In the present report it has been shown that the Nordic comprehensive model for straight
line propagation elaborated previously can be modified to include the effect of a moder-
ately refracting atmosphere by replacing the straight rays by curved rays. The curved
rays are predicted according to the heuristic model principle.

In the case of a strongly refracting atmosphere the effects of multiple ground reflections
(downward refraction) and shadow zones (upward refraction) may turn up. In these
cases the refraction problem can no longer be solved by simple geometrical modifica-
tion of rays but calls for a real extension of the models. Such an extension has been pro-
posed in the present report.

In the heuristic model the actual sound speed profiles has to be approximated by a linear
sound speed profile. The task of elaborating a procedure for approximating a non-linear
sound speed profile by an equivalent linear profile has been described in a parallel re-
port.

The method for including effects of refraction has not been validated by comparison
with measurements but comparison with results predicted by the Parabolic Equation
method for flat terrain has been carried out satisfactorily. However, further validation by
measurements or accurate prediction methods is strongly needed and might lead to
model adjustments.

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Introduction

In 1998 a comprehensive model for propagation of sound in an atmosphere without sig-
nificant refraction has been elaborated and is described in [1]. In this model, which is
based on geometrical ray theory combined with theory of diffraction, the sound rays are
assumed to follow straight lines.

A simple approach to include the influence of weather is to model the effect of refrac-
tion by curved sound rays. For simple sound speed profiles it may be assumed that the
sound speed varies linearly with the height above the ground in which case the sound
rays will travel along circular arcs. Such a simplification is the basis of the "heuristic"
model proposed by L'Espérance [2]. Via the ray curvature the heuristic model in a sim-
plified way combines the principles of linear ray acoustics with the effect of the
weather.

The heuristic model by L'Espérance [2] has been investigated during 1998 as described
in [3]. However, in the model only propagation over a flat terrain has been considered.
Therefore, the possibilities of using the heuristic principle in case of screens has been
studied in [4] leading to a proposed model.

The aim of the Nordic project is to develop prediction models with sufficient accuracy
for "good-natured" weather. Good-natured weather is weather where the sound speed as
a function of the altitude is either decreasing or increasing monotonically without sig-
nificant jumps in the sound speed gradient. Most often good-natured weather is repre-
sented by an approximately logarithmic sound speed profile. The heuristic model con-
cept is not expected to be applicable in case of irregularly shaped sound speed profiles.
The range of application of the heuristic approach is more thoroughly discussed in [5].

Considerations concerning the possibility of using the heuristic principle in the compre-
hensive model was initially discussed in 1998 [6]. The present report contains the final
principles elaborated during 1999 for including weather effects in the comprehensive
model.

The very crucial point in the heuristic model concept is the procedure for approximating
a non-linear sound speed profile by an equivalent linear profile. It has been decided to
keep the description of the elaborated procedure in a separate report [5] as the procedure
still is object of discussions and possible future improvement. Comparison between pre-
dictions made by the method outlined below and predictions by the Parabolic Equation
method can also be found in [5].

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Air Absorption

In [1] the effect of air absorption is determined for a homogeneous atmosphere based on
air temperature and relative humidity. For an inhomogeneous refractive atmosphere the
two parameters may vary along the propagation path. In the method described in [7]
which is used in [1] a method is presented for a layered atmosphere but this method has
been found too advanced, partly because predictions shall be made for good-natured
weather only, partly due to the great uncertainty of weather parameters.

As a simple approach it is proposed to use the average values of weather parameters
along the rays calculated according to the heuristic model extended to include screens.
Spatial information on relative humidity will seldom be available and a value obtained
in a single or a few points will therefore often be used as representative of the entire
propagation path. However, the average temperature along the ray path can easily be
obtained from parameters which are already available in the heuristic model for other
purposes. This concerns the travel time

and the length of the ray R between source and

receiver (or screen tops). The average sound speed



c along the path can be calculated

by Equation (1) and hence the average temperature



t in

C corresponding to this sound

speed can be determined by Equation (2).

(1)

273

−

÷÷ø

ççè

(2)

Terrain and Screen Effects

The comprehensive model for terrain and screen effects assuming straight line propaga-
tion described in [1] can easily be modified to include curved rays according to the heu-
ristic model principle as long as the number of rays in the base models remains un-
changed. This is fulfilled when the weather conditions are not causing multiple ground
reflections (strong downwind) or shadow zones (strong upwind). In these cases the re-
fraction problem can no longer be solved by simple geometrical modification of rays but
calls for a real extension of the models.

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In the comprehensive model a prediction for a complex terrain is carried by one of three
propagation models named “flat terrain”, “valley-shaped terrain” or “hill-shaped terrain”
or by a combination of the three models.

The three propagation models are according to [1] founded on three base models:

1) Flat terrain

2) One screen with a flat reflecting surface before and after the screen

3) Two screens with a flat reflecting surface before, after and between the screens

Therefore, prediction for any complex terrain is always the result of predictions by the
three base models combined in a suitable manner by the Fresnel-zone interpolation prin-
ciple and the model transition principles described in [1]. This implies that, if the prob-
lem of using curved rays is solved for each of these basic models, and the Fresnel-zone
interpolation principle and model transition principles are modified to deal with curved
rays, the problem has been solved for the entire comprehensive terrain and screen effect
model.

3.1

Flat Terrain

As in the non-refraction case the model for flat terrain and moderate refraction contains
two rays (a direct and a reflected ray) as shown in Fig. 1 and 2.

Figure 1
Ray model for flat terrain and downward refraction.

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Figure 2
Ray model for flat terrain and upward refraction.

Modifications according to the heuristic principle concern the travel time

and distance

R from the source to the receiver (

and R

for the direct ray and

and R

for the re-

flected ray) and the ground reflection angle

as described in [2]. The free-space

Green's function R

-1

jkR

where k is the wave number, is modified by replacing kR by

where f is the frequency. The spherical reflection coefficient Q is calculated based

on the modified values of R

and

In case of strong downward refraction additional rays will occur in the model for flat
terrain. A method for including the effect of multiple rays in excess of the two rays al-
ready included in the modified ray model has been developed in [8]. The contribution
from the multiple reflection model is added incoherently to the contribution of the modi-
fied comprehensive model.

In the case of strong upward refraction no ray reaches the receiver in the model for flat
terrain, resulting in an acoustical shadow zone as shown in Fig. 3.

Shadow
zone

Figure 3
Acoustical shadow zone.

In [4] it has been proposed that the difficult problem of a meteorologically generated
shadow zone in the case of propagation over flat terrain is considered analogous to the
problem of predicting sound levels in the shadow zone behind a diffracting wedge in

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case of a non-refracting atmosphere. The task has been to elaborate a method based on
this idea which fits into the structure of the comprehensive model and which does not
introduce a discontinuity at the edge of the shadow zone. It has also been necessary in
the elaboration of a method to consider the applicability in the presence of screens.

At the edge of the shadow zone the direct and reflected ray become identical and graze
the ground. The grazing angle of the reflected ray becomes 0 in this case. Therefore the
sound pressure at the receiver can be expressed by Equation (3) as p

= p

. Q is deter-

mined using the curved path length R of the direct ray and

= 0. The horisontal dis-

tance d

from the source to the reflection point when the receiver is at the edge of the

shadow zone can be calculated by Equation (4) given that the total horisontal propaga-
tion distance is substantially less than the ray radius of curvature.

(

)

(3)

(4)

To avoid discontinuities it has been decided to divide the ground effect in the shadow
zone into a reflection effect contribution

∆

and a shadow zone shielding effect contri-

bution

∆

as shown in Equation (5).

∆

(5)

∆

is calculated based on Equation (3). p

is calculated using the path length R for the

ray from the source to the receiver disregarding the ground as shown in Fig. 4. Q is cal-
culated on the basis of R and

= 0. Although R will increase somewhat the more the

receiver moves into the shadow zone, the effect on p

and Q and therefore on

∆

will

be almost negligible.

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Figure 4
Ray used to calculate the reflection effect contribution in the shadow zone.

The shielding effect

∆

is determined using a modification of the Hadden/Pierce so-

lution applied in the in the Nordic propagation model for wedge-shaped screens [9]. The
Hadden/Pierce solution is a four rays model where the finite impedance of the wedge is
taken into account by applying a spherical reflection coefficient to the image rays. The
first term in the Hadden/Pierce solution (n = 1 in Equation 3.16 of [9]) is interpreted as
the direct diffraction ray and the other three terms (n = 2, 3, and 4) are interpreted as
reflected diffraction rays. As the ground reflection has already been included in the term
1+Q in the calculation of

∆

the contributions of the reflected rays should be omitted

in the calculation of

∆

. Furthermore the equation used to calculate

∆

should be

normalized to produce a value of 0 for at 180° wedge.

When the receiver is in the shadow zone, the vertical distance h

between the ground

surface and the ray from the source S to the receiver R at the horisontal distance d

from the source as shown in Fig. 4 is used to define an equivalent wedge. The top of the
wedge is placed h

above the line from the S to R at the distance d

from the source

measured along the line, and the wedge legs are passing through S and R as shown in
Fig. 5.

Figure 5
Wedge used to calculate the shielding effect contribution in the shadow zone.

The diffracted sound pressure p

for the wedge in Fig. 5 can be predicted by Equation

(6) where D

is the diffraction coefficient defined in [9] but including only the first

term in the Hadden/Pierce solution (n = 1). If

∆

is defined as the diffracted sound

pressure level determined on the basis of Equation (6) relative to the level for a 180°

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wedge,

∆

can be calculated by Equation (7). R is the direct distance from the source

to the receiver and R

is the distance over the top of the wedge. The constant 2 ensures

that

∆

is 0 for a 180° wedge where D

becomes 0.5.

jkR

(6)

÷÷ø

ççè

log

∆

(7)

3.2

Terrain with a Single Screen

For moderate downward and upward refraction the model for a single screen consists of
4 rays as shown in Fig. 6 and 7 in the same way as in the model for non-refracting at-
mosphere.

p ,p

G,1

G,2

Figure 6
Ray model for a single screen and downward refraction.

p ,p

G,1

G,2

Figure 7
Ray model for a single screen and upward refraction.

The screen model for a refracting atmosphere proposed in [4] which is based on the heu-
ristic model approach, concerns modification of the travel time

and distance R be-

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tween source, screen top and receiver and of the ground reflection angles before and
after the screen in the same way as for flat terrain. In addition to that modified diffrac-
tion angles are determined on the basis of curved rays for use in the Hadden/Pierce dif-
fraction solution [9].

In the case of strong downward refraction additional rays will occur due to multiple re-
flections before or after the screen. A method for including the effect of multiple rays in
excess of the 4 rays already included in the ray model has been developed in [8]. The
contribution from the multiple reflection model is added incoherently to the contribution
of the modified comprehensive model.

In the case of strong upward refraction the ray from the source to the top of the screen or
from the top of the screen to the receiver may be blocked by the ground, resulting in
acoustical shadow zones. The problem of shadow zones in the case of a screen has been
solved analogously to flat terrain. If a shadow zone occurs before, after, or on both sides
of the screen, the combined ground and screen effect is determined by Equation (8) to
(10), respectively. The sound pressures p

to p

correspond to the rays defined in Fig. 7

and p

is the free-field sound pressure. Q

and Q

are the spherical reflection coefficients

before and after the screen.

∆

SZ,1

is the excess shielding effect on the source side of the

screen and is calculated using the same procedure as for flat terrain but with the receiver
placed on the top of the screen.

∆

SZ,2

is in the same way the excess shielding effect on

the receiver side of the screen but with the source placed on the top of the screen.

(

)

∆

log

∆

÷÷ø

ççè

(8)

(

)

∆

log

∆

÷÷ø

ççè

(9)

(

)(

)

∆

log

∆

(10)

3.3

Two Screens

For moderate downward and upward refraction the model for double screens includes 8
rays as in the model for a non-refracting atmosphere.

The heuristic modifications in the double screen model concern the travel time

and

distance R between source, screen tops and receiver, the ground reflection angles be-

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fore, after and between the screen and the diffraction angles in the same way as for the
single screen case.

In case of strong downward refraction the effect of additional rays in excess of the 8
rays already included in the ray model will follow the principle outlined in [8]. In the
same way as for the single screen case the contribution from the multiple reflections is
added incoherently to the contribution of the modified comprehensive model.

In case of strong upward refraction shadow zones may occur before, after, or between
the screens. The combined ground and screen effect is determined by equations follow-
ing the same principle given by Equation (8) to (10). Instead of the three possible com-
binations for a single screen expressed by Equation (8) to (10), there will be seven pos-
sible combinations in case of a double screen. The excess shielding effect before the
first screen and after the second screen is calculated corresponding to a single screen
and the excess shielding effect between the screens is calculated by replacing source and
receiver by the screen tops.

3.4

Fresnel Zone

As mentioned earlier, prediction by the propagation model for a valley- and hill-shaped
terrain is the result of predictions by the three base models combined according to the
Fresnel-zone interpolation principle. This implies that the Fresnel-zone interpolation
principle has to be modified to deal with curved rays. It is assumed that the Fresnel-zone
ellipsoid has to be bent so that its axis of rotation follows the curved ray. As shown in
Fig. 8 the size of the Fresnel-zone is calculated using the algorithms for straight line
propagation but for a modified image source position S’’ and a modified receiver posi-
tion R’. The angle between the line S’’R’ and the ground surface is equal to the ground
reflection angle

of the curved ray. The distance from the reflection point to S’’ is

equal to the distance to the source S measured along the curved ray, while the distance
from the reflection point to R’ in the same way is equal to the curved distance to the re-
ceiver R.

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S’

S’’

R’

Figure 8
Determination of the Fresnel-zone in case of curved rays.

3.5

Transition between Propagation Models

The transition between the models for flat terrain and non-flat terrain depends in the
non-refraction case on the change in

∆

corresponding to the average deviation of the

terrain

∆

h from the equivalent flat terrain (as described in [1]). Same principle is used in

the refraction case but

∆

is determined on the basis of curved rays.

The transition between screened and unscreened terrain as well as between the single
and double screen cases are in the non-refraction case based on the path length differ-
ence

∆

l and on the height of the screen above the ground compared to the wavelength

and to effective width of the sound field at the screen as described in [1]. The path
length difference is defined as the difference in length of the path from source to the re-
ceiver via the top of the screen and the length of the direct path. If the top of the screen
is below the line-of-sight the path length difference is represented by a negative value.

In the refraction case, the transition terms which are based on the height of the screen
are unchanged whereas the term based on the path length difference is modified to take
the curved rays into account. In the light of the very approximative nature of the transi-
tion equations, the calculation of the path length difference has been simplified as de-
scribed in the following.

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The geometrical parameters used when predicting the path length difference

∆

l are de-

fined in Figure 9 (for downward refraction). The top and bottom point of the screen are
denoted T and G, respectively. The intersection between the screen (or vertical exten-
sion of the screen) and the direct ray path (curved due to refraction) is denoted P
whereas the intersection with the line-of-sight is denoted L.

Figure 9
Definition of geometrical parameters in the calculation of path length difference.

In the case of downward refraction (P above L)

∆

l is calculated by Equation (11) if T is

above L and by Equation (12) if T is below L. In the case of upward refraction (P below
L)

∆

l is calculated by Equation (13) if T is above L and by Equation (14) if T is below

∆

−

≥

(11)

∆

−

(12)

∆

−

≥

(13)

∆

−

(14)

If the relative sound speed gradient a’ [2] is positive (downward refraction) and h

less than or equal to h



is determined by Equation (15).

’

is the angle between

the ray and the horizontal direction at the source or receiver which ever is the lowest [2]
and d

SCR

is the horizontal distance from the source to the screen. If h

is greater than h

the same equation is used except d

SCR

is replaced by d-d

SCR

and h

is replaced by h

. d is

the horizontal distance from source to receiver.

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∧

≤

SCR

tan

cos

−

÷÷ø

ççè

(15)

If a’ is negative (upward refraction) and h

is greater than or equal h



is deter-

mined by Equation (16). If h

is less than h

the same equation is used except d

SCR

replaced by d-d

SCR

and h

is replaced by h

. d is the horizontal distance from the source

to the receiver.

∧

≤

SCR

tan

cos

−

÷÷ø

ççè

−

(16)

Equations (15) and (16) are taken from [10].

3.6

Atmospheric Turbulence

A method for taking into account the energy scattered from atmospheric turbulence into
the shadow zone of a screen has been proposed in [4] and adopted in the comprehensive
propagation model for an atmosphere without refraction [1]. In the method the contri-
bution of scattered energy is added incoherently to the contribution from the screen
model. The prediction is based on geometrical parameters determined for the top of the
screen in relation to the line-of-sight.

It has been found that no modification should in principle be applied to the model of [4]
to account for propagation in a refractive atmosphere. However, the model in [4] is pre-
scribed to be used only when the top of the screen is above or at the line-of-sight be-
cause the contribution has been assumed to be insignificant when the screen is below the
line-of-sight. In case of upward refraction significant shadow zones may occur behind a
screen lower than the line-of-sight or even in the case of flat terrain. It will therefore be
necessary to take into account the contribution from turbulent scattering also in these
cases.

It is proposed that when the top of the screen is below the line-of-sight, the contribution
is predicted as if the screen top was at the line-of-sight (h

= 0 in [4]).

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3.7

Finite Screens

Finite screens are in the comprehensive model taken into account by applying the solu-
tion for an infinite screen but adding the contribution from sound diffracted around the
vertical edges of the screen. This is done practically by adding an extra propagation path
from the source via the vertical edge of the screen to the receiver. The diffracted sound
pressure for each side of the screen is calculated by multiplying the sound pressure cal-
culated using the propagation conditions along the path by the diffraction coefficient of
the vertical edge. The sound pressures diffracted around the vertical edges are added
incoherently to the sound pressure diffracted over the top of the screen.

As the path around the end of the screen is a breakline, the relative sound speed gradient
used in the heuristic modifications may vary along the path. This is solved by calculat-
ing a weighted average of relative sound speed gradient along the path by the same
method as introduced for a reflected ray path (described in Section 5).

Scattering Zones

In the model for predicting the effect of scattering zones one of the basic parameters
will be the distance through the scattering zone measured along the source-receiver ray
path. In the case of a refracting atmosphere this distance has to be measured along the
curved path in stead. Otherwise the model will be identical to the model for a non-
refracting atmosphere.

Reflections

When the propagation path is a path generated by the reflection from an obstacle the
direction of propagation before and after the reflection will change relative to the direc-
tion of the wind. The implication is that the relative sound speed gradient (denoted “a”
in [2]) and consequently the radius of curvature of the circular rays may change at the
reflection. If more than one reflection is involved the curvature of the rays may change
each time the ray is reflected. In case of strong sound speed gradients the propagation
may involve complicated cases of simple downward and upward reflection combined
with multiple ray propagation and shadow zone effects.

Initially it was expected that the effect of refraction could be taken into account for a
reflected path by changing the ray curvature piecewise depending of the direction of
propagation [6]. However, it has been found that the geometrical complexity increases

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very fast and the conclusion is that this approach in many cases hardly would produce
reliable results.

Considering the complexity of the problem combined with a complete lack of experi-
mental knowledge a simple straight-forward approach has been proposed. The value of
the relative sound speed gradient "a" and c(0) is determined for each segment of the re-
flected ray path and the average value of "a" and c(0), weighted by the horizontal length
of the segment d

, is determined by Equations (17) and (18). Then, the propagation ef-

fect for the entire reflected path is calculated assuming that the effect of refraction is
represented by the average value of "a" and c(0). The variables of Equation (17) are de-
fined in Fig. 19 in the case of a single reflecting surface. If the reflecting surface is close
to the source or receiver or the reflected ray path is close to the direct path this assump-
tion appear to be quite reasonable, and in most cases where a reflection is contributing
significantly compared to the direct sound these requirements are fulfilled. However, the
approximation in cases with downward refraction half the way and upward refraction
the rest of the way by no refraction will be very rough. Fortunately, such cases are of
little practical importance unless the direct path is affected by excessive attenuation
while the reflected path is not.

(17)

)

(

)

(

(18)

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S’

R’

Figure 10
Plan view illustrating different refrac-
tion before and after reflection from
obstacle.

Conclusions

In the present report it has been shown that the comprehensive model for straight line
propagation described in [1] can be modified to include the effect of a moderately re-
fracting atmosphere by replacing the straight rays by curved rays. The curved rays are
predicted according to the heuristic model principle.

In a strongly refracting atmosphere the effects of multiple ground reflections (downward
refraction) and shadow zones (upward refraction) may turn up. In these cases the refrac-
tion problem can no longer be solved by simple geometrical modification of rays but
calls for a real extension of the models. Such extensions have been proposed.

In the heuristic model the actual sound speed profile has to be approximated by a linear
sound speed profile. The task of elaborating a procedure for approximating a non-linear
sound speed profile by an equivalent linear profile has been described in a parallel re-
port.

The method for including effects of refraction has not yet been validated by comparison
with measurements but comparison with results predicted by the Parabolic Equation
method for flat terrain has been carried out satisfactorily. Further validation by meas-
urements or accurate calculation is strongly needed and might lead to adjustments.

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The content of the report is a proposal which has not been discussed by the Technical
Committee. After such a discussion alterations may take place.

References

[1] B. Plovsing, J. Kragh: ‘Prediction of Sound Propagation in an Atmosphere with-

out Significant Refraction, Outline of a Comprehensive Model’, DELTA Acous-
tics & Vibration Report AV 1818/98, Lyngby 1998.

[2] A. L'Espérance, P. Herzog, G. A. Daigle, and J. R. Nicolas: ‘Heuristic model for

outdoor sound propagation based on an extension of the geometrical ray theory in
the case of a linear sound speed profile’, Appl. Acoust. 37, 111-139, 1992.

[3] S. Å Storeheier: ‘Nord2000: Sound propagation under simplified meteorological

models: a heuristic approach’, SINTEF Report STF40 A98075, Trondheim 1998.

[4] M. Ögren: ‘Propagation of Sound - Screening and Ground Effect, Part 2: Re-

fracting Atmosphere’, Swedish National Testing and Research Institute, SP RE-
PORT 1998:40, ISBN: 91-7848-746-3, Borås 1998.

[5] B. Plovsing and J. Kragh: ‘Approximation of a Non-Linear Sound Speed Profile

by an Equivalent Linear Profile’, DELTA Acoustics & Vibration Report AV
2005/99, Lyngby 1999.

[6] B. Plovsing and J. Kragh: ‘Prediction of Sound Propagation in an Atmosphere

with Significant Refraction, Considerations concerning the Comprehensive
Model’, DELTA Acoustics & Vibration Report AV 1819/98, Lyngby 1998.

[7] P. D. Joppa, L. C. Sutherland and A. J. Zuckerwar: ‘Representative frequency ap-

proach to the effect of bandpass filters on evaluation of sound attenuation by the
atmosphere’, Noise Control Eng. J. 44, 261-273, 1996.

[8] M. Ögren: ‘Multi reflected rays in a refracting atmosphere - Nord 2000 Progress

report’. SP Technical Note 1999:28, Borås 1999.

[9] M. Ögren: ‘Propagation of Sound - Screening and Ground Effect. Part I: Non-

refracting Atmosphere’. Swedish National Testing and Research Institute, SP Re-
port No. 1997:44, ISBN: 91-7848-703-X, Borås 1997.

[10] B. Plovsing and J. Kragh: ‘Wind Turbine Noise Propagation Model’, DELTA

Acoustics & Vibration Report AV 1119/98, Lyngby 1998.

Document Outline

Summary
Introduction
Air Absorption
Terrain and Screen Effects
Scattering Zones
Reflections
Conclusions
References

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