencies.

### ACKNOWLEDGMENTS

- This work began with sponsorship by the Europe Commission through its RTD-program, Contract No. MAS CT95-0031~BASS!.
- This is Woods H Oceanographic Institution Contribution No. 10437. 1R. H. Love, ‘‘Measurements of fish target strength: A review,’’ Fish.

Did you find this useful? Give us your feedback

...read more

##### Citations

75 citations

### Cites methods from "Comparing Kirchhoff-approximation a..."

...New models such as the FMM and BEM (Foote and Francis, 2002) have potential for modeling the scattering over a much larger range of orientation angles and frequencies, including near the resonance frequency of the swimbladder....

[...]

51 citations

### Cites methods from "Comparing Kirchhoff-approximation a..."

...Matches of KRM model predictions to empirical backscatter measurements of pollack (Pollachius pollachius) and saithe (Pollachius virens) were comparable to those obtained using the boundary element model (BEM) over a frequency range of 38.1 to 120.4 kHz (Foote and Francis 2002)....

[...]

36 citations

### Cites background from "Comparing Kirchhoff-approximation a..."

...The form that this integral takes for the far-field backscattering amplitude may be found in Foote (1985) and Foote and Francis (2002)....

[...]

30 citations

### Cites methods from "Comparing Kirchhoff-approximation a..."

...Kirchhoff approximations have been used to predict backscatter of several species, ranging from fish with large, air-filled swimbladders (Jech et al., 1995; Horne et al., 2000; Foote and Francis, 2002) to deepwater species without airfilled bladders (Barr, 2001; Kloser and Horne, 2003)....

[...]

27 citations

##### References

17,323 citations

1,072 citations

247 citations

^{1}

203 citations

164 citations

##### Related Papers (5)

##### Frequently Asked Questions (2)

###### Q2. What have the authors stated for future works in "Comparing kirchhoff-approximation and boundary-element models for computing gadoid target strengths" ?

Some of the effects mentioned here may be addressed in a future work.