The Kirchhoff-Ray-Mode (KRM) model has been used in fishery acoustics for more than two decades to predict backscatter by individuals or aggregations of swimbladdered fish. Backscatter contributions from a fluid-like fish body and gas-filled swimbladder are coherently summed to estimate backscatter from the whole fish. For field applications, where morphologies and orientations of individual fish and their swimbladders are unknown, the coherent KRM model based on a limited number of X-ray images containing precise morphological information on swimbladders, body sizes and orientations may not provide statistically stable and robust backscatter predictions. An incoherent KRM model is proposed, which is independent of the exact knowledge of body–swimbladder acoustic interactions. When backscatter contributions from fish bodies and swimbladders are comparable, differences in coherent and incoherent KRM model target strength (TS) predictions can be as high as a few decibels. Incoherent TS predictions as a function of frequency and orientation are statistically more stable and less sensitive to variations in anatomy among individual fish than those predicted using the traditional coherent KRM model.

Acoustic models are used to characterize acoustic scattering (i.e., reflected energy) from fish and other aquatic organisms. Reflected energy amplitudes from single animals, termed target strength (TS), have been studied over the past 30 years [

In 1994, Clay and Horne proposed the Kirchhoff-Ray-Mode model (KRM) and calculated the TS of cod [

The KRM model calculates backscatter by computing scattering contributions (i.e., scattering amplitude or scattering length) from a series of circular disks along the fish axis (i.e., sagittal axis lengthwise along fish body) as a function of length (

As an alternate approach, an incoherent KRM backscatter model eliminates the coherent interaction between the fish body and the swim bladder and may be more appropriate when estimating backscatter from aggregations of animals. To illustrate the application to aggregations of fish, we calculate and average predicted TS of many fishes that are similar in length and compare the results obtained from the proposed incoherent KRM model with those from the coherent KRM model.

For an individual (whole) fish (

For a simultaneously insonified acoustic volume that includes

The above equation is the coherent KRM model (Clay, [

By dropping the 3rd term, the average differential cross section can be approximated by an incoherent average or sum:

This incoherent KRM model can use

Eighty dorsal and lateral radiograph images (cf. Henderson and Horne [

We will provide a brief summary of how to calculate the TS here. A more detailed description can be found in Clay and Horne [

Many factors influence backscatter amplitude or scattering length, such as acoustic frequency and shape, length, and orientations of the

Equation (6) is the reflection coefficient of the

The scattering length of fish body

TS is a logarithmic representation of the backscattering cross section

TS will be used in simulations to quantify backscatter by individuals and by aggregations of Pacific hake. Simulations investigating differences between coherent and incoherent scattering models are presented in the next section, where two scenarios, one involving single fish and the other involving multiple fish, are evaluated.

Ten fish (200 ± 10 mm) are used to demonstrate differences between coherent and incoherent KRM model predictions as a function of the number of fish used in calculating backscatter amplitudes for a group (see

Incoherent TS predictions of small fish are significantly different from results estimated using the coherent model. Backscattered waves from fish

Differences between predictions from the two models (^{−1}, the wavelength at 70 kHz is about 0.021 m, which is the same order of magnitude as the lengths of the

In large fish, most backscattered energy is derived from the

To demonstrate differences in coherent and incoherent model predictions for aggregations of fish, ten fish of similar length (200 ± 10 mm) were used to demonstrate differences between coherent and incoherent KRM model predictions as a function of the number of fish used in calculating backscatter amplitudes for a group. Groups consist of three, five, seven, and ten fish, with members of each group picked randomly from the ten digitized fish and the modeling exercise repeated ten times for each group size. Average TS values are calculated using coherent and incoherent KRM models separately and then compared.

TS curves calculated with the incoherent KRM model always resulted in smoother frequency response curves and smaller amplitude fluctuations, indicating that the incoherent TS predictions are less variable and statistically more robust across the frequency range. As shown in

For fish aggregations, where the number of fish is very large, the cross term in Equation (2) is expected to approach 0. As a result, average TS curves calculated using coherent and incoherent KRM models are expected to converge. In addition, the incoherent KRM model does not need to compute the cross term, which makes it more computationally efficient, although, given the computational capability of modern computers, this difference is very small.

The incoherent KRM model is a revised version of the original KRM model proposed by Clay and Horne [

Both coherent and incoherent KRM models can be used to estimate fish density. When estimating TS of fish within aggregations, both models are effective. The choice of model formulation is dependent on the objectives. A coherent KRM model may be more appropriate for single fish model and ex situ measurement comparisons if the morphologies of the fish

Conceptualization, D.C. and C.L.; methodology, D.C. and C.L.; software, D.C., J.H. and C.L.; validation, D.C., J.H. and C.L.; formal analysis, D.C. and C.L.; investigation, H.L.; resources, H.L. and C.L.; data curation, J.H.; writing—original draft preparation, C.L.; writing—review and editing, D.C. and J.H.; visualization, D.C. and J.H.; supervision, D.C., J.H. and H.L.; project administration, H.L.; funding acquisition, H.L. All authors have read and agreed to the published version of the manuscript.

Not applicable.

Not applicable.

Data is unavailable due to college regulation.

This work was supported, in part, by the NFSC-United Fusion Fund of Zhejiang under Grant U1809212 and supported, in part, by the NFSC-United Fusion Fund of Shandong under Grant U1906218. It was also supported, in part, by the Heilongjiang Natural Science Fund under Grant ZD2020D001.

The authors declare no conflict of interest.

Representative X-ray images of Pacific hake: (

Coordinate rotation from x-y plane to u-v coordinates (left) and a lateral view of a digitized fish image (fish body and

TS backscatter curves for three individual (fish (

Digital images and predicted backscatter curves of two 200 mm-length Pacific hake (

Comparison of TS frequency response for four groups of 200 mm Pacific hake using the coherent KRM model (red) and incoherent KRM model (blue). (

Comparison of ranges of TS from multiple fish using coherent and incoherent Kirchhoff-ray-mode backscatter models.

Box and whisker plots of (

Fish lengths (FL) and

FL (mm) | ||
---|---|---|

1 | 191 | 28 |

2 | 198 | 42 |

3 | 200 | 38 |

4 | 198 | 35 |

5 | 202 | 43 |

6 | 208 | 43 |

7 | 205 | 46 |

8 | 203 | 42 |

9 | 205 | 41 |

10 | 210 | 42 |

Mean | 202 | 40 |

Std | 5.25 | 4.90 |

Differences between coherent and incoherent KRM model predictions for three Pacific hake (

Fish A (TS_{SWB}_{fb} |
Fish B (TS_{SWB}_{fb} |
Fish C (TS_{SWB}_{fb} |
||||
---|---|---|---|---|---|---|

KRM (coh) | KRM (inc) | KRM (coh) | KRM (inc) | KRM (coh) | KRM (inc) | |

TS_{fb} |
<−58> | <−59> | <−59> | |||

TS_{SWB} |
<−58> | <−58> | <−55> | |||

TSmax (dB) | −53 | −55 | −53 | −55 | −52 | −54 |

TSmin (dB) | −64 | −58 | −62 | −58 | −60 | −60 |

Span (dB) | 11 | 3 | 9 | 3 | 8 | 6 |

Differences in predicted target strengths (dB) predicted using a coherent and incoherent KRM model.

Fish A | Fish B | |||
---|---|---|---|---|

Coherent KRM | Incoherent KRM | Coherent KRM | Incoherent KRM | |

TS max (dB) | −35 | −37 | −34 | −36 |

TS min (dB) | −59 | −54 | −46 | −46 |

Range (dB) | 24 | 17 | 12 | 10 |

mean (dB) | −42 | −42 | −39 | −39 |

median (dB) | −40 | −40 | −39 | −38 |

Std | 6.003 | 4.858 | 3.402 | 3.151 |

Target strength (TS) value differences of four group sizes from Coherent (Coh) and Incoherent (Incoh) Kirchhoff-ray-mode (KRM) backscatter models.

Three Fish | Five Fish | Seven Fish | Ten Fish | |||||
---|---|---|---|---|---|---|---|---|

Coh KRM | Incoh KRM | Coh KRM | Incoh KRM | Coh KRM | Incoh KRM | Coh KRM | Incoh KRM | |

TS max (dB) | −36 | −37 | −36 | −38 | −36 | −38 | −46 | −40 |

TS min (dB) | −62 | −48 | −50 | −45 | −49 | −44 | −38 | −44 |

Range (dB) | 26 | 11 | 14 | 7 | 13 | 6 | 8 | 4 |

Mean (dB) | −42 | −41 | −41 | −41 | −42 | −41 | −41 | −41 |

Median (dB) | −40 | −40 | −41 | −42 | −41 | −42 | −41 | −41 |

Std (dB) | 14.786 | 9.634 | 13.91 | 6.939 | 12.96 | 6.393 | 7.891 | 3.781 |