# Deep learning for low frequency extrapolation of multicomponent data in
elastic full waveform inversion

Hongyu Sun and Laurent Demanet  Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139,
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(December, 2020)

###### Abstract

Full waveform inversion (FWI) strongly depends on an accurate starting model
to succeed. This is particularly true in the elastic regime: The cycle-
skipping phenomenon is more severe in elastic FWI compared to acoustic FWI,
due to the short S-wave wavelength. In this paper, we extend our work on
extrapolated FWI (EFWI) by proposing to synthesize the low frequencies of
multi-component elastic seismic records, and use those “artificial” low
frequencies to seed the frequency sweep of elastic FWI. Our solution involves
deep learning: we separately train the same convolutional neural network (CNN)
on two training datasets, one with vertical components and one with horizontal
components of particle velocities, to extrapolate the low frequencies of
elastic data. The architecture of this CNN is designed with a large receptive
field, by either large convolutional kernels or dilated convolution. Numerical
examples on the Marmousi2 model show that the 2-4Hz low frequency data
extrapolated from band-limited data above 4Hz provide good starting models for
elastic FWI of P-wave and S-wave velocities. Additionally, we study the
generalization ability of the proposed neural network over different physical
models. For elastic test data, collecting the training dataset by elastic
simulation shows better extrapolation accuracy than acoustic simulation, i.e.,
a smaller generalization gap.

##  1 Acknowledgments

The authors thank Total SA for support. Laurent Demanet is also supported by
AFOSR grant FA9550-17-1-0316. Hongyu Sun acknowledges SEG scholarships (SEG
75th Anniversary Scholarship and Michael Bahorich/SEG Scholarship) for
funding. Tensorflow (Abadi et al., 2015) and Keras (Chollet et al., 2015) are
used for deep learning. Elastic FWI in this note is implemented using the open
source code DENISE (https://github.com/daniel-koehn/DENISE-Black-Edition).
Acoustic training datasets are simulated using Pysit (Hewett & Demanet, 2013).

##  2 Introduction

Full waveform inversion is well-known for its great potential to provide
quantitative Earth properties of complex subsurface structures. Acoustic FWI
is widely used and has been successfully applied to real seismic data.
However, most seismic data have strong elastic effects (Marjanović et al.,
2018). The acoustic approximation is insufficient to estimate correct
reflections and introduces additional artifacts to FWI results (Plessix et
al., 2013; Stopin et al., 2014). Therefore, it is desirable to develop a
robust elastic FWI method for high-resolution Earth model building.

Foundational work has shown the ability of elastic FWI to retrieve realistic
properties of the subsurface (Tarantola, 1986; Mora, 1987). However, it has
difficulty handling real data sets. Elastic FWI is very sensitive to: accuracy
of the starting model; correct estimation of density; proper definition of
multi-parameter classes; and noise level (Brossier et al., 2010). The complex
wave phenomena in elastic wavefields bring new challenges to FWI.

Among the many factors that affect the success of elastic FWI, the lowest
starting frequency is an essential one, given that an accurate starting model
is generally unavailable. Compared to acoustic FWI, the nonlinearity of
elastic FWI is more severe due to the short S-wave propagating wavelength.
Therefore, elastic FWI always requires a lower starting frequency compared to
acoustic FWI. Additionally, the parameter cross-talk problem exists in elastic
FWI and becomes more pronounced at higher frequencies, so ultra-low
frequencies are required for a successful inversion of S-wave velocity and
density.

In synthetic studies of elastic FWI, Brossier et al. (2009) invert the
overthrust model (Aminzadeh et al., 1997) from 1.7Hz. Brossier et al. (2010)
invert the Valhall model (Sirgue et al., 2009) from 2Hz. Both inversion
workflows start from Gaussian smoothing of true models. Moreover, Choi et al.
(2008) invert the Marmousi2 model (Martin et al., 2006) using a velocity-
gradient starting model but a very low frequency (0.16Hz). For a successful
inversion of the Marmousi2 density model, Köhn et al. (2012) use 0-2Hz in the
first stage of multi-scale FWI. Jeong et al. (2012) invert the same model from
0.2Hz.

Few applications of elastic FWI to real data sets are reported in the
literature (Crase et al., 1990; Sears et al., 2010; Marjanović et al., 2018).
Vigh et al. (2014) use 3.5Hz as the starting frequency of elastic FWI given
that the initial models are accurate enough. Raknes et al. (2015) apply 3D
elastic FWI to update P-wave velocity and obtain S-wave velocity and density
using empirical relationships. Borisov et al. (2020) perform elastic FWI
involving surface waves in the band of 5-15Hz for a land data set.

New developments in acquisition enhance the recent success of FWI by measuring
data with lower frequencies and longer offsets (Mahrooqi et al., 2012;
Brenders et al., 2018). However, only acoustic FWI was applied to the land
data set with low frequencies down to 1.5 Hz (Plessix et al., 2012). In
addition to the expensive acquisition cost for the low-frequency signals,
direct use of the field low-frequency data requires dedicated pre-processing
steps, including travel-time tomography, for an accurate enough model to
initialize FWI. The final inversion results strongly rely on the starting
tomography model. Hence, attempting to retrieve reliable low-frequency data
offers a sensible pathway to relieve the dependency of elastic FWI on starting
models.

Deep learning is an emerging technology in many aspects of exploration
geophysics. In seismic inversion, several groups have experimented with
directly mapping data to model using deep learning (Araya-Polo et al., 2018;
Yang & Ma, 2019; Wu & Lin, 2019; Zhang & Lin, 2020; Kazei et al., 2020).
Within Bayesian seismic inversion framework, deep learning has been applied
for formulating priors (Herrmann et al., 2019; Mosser et al., 2020; Fang et
al., 2020). Other groups use deep learning as a signal processing step to
acquire reasonable data for inversion. For instance, Li et al. (2019) use deep
learning to remove elastic artifacts for acoustic FWI. Siahkoohi et al. (2019)
remove the numerical dispersion of wavefields by transfer learning.

Computationally extrapolating the missing low frequencies from band-limited
data is the cheapest way for FWI to mitigate the cycle-skipping problem. Li &
Demanet (2015, 2016) seperate the shot gather to atomic events and then change
the wavelet to extrapolate the low frequencies. Li & Demanet (2017) extend the
frequency spectrum based on the redundancy of extended forward modeling.
Recently, Sun & Demanet (2018); Ovcharenko et al. (2018); Jin et al. (2018)
have utilized CNN to extrapolate the missing low frequencies from band-limited
data. They have proposed different architectures of CNN to learn the mapping
between high and low frequency data from different features in the training
datasets. However, only acoustic data are considered in these studies.

Although the mechanism of deep learning is hard to explain, the feasibility of
low frequency extrapolation has been discussed in terms of sparsity inversion
(Hu et al., 2019) and wavenumber illumination (Ovcharenko et al., 2019). With
multiple-trace extrapolation, the low wavenumbers of far-offset data have been
proposed as the features in the frequency domain detected by CNN to
extrapolate the missing low frequencies (Ovcharenko et al., 2019). In
contrast, for trace-by-trace extrapolation (Sun & Demanet, 2020), the features
to learn are the structured time series themselves. The feasibility of trace-
by-trace frequency extrapolation has been mathematically proved in simple
settings in Demanet & Nguyen (2015); Demanet & Townsend (2019), as a by-
product of super-resolution.

In this paper, we extend our workflow of extrapolated FWI with deep learning
(Sun & Demanet, 2020) into the elastic regime. We separately train the same
neural network on two different training datasets, one to predict the low-
frequency data of the horizontal components ($v_{x}$) and one to predict the
low frequencies of the vertical components ($v_{y}$). The extrapolated low
frequency data are used to initialize elastic FWI from a crude starting model.
For the architecture design of CNN, a large receptive field is achieved by
either large convolutional kernels or dilation convolution. Moreover, to
investigate the generalization ability of neural networks over different
physical models, we compare the extrapolation results of the neural networks
trained on elastic data and acoustic data to predict the elastic low-frequency
data. We also investigate several hyperparameters of deep learning to
understand its bottleneck for low frequency extrapolation, such as mini-batch
size, learning rate and the probability of dropout.

The organization of this article is as follows. In Section 3, we first briefly
review elastic FWI and its implementation in this paper. Then, we present the
architecture of neural networks and training datasets for low frequency
extrapolation of elastic data. Section 4 describes the numerical results of
low frequency extrapolation, extrapolated elastic FWI and investigation of
hyperparameters. Section 5 discusses the limitations of the method. Section 6
comes to conclusions and future directions.

##  3 Method

We first give a brief review of elastic FWI as implemented in this paper. Then
we illustrate the feasibility of low frequency extrapolation, and design two
deep learning models for this purpose. Afterwards, the training and test
datasets are provided to train and verify the performance of the proposed
neural networks.

###  3.1 Review of elastic FWI

Elastic FWI is implemented in the time domain to invert the P-wave velocities
($\mathbf{v}_{p}$), S-wave velocities ($\mathbf{v}_{s}$) and density ($\rho$)
simultaneously. The object function $E$ is formulated as

$E=\frac{1}{2}\delta\mathbf{d}^{T}\delta\mathbf{d}=\frac{1}{2}\sum\limits_{s}\sum\limits_{r}\int[\mathbf{u}_{cal}-\mathbf{u}_{obs}]^{2}dt,$
(1)

where $\mathbf{d}$ are the residuals between observed wavefields
$\mathbf{u}_{obs}$ and calculated wavefields $\mathbf{u}_{cal}$. In 2D, both
$\mathbf{u}_{obs}$ and $\mathbf{u}_{cal}$ contain the $v_{x}$ and $v_{y}$
components of elastic wavefields. The gradient
$\frac{\delta{E}}{\delta{\mathbf{m}}}$ relative to the model parameters
$\mathbf{m}$ is calculated in terms of $\mathbf{v}_{p}$, $\mathbf{v}_{s}$ and
$\rho$ using the velocity-stress formulation of the elastic wave equation Köhn
et al. (2012). The starting models $\mathbf{m_{0}}$ are updated using the
L-BFGS method Nocedal & Wright (2006).

###  3.2 Deep learning models for low-frequency extrapolation

We choose CNN to perform the task of low-frequency extrapolation. By trace-by-
trace extrapolation, the output and input are the same seismic recording in
the low and high frequency band, respectively. In 2D, the elastic data contain
horizontal and vertical components. As a result, we propose to separately
train the same neural network twice on two different training datasets: one
contains $v_{x}$ and the other contains $v_{y}$.

We design two kinds of CNN architectures with large receptive field for low-
frequency extrapolation. A very large receptive field enables each feature in
the final output to include a large range of input pixels. Since any single
frequency component is related to the entire waveform in the time domain,
extrapolation from one frequency band to the other requires a large receptive
field to cover the entire input signal. Typically, the receptive field is
increased by stacking layers. For example, stacking two convolutional layers
(without pooling) with $3\times 3$ filter results in a layer of $5\times 5$
filter. However, it requires many layers to result in a large enough receptive
field and is computationally inefficient. Therefore, we design the CNN
architecture with two methods: directly use a large filter or a small filter
with dilated convolution.

Figure 1: ARCH1. The first choice of deep learning architecture that directly
employs a large filter on each convolutional layer (Sun & Demanet, 2020). The
size and number of filters are labeled on the top of each convolutional layer.

The first CNN architecture (ARCH1) directly employs a large filter on
convolutional layers, which is the same as Sun & Demanet (2020) (Figure 1).
Recall that ARCH1 is a feed-forward stack of five convolutional blocks. Each
block is a combination of a 1D convolutional layer, a PReLU layer and a batch
normalization layer. The length of all filters is 200. On each convolutional
layer, the channels number 64, 128, 64, 128 and 32, respectively. Although
only one trace is plotted in Figure 1, the proposed neural network can easily
explore multiple traces of the shot gather for multi-trace extrapolation by
increasing the size of the kernels from $200\times 1$ to $200\times ntr$,
where $ntr$ is the number of the input traces.

Figure 2: (a)ARCH2. The second choice of deep learning architecture with
dilated convolution. The filter length is 2 on each convolutional layer but
two dilated convolutional blocks are stacked to increase the receptive field
exponentially with depth. (b)The dilated causal convolutional block. Each
block has 11 convolutional layers. The filter size, number of channel and
dilation of each convolutional layer are labeled on the top.

The second CNN architecture (ARCH2) uses dilated convolution to increase the
receptive field by orders of magnitude. A dilated convolution (convolution
with holes) is a convolution where the filter is applied over an area larger
than its length by skipping input values with a certain step (dilation). It
effectively allows the network to operate on a coarser scale than with a
normal convolution. This is similar to pooling or stride, but here the output
has the same size as the input. As a special case, dilated convolution with
dilation of one yields the standard convolution. Stacked dilated convolutions
enable networks to have very large receptive fields with just a few layers. In
addition to save computational cost, this method helps to preserve the input
resolution throughout the network (Oord et al., 2016). Moreover, we use causal
convolution to process time series (Moseley et al., 2018), although this
choice does not appear to be essential in our case.

The architecture of ARCH2 (Figure 2) has two dilated convolutional blocks.
Each block consists of 10 1D convolutional layers. Each convolutional layer is
followed by a PReLU layer and a batch normalization layer. On each
convolutional layer, there are 64 causal convolutional filters with a length
of 2 (Figure 2b). The dilations of the ten convolutional layers are $2^{0}$,
$2^{1}$, …, $2^{10}$, respectively. The exponential increase in dilation
results in exponential growth, with depth, of the receptive field (Yu &
Koltun, 2015).

With the two proposed architectures, we can compare the specific receptive
fields of both ARCH1 and ARCH2. Without pooling layer, the size of the
receptive field ${RF}_{l+1}$ on the $l+1$ layer is

${RF}_{l+1}={RF}_{l}+(k_{l+1}-1)\times s_{l+1}\times d_{l+1},\quad
l=0,...,n\quad,$ (2) ${RF}_{0}=1,$ (3)

where $k_{l+1}$ is the kernel size of the $l+1$ convolutional layer. $s_{l+1}$
is the stride size. $d_{l+1}$ is the dilation on the $l+1$ layer if the layer
contains a dilated convolution. Otherwise, $d_{l+1}$ equals 1 for regular
convolutional layers.

The receptive field is 996 for ARCH1 and 4095 for ARCH2. Since a smaller
kernel is used in ARCH2, the number of trainable parameter in ARCH2
($p_{2}$=148,453,480) is much less than ARCH1 ($p_{1}$=294,602,648). Although
both neural networks are able to perform low frequency extrapolation,
convolution with dilation is more efficient than directly using a large
convolutional kernel.

###  3.3 Training and test datasets

The training and test datasets are simulated on the elastic training and test
models. The Marmousi2 elastic model (Figure 3) is referred to as the test
model in deep learning. This is also the true model in the subsequent elastic
FWI. The training models (Figure 4) are six batches randomly extracted from
the Marmousi2 model. Our previous work (Sun & Demanet, 2020) has shown that
the random selection of the training models on the Marmousi model provides
enough generalization ability for the neural network to extrapolate low
frequencies on the full-size Marmousi model. Here in the elastic regime, each
model consists of three parameters: $\mathbf{v}_{p}$, $\mathbf{v}_{s}$ and
$\mathbf{\rho}$. The size of each model is $500\times 174$ with a grid spacing
of 20m, including a water layer on the top of each model with a depth of 440m.

Figure 3: The true Marmousi2 model: (a)$\mathbf{v}_{p}$, (b)$\mathbf{v}_{s}$
and (c)$\rho$. Figure 4: The six training models randomly extracted from the
Marmousi2 model. The size of each training model is equally $500\times 174$
with a grid spacing of 20m including a 440m depth water layer. Each training
model contains three parameters: $\mathbf{v}_{p}$, $\mathbf{v}_{s}$ and
$\mathbf{\rho}$.

Both training and test datasets are simulated using a 2D time domain stress-
velocity P–SV finite-difference (FD) code (Virieux, 1986; Levander, 1988) with
an eighth-order spatial FD operator. A Ricker wavelet with a dominant
frequency of 10Hz is used as the source signal. The sampling rate and the
recording time is 0.02s and 6s, respectively. It is not necessary to collect
the training and test datasets using the same acquisition geometry. To collect
the test dataset, 50 shots are excited evenly from 800m to 8640m in the water
layer at the same depth of 40m. 400 receivers are placed from 800m to 8780m
under the water layer with a depth of 460m to record $v_{x}$ and $v_{y}$ of
the elastic wavefields. However, for the training model, there are 100 shots
excited evenly from 500m to 8420m with a depth of 40m on each training model.
400 receivers are placed from 480m to 8460m.

After the forward modeling, two training datasets are collected, one with a
dataset of horizontal components and one with a dataset of vertical
components. The 2D elastic data on the test model is also separated into two
test datasets to process each component individually. By trace-by-trace
extrapolation setup, there are $6\times 100\times 400=240,000$ training
samples in each training dataset and $1\times 50\times 400=20,000$ test
samples in each test dataset.

A simple preprocessing step can be used to improve the deep learning
performance. Each sample in the training and test datasets is normalized to
one by dividing the raw signal by its maximum. Then all the data are scaled
with a constant (for instance, 100) to stabilize the training process. The
values used to normalize and scale the raw data are recorded to recover the
original observed data for elastic FWI. After this process, each sample in the
training and test dataset is separated into a low-frequency signal and a high-
frequency signal using a smooth window in the frequency domain. Then, each
time series in the high-frequency band is fed into the neural network to
predict the low-frequency time series.

##  4 Numerical Examples

The numerical examples section is divided into four parts. In the first part,
we train ARCH1 to extrapolate the low-frequency data of bandlimited multi-
component recordings simulated on the Marmousi2 model (Figure 3). In the
second part, we study the generalization ability of the proposed neural
network over different physical models (acoustic or elastic wave equation).
Then, we use the extrapolated low-frequencies of multi-component band-limited
data to seed the frequency sweep of elastic FWI on the Marmousi2 model. In the
last part, we investigate the hyperparameters of deep learning for low-
frequency extrapolation using the proposed ARCH1 and ARCH2.

###  4.1 Low frequency extrapolation of multicomponent data

We first extrapolate the low frequency data below 5Hz on the Marmousi2 model
(Figure 3) using 5-25Hz band-limited data. Each sample in the training and
test datasets is separated into a 0-5Hz low-frequency signal and a 5-25Hz
high-frequency signal using a smooth window in the frequency domain. The time
series in the high-frequency band is directly fed into the neural network to
predict the 0-5Hz low-frequency time series. To deal with the multicomponent
data, the neural network ARCH1 is trained twice: once on the training dataset
of $v_{x}$ and once on the training dataset of $v_{y}$. Both training
processes use the ADAM method with a mini-batch of 32 samples. We refer
readers to Sun & Demanet (2020) for more details about training. Figures 5(a)
and 5(b) show the training processes over 40 epochs to predict the low
frequencies of $v_{x}$ and $v_{y}$, respectively. The curves of training loss
decay over epochs on both the training and test datasets, which indicate that
the neural network does not overfit.

(a) Figure A

(b) Figure B

Figure 5: The learning curves of ARCH1 trained to extrapolate the 0-5Hz low
frequencies of (a) $v_{y}$ and (b) $v_{x}$ from the 5-25Hz band-limited
elastic recordings. Figure 6: The extrapolation results of ARCH1 for Marmousi2
model: comparison among the (a) band-limited recordings ($5.0-25.0$Hz), (b)
predicted and (c) true low-frequency recordings ($0.1-5.0$Hz) of $v_{y}$ and
(d) band-limited recordings ($5.0-25.0$Hz), (e) predicted and (f) true low-
frequency recordings ($0.1-5.0$Hz) of $v_{x}$.

(a) Figure A

(b) Figure B

Figure 7: Extrapolation results of ARCH1 trained on elastic data: comparison
of the amplitude and phase spectrum of (a) $v_{y}$ and (b) $v_{x}$ at
$x=6.1$km among the bandlimited recording ($5.0-25.0$Hz), the recording
($0.1-25.0$Hz) with true and predicted low frequencies ($0.1-5.0$Hz).

Figure 6 shows the extrapolation results of both $v_{x}$ and $v_{y}$ where the
source is located at 7.04km. Figures 7(a) and 7(b) compare the amplitude and
phase spectrum of $v_{y}$ and $v_{x}$ at $x=6.1$km among the band-limited
recording ($5.0-25.0$Hz), the fullband recording with true and predicted low
frequencies ($0.1-5.0$Hz). Despite minor prediction errors on both amplitude
and phase, the neural network ARCH1 can successfully recover the low
frequencies of $v_{x}$ and $v_{y}$ recordings with satisfactory accuracy.

(a) Figure A

(b) Figure B

Figure 8: Extrapolation results of ARCH1 trained on acoustic data: comparison
of the amplitude and phase spectrum of (a) $v_{y}$ and (b) $v_{x}$ at
$x=6.1$km among the bandlimited recording ($5.0-25.0$Hz), the recording
($0.1-25.0$Hz) with true and predicted low frequencies ($0.1-5.0$Hz).

###  4.2 Generalization over physical models

To study the generalization ability of the proposed neural network over
different physical models (acoustic wave and elastic wave), we train ARCH1
once on an acoustic training dataset and simultaneously predict the low
frequencies of both $v_{x}$ and $v_{y}$ in the same elastic test dataset. The
acoustic training dataset is simulated using the acoustic wave equation on
only the P-wave velocity models in Figure 4. Figures 8(a) and 8(b) show the
amplitude and phase spectrum of $v_{y}$ and $v_{x}$ at $x=6.1$km after
training with the same procedure. Compared with the results in Figure 7, the
extrapolation accuracy of the same trace in the test data is much poorer on
acoustic training dataset than elastic training dataset. Even the
extrapolation of the vertical component is not successful when the neural
network is trained on acoustic data. This is an indicator that the neural
network has difficulty generalizing to different physical models.

###  4.3 Extrapolated elastic full waveform inversion

Figure 9: Comparison among (a) the initial $\mathbf{v}_{p}$ model, the
inverted low-wavenumber velocity models using (b) $2.0-4.0Hz$ extrapolated
data and (c) $2.0-4.0Hz$ true data. The inversion results in (b) and (c) are
started from the initial model in (a). Figure 10: Comparison among (a) the
initial $\mathbf{v}_{s}$ model, the inverted low-wavenumber velocity models
using (b) $2.0-4.0Hz$ extrapolated data and (c) $2.0-4.0Hz$ true data. The
inversion results in (b) and (c) are started from the initial model in (a).
Figure 11: Comparison among (a) the initial $\rho$ model, the inverted low-
wavenumber density models using (b) $2.0-4.0Hz$ extrapolated data and (c)
$2.0-4.0Hz$ true data. The inversion results in (b) and (c) are started from
the initial model in (a). Figure 12: Comparison of the inverted
$\mathbf{v}_{p}$ models from elastic FWI using $4-20Hz$ band-limited data. (a)
The resulting model starts from the original initial model. (b) The resulting
model starts from the inverted low-wavenumber velocity model using $2.0-4.0Hz$
extrapolated data. (c) The resulting model starts from the inverted low-
wavenumber velocity model using $2.0-4.0Hz$ true data. Figure 13: Comparison
of the inverted $\mathbf{v}_{s}$ models from elastic FWI using $4-20Hz$ band-
limited data. (a) The resulting model starts from the original initial model.
(b) The resulting model starts from the inverted low-wavenumber velocity model
using $2.0-4.0Hz$ extrapolated data. (c) The resulting model starts from the
inverted low-wavenumber velocity model using $2.0-4.0Hz$ true data. Figure 14:
Comparison of the inverted $\rho$ models from elastic FWI using $4-20Hz$ band-
limited data. (a) The resulting model starts from the original initial model.
(b) The resulting model starts from the inverted low-wavenumber density model
using $2.0-4.0Hz$ extrapolated data. (c) The resulting model starts from the
inverted low-wavenumber density model using $2.0-4.0Hz$ true data.

We perform extrapolated elastic FWI using 4-20Hz band-limited data on the
Marmousi2 model. The lower band of the band-limited data is 4Hz. Figures 9a,
10a and 11a show the initial models of $v_{p}$, $v_{s}$ and $\rho$,
respectively. Unlike in the previous examples, a free surface boundary
condition is applied to the top of the model to simulate the realistic marine
exploration environment. The free surface condition damages the low frequency
data and thus the energy in the low frequency band 0-2Hz is close to zero in
the simulated full-band data. This brings a new challenge to the low frequency
extrapolation and introduces prediction errors to the extrapolated data. For
this reason, we start elastic FWI using 2-4Hz extrapolated data before
exploring the band-limited data.

Starting from the crude initial model in Figure 9(a), Figures 9(b) and 9(c)
show the resulting P-wave velocity models after 30 iterations using
extrapolated and true 2-4Hz low-frequency data, respectively. The inverted
S-wave velocity models using extrapolated and true low frequencies are shown
in Figures 10(b) and 10(c). Also, Figures 11(b) and 11(c) compare the inverted
density models using the extrapolated and true 2-4Hz low frequencies. The
inverted low-wavenumber models of $v_{p}$, $v_{s}$ and $\rho$ using
extrapolated data are roughly the same as those using true data. However, the
inversion of density model is not successful since 2-4Hz data are relatively
high frequencies for the inversion of density model.

Then the inversion is continued with the 4-20Hz band-limited data. We utilize
a multiscale method (Bunks et al., 1995) and sequentially explore the 4-6Hz,
4-10Hz and 4-20Hz band-limited data in the elastic FWI. In each frequency
band, the number of iterations is 30, 30 and 20, respectively. Figures 12 and
13 show the resulting $v_{p}$ and $v_{s}$ models started from different low
wavenumber models. The inversion results of $v_{p}$ and $v_{s}$ started from
2-4Hz extrapolated data are very close to the results started from 2-4Hz true
data. Conversely, elastic FWI directly starting from the crude initial models
using the band-limited data shows large errors.

Figure 14 shows the resulting density models using the 4-20Hz band-limited
data but started from different models in Figure 11. Since the starting
frequency band (2-4Hz) is relatively high for the inversion of density on the
crude initial model (Figure 11(a)), the inverted models using 2-4Hz data
(Figures 11(b) and (c)) only show high-wavenumber structure of the density
model. With band-limited data involved in the inversion, the inverted density
models (Figure 14) resemble migration results but show the density
perturbation (Mora, 1987). For a successful inversion of the density model, a
much lower starting frequency band is required to recover the low-wavenumber
structures.

###  4.4 Investigation of hyperparameters

The performance of deep learning is very sensitive to the hyperparameters of
training. However, choosing appropriate hyperparameters requires expertise and
extensive trial and error. Here we discuss the influence of several
hyperparameters on the training process, including mini-batch size, learning
rate, and a layer-specific hyperparameter, i.e., dropout. We also compare the
performance of ARCH1 and ARCH2 in terms of training cost and prediction
accuracy. In each case, we compare the model performance of the new
hyperparameter setting with the results predicted by the neural network ARCH1
in Section 4.3. The neural network is trained using a mini-batch of 32,
learning rate of $10^{-3}$. The probability of dropout is $50\%$.

The first hyperparameter is the mini-batch size. A batch is a small subset of
training data randomly selected by the optimizer to calculate the gradient.
Choosing a suitable mini-batch size is a trade-off between training speed and
test accuracy. Fewer samples in a batch slow down the training process but a
large batch increases the instability of the neural network and lead to a poor
performance. Figures 15 and 15 show the learning curves of ARCH1 when
processing the $v_{y}$ and $v_{x}$ components using a batch size of 16, 32, 64
and 128. A mini-batch of 32 gives more reasonable decrease of the training
loss among others. Therefore, we choose it as the mini-batch size in our
experiments.

The second essential hyperparameter is the learning rate of the optimizer.
Figure 16 compares the learning curves when the learning rates equal to
$10^{-2}$, $10^{-3}$ and $10^{-4}$, respectively. If the learning rate is
small, training is more reliable, but it will take significant time because
steps towards the minimum of the loss function are tiny. The model may also
miss the important patterns in the training data; Conversely, if the learning
rate is high, training may not converge. Weight changes are so large that the
optimizer overshoots the minimum and makes the loss worse. We observe that a
learning rate of $10^{-3}$ seems to be an optimal learning rate and can
quickly and stably find the minimum loss.

Moreover, we study a model-related hyper-parameter, i.e., dropout in ARCH1.
Dropout prevents neural networks from overfitting by randomly dropping neurons
during training. Each neuron is retained with a fixed probability $p$
independent of other neurons. $p$ can be chosen using a validation set or can
simply be set at 0.5, which seems to be close to optimal for a wide range of
networks and tasks (Srivastava et al., 2014). Figure 17 compares the learning
curves of ARCH1 with a dropout rate of $0\%$ (no dropout), $20\%$ and $50\%$,
respectively. It seems that for this training dataset, a case without dropout
layer does not hurt the performance of ARCH1. All of the three cases give
reasonably right learning curves on both training and test datasets.

Finally, we compare the performance of ARCH1 and ARCH2 using the same training
dataset and hyperparameter. Each network is trained twice with the training
dataset of $v_{x}$ and the training dataset of $v_{y}$. According to the
learning curves in Figure 17, the training of ARCH2 is much more stable than
that of ARCH1. ARCH2 also requires less training time, due to the less
trainable parameters compared with ARCH1. Figure 18 shows the extrapolated
elastic FWI results started from the 2-4Hz extrapolated low frequency data
using ARCH2. According to the comparison of the inverted models started from
the extrapolated data using ARCH1 and ARCH2, both neural networks are able to
provide sufficient accuracy for the inversion of $v_{p}$ and $v_{s}$.

Figure 15: The learning curves of ARCH1 trained using different batch sizes to
extrapolate the 0-4Hz low frequencies of (a) $v_{y}$ and (b) $v_{x}$ from the
4-25Hz band-limited elastic recordings.

Figure 16: The learning curves of ARCH1 trained using different learning rates
to extrapolate the 0-4Hz low frequencies of (a) $v_{y}$ and (b) $v_{x}$ from
the 4-25Hz band-limited elastic recordings.

Figure 17: The learning curves of ARCH1 and ARCH2 trained to extrapolate the
0-4Hz low frequencies of (a) $v_{y}$ and (b) $v_{x}$ from the 4-25Hz band-
limited elastic recordings. Figure 18: Comparison of the inverted (a)
$\mathbf{v}_{p}$, (b) $\mathbf{v}_{s}$ and (c) $\rho$ models from extrapolated
elastic FWI. The resulting models start from the inverted low-wavenumber
models using $2.0-4.0Hz$ data extrapolated using ARCH2.

##  5 Discussions and Limitations

Recovering the density using FWI is very challenging, independently of the
bandwidth extension question, for the following reasons. (1) Cross-talk
happens using short-offset data since P-wave velocity and density have the
same radiation patterns at short apertures. (2) The variations in density are
smaller than those in velocities. (3) Inversion of density requires ultra-low
frequencies. Although elastic FWI does not always allow to correctly estimate
density, it stands a better chance of properly reconstructing velocities, with
either extrapolated or true low frequencies.

In extrapolated FWI, the choice of starting frequency is a trade-off between
the accuracy of extrapolated low frequency data and the lowest frequency to
mitigate the cycle-skipping problem. We start elastic FWI with 2-4Hz
extrapolated low frequency data due to the insufficient extrapolation accuracy
in the near-zero frequency range. The accuracy of the 2-4Hz extrapolated low
frequency data is sufficient for elastic FWI of P-wave and S-wave velocities
when starting from 4Hz band-limited data. However, the inversion of density
model still surfers from the cycle-skipping problem and lack of the low-
wavenumber structure.

The numerical example shows that the neural network cannot meaningfully
generalize from the acoustic training dataset to the elastic test dataset. In
addition to the wave propagation driven by different physics, another factor
that makes the generalization fail could be numerical modeling. The synthetic
elastic training and test datasets are simulated by solving the stress-
velocity formulation of the wave equation using standard staggered grid with
an eighth order FD operator. However, the acoustic training dataset is
simulated by solving the stress-displacement formulation using a sixth order
FD operator.

The source signal is assumed to be known for extrapolated elastic FWI in this
paper. However, for field data, the source signal may vary shot by shot. One
solution could be to retrieve the source wavelet of the field dataset firstly,
and then artificially boost the low-frequency energy after denoising. The new
source signal can be used to synthesize the training dataset for low-frequency
extrapolation. It can also be the source wavelet in the following elastic FWI
using the extrapolated low-frequency data. In this way, the uncertainty of the
source can be controlled to some extent.

We do not provide the numerical results of other factors that affect the
performance of the deep learning models. For example, regularization of
training loss, number of iterations, parameters of neural network, number of
training samples and even inverse crime. Deep learning models contain many
hyper-parameters and finding the best configuration for these parameters in a
high dimensional space is challenging.

Finally, neural networks are trained using a stochastic learning algorithm.
This means that the same model trained on the same dataset may result in a
different performance. The specific results may vary, but the general trend
should be the same, as reported in Section 4.4.

##  6 Conclusions

To relieve the dependency of elastic FWI on starting models, low-frequency
extrapolation of multi-component seismic recordings is implemented to
computationally recover the missing low frequencies from band-limited elastic
data. The deep learning model is designed with a large receptive field in two
different ways. One directly uses a large filter on each convolutional layer,
the other utilizes dilated convolution to increase the receptive field
exponentially with depth. By training the neural network twice, once with a
dataset of horizontal components and once with a dataset of vertical
components, we can extrapolate the low frequencies of multi-component band-
limited recordings separately. The extrapolated 0-5Hz low frequencies match
well with the true low-frequency data on the Marmousi2 model. Elastic FWI
using 2-4Hz extrapolated data shows similar results to the true low
frequencies. The accuracy of the extrapolated low frequencies is enough to
provide low-wavenumber starting models for elastic FWI of P-wave and S-wave
velocities on data band-limited above 4Hz.

The generalization ability of the neural network over different physical
models is studied in this paper. The neural network trained on purely acoustic
data shows larger prediction error on elastic test dataset compared to the
neural network trained on elastic data. Therefore, collecting more realistic
elastic training dataset will help to process the field data with strong
elastic effects.

## References

  * Abadi et al. (2015) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mané, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viégas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., & Zheng, X., 2015.  TensorFlow: Large-scale machine learning on heterogeneous systems, Software available from tensorflow.org.
  * Aminzadeh et al. (1997) Aminzadeh, F., Jean, B., & Kunz, T., 1997.  SEG/EAGE 3-D Salt and Overthrust Models. SEG/EAGE 3-D Modeling Series, No. 1: Distribution CD of Salt and Overthrust models, SEG book series.
  * Araya-Polo et al. (2018) Araya-Polo, M., Jennings, J., Adler, A., & Dahlke, T., 2018.  Deep-learning tomography, The Leading Edge, 37(1), 58–66.
  * Borisov et al. (2020) Borisov, D., Gao, F., Williamson, P., & Tromp, J., 2020.  Application of 2D full-waveform inversion on exploration land data, Geophysics, 85(2), R75–R86.
  * Brenders et al. (2018) Brenders, A., Dellinger, J., Kanu, C., Li, Q., & Michell, S., 2018.  The wolfspar® field trial: Results from a low-frequency seismic survey designed for FWI, in in 88th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 1083–1087.
  * Brossier et al. (2009) Brossier, R., Operto, S., & Virieux, J., 2009.  Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion, Geophysics, 74(6), WCC105–WCC118.
  * Brossier et al. (2010) Brossier, R., Operto, S., & Virieux, J., 2010.  Which data residual norm for robust elastic frequency-domain full waveform inversion?, Geophysics, 75(3), R37–R46.
  * Bunks et al. (1995) Bunks, C., Saleck, F. M., Zaleski, S., & Chavent, G., 1995.  Multiscale seismic waveform inversion, Geophysics, 60(5), 1457–1473.
  * Choi et al. (2008) Choi, Y., Min, D.-J., & Shin, C., 2008.  Frequency-domain elastic full waveform inversion using the new pseudo-hessian matrix: Experience of elastic Marmousi-2 synthetic data, Bulletin of the Seismological Society of America, 98(5), 2402–2415.
  * Chollet et al. (2015) Chollet, F. et al., 2015.  Keras.
  * Crase et al. (1990) Crase, E., Pica, A., Noble, M., McDonald, J., & Tarantola, A., 1990.  Robust elastic nonlinear waveform inversion: Application to real data, Geophysics, 55(5), 527–538.
  * Demanet & Nguyen (2015) Demanet, L. & Nguyen, N., 2015.  The recoverability limit for superresolution via sparsity, arXiv preprint arXiv:1502.01385.
  * Demanet & Townsend (2019) Demanet, L. & Townsend, A., 2019.  Stable extrapolation of analytic functions, Foundations of Computational Mathematics, 19(2), 297–331.
  * Fang et al. (2020) Fang, Z., Fang, H., & Demanet, L., 2020.  Deep generator priors for bayesian seismic inversion, Advances in Geophysics, 61, 179–216.
  * Herrmann et al. (2019) Herrmann, F. J., Siahkoohi, A., & Rizzuti, G., 2019.  Learned imaging with constraints and uncertainty quantification, arXiv preprint arXiv:1909.06473.
  * Hewett & Demanet (2013) Hewett, R. & Demanet, L., 2013.  the pysit team, 2013, PySIT: Python seismic imaging toolbox v0.5.
  * Hu et al. (2019) Hu, W., Jin, Y., Wu, X., & Chen, J., 2019.  Progressive transfer learning for low frequency data prediction in full waveform inversion, arXiv preprint arXiv:1912.09944.
  * Jeong et al. (2012) Jeong, W., Lee, H.-Y., & Min, D.-J., 2012.  Full waveform inversion strategy for density in the frequency domain, Geophysical Journal International, 188(3), 1221–1242.
  * Jin et al. (2018) Jin, Y., Hu, W., Wu, X., Chen, J., et al., 2018.  Learn low wavenumber information in FWI via deep inception based convolutional networks, in in 88th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 2091–2095.
  * Kazei et al. (2020) Kazei, V., Ovcharenko, O., Plotnitskii, P., Peter, D., Alkhalifah, T., Silvestrov, I., Bakulin, A., & Zwartjes, P., 2020.  Elastic near-surface model estimation from full waveforms by deep learning, in in 90th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 3872–3876.
  * Köhn et al. (2012) Köhn, D., De Nil, D., Kurzmann, A., Przebindowska, A., & Bohlen, T., 2012.  On the influence of model parametrization in elastic full waveform tomography, Geophysical Journal International, 191(1), 325–345.
  * Levander (1988) Levander, A. R., 1988.  Fourth-order finite-difference P-SV seismograms, Geophysics, 53(11), 1425–1436.
  * Li et al. (2019) Li, D., Gao, F., & Williamson, P., 2019.  A deep learning approach for acoustic FWI with elastic data, in in 89th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 2303–2307.
  * Li & Demanet (2017) Li, Y. & Demanet, L., 2017.  Extrapolated full-waveform inversion: An image-space approach, in in 87th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 1682–1686.
  * Li & Demanet (2015) Li, Y. E. & Demanet, L., 2015.  Phase and amplitude tracking for seismic event separation, Geophysics, 80(6), WD59-WD72.
  * Li & Demanet (2016) Li, Y. E. & Demanet, L., 2016.  Full-waveform inversion with extrapolated low-frequency data, Geophysics, 81(6), R339–R348.
  * Mahrooqi et al. (2012) Mahrooqi, S., Rawahi, S., Yarubi, S., Abri, S., Yahyai, A., Jahdhami, M., Hunt, K., & Shorter, J., 2012.  Land seismic low frequencies: acquisition, processing and full wave inversion of 1.5-86 Hz, in in 82nd Annual SEG meeting, SEG Technical Program Expanded Abstracts, 1–5.
  * Marjanović et al. (2018) Marjanović, M., Plessix, R.-É., Stopin, A., & Singh, S. C., 2018.  Elastic versus acoustic 3-D full waveform inversion at the east pacific rise 9∘ 50’ N, Geophysical Journal International, 216(3), 1497–1506.
  * Martin et al. (2006) Martin, G. S., Wiley, R., & Marfurt, K. J., 2006.  Marmousi2: An elastic upgrade for marmousi, The Leading Edge, 25(2), 156–166.
  * Mora (1987) Mora, P., 1987.  Nonlinear two-dimensional elastic inversion of multioffset seismic data, Geophysics, 52(9), 1211–1228.
  * Moseley et al. (2018) Moseley, B., Markham, A., & Nissen-Meyer, T., 2018.  Fast approximate simulation of seismic waves with deep learning, arXiv preprint arXiv:1807.06873.
  * Mosser et al. (2020) Mosser, L., Dubrule, O., & Blunt, M. J., 2020.  Stochastic seismic waveform inversion using generative adversarial networks as a geological prior, Mathematical Geosciences, 52(1), 53–79.
  * Nocedal & Wright (2006) Nocedal, J. & Wright, S., 2006.  Numerical optimization, Springer Science & Business Media.
  * Oord et al. (2016) Oord, A. v. d., Dieleman, S., Zen, H., Simonyan, K., Vinyals, O., Graves, A., Kalchbrenner, N., Senior, A., & Kavukcuoglu, K., 2016.  Wavenet: A generative model for raw audio, arXiv preprint arXiv:1609.03499.
  * Ovcharenko et al. (2018) Ovcharenko, O., Kazei, V., Peter, D., Zhang, X., & Alkhalifah, T., 2018.  Low-frequency data extrapolation using a feed-forward ANN, in 80th EAGE Conference and Exhibition 2018, European Association of Geoscientists & Engineers, 1–5.
  * Ovcharenko et al. (2019) Ovcharenko, O., Kazei, V., Kalita, M., Peter, D., & Alkhalifah, T., 2019.  Deep learning for low-frequency extrapolation from multioffset seismic data, Geophysics, 84(6), R989–R1001.
  * Plessix et al. (2013) Plessix, R., Milcik, P., Rynja, H., Stopin, A., Matson, K., & Abri, S., 2013.  Multiparameter full-waveform inversion: Marine and land examples, The Leading Edge, 32(9), 1030–1038.
  * Plessix et al. (2012) Plessix, R.-É., Baeten, G., de Maag, J. W., ten Kroode, F., & Rujie, Z., 2012\.  Full waveform inversion and distance separated simultaneous sweeping: a study with a land seismic data set, Geophysical Prospecting, 60(4), 733–747.
  * Raknes et al. (2015) Raknes, E. B., Arntsen, B., & Weibull, W., 2015.  Three-dimensional elastic full waveform inversion using seismic data from the sleipner area, Geophysical Journal International, 202(3), 1877–1894.
  * Sears et al. (2010) Sears, T. J., Barton, P. J., & Singh, S. C., 2010.  Elastic full waveform inversion of multicomponent ocean-bottom cable seismic data: Application to Alba Field, UK North Sea, Geophysics, 75(6), R109–R119.
  * Siahkoohi et al. (2019) Siahkoohi, A., Louboutin, M., & Herrmann, F. J., 2019.  The importance of transfer learning in seismic modeling and imaging, Geophysics, 84(6), A47–A52.
  * Sirgue et al. (2009) Sirgue, L., Barkved, O., Van Gestel, J., Askim, O., & Kommedal, J., 2009.  3D waveform inversion on Valhall wide-azimuth OBC, in 71st EAGE Conference and Exhibition incorporating SPE EUROPEC 2009, European Association of Geoscientists & Engineers, pp. cp–127.
  * Srivastava et al. (2014) Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R., 2014.  Dropout: a simple way to prevent neural networks from overfitting, The journal of machine learning research, 15(1), 1929–1958.
  * Stopin et al. (2014) Stopin, A., Plessix, R.-É., & Al Abri, S., 2014.  Multiparameter waveform inversion of a large wide-azimuth low-frequency land data set in Oman, Geophysics, 79(3), WA69–WA77.
  * Sun & Demanet (2018) Sun, H. & Demanet, L., 2018.  Low frequency extrapolation with deep learning, in in 88th Annual SEG meeting, SEG Technical Program Expanded Abstracts, 2011–2015.
  * Sun & Demanet (2020) Sun, H. & Demanet, L., 2020.  Extrapolated full-waveform inversion with deep learning, Geophysics, 85(3), R275–R288.
  * Tarantola (1986) Tarantola, A., 1986.  A strategy for nonlinear elastic inversion of seismic reflection data, Geophysics, 51(10), 1893–1903.
  * Vigh et al. (2014) Vigh, D., Jiao, K., Watts, D., & Sun, D., 2014.  Elastic full-waveform inversion application using multicomponent measurements of seismic data collection, Geophysics, 79(2), R63–R77.
  * Virieux (1986) Virieux, J., 1986.  P-SV wave propagation in heterogeneous media; velocity-stress finite-difference method, Geophysics, 51(4), 889–901.
  * Wu & Lin (2019) Wu, Y. & Lin, Y., 2019.  InversionNet: An efficient and accurate data-driven full waveform inversion, IEEE Transactions on Computational Imaging, 6, 419–433.
  * Yang & Ma (2019) Yang, F. & Ma, J., 2019.  Deep-learning inversion: A next-generation seismic velocity model building method, Geophysics, 84(4), R583–R599.
  * Yu & Koltun (2015) Yu, F. & Koltun, V., 2015.  Multi-scale context aggregation by dilated convolutions, arXiv preprint arXiv:1511.07122.
  * Zhang & Lin (2020) Zhang, Z. & Lin, Y., 2020.  Data-driven seismic waveform inversion: A study on the robustness and generalization, IEEE Transactions on Geoscience and Remote Sensing, 58(10), 6900–6913


# Time as change

Marcello Poletti<EMAIL_ADDRESS>San Giovanni Bianco, Italy

###### Abstract

According to Aristotle ”time is the number of change with respect to the
before and after”[1]. That’s certainly a vague concept, but at the same time
it’s both simple and satisfying from a philosophical point of view: things do
not change along time, but they do change and the measurement of such changes
is what we call time. This deprives time of any attribute of substantiality,
meanwhile depriving it of all problems in defining the properties of time as a
substance. With the rise of Classical Mechanics, Aristotle’s view is abandoned
and Newton’s concept of ”true”[2] and absolute time imposes itself; time flows
independently on changes of any kind. Relativity will then radically modify
our concept of time, but won’t actually modify the fundamental idea: things
keep changing along time - changes do not make time. This work will argue
Aristotle’s thesis, showing how such an approach automatically leads to the
principles of Special Relativity. An interesting consequence and, at least
virtually, measurable will also be highlighted: the fact that synchronizing
two clocks with a precision greater than a certain scale is impossible,
estimating such scale around $10^{-22}s$.

##  I Introduction

The most evident difference between an Aristotle’s approach, according to
which time is constituted by the totality of changes, and the current approach
used in Physics, according to which things change along time, is probably that
the previous one doesn’t admit, by definition, a permanent state of rest and
the second approach does admit it. The first law of motion states that ”a body
continues in its state of rest […]” and this is in clear contradiction of
Aristotle’s approach, unless we envisage that the rest state is only,
implicitly or explicitly, an approximation for disregarding a series of
changes which ”maintain active” the time stream.

This is actually compatible with our usual idea of matter. A stone is in a
state of rest, according to the current meaning, as its mass center doesn’t
clearly move (and the body has not angular momentum) but we imagine it,
although at rest, being made of molecules and atoms which aren’t actually in a
state of rest but on the contrary shake, having a complex internal structure,
and constantly exchange electromagnetic radiation by absorption and emission
with the surrounding environment. A body in its ”state of rest” is actually a
continuous succession of events.

Einstein originally writes[3] his most famous equation opposite to the common
use:

$m=E/c^{2}$

explicitly stating ”The mass of a body is a measure of its energy content” as
if the mass itself, even if in the state of rest, was thought as containing
constant changes.

Penrose[4] combines Einstein’s equation with the equally famous De Broglie’s
equation[5], achieving:

$m=h\nu/c^{2}$

and he comments even more explicitly: ”The mass of any stable particle,
therefore, determines a very precise clock rate, as given by this frequency.”.

According to this view Newtonian and Aristotelian time can be reconciled,
considering the first one an approximation, so that the majority of changes
which ”measure time” are ignored and changes caused by the motion studied by
the mechanics (e.g. a falling stone) have no perceived influence on the total
changes which are taking place.

##  II Time as change in Classical Mechanics

### Definition

The Classical Mechanics typically studies the motion of pointlike particles
and, by extension, of spatially extended bodies, and fluids. A typical point
particle is an ideal body with no size and a non-zero mass. In order to
organize a Classical Mechanics based on Aristotle’s time we’ll consider
Democritus’ massless atoms, in order not to face difficulties previously
mentioned in the introduction, atoms in motion inside of a Newton’s Euclidean
space.

The only dynamic properties for these point particles are their relative
positions, therefore the Aristotle’s definition of time can be easily
expressed as:

$\Delta T=\Delta S/c$ (1)

That means, time interval is identified with displacement, up to an arbitrary
constant c (c being a velocity) whose only aim here is to lead to a change of
unit of measurement. The previous definition 1 rewritten in differential form
can be expressed as:

$dx^{2}+dy^{2}+dz^{2}-cdt^{2}=0$

Which corresponds to the metric of light-like paths in a Minkowski space.

Regarding the speed we notice clearly that the only accepted value is:

$v=\frac{\Delta S}{\Delta T}=c$

### Moving Reference Frames

Let P a path traced by the point, it can be expressed in a parametric form:

$P=\begin{cases}x(t)\\\ y(t)\\\ z(t)\end{cases}$

And such form will be called normal if

$|\dot{P}|=\sqrt{\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2}}=c$

For a path in a normal form we have:

$\Delta S(t)=\mathcal{L}(P,t)=\int_{0}^{t}{|\dot{P}|}=ct$

And therefore, for a path in a normal form, $t$ parameter corresponds with the
given definition of time.

The equivalence space-time defined in 1 makes these two concepts
interchangeable. Inter alia a Cartesian axis can be identified by the path
which travels it, identifying its points with the necessary time for the path
to reach them.

Consider then the following normal linear path in a reference frame $k$, with
its origin in $O$.

$P=\begin{cases}ct\\\ 0\\\ 0\end{cases}$

Apply therefore a linear transformation of the reference frame $k$
parameterized with the same time $t$ and depending on the parameters $v$ and
$\gamma$:

$k^{\prime}=\begin{cases}x^{\prime}=\gamma(x-vt)\\\ y^{\prime}=y\\\
z^{\prime}=z\end{cases}$

Hence, $P$ reaches the point $(x,0,0)$ at time $t_{x}=x/c$; we have therefore

$k^{\prime}=\begin{cases}x^{\prime}=\gamma x\dfrac{c-v}{c}\\\ y^{\prime}=y\\\
z^{\prime}=z\end{cases}$

Applying now the inverse transformation on $k^{\prime}$, it is required by
symmetry that

$x^{\prime\prime}=\gamma\left[\gamma x\dfrac{c-v}{c}\right]\dfrac{c+v}{c}=x$

Hence

$\gamma=\dfrac{1}{\sqrt{1-v^{2}/c^{2}}}$

Finally, time will have a formal transformation in $k^{\prime}$

$t^{\prime}=\dfrac{x^{\prime}}{c}=\gamma(t-\dfrac{v}{c^{2}}x)$

The reference frame $k^{\prime}$ will be called moving with velocity $v$
relative to $k$.

A moving reference frame follows Lorentz transformations.

### Moving paths

Given now a generic normal path:

$P=\begin{cases}x(t)\\\ y(t)\\\ z(t)\end{cases}$

That can be expressed in an integral form as:

$P=\begin{cases}\int\dot{x}(t)+x_{0}\\\ \int\dot{y}(t)+y_{0}\\\
\int\dot{z}(t)+z_{0}\\\ \end{cases}$

The infinitesimal shift is:

$dP=\begin{cases}\dot{x}dt\\\ \dot{y}dt\\\ \dot{z}dt\\\ \end{cases}$

and, if placed within a reference frame $k^{\prime}$ moving at a $-v$
velocity, it becomes

$dP^{\prime}=\begin{cases}\gamma(\dot{x}+v)dt\\\ \dot{y}dt\\\ \dot{z}dt\\\
\end{cases}$

hence

$P^{\prime}=\begin{cases}\gamma(x+vt-x_{0})+x_{0}\\\ y\\\ z\\\ \end{cases}$

$P^{\prime}$ path is equivalent to $P$ within the reference frame $k^{\prime}$
moving with velocity $-v$, vice versa we will affirm that $P^{\prime}$ is
equivalent to $P$ moving with speed $v$.

### Clocks

$P^{\prime}$ path obtained in the previous paragraph is a non-normal path, we
have

$\dot{P}^{\prime}=\begin{cases}\gamma(\dot{x}+v)\\\ \dot{y}\\\ \dot{z}\\\
\end{cases}$

Hence

$\mathcal{L}(P^{\prime},t)=\int_{0}^{t}{|\dot{P}^{\prime}|}=\int_{0}^{t}{\sqrt{\gamma^{2}(\dot{x}+v)^{2}+\dot{y}^{2}+\dot{z}^{2}}}$

By using the normality of $P$, we have:

$\mathcal{L}(P^{\prime},t)=\int_{0}^{t}{\sqrt{\dfrac{(\dot{x}+v)^{2}}{1-v^{2}/c^{2}}+c^{2}-\dot{x}^{2}}}$

By simplifying

$\mathcal{L}(P^{\prime},t)=\gamma\int_{0}^{t}{c+\dfrac{\dot{x}v}{c}}=\gamma(ct+v(x-x_{0})/c)$

Specifically, if $P$ is cyclic, that is if $P(t)=P(t+T)$, we obtain for
$P^{\prime}$ a length of a path of one cycle equal to

$\mathcal{L}(P^{\prime},t)=\gamma cT$

The time of a $T$ cycle is therefore dilated of a Lorentz factor

$T^{\prime}=\gamma T$

We might consider a cycling path as a clock marking time in relation to its
own cycles; according to this, a clock’s mechanism is slowed down of a Lorentz
factor.

A clock or a cyclic path can be thought as ”objects themselves”, choosing to
ignore their generating path and studying their overall dynamics.

Such dynamics, as shown above, are subject to the laws of Special Relativity.

### Mass

In order to introduce a mass concept, we consider a circular oscillator of
radius $r$, normal, within the plane $(x,y)$:

$C=\begin{cases}rcos(ct/r)\\\ rsin(ct/r)\\\ 0\\\ \end{cases}$

$C$ is an oscillator of radius $r$, pulse $\omega=c/r$ and frequency
$\nu=c/2\pi r$.

Using Penrose’s interpretation shown above, we get the definition:

$m=h\nu/c^{2}=\hbar/cr$

Vice versa

$r=\hbar/cm$

In order to get a real world example, if an electron was shaped as a circular
path, its radius would be

$r_{e}=3.87\cdot 10^{-13}$

a value lower than Bohr’s radius $a_{0}=\hbar/(cm_{e}\alpha)$ for a factor
identical to the inverse of the fine-structure constant, equal to
approximately 137.

### Scholium

Aristotle’s definition of time, applied to Classical Mechanics via the simple
definition $\Delta T=\Delta S$ (in natural units), automatically leads to a
definition of Moving Reference Frame subject to Lorentz transformations,
leading therefore to the definition of moving path and from there to Special
Relativity. Note that this happens with no mention of properties of light or
of the electromagnetic field. Such a concept leads to two symmetrical
observations: on one side, this concept is enough of a prove that defining
time as a measure of change is a hypothesis, mainly abandoned by modern
Physics, really worth being reconsidered and further explored in different
fields of Physics; on the other side, such an approach allows re-reading the
Relativity itself under a new light which considers both existence and
constancy of a maximum speed not as postulates but as simple consequences of
the definition of time itself. Therefore time and space turn out to be deeply
interconnected and interchangeable, substantially, by definition.

Through this approach, time loses any substantial characterization (one might
say that ”time does not exist”[6] as well) and is replaced by the weaker but
philosophically more gratifying concept of change.

##  III Multiple paths

Given two generic paths, $P_{1}$ e $P_{2}$, a number of different applications
of the concepts of displacement exist; there are therefore various different
definitions of the concept of time, as shown thus far.

On one side, $P_{1}$ and $P_{2}$ separately define two proper times $t_{1}$
and $t_{2}$, as if the other path was negligible;

on the other side, it is altogether possible to consider the total
displacement as time $t_{12}$ of the couple, hence:

$t_{12}=t_{1}+t_{2}$

Clearly, as a consequence of this dissertation’s assumptions, no mechanisms
synchronizing $t_{1}$ and $t_{2}$ can be accepted. That is, it isn’t
acceptable that any absolute clock establishing a defined relationship between
the two paths exists. In other words, the idea of ”time as change” necessarily
implies that knowing $t_{1}$ is not enough to gather exact information about
$t_{2}$.

Suppose to discretize the frame, postulating that every move (therefore every
time) happens along discrete steps. Assigning such steps to any of the two
paths randomly seems necessary.

Hence arises the actual problem of computing the probability that when the
time of $P_{1}$ path is $t_{1}$, then the time of $P_{2}$ path is $t_{2}$.

The Classic Probability Theory answers this question with the negative
binomial distribution (or Pascal distribution):

$P(t_{1},t_{2})=\dfrac{\binom{t_{1}+t_{2}-1}{t_{1}-1}}{2^{t_{1}+t_{2}}}$

Hence the probability density that $t_{2}=xt_{1}$ can be deduced

$\tilde{P}(n,x)=\dfrac{\binom{n+nx-1}{n-1}}{2^{n+nx}}=nP(n,nx)$

Where $n$ is a whole number expressing the size of time quantum $dt=1/n$.

By applying the central limit theorems (De Moivre-Laplace) we get

$\tilde{P}(n,x)\approx\frac{ne^{-\frac{(n-nx-1)^{2}}{2(n+nx-1)}}}{\sqrt{2\pi(n+nx-1)}}$

Given $\tau=1/n$ we have

$\tilde{P}(n,x)\approx\frac{e^{-\frac{(1-x-\tau)^{2}}{2(1+x-\tau)}}}{\sqrt{2\pi\tau(1+x-\tau)}}$

And we can approximate this with a normal distribution with standard deviation

$\sigma=\sqrt{2\tau}$

If $\tau$ is infinitesimal, so changes happen in continuity, $\sigma$ is
infinitesimal as well and times $t_{1}$ and $t_{2}$ have a natural tendency to
line up. On the contrary, if $\tau$ was subject to a form of quantization,
both times would be statistically out of phase of a factor $\sigma$.

In particular, choosing for $\tau$ the value of Planck time ($10^{-44}s$), we
obtain

$\sigma\approx 10^{-22}s$

In other words, two clocks couldn’t be synchronizable with a precision greater
than $10^{-22}s$ which certainly is an extremely small time value, but not too
far from the scale of atomic phenomena.

### Scholium

If the ”measure of change” requested by the Aristotle’s approach is meant as a
discrete computing of changes, the limited precision with which two clocks are
synchronizable arises. Therefore below such limit the notion of time becomes
deeply messy and presumably inapplicable in its traditional meaning.

The usual chance to order events according a before and an after would only be
possible from a certain scale up and such scale, as small as it may be, would
be much larger than the quantization scale anyway, to the point of being
comparable to the shortest time intervals so far measured [7].

##  IV Conclusions

Nature of time is still an important enigma in Natural Philosophy but probably
time is ”The Enigma” for most physicians and philosophers. I tried to show the
great richness that a thousands years old idea might hold.

This essay is an invitation to try and stop thinking of how things have been
changing over time, considering instead how things change and the measurement
of such changes is the measurement of time. This time is structurally
relativistic and local, and it has a peculiar extensive nature: the time of
two similar objects is approximately twice as much the time of a single
object. Everyday time becomes somehow thermic, a sort of an almost constant
average of the number of local changes. That’s more than an analogy, both
Aristotle’s time and temperature measure change, agitation. According to this
view, similar to the state of quiet, also the absolute thermic zero cannot be
a persistent absolute state as this condition would stop the flowing of time.

## Acknowledgements

Thanks to Alessandra and Carlo for helping me with the translation of this
article.

To Deborah, who challenged time.

## References

  * [1] Aristotele, Fisica IV
  * [2] I. Newton, ”Philosophiae naturalis principia mathematica”, 1687
  * [3] A. Einstein, ”Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, 1905
  * [4] R. Penrose, ”Fashion, Faith, and Fantasy in the New Physics of the Universe”, 2016
  * [5] L. De Broglie,” Recherches sur la théorie des Quanta”, 1925
  * [6] C. Rovelli, ”Forget Time”, https://arxiv.org/pdf/0903.3832.pdf
  * [7] https://www.science.org/doi/full/10.1126/science.abb9318