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Estimation error in sound velocity depending on size of target scatterer
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REGULAR PAPER

Japanese Journal of Applied Physics 60, SDDE17 (2021)
https://doi.org/10.35848/1347-4065/abf39e

Estimation error in sound velocity depending on size of target scatterer
Aoi Nakayama1, Shohei Mori2*, Mototaka Arakawa1,2, and Hiroshi Kanai1,2
1Graduate School of Biomedical Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan
Graduate School of Engineering, Tohoku University, Sendai, Miyagi 980-8579, Japan2*

E-mail: mori@ecei.tohoku.ac.jp

Received December 7, 2020; revised February 18, 2021; accepted March 30, 2021; published online April 28, 2021
The quality of ultrasonic images can be improved by estimating the sound velocity accurately. Our previous study proposed a method to estimate
the sound velocity based on the difference between the reception times of radiofrequency signals received by elements in an ultrasonic probe.
Because the method assumed an ideal point scatterer as the target, the estimation error in the sound velocity increased with an increase in thetarget scatterer size. In the present study, the effect of the target scatterer size on the estimation method was examined, and the relationshipbetween the size of the target scatterer and the estimation error in the sound velocity was quantified. Through simulations and basic experiments, it
was confirmed that the estimation error was caused by the change in the reception time from the target surface and that the estimation error
depended on the depth and size of the target scatterer. © 2021 The Japan Society of Applied Physics

1. Introduction

Medical ultrasound is useful in the diagnosis of diseases andthe observation of various organs.1–3) Array probes are widelyused in ultrasonic medical diagnosis, and the transmitting andreceiving beams are formed by applying an appropriate delay
time to each element of the probe,4,5) assuming that the soundvelocity in vivo is 1540 m s−1.6–9) However, the actual soundvelocity in vivo depends on each tissue, therefore, the delay
time cannot be corrected accurately. This difference in soundvelocity deteriorates the image quality of the ultrasonictomographic image.7,10,11) The average sound velocities in fat(1450 m s−1) and muscle tissues (1647 m s−1) are signiﬁcantly
different from those in other tissues (1540 m s−1).12,13)
Therefore, deterioration of the image quality is especiallysigniﬁcant in the ultrasonic diagnosis of obese patients andbreasts.14) This problem can be solved by estimating the soundvelocity in vivo and reconstructing the received beam with anappropriate delay time.
Furthermore, it is expected that non-invasive diagnosis,
such as the diagnosis of liver disease, will be realized byestimating the sound velocity in vivo. Recently, the numberof patients with fatty liver has been increasing.15) It is known
that there is a correlation between the fat content of the livertissues and the tissue sound velocity.16–18) Therefore, if thesound velocity in the liver tissue can be estimated accurately,
non-invasive and quantitative diagnosis of fatty liver19) is
expected.
Many studies have been conducted to solve the problem of
quality deterioration of ultrasonic tomographic images owing
to the tissue dependence of the sound velocity in vivo. One of
the methods is the phase aberration correction method,20,21)
in which the delay set for each received beam is corrected bydetecting the phase shift of the radiofrequency (RF) signals
received by each element in the probe. While this method isadvantageous for correcting the time delay error with highaccuracy, it has the disadvantage of requiring a highcomputational capacity.19) In other methods,22–25) the focus
quality factor (FQF) is calculated in the region of interest set
in the measurement area. By changing the sound velocity
used to form the received beams, the optimal sound velocity
is determined such that the FQF is maximum. These methods
require a signiﬁcant computational load.

Methods for estimating not only the average soundvelocity but also the local sound velocity distribution havebeen proposed.26–28) In the method proposed by Ali et al.,26)
the average sound velocity was determined using thecoherence factor (CF), and the local sound velocity wasestimated using the average sound velocity. Sanno et al.27)
developed a method to estimate the sound velocity distribution using the signal-to-noise ratio factor in addition to the
CF. They demonstrated the effectiveness of each method
through experiments with a wire phantom.
Our group has proposed a method to estimate the
distribution of local sound velocity in each multi-layered
medium based on the difference among the reception times of
RF signals received by the elements in an ultrasonic probe.28)
However, this method assumed the ideal point scatterer as thetarget. In a previous study, the estimation error in the soundvelocity was always positive and increased as the diameter of
the target scatterer became larger.29) In our previous study,29)
the effect of the target scatterer size on the previously
proposed sound velocity estimation method28) was conﬁrmed
by simulations and experiments. However, the relationshipbetween the size of the target scatterer and the estimation
error was not quantiﬁed.
In the present study, we quantiﬁed the estimation error
inﬂuenced by the size of the target scatterer on the soundvelocity estimation method28) by approximating that the ﬁrst
arrival wave from the surface of the target scatterer was
dominant in the sound velocity estimation. From simulated
and basic experiments, the relationship between the target
scatterer size and the estimation error in the sound velocity
was conﬁrmed.
2. Principle and methods
2.1. Previous estimation method of sound velocity
assuming ideal point scatterer as target

We now brieﬂy explain the sound velocity estimation method
in a homogeneous medium using scattered waves from an
ideal point scatterer. In the previously proposed method, thesound velocity was estimated in each of the multi-layered
mediums.28) However, in the present study, the target
medium is assumed to be homogeneous to concentrate thediscussion on the effect of the size of the target scatterer. A
target ideal point scatterer is set at a depth d below the central

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A. Nakayama et al.

element of the linear probe, as shown in Fig. 1(a). The wave
scattered from the point scatterer is received by each elementin the probe. We describe the propagation time, T0 (xk ), as thetotal time during which a plane ultrasonic wave is transmitted
from the probe, scattered by the ideal point scatterer at (0, d ),and received by the kth element at ( xk , 0 ). T0 (xk ) isgeometrically given by
T0 (xk ) =d+

xk2 + d 2
c

,

(a)

(1 )

where c is the sound velocity in the medium. The square{T0 (xk ) - T0 (0) /2}2 is described by the following quadratic
function:{

T (0 )T0 (xk ) - 0
2

2

}

d2
1
= 2 xk2 + 2 = axk2 + b ,
c
c

(b)

(2 )

where a and b are the coefﬁcients of the quadratic function.
From the quadratic approximation applied to the square of themeasured propagation time {t (xk ) - t (0) /2}2 , the soundvelocity c and the depth d are estimated as follows:
cˆ =

1
,
a

(3 )

dˆ =

b
.
a

(4 )

2.2. Effect of scatterer size on sound velocity estimation

The target scatterer has a ﬁnite size in vivo. We considered theeffect of a ﬁnite-sized target scatterer with radius r on theestimated sound velocity using the aforementioned method. We
assumed that the focused wave was approximated by a plane
wave near the focal region, and only the scattered wave fromwithin the focal region was received by the ultrasonic element.
A schematic of the ﬁrst arrival wave from the surface of acylindrical scatterer with radius r is shown in Fig. 1(b). The
blue straight lines show the propagation paths of the scatteredwaves, each of which is transmitted from the central element,
scattered from the scatterer at point A (0, d), and received by thecentral element or kth element. The red straight lines show the
propagation paths of the waves scattered at the surface otherthan point A and received by the central element or kth
element.
The RF wave received by the kth element is composed of
scattered waves. When the target is an ideal point scatterer,that is, r is considerably smaller than the wavelength, the ﬁrst
arrival wave among the RF waves received by the kthelement is shown by the blue path. However, when the target
scatterer has a ﬁnite size, there is a ﬁrst arrival wave otherthan the ones indicated by the blue path, for example, shownby the red path in Fig. 1(b), and the ﬁrst arrival wave
indicated by the red path is received earlier than thatindicated by the blue path as r increases.
The curvature coefﬁcient a of the quadratic function in
Eq. (2) is determined by the difference between the arrival
times to the elements. When the scatterer has a ﬁnite size, abecomes smaller than that of the ideal point scatterer, as
shown by the dashed blue line in Fig. 1(b). Therefore, thesound velocity, estimated using Eq. (3), is higher than the(c)Fig. 1. (Color online) (a) Schematic of the reception of scattered waves
from an ideal point scatterer.29) (b) Schematic of the reception of scatteredwaves from the surface of a scatterer with a ﬁnite size. (c) The simulation
model in which multiple ideal point scatterers are on the scatterer surface.29)

true value as the scatterer size increases, and the estimation
error increases positively.
2.3. Quantification of relationship between target
scatterer size and error in sound velocity estimation
To quantify the estimation error described in the previous
section, we consider the ﬁrst arrival wave received by the kthelement, as shown in Fig. 1(c). For this purpose, let us
assume that the surface of the scatterer with a radius r iscomposed of many ideal point scatterers set on the surface ofthe scatterer at circumferential intervals Dl, as shown in
Fig. 1(c).
The blue straight lines show the propagation paths of the
wave transmitted from the central element, scattered at point

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A. Nakayama et al.

S0 , and received by the central element or kth element. The
blue dashed line shows the parabola of the propagation time
distribution when the scatterer is an ideal point. The length ofthe path L 0 (xk ) is geometrically expressed as
L 0 (x k ) = d 0 +xk2 + d 02 .

(5 )Meanwhile, the red straight lines show the propagation
path of the wave transmitted from the element above thepoint scatterer Sm , scattered at point Sm , and received by thekth element. The red dashed line shows the parabola of the
propagation time distribution of the ﬁrst arrival wave when
the scatterer has a radius r. The length of the red path
L (xk , xm¢ ; r , d 0 ) is geometrically expressed as
L (xk , x m¢ ; r , d 0) = d 0 + r +r 2 - x m¢ 2 ) 2 ,where xm¢ is the lateral position of the mth ideal point scatterer
Sm and is given bymDl ⎞xm¢ = r sin ⎛.
⎝ r ⎠The minimum of L min (xk ; r , d 0 ) is determined by
L min (xk ; r , d 0) = min L (xk , xm¢ ; r , d 0).

(7 )

(8 )

xm¢

We deﬁne the minimum propagation time of the red paths byTmin (xk ; r , d 0 ),Tmin (xk ; r , d 0) = min T (xk , xm¢ ; r , d 0) ,where T (xk , xm¢ ; r , d 0 ) represents the propagation time of the
red path in Fig. 1(c). The scatterer with r » 0 is an ideal point
scatterer. The square {Tmin (xk ; r » 0, d 0 ) - T0 (0; r » 0, d 0 ) /2}2
is approximated byTmin (0; r » 0, d 0 )2
d2
T0 (0) 2
1 2= 2 xk + 02 ,T0 (xk ) 2
c0
c0

{

2

}

}

Err (xk ; r, d 0 ) ^c - c0=c0r-

1
L min (xk ; r , d 0).
c01{L (xk , xm¢ ; r , d 0)c0
- L (0, 0; r , d 0)}.

tmk (xk , xm¢ ; r , d 0) =(14)

The received waveform at the kth element, composed of thescattered waves from multiple positions on the scatterer
surface, is obtained byM

å h {t - tmk (xk , xm¢ ; r , d 0)}.

(15)

m =-M

(10)

The ultrasonic propagation time t (xk ; r , d 0 ) is determined bydetecting the peak of the simulated waveformhk (t; xk , r , d 0 ) at the kth element. By approximating eachof the resultant distributions {tmin (xk ; r , d 0 )} and{t (xk ; r , d 0 )} by the quadrature function, the sound velocity
is estimated using Eq. (3).
2.5. Experimental method
Using wires of different sizes as the target scatterers, theeffect of the target scatterer size on the sound velocity
estimation was conﬁrmed through a basic experiment. In2
xk2 + d^r 2 - {xm’ ,min (xk ; r, d 0 )}2 +(13)

The simulated experiment was conducted under the following
conditions in the model shown in Fig. 1(c). Dl was set to
1 mm. The true value of the sound velocity was set to 1495 m
s−1. To simulate the wave scattered at the surface of thetarget scatterer, the experimentally measured waveform froma tungsten wire with a diameter of 0.03 mm in a water tank
was received at the central element and used as the ultrasoundpulse waveform h 0 (t ) scattered from the ideal point scatterer
and received by the 0th element at the position (0, 0).
The difference between the delay time of the scatteredwave from the mth point scatterer Sm and that of the scatteredwave from the 0th point scatterer S0 , received at the kthelement, is determined byh k (t ; x k , r , d 0 ) =where c0 is the true sound velocity in the medium.
Equation (10) is equal to Eq. (2) in the case where c = c0 ,and the sound velocity can be estimated using Eq. (3) as thetarget scatterer is an ideal point scatterer. For the scatterer
with r  0, {Tmin (xk ; r , d 0 ) - Tmin (0; r , d 0 ) /2}2 does notcoincide with Eq. (2).
From Eqs. (6)–(10), the relative estimation error
Err (xk ; r , d 0 ) of the sound velocity is given by:(12)

Equation (11) shows that the relative estimation error
Err (xk ; r , d 0 ) does not depend on the true sound velocity c0 ,and it is a function of the element position xk , the radius of
the scatterer r, and the depth of the scatterer d 0.
2.4. Simulated experimental method
From the simulated experiment, the minimum value,
tmin (xk ; r , d 0 ), of the propagation times of the ﬁrst arrival
waves that are transmitted from the position (xk , 0), scatteredat the target scatterers, and received by the kth element, isquantitatively determined by

(9 )

xm¢

»

xm¢

tmin (xk ; r , d 0) =(6 ){xm¢ , min (xk ; r , d 0) = argmin L (xk , xm¢ ; r , do).

r 2 - x m¢ 2

(xk - x m¢ ) 2 + (d 0 + r -Tmin (xk ; r » 0, d 0) -

where

- 1,

(xk - xm’ ,min (xk ; r, d 0 ))2 + (d 0 + r -SDDE17-3

r 2 - {xm’ ,min (xk ; r, d 0 )}2 )2(11)© 2021 The Japan Society of Applied Physics

Jpn. J. Appl. Phys. 60, SDDE17 (2021)

A. Nakayama et al.

Fig. 2. (Color online) Experimental conﬁguration.

the simulated experiment, the focused wave was approximated by a plane wave near the focal region; however, thefocused wave was transmitted in this basic experiment using
wires. The experimental setup is illustrated in Fig. 2.
Ultrasonic diagnostic equipment (Prosound α10, Hitachi
Aloka, Japan) was used with a linear probe (UST-5412,
Hitachi Aloka, Japan). The transmission frequency was set at
7.5 MHz, the sampling frequency was set at 40 MHz, and 96
elements were used for transmitting and receiving ultrasonicbeams.
A tungsten wire with a diameter of 0.03 mm, nylon wires
with diameters of 0.25, 0.50, 0.75, and 0.90 mm, and silicone
rubber wires with diameters of 2.0, 3.0, 4.0, 5.0, and 6.0 mm
were used as the target scatterers. The top surface of eachwire was set at a depth of 30 mm from the probe surface inthe water. The RF signals were acquired 10 times byrearranging the position of the wire for each measurement.
The focal point was also set to a depth of 30 mm.
The depth dependence of the relative estimation error wasinvestigated using a similar basic experiment by changing thetarget depth. Several wires with diameters of 0.03, 0.50, 0.90,
3.0, and 6.0 mm were used as the target scatterers. The top
surface of each wire was set at a depth of 40 mm from theprobe surface in the water. The focal point was set to a depthof 40 mm. Other conditions were the same as in the basic
experiment when the depth was set at a depth of 30 mm fromthe probe surface in the water.
3. Results and discussion
3.1. Determination of propagation time in basic experiment

The ultrasonic propagation time t (xk ) was determined bydetecting the positive peak of the received waveform at each
receiving element. The detected nth positive peak time
tpeak (xk ) of the waveform was corrected to the rise timet (xk ) of the pulse wave as follows:t (xk ) = tpeak (xk ) -n - 0.75
,
f0

(16)

where f0 is the transmission frequency. Figure 3(a) shows the
waveform of the scattered wave from the scatterer received
by the central element in the case of the scatterer with adiameter of 0.25 mm. Figure 3(b) shows the waveforms h (t )and h (t - tAP ), which simulated the scattered waves fromthe anterior and posterior walls of the scatterer, respectively,
where tAP is the delay in the reception time of the scatteredwaves from the posterior wall concerning that of the scattered

(a)

(b)

Fig. 3. (Color online) Received waveform at the central element (0.25 mm)(a) measured in the basic experiment and (b) generated by the simulated
experiment (red line: from the anterior wall, blue line: from the posteriorwall).29)

waves from the anterior wall. By comparing the waveforms
in Figs. 3(a) and 3(b), it was observed that the scatteredwaves from both the walls interfered.
Therefore, to suppress the interference, the position of thepeak (O) in Fig. 3(a) was determined for the scattered waves
from the anterior wall for a diameter of 2r ≥ 0.25 mm. For adiameter of 2r = 0.03 mm, however, the delay time between
the scattered waves from the anterior and posterior walls wasapproximately only a 0.1 period of the transmitted wave.
Therefore, the scattered waves from the anterior and posterior
walls could not be separated. Thus, the position of the peak
was determined for the interfered waveform of the wire. The
effect of the interference of the scattered waves on thedetermination of the propagation time for the thin wire
should be examined in the future.
3.2. Propagation time for ideal point scatterer andlarger scatterers
Figure 4 shows the propagation time distributions determinedusing Eq. (14) when the scatterer is an ideal point scatterer

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A. Nakayama et al.

with a diameter of 2r » 0 and when the diameter 2r is
6.0 mm. From Fig. 4, it was observed that the coefﬁcient a ofthe parabola of the propagation time distribution became
small when the diameter was 6.0 mm, which corresponded tothe cause of the increase in the estimation error of the soundvelocity described in Sect. 2.2.
3.3. Estimated sound velocities
Figure 5 shows the sound velocities estimated in thesimulated experiments and the basic experiments using wires.
The true sound velocity c0 was determined from thetemperature of the water.30) Since the water temperature
was different in each measurement, the true sound velocities
were determined as 1495 m s−1 (for 2r = 0.90– 6.0 mm),
1496 m s−1 (for 2r = 0.50, 0.75 mm), and 1497 m s−1 (for2r = 0.03, 0.25 mm). The averages and standard deviationsof the estimated sound velocities in 10 measurements areshown by red dots and error bars, respectively.
The estimation error in the sound velocity increased withthe diameter of the scatterer for both basic and simulated
experiments. The results from the simulated and basic
experiments were within 0.3%. From these results, it was
conﬁrmed that the increase in the sound velocity estimation
error, because of the increase in the diameter of the scatterer,
was caused by the ﬁrst arrival wave from the surface of thescatterer.
3.4. Relationship between size of target scatterer andestimation error in sound velocity
Figure 6 shows the relative estimation error Err (xk ; r , d 0 ) of
the sound velocity calculated using Eq. (11), and themeasured estimation error ratio (cˆ - c0 ) /c0 for the experimental results in Fig. 5. The averages and standard deviationsof the estimated sound velocities for the 10 measurements areindicated by red dots and error bars, respectively. For
Err (xk ; r , d 0 ), the results {Err (xk ; r , d 0 )} for various {xk }values were averaged over xk , and the average value is shownby the solid line.
The relative estimation error {Err (xk ; r , d 0 )} corresponded
well with the measured estimation error ratio (csim - c0 ) / c0in the simulated results considering the ﬁrst arrival wave, andthe difference between {Err (xk ; r , d 0 )} and (csim - c0 ) / c0
was within 0.024%. However, in the basic experiment, themeasured estimation error (cexp - c0 ) / c0 did not match wellwith the averaged value {Err (xk ; r , d 0 )}, and the difference

Fig. 4. (Color online) The propagation time distributions for the ideal point
scatterer (blue) and the scatterer with a size (red) in the simulated experiment.

Fig. 5. (Color online) Estimated results of the sound velocity in thesimulated experiment and the basic experiment using wires. The red dots andthe error bars show the average values and the standard deviations of theestimated sound velocities in the basic experiments, respectively; the blue
dots show the estimated sound velocities in the simulations with theinterference waveform; the diamond symbols show the estimated soundvelocities in the simulations with the ﬁrst arrival wave, and black “+”symbols show the true values of the sound velocity.

Fig. 6. (Color online) Estimation error ratio of the sound velocity. The red
dots and the error bars show the average values and the standard deviationsof the estimation error ratios in the basic experiment, respectively; the blue
dots show the estimation error ratios in the simulations with the same
condition of the experiment, rectangle symbols show the estimation error
ratios in simulations with c0 = 1400 m s−1, the diamond symbols show
estimation error ratios in the simulations with c0 = 1600 m s−1, and thestraight line represents the estimation error ratios calculated by Eq. (11).

between {Err (xk ; r , d 0 )} and (cexp - c0 ) / c0 was within
0.31%. From the results, it was conﬁrmed that the relative
estimation error depends on not the true sound velocity but theradius of the target scatterer at a certain depth.
Figure 7 shows the relative estimation errors Err (xk ; r , d 0 )of the sound velocity when the depths of the scatterer were
set to d 0 = 30 mm and 40 mm. When the target depth d 0 was
40 mm, the relative estimation error {Err (xk ; r , d 0 )} corresponded well with the measured estimation error ratio(c
sim - c0 ) / c0 in the simulated results considering the ﬁrst
arrivalwave,
althoughthedifference
betweenc)/cwaswithin{Err (xk ; r , d 0 = 40 mm)} and (cexp
000.21%. The relative estimation error Err (xk ; r , d 0 = 40 mm)
of
the
sound
velocitywasdifferentfrom
Err (xk ; r , d 0 = 30 mm). These results conﬁrmed the depth

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Acknowledgments

This work was partially supported by JSPS KAKENHI,
Grant No. 20H02156.

Fig. 7. (Color online) Estimation error ratio of sound velocity at different
depths (d 0 = 30 and 40 mm ). The red dots and the error bars show theaverage values and the standard deviations of the estimation error ratios inthe basic experiments (d 0 = 30 mm ), respectively, the orange diamondsymbols show the estimation error ratios in simulations (d 0 = 30 mm ), theblue dots and error bars show the average value and the standard deviationsof the estimation error ratios in the basic experiments (d 0 = 40 mm ),respectively; the green diamond symbols show the estimation error ratios inthe simulations (d 0 = 40 mm ), and the straight-lines represent the estimation
error ratios calculated by Eq. (11).

dependence of the relative estimation error Err (xk ; r , d 0 ) of thesound velocity.
From these results, it was conﬁrmed that the relative
estimation error of sound velocity Err (xk ; r , d 0 ) can bequantiﬁed as a function of the depth and radius of the target
scatterer. However, the relative estimation error
Err (xk ; r , d 0 ) in the basic experiment was lower than thatin the simulated experiment and that calculated using
Eq. (11) for scatterers with diameters greater than 2 mm. It
is a future task to clarify the cause of this difference.
4. Conclusions

In the present study, we quantiﬁed the effect of the size of thescatterer on the sound velocity estimation method. To investigate the effect, we considered a model in which multiple ideal
point scatterers were lined up on the surface of the scatterer, andconducted experiments simulating a situation in which ultrasonic waves were scattered on the surface. We also conducted abasic experiment using wires. It was conﬁrmed that the relative
estimation error of the sound velocity does not depend on thetrue value of the sound velocity, and this estimation error ratio
can be quantiﬁed as a function of the depth and radius of thetarget scatterer. The results of the present study will lead to the
proposal of an estimation method for sound velocity considering
the effect of the target scatterer size in the future.

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