Prepared for DISS_19 in L’Aquila on Nov. 28-29, 2019

Assessing the Wave Propagation Velocity and Damping
at Commodus Passage of Colosseum Using Microtremor
Applying the CERS methods
Yutaka Nakamura1, Jun Saita1 and Tsutomu Sato1

Abstract
The foundation of Colosseum is 12 m-thick spread foundation
divided into upper and lower 6 m-thick layers. Commodus Passage
was constructed with cutting the upper foundation about 60 m-long asa private passage to connect between the residence of emperor andColosseum between 81 AD and 96 AD. Various investigations havebeen conducted for this passage to analyze directly the foundation
structure. This paper describes the assessed wave propagation velocity
and other characteristics applying the CERS <səːrz> methods withmeasuring microtremor on the floor of the passage. The CERS
methods is an inclusive term of four methods as the C, E, R and S
methods. The C or E methods are to realize realtime calculation of the
wave propagation time and damping factor between a free end and apoint in a media or reflecting plane, respectively. The R and S
methods can calculate in realtime generally the time difference
between two points separated spatially based on the basis of maximum
cross-correlation as the R method or the basis of simple least error as
the S method. The calculated result of the CERS methods can bechecked mutually. Please see Nakamura [1] in detail. As a result, it
can be estimated the wave propagation velocity of various wave
motion component for every about 10 m on the floor. This result isagreeable with the P wave velocity structure by previous research.
Keywords: CERS methods, Foundation of Colosseum, Wave
Propagation Velocity, Microtremor_______________________________
1 System and Data Research Co., Ltd., Tokyo, Japan.

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Y. Nakamura, J. Saita and T. Sato

1. Introduction
The foundation of Colosseum built between 72 AD and 80 AD is 12
m-thick spread foundation divided into upper and lower 6 m-thicklayers. Commodus Passage was constructed with cutting about 60
m-long as a private passage to connect between the residence ofemperor and Colosseum during the reign of emperor Domitianus
between 81 AD and 96 AD. Various investigations have been
conducted for this passage because it is possible to analyze directly
the foundation structure of Colosseum. This paper describes theestimated wave propagation velocity and other characteristics
applying the CERS <səːrz> methods with measuring microtremor on
the floor of the passage. The CERS methods is an inclusive name offour methods, C, E, R and S methods, as Figure 1. The C method cancalculate in realtime the wave propagation time and damping between
a free end and a point among medium till the reflecting plane
assuming a one dimensional wave field, and the E method can also
calculate them between a free field and reflecting plane. Additionally,
the R and S methods can generally calculate in realtime the timedifference of waveforms between two points separated spatially basedon the maximum cross-correlation basis and minimum simple error
basis, respectively. The result of these four methods can be checkedmutually. Please see Nakamura [1] for the detail of the CERS
methods.

Figure 1. Concept of the CERS methods
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2. Estimation method of the wave propagation velocity(1) The R method
At first, the R method, estimating the wave propagation time
between two points based on the maximum correlation function
calculated from the microtremor measured at two points, is applied.
Because it is considered that the source of microtremor is unclear andvaries from hour to hour, the wave propagation time was presumed as
the maximum value between estimated maximum time differenceschoosing as the large cross-correlation coefficient as possible in thisinstance. The detail is explaned in next section and also will bementioned later.

(2) Concept of wave propagation time estimation applying
the R or S methods for microtremor
Microtremor must be recorded for XYZ components setting the Y
component as the direction of measurement line with sensors put on
the measurement line. Figure 2 shows the image of the geologicalstructure, sensor arrangement and measuring components.

Figure 2. Image of sensor arragement
Recorded wave motion by the sensors at measurement points #1
and #2 is considered not only as direct delivered wave but also as awave delivered as an refracted wave propagating at the lower layer.
According to the wave direction and propagation direction, the sensor
of the X direction is considered as measuring SH wave. In this case,although the both waves propagating at the upper and lower layers are
SH wave, the wave propagating at the lower layer seems to become
dominant because the lower layer has larger velocity and lower

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Y. Nakamura, J. Saita and T. Satodamping. In case of the Y or Z component, it is also possible to exist
both a direct delivered wave and refracted wave, but the direct
delivered wave must be dominant because the refracted wave
accompanies with conversion of wave kind. In case of microtremor,
wave motion comes from various directions. On the estimation of the
wave propagation time, the maximum absolute time difference isconsidered with large cross correlation for the R method or small error
for the S method.
Figure 3 shows the targeted wave motion of the direct delivered or
refracted wave for each component, and the wave motion which tends
to arise is marked pink. Here the wave propagation time is estimatedas approximating the critical angle as vertical. And t*12 means thewave propagation from the measurement points #1 to #2.

Figure 3. Variation of wave propagation between sensors #1 and #2(3) The C and E methods
Because the wave motion propagating in the foundation
horizontally is considered as reflecting multiply in the foundation, it ispossible to set #1 as a measurement point at the free end and #2 as ameasurement point in the medium with considering the wave motion
field toward to the outer edge of the foundation from the arena. So thewave propagation time and the damping factor between these points
are estimated with applying the C method. Then, if it is able to assume
the foundation as the wave motion field with horizontal multiple
reflection, it may be possible to estimate the wave propagation time4Assessing the Wave Propagation Velocity and Damping…

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and the damping factor of the foundation from the arena side to theouter edge applying the E method.

3. Microtremor measurement and result of its analysis(1) Distribution of the measurement points
As seen in Figure 4, Commodus Passage of Colosseum passes
linearly the foundation from the south side of the arena with cutting
the upper foundation and then it is directly connected to the tunnel toeast leaving from the foundation. Microtremor measurement was
conducted at every about 10 m of the part along the outer side of thearena to the entrance of Commodus Passage with 10 m long, from theentrance to the terminal of the tunnel with 60 m long and from theedge of the foundation to the terminal of the connected tunnel with 10
m long. Microtremor was recorded simultaneously at two points
among points #0 to #8, totally nine points. The measurement was
conducted in the morning on Wednesday, 18th October, 2017 in local
time and Figure 5 shows the distribution of the measured points withphotographs during measurement. Figure 6 shows the location of
measurement points from #1 to #7 in outline adding a verticalcross-section figure appeared on L. Orlando et al. [2]. The
measurement instrument is NewPIC++ manufactured by System andFigure 4. Commodus Passage of Colosseum

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Y. Nakamura, J. Saita and T. Sato

Figure 5. Measuring points for microtremor at Commodus Passage(Map adapted from L. Orlando et al. [2])Figure 6. Measuring points at the passage(adapted from L. Orlando et al. [2])6

Photo. 1. Instulment

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Data Research, which can record accurate three-component waveformof two points simultaneously (see Photograph 1). Microtremor wasrecorded as eight sets of measurement point, #0 and #1, #2 and #1, #2
and #3, #4 and #3, #4 and #5, #6 and #5, #6 and #7, and #8 and #7, forfive minutes waveform in 100 Hz sampling.

(2) The wave propagation time estimation applying the R
methodFigure 7 represents an example for the time variation of timedifference and cross-correlation coefficient between the points #2 and#1 applying the R method using exponential smoothing for averaging
with half-life period of 10 seconds. As this figure, the time difference
between these two points fluctuates widely corresponding to thevarious coming direction of the wave motion. The wave propagation
time between the two points corresponds to the case when the coming
direction of each wave motion coincides with the direction of
measured line, and it corresponds to the maximum or minimum
measured time difference. At this time, it becomes important to findthe time difference with the maximum cross-correlation coefficient.

Figure 7. An example of measured time difference and cross-correlation coefficient between points #2 and #1 applying the R method
Figure 8 represents the relationship between the time difference andcross-correlation coefficient in case of the measurement set #2 and #1
when the horizontal axis is time difference and the vertical axis is the
cross-correlation coefficient. This figure shows that the cross-correlation coefficient is basically over 0.6 for each direction as X(T), Y(R)and Z(V) without the tunnel on the foundation in the outer ground.
Especially the coefficient becomes close to 1.0 for the R direction

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Y. Nakamura, J. Saita and T. Satocorresponding to the P wave. Also, this analysis used the calculation
result of 290 seconds omitting the first and last 5 seconds data withconsideration for existing of the transitional result at the first and lastpart of the cross-correlation calculation.

Figure 8. Examples of relationship between measured timedifference and cross-correlation coefficient(3) The wave propagation time estimation applying the C
and E methods
It is possible to estimate the wave propagation time between the
points #1 and #2 applying the C method assuming the point #1 as thepoint at the free end and point #2 as the point in the media. And it ispossible to estimate the wave propagation time between the free end
and the reflecting plane applying the E method with waveforms ateach point. The latter can estimate the wave propagation time at sevenpoints on the foundation as points #1 to #7 and it is expected to know
about the homogeneity on the materials of the upper foundation
confirming with the behavior of the estimation result at each point.
Here considers only about the R direction because it is possible toexist stably only the P wave motion of the R direction as orthogonal
direction against the circumference considering a situation of the wave
motion field.

4. Estimated result and discussion
Table 1 shows the estimated wave propagation velocity of eachdirection by the R method arranging with each set of the measurement.
This table also shows the estimated maximum wave propagation timeand cross-correlation coefficient of that time for each direction of each8Assessing the Wave Propagation Velocity and Damping…

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set. The V direction of each set corresponds to the SV wave or thebending wave, SB wave. The T and R directions of the horizontal
component correspond to the P wave and SH wave, respectively, forthe sets of #0 and #1, and #8 and #7, and SH wave and P wave,
respectively, for the other sets. In any case, the cross-correlation
coefficient becomes small in order of P wave, SH wave and SV wave
or SB wave. The waveforms of two points show high correlativity as
exceeding 0.9 of the coefficient during the P wave. Here, “set of #A
and #B” means the propagation from the point #A to #B in case of theplus value of the time difference, and, conversely, that from the point#B to #A in case of the minus value of the time difference.

Table 1. Estimated results applying the R method

Without some exceptions, the result of the estimation basicallyshows the propagation time corresponding to one direction. It isinferable that the wave motion propagates mainly from the arena, or
northern side, in case of the direction of wave propagation at the
points #1 to #7 on the foundation. In case of the sets of #0 and #1, and#8 and #7 as the measurement line of east-west direction, bidirectional
wave propagation can be confirmed. It may be related to the locationof the wave source as road traffic at three directions on north, east and10

Y. Nakamura, J. Saita and T. Satosouth, subway at north (under construction) and west or tram at eastand south as Figure 9.

Figure 9. Traffic noise sources around Colosseum
Figure 10 shows the estimated wave propagation velocity
corresponding to the middle of the set of measurement point. The
wave propagation velocity of the SH wave becomes higher than thatof the P wave at the section including the point #4 and #5 from thisfigure. It suggests that the SH wave of microtremor mainly propagatesfrom high velocity layer in underground, maybe lower foundation, toeach point and there is little constituent propagating the surface layer.

Figure 10. Estimated wave propagation velocities by the R method
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On the P wave, it seems that there is little propagation fromunderground considering the direction of propagation and vibrationand it mainly consists of direct wave propagating as the P wave in thesurface layer. If there is high velocity layer behind the wall, the SV or
SB wave may behave mainly as the wave propagation from the layer
with relationship between the direction of the vibration andpropagation. However, because there is basically upper foundation
behind the wall and the propagation velocity of the upper foundation
seems to be lower than that of the lower foundation, the estimatedpropagation velocity can be considered consequently as the SV or SB
wave propagating on the floor. So, if there is cavity under the floor, it
is expected that the SB wave becomes slower at such point.
Additionally, in case of the wave propagation time of the SH wave forthe set of #6 and #7, because the cross-correlation coefficient between
the waveforms of each point becomes extremely low, there is apossibility that the propagation time is not estimated with proper
accuracy. Although it is possible from the P wave velocity to estimatethat the foundation close to arena is very hard, the P wave velocity at
the main part of the upper foundation is almost less than 2000 m/s
without rather high value between the points #4 and #5 slightly outer
than the center. The SH wave distribution suggests the existence of thehigh velocity layer in underground around the site between the points#4 and #5 because of higher propagation velocity than the P wave
around there. This portion is where the structure still remains and it
can be considered that the wave propagation velocity becomes large
because of the affection of confined stress.
Figure 11 shows the estimated SH wave velocity at the bothfoundation using microtremor Nakamura et al. [3]. The red marked
point is corresponding to the measurement point #5 of Commodus
passage, and at this point, the SH wave velocity was estimated 600
m/s for the upper foundation and 2000 m/s for the lower foundation.
The result agrees with that of measurement in this time.
And it is confirmed that the wave propagation velocity of the SV or
SB wave becomes significantly lower than that of the other wave
component, between #0 and #1, #1 and #2, #4 and #5, and #7 and #8.
Because it is estimated that the wave propagation velocity of the SV
or SB wave becomes generally low when a cavity exists under the

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Y. Nakamura, J. Saita and T. Sato

Figure 11. Estimated SH wave velocities in upper and lower
foundation of Colosseum after Nakamura et al. [3]floor, it suggests the existence of a cavity, horizontal crack or a soft
layer under the floor around these points.
Figure 12 shows a longitudinal section of the P wave velocity along
Commodus Passage derived from a Vp tomographic image based on
the inversion of the first arrival traveltimes by Salomon Hailemikael etal. [4]. This analysis was conducted as reading totally 1395 wave
arrival time of stacked waveforms in 1/1000 second accuracy withwaveforms recording vibrations three times hitting the floor every 2 musing 5 kg-hammer with installing 59 vertical sensors (geophone) on

Figure 12. 2D tomographic model obtained by the seismic active
data acquired along Commodus Passage [4]12Assessing the Wave Propagation Velocity and Damping…

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the center of the floor for every 1 m. It can be confirmed from thisfigure that there exists high velocity layer at the underground aroundthe point #4 or #5 and the research estimates the P wave velocity as
3500 to 4000 m/s and it is agreeable with the estimated SH wave
velocity in this research as 1900 to 2900 m/s. The P wave velocity
around the floor surface read from Figure 12 is shown on Figure 10 asa light colored circle and is also agreeable with the result of estimationin this research although differing in detail.
There are caves at underground between the points #0 and #1 as
Photograph 2. And exploration result using the ground penetratingradar (GPR) by Luciana Orlando et al. [2] confirms an existence oflayer constructions around the point #2 and in the undergroundbetween the points #4 and #5, and a similar construction in theunderground between the points #7 and #8 to that between the points#1 and #2. These results agree with the result of the analysis applying
the R method for microtremor.

Photograph 2. A view around the entrance of Commodus Passage
Next, the result of the wave propagation velocity estimation
applying the C and E methods will be described. Figure 13 compares
the propagation time or distance attenuation factor every time withrelative error. Estimating the waveform as in the medium at the point#2 from the waveform observed at the point #1 as the free end
applying the C method for the measurement set of #1 and #2 with
distance of 10.7 m, the wave propagation time and distanceattenuation can be read as 0.724 (in 1/100 sec) and 11.5/1000 (in 1/m)

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Y. Nakamura, J. Saita and T. Sato

Figure 13. Relationship between travel time or attenuation andrelative error derived from the simultaneous record at the points#1 and #2 applying the C method assuming the minimum
relative error as the estimated value
for T direction corresponding to the SH wave, 0.458 (in 1/100 sec)
and 14.4/1000 (in 1/m) for R direction corresponding to the P wave
and 1.59 (in 1/100 sec) and 33/1000 (in 1/m) for V direction maybe
corresponding to the bending wave, as the minimum cross-correlation
between the estimated and observed waveform. It is estimated fromthis result that the SH wave velocity is 1480 m/s, the P wave velocity
is 2340 m/s, and the bending wave velocity is 673 m/s. it is consideredthat the velocity estimated by the C method averagely shows the
behavior of entire the layer and the velocity estimated by the R
method reflects the local behavior.
And, although not indicating here, because the predominantfrequency of the R direction is around 10Hz, the wavelength
predominating at the wave field seems to become over 200 m. This
wavelength is considerably longer than the dimension of thefoundation as 12 m thick and 60 m width, and the measured P wave
velocity can be considered close to that in a thin rod instead of that in
an infinite media. Assuming the measured P wave velocity between
the points #1 and #2 as that in a thin rod, Poisson’s ratio of these
section is estimated to be 0.250 in relation to the measured S wave
velocity. Additionally, assuming as in an infinite media, Poisson’s
ratio becomes 0.167. In either case, it is interesting that they are in asame range of Poisson’s ratio of modern concrete.
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The propagation time estimated by the E method can be considered
as giving the wave propagation time between the free end (the arenaside end of the foundation plate) and the reflecting plane (outer side of
the foundation plate), and this research considers on the radial
direction corresponding to the P wave. Figure 14 shows the
propagation velocity of the P wave horizontally propagating in thefoundation plate estimated at each measurement point. Here, the P
wave propagation distance is assumed as the distance excluding thewidth of the retaining walls at the end of the arena and periphery of
the foundation from the width of entire the foundation, 53 m from
Figure 5. These retaining walls had worked as a formwork whenconstructing Colosseum. The P wave velocity between the points #1
and #2 derived from the C method is indicated as a mark “⋆” in Figure
14 and the velocity agrees with that derived from the E method. What
the wave propagation velocity becomes higher toward out side isconsidered that the measured floor descends gradually after the point#3 and approach the lower foundation.

Figure 14. Estimated P wave propagation velocities on the radial
direction at each point applying the E method

5. Conclusion

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Y. Nakamura, J. Saita and T. Sato

Microtremor was measured at the floor of Commodus Passage
constructed by open-cut the foundation of Colosseum and the physicalproperties of the foundation were confirmed. As a result, it can be
estimated the wave propagation velocity of various wave motion
component for every about 10 m on the floor. This result is agreeablewith the P wave velocity structure by Salomon Hailemikael et al. [4]or the search result using the ground penetrating radar (GPR) by
Luciana Orlando et al. [2]. It is expected that the CERS methods makepossible in the future to confirm the physical properties of entire thefoundation with simultaneous microtremor measurement not only at
horizontally distanced two points but also at vertically distanced two
points.

Acknowledgement
With corporation of Professor Gianfranco Valente of L’Aquila
University and Dr. Fabio Fumagalli of special superintrndence for
Colosseum, the author Nakamura conducted this measurement on 18th
October, 2017, at the occasion of DISS_17. The authors would like toexpress our thanks to these two parsons.

References[1] Yutaka Nakamura: On the CERS methods, realtime assessment
methods for a travel time and a damping (in Japanese), Proceedings of
JAEE annual meeting, P4-20, December, 2017.
[2] Luciana Orlando, Giorgio De Donno, Luca Di Giambattista andLucia Palladini: Investigating the foundation of the Amphitheatrum
Flavium through the Passage of Commodus, ANNALS OF
GEOPHYSICS, 60 (4), 2017.
[3] Yutaka Nakamura, Jun Saita, Tsutomu Sato and Gianfranco
Valente: Attempts to Estimate the Physical Property of Surrounding
Ground and Foundation Concrete of Colosseum using Microtremor,
Proceedings of the 4th International Workshop on “Dynamic
Interaction of Soil and Structure (DISS_15)” Rome, 12-13 November
2015, 1st edition: October 2016.
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17[4] Salomon Hailemikael, Giuliano Milana, Fabrizio Cara, Maurizio
Vassallo, Marta Pischiutta, Sara Amoroso, Paola Bordoni, Luciana
Cantore, Giuseppe Di Giulio, Deborah Di Naccio, Daniela Famiani
and Alessia Mercuri: Sub-surface characterization of theAmphiteatrum Flavium area (Rome, Italy) through single-station
ambient vibration measurements, ANNALS OF GEOPHYSICS, 60(4), 2017.