Bellhop3D User Guide 2016 7 25

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Heat, Light, and Sound Research, Inc.

BELLHOP3D User Guide
Michael B. Porter
July 25, 2016

Heat, Light, and Sound Research, Inc.
com www.hlsresearch.com

12625 High Bluff Dr. Suite 211 San Diego, CA 92130

T +001 858 457 0800

mikeporter@hlsresearch.-

Heat, Light, and Sound Research, Inc.

Table of Contents

BELLHOP3D User Guide

Heat, Light, and Sound Research, Inc.

BELLHOP3D USER GUIDE

I. Introduction

II. Under the Hood: the BELLHOP algorithm

A. Introduction: Beam types and overview of beam tracing

B. Theory

C. Geometric Beams

D. The Compound Matrix Method

E. Beam changes across interfaces

III. Running BELLHOP3D

A. Environmental Information: Basic input file

B. Environmental Information: Bathymetry

C. Environmental Information: Oceanography

D. Ray trace run: Nx2D

E. Ray trace run: Full 3D Mode

F. Transmission Loss

IV. Test Cases

A. Free-Space and Half-Space Propagation

B. The Perfect Wedge

C. Truncated Wedge

D. Seamount

E. The Harvard Case Eddy Scenario

F. Rotated Munk Profile

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G. Korean Seas

H. Incoherent and Semi-Coherent TL options

V. Summary

Acknowledgments

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BELLHOP3D USER GUIDE
I. Introduction
BELLHOP3D is a beam tracing model for predicting acoustic pressure fields in ocean
environments. It is an extension to 3D environments of the popular BELLHOP model and includes
(optionally) horizontal refraction in the lat-long plane. Of course, 3D pressure fields can be
calculated by a 2D model simply by running it on a series of radials (bearing lines) from the source.
This is the so-called Nx2D or 2.5D approach. However, that approach neglects the refraction of
sound energy out of the vertical plane associated with each bearing line. Such out-of-plane effects
can be important when there are significant horizontal gradients in the environment. These occur
with strong oceanographic features such as nonlinear internal waves, or in areas with strong
bathymetric features. This is currently an active area of research.
A very preliminary research version of BELLHOP3D was written in 1985 in FORTRAN. However, it
did not allow environmental information (sound speed and bathymetry) to be read in from input
files. Instead, the research version required a user-defined analytic function for the sound speed,
c( x, y, z ), as a function of latitude, longitude, and depth. The research version also did not allow
variable bathymetry.
Separately a Matlab conversion of the original research code was done around 2004. The Matlab
environment is much easier to develop in; however, we have chosen not to build off of that here,
since the Matlab code runs much more slowly (about 50 times slower).
The beam tracing structure leads to a particularly simple algorithm. Several types of beams are
implemented including Gaussian and hat-shaped beams, with both geometric and physics-based
spreading laws. BELLHOP3D can produce a variety of useful outputs including transmission loss,
eigenrays, arrivals, and received time-series. It allows for lat-long variation in the top and bottom
boundaries (altimetry and bathymetry), as well as full 3D variation in the sound speed profile.
Additional input files allow the specification of directional sources as well as geoacoustic properties
for the bounding media. Top and bottom reflection coefficients may also be provided. BELLHOP3D
is implemented in Fortran with Matlab wrappers to display the input and output. The code

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conforms closely to standards and can be used on a variety of platforms (Mac, Windows, and
Linux). This report describes the code and illustrates its use.

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II. Under the Hood: the BELLHOP algorithm
A. Introduction: Beam types and overview of beam tracing

BELLHOP3D includes 4 different types of beams:
• Cerveny Beams: We use this term to refer to the original beams derived in the 1982 paper by
Cerveny, Popov, and Psencik. These beams evolve roughly as a true propagating beam would;
however, the beam field is only accurate near the axis of the ray which can cause problems.
• Geometric Hat-Beams: Much more reliable than the Cerveny beams; recovers conventional ray
theory but with the advantages of a beam-tracing algorithm. Good framework for algorithmic
testing because any flaws are obvious.
• Geometric Gaussian-Beams: Much improved accuracy relative to the hat beams with some
increase in run time.
• Geometric Hat-Beams in Cartesian Coordinates: Improved efficiency compared to the hat
beams in ray-centered coordinates. Results are essentially identical to the ray-centered beams.
These last two options should generally be the standard for use with BELLHOP3D; however, we
provide the other choices for research purposes.
The initial version of BELLHOP3D has been developed using what we call ‘Cerveny’ beams. This
style of Gaussian beam follows the classic 1983 paper by Cerveny, Popov, and Psencik in which
the beams evolve according to physical laws. In other words, they focus and defocus roughly as a
real beam in space would do. The reason for starting with this formulation is essentially historical.
In practice, with our 2D version of BELLHOP we had implemented quite a few different types of
beams. Originally BELLHOP (2D) also started off with the Cerveny beams. The first change we did
was to implement a formulation of the Cerveny beams is Cartesian coordinates rather than raycentered coordinates. Ray-centered coordinates are the arclength, s, and normal distance from the
ray, n. Cartesian coordinates are the range coordinate, r, and the vertical distance, delta z, from the
ray. The representation in Cartesian coordinates leads to a much more efficient code for evaluating
the contribution of a given beam to the total acoustic field.
The next stage in the evolution of BELLHOP was to implement ‘geometric’ hat-shaped beams.
These beams are not physical per se, but instead spread according to the expansion of a ray-tube.
By using a hat-shape (triangle) for each beam one obtains a construction inspired by the finiteelement literature. The result recovers precisely the ray-theoretic result. However, the beam

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implementation provides an elegant framework for the implementation and yielded a ray-tracing
result that was free of many of the traditional problems of ray models.
Finally, we replaced the hat-shaped beams with Gaussian beams but still implement with a
geometric spreading function. Further, we implemented a stop on the minimum width of the beams
to eliminate caustics in the ray field.
In summary, the 2D version of BELLHOP has beams implemented in both ray-centered and
Cartesian coordinates. It also has beam types that spread physically (Cerveny) or geometrically.
The geometric beams are either hat-shaped or Gaussian. Finally, they may optionally have a limit
on the minimum width.
The obvious question then is which is the best? The answer is that we generally obtain the best
accuracy with geometrically-spreading Gaussian beams that have the minimum-width constraint.
However, we also often use the geometrically-spreading hat-shaped as they are faster. They are
faster because with the Gaussian beams one gets many contributing beams of a given class at a
specific receiver location; with hat-shaped beams, exactly two beams of a given class contribute.
To get the hat-shaped geometric beams to work, one must be extremely careful in the
implementation. If the beams are too narrow, then gaps between the beams will show up in the
field plots. If they are too wide, then that is also easy to observe in the beam plots. Once they are
correctly implemented, then the Gaussian beam version is trivial to implement and smooths out
many of the flaws of ray theory.
A further subtlety that is unique to the 3D version of Gaussian beams is that they can be
implemented two ways. The traditional form from the literature solves the beam spreading
equations twice. The first solution gives the spreading in the vertical plane. The second solution
gives the spreading in the horizontal plane. These are called fundamental solutions.
In our original formulation we combined these using a technique that is not well-known, called
variously the delta-matrix formulation or compound-matrix formulation. This technique recognizes
that all the final quantities of interest to construct the Gaussian beams involve determinants of the
fundamental solutions. Rather than solve differential equations for the fundamental solutions, we
solve a new set of differential equations for the determinants of the fundamental solutions.
However, to simplify our development and debugging, we have gone back during to a formulation
using the fundamental solutions. This allows us to check the spreading behavior of the beams in
the various planes.
To illustrate the features of the different beam types we consider a canonical deep water sound
speed profile. First, we show in Fig. 1 how a Cerveny beam in BELLHOP3D looks in side view.
Note that the beam refracts within the water column and focuses and defocuses due to those
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same refractive effects. At the frequency used (50 Hz) the wavelength is about 30 m so it is difficult
to get beams narrower than a few hundred meters. (This is discussed in more detail in the latest
edition of Computation Ocean Acoustics.) The fact that the beams are this large is in fact one of
the limitations of the method, since the beam field is only accurate near the axis of the beam.

Figure 1: A Cerveny-style beam as first implemented in BELLHOP3D.

Figure 2 shows the new implementation of geometric, hat-shaped beams. In this case, the width of
the beams is determined entirely by the spacing of the fan of rays. As we increase the number of
rays that are traced, the beams get narrower and narrower. To check the implementation, we
compare the result to the geometric, hat-shaped beams in BELLHOP3D, but using the Nx2D
option (Figure 3). This is essentially the same as what is in the standard 2D version of BELLHOP
and has been tested extensively. We see that there is an excellent correspondence between the
two.
Finally, we compute the entire acoustic field by summing up the contributions of a fan of such
beams. The comparison of BELLHOP3D with geometric beams using the full 3D option and the
Nx2D option is shown in Figure 4 and is excellent. These results have also been checked again
normal mode and wavenumber integration solutions and are of high quality.

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Figure 2: A geometric beam in the full 3D option.

Figure 3: A geometric beam in the Nx2D option.

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Figure 4: Transmission loss calculated with geometric beams with the full 3D option.

Figure 5: Transmission loss calculated using geometric beams in the Nx2D option.

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B. Theory

The extension of Gaussian beams to three dimensions is described by Babich and Popov and
requires fairly minor modifications to the 2D algorithm. Let us go through the three steps required
for constructing the beam solution. As usual we begin by tracing a set of rays however in the 3D
case the rays form a fan over both azimuthal and elevation-declination angles. The ray equations
in 3D are given by

dx
= c (s),
ds
dy
= c (s),
ds
dz
= c (s),
ds

d
=
ds
d
=
ds
d
=
ds

1 dc
,
c2 dx
1 dc
,
c2 dy
1 dc
,
c2 dz

where c( x, y, z, ) is the ocean sound speed ( x( s), y( s ), z( s ) ) is the ray trajectory. This first-order
system of ordinary differential equations (ODEs) is integrated using a simple second-order RungeKutta method. The initial conditions prescribe that the rays emanate from the source position (xs,
ys, zs) and with take-off angles alpha and beta corresponding to the declination angle and the
azimuthal angle of the ray:

x(0) = xs ,
y(0) = ys ,
z(0) = zs ,

1
cos cos ,
c(0)
1
=
cos sin ,
c(0)
1
=
sin .
c(0)
=

The Gaussian beam is constructed around a central ray and defined in terms of ray-centered
coordinates (s, m, n) as indicated in the figure. Here s is the arclength along the ray, and (m,n) are
normal distances from a field point to the central ray. It is usually not an important issue; however,
this coordinate system has a zone of regularity around the central ray where a receiver point has a
well-defined ray-centered coordinate. However, for some other receiver points there is more than
one normal from the ray to the receiver. For instance if the ray is a circular arc, then radials from the
center of the circle are all valid normals.

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To be specific, the values of (m,n) are defined as normal distances in the direction of the following
two normal vectors to the ray:

ẽm =(L
L
ẽn =(L
L
and L
"

1
1

[c
[c

cos + sin ],
cos
sin

sin ], cL cos ),
cos ],

sin + cos ], cL cos ),

. (These formulas are derived in Cerveny and Hron. This ray-centered

coordinate system (" t̃, ẽm , ẽn ) is a rotating trihedral with rotation angle "

satisfying the differential

equation:

d
1 ( cx
=
2+
ds
c(s)

cy )
2

which must also be integrated along the central ray of the beam. This is one of many ways of
calculating the rotating trihedral.

The coordinate system defined by the rotating trihedral is magical. As discussed by Popov (1977)
we can use Hamilton’s Principle to derive equations that characterize how a ray displaces as we
make infinitesimal changes to its original position or take-off angles. These equations are amazingly
simple when presented in the coordinate system of the rotating trihedral. Furthermore, the way the
ray displaces characterizes the spreading of the ray tube and therefore the intensity along the
central ray.
The construction of beams in the neighborhood of the central rays requires the integration of a
system of auxilliary equations

dP
1
=
VQ
ds
c2
dQ
= cP
ds
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where V is a matrix of curvatures of the sound speed taken in two normal directions:

V =

cnn
cmn

cnm
cmm

q̃1
q̃2

q̂1
q̂2

P =

p̃1
p̃2

p̂1
p̂2

The derivatives cnn, cmm, etc indicate derivatives in direction of the normal vectors ẽm , ẽn. We
can rewrite these in terms of the x, y, z derivatives of the sound speed as:

cnn =cxx e21x + cyy e21y + czz e21z + 2cxy e1x e1y + 2cxz e1x e1z + 2cyz e1y e1z
cmn =cxx e1x e2x + cyy e1y e2y + czz e1z e2z +
cxy (e1x e2y + e2x e1y ) + cxz (e1x e2z + e2x e1z ) + cyz (e1y e2z + e2y e1z )
cmm =cxx e22x + cyy e22y + czz e22z + 2cxy e2x e2y + 2cxz e2x e2z + 2cyz e2y e2z

The P-Q differential equation tells us how the ray is perturbed due to a change in the ray initial
condition (either by displacing the source position or changing the ray angle). To obtain Gaussian
beams, we use the initial conditions:

p̃1
p̃2
q̃1
q̃2

BELLHOP3D User Guide

p̂1
p̂2
q̂1
q̂2

1
0
1

0
1
0
2

"10

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where " 1,2 are the beam constants that control the initial beam widths in the two normal directions
to the ray. The epsilons are in general complex numbers and their real and imaginary parts allow
independent control of both the beam width and the beam curvature (that is, the curvature of its
wavefronts).
Once we have integrated these equations along the ray we form a Gaussian beam as:

ubeam (s, m, n) =

= PQ

The expression for "

p̃1 q̂2
p̃2 q̂2

1
|Q(s)|
p̂1 q̃2
p̂2 q̃2

ei [

(s)+ 21 q t (s)q ]

p̃1 q̂1 + p̂1 q̃1
p̃2 q̂1 + p̂2 q̃1

/|Q|

has been obtained here by using the cofactor representation of the inverse of

Q. In addition |Q| denotes the determinant of Q. In general the determinant of Q is a complex
number that wraps around the origin. To get the correct branch of the square root we need to track
these rotations. This produces the so-called KMAH index that accounts for phase changes at
caustics.

C. Geometric Beams

As discussed earlier the geometric beam is designed to have a beam width that expands and
contracts in proportion to the ray tube. The hat-shaped beam decays linearly away from the central
ray. One may envision a beam that has an amplitude decay in the form of a pyramid; however,
since the ray tube does not necessarily have a square cross-section the beam shape is really
tetrahedral. The formula for the hat-shaped beam is simply:

ubeam (s, m, n) =

1
|Q(s)|

(s) [L1 (s)

n] [L2 (s)
L1 (s)L2 (s)

for |n|< L1 and |m|
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