# Spatial Deep Deconvolution U-Net for Traffic Analyses with Distributed
Acoustic Sensing

Siyuan Yuan, Martijn van den Ende, Jingxiao Liu, Hae Young Noh, Robert Clapp,
Cédric Richard, Biondo Biondi Siyuan Yuan, Jingxiao Liu, Hae Young Noh, Robert
Clapp, and Biondo Biondi are with Stanford University, CA, U.S.A.Martijn van
den Ende, and Cédric Richard are with the Université Côte d’Azur,, OCA, UMR
Lagrange, France. This work has been submitted to the IEEE for possible
publication. Copyright may be transferred without notice, after which this
version may no longer be accessible.

###### Abstract

Distributed Acoustic Sensing (DAS) that transforms city-wide fiber-optic
cables into a large-scale strain sensing array has shown the potential to
revolutionize urban traffic monitoring by providing a fine-grained, scalable,
and low-maintenance monitoring solution. However, there are challenges that
limit DAS’s real-world usage: noise contamination and interference among
closely traveling cars. To address the issues, we introduce a self-supervised
U-Net model that can suppress background noise and compress car-induced DAS
signals into high-resolution pulses through spatial deconvolution. To guide
the design of the approach, we investigate the fiber response to vehicles
through numerical simulation and field experiments. We show that the localized
and narrow outputs from our model lead to accurate and highly resolved car
position and speed tracking. We evaluate the effectiveness and robustness of
our method through field recordings under different traffic conditions and
various driving speeds. Our results show that our method can enhance the
spatial-temporal resolution and better resolve closely traveling cars. The
spatial deconvolution U-Net model also enables the characterization of large-
size vehicles to identify axle numbers and estimate the vehicle length.
Monitoring large-size vehicles also benefits imaging deep earth by leveraging
the surface waves induced by the dynamic vehicle-road interaction.

###### Index Terms:

Traffic monitoring, intelligent transportation, Distributed Acoustic Sensing,
deconvolution, U-Net.

##  I Introduction

Traffic monitoring systems, which automatically and continuously detect,
track, and characterize vehicles in moving traffic, provide valuable
information for urban management, maintenance, and planning. Conventional
monitoring systems include vision-based [1, 2, 3] and pavement sensing
technologies (e.g., inductive loops [4, 5, 6] and piezoelectric sensors [7, 8,
4]). These approaches are well-developed but have several drawbacks. For
example, camera systems bring individual-privacy concerns and are sensitive to
weather conditions; point pavement sensing systems provide spatially sparse
sampling and are challenging to maintain. Recent mobile sensing methods
overcome the on-site installation and maintenance challenge, but such systems
require users to opt-in location tracking and rely on cellular connectivity.

An emerging fiber-optic sensing technology, Distributed Acoustic Sensing
(DAS), which turns optical fibers into dense-sampling seismic recording
arrays, has the potential to revolutionize urban traffic monitoring by
providing fine-grained and ubiquitous traffic flow and accident information.
DAS-based monitoring is performed by connecting an optoelectronic DAS
instrument called the interrogator to one end of a standard
telecommunications‐grade optical fiber. The interrogator sends short laser
pulses into the optical fiber and measures the subtle phase shifts of Rayleigh
scattered light returning to the detector at a predicted two‐way travel time
[9, 10]. In this way, the strain field induced by natural processes (e.g.
earthquakes) and urban activities (e.g. moving vehicles, construction, pumps)
acting on the fiber coupled to the Earth can be sampled at a meter‐scale
spatial resolution over tens of linear fiber kilometers. More recently, DAS
has been proven effective even on pre-existing telecommunication fiber
infrastructure (“dark fiber”) to record broadband signals for earthquake
monitoring and near-surface imaging [11, 12, 13]. As fibers installed near
roads capture deformation induced by moving cars, DAS technology has provided
a geophysical solution for real-time traffic monitoring including average
speed monitoring, congestion time estimation, and queue detection [14].
However, due to the hardware constraints, their system only has intensity
measurements without the phase information limiting its practical value. With
the DAS technology advancement, new interrogators (e.g. OptaSense QuantX [15])
capable of recording full waveform information for up to 50 km with 1 m
channel spacing have become practical for traffic monitoring applications.
Thanks to their broadband nature, QuantX-like DAS recordings comprise rich
traffic information, including high-frequency ($>2$ Hz) surface waves due to
vehicle-road dynamic interaction and the low-frequency quasi-static
deformation caused by vehicle loading [16, 17, 18]. The surface-wave
components have been used to image the subsurface velocity structure down to
hundreds of meters beneath the road [16]. The quasi-static components
reflecting the vehicle trajectories are much more compact and have a simpler
waveform than the surface waves. [17, 19, 20] demonstrate the feasibility of
using the quasi-static signals to estimate vehicle count and speeds, and
illustrate several advantages for a DAS traffic monitoring system: first, DAS
records absolute signals of human-induced deformation that cannot be easily
tied to any individual; second, DAS monitoring can be cost-efficient for city-
scale monitoring with pre-existing “dark fiber”; last, DAS has low day-to-day
operation and maintenance costs with only a single interrogator being deployed
in a secure and easily accessible location, as opposed to having numerous
individually powered instruments deployed across a city (exposed to
meteorological conditions and detrimental interactions with humans, plants,
and animals).

Despite the aforementioned advantages, analyzing data from a DAS array located
in an urban environment is challenging because it records a complex mixture of
inherently unlabeled signals [21]. A robust and accurate car-signal detection
algorithm is essential to enable the real-world usage of a DAS-based traffic
monitoring system. To detect car-induced quasi-static signals, [17] applied a
common seismological method, the short-time-average through long-time-average
trigger (STA/LTA). To exploit the array geometry of DAS, beamforming
algorithms have been applied to detect cars and measure their speed [19, 20].
These simple methods performed well on roads with relatively light traffic and
without complicated traffic patterns. But they are likely to become inaccurate
if many vehicles transit simultaneously close to the same segment of the fiber
cable, as signals from different cars start overlapping leading to complex
patterns.

To reduce the interference among closely traveling cars, [22] proposed a self-
supervised time-domain deconvolution Auto-Encoder (time-domain DAE) with the
vehicles’ impulse response from the quasi-static recordings. Compared to a
conventional channel-wise deconvolution algorithm, the time-domain DAE model
has the benefits of incorporating the spatial-temporal characteristics of car
signals, leading to much sharper and localized outputs. Applying a beamforming
algorithm to the localized outputs rather than the original inputs shows
significant improvements in terms of the resolution in car-speed estimation
and detection accuracy. Furthermore, as shown in [22], the time-domain DAE
model, once trained, can deconvolve 24-hour recordings in less than 30
seconds, achieving $>$ 400 times speedup compared to a conventional iterative
approach, which makes the method promising for real-time processing. However,
the time-domain DAE model assumes a stationary Ricker wavelet in the time
domain as the vehicle’ impulse response. As the following sections of the
paper will show, vehicle’s temporal wavelet is non-stationary to speed
variation. Due to the wavelet mismatch, a time-domain DAE model that works
well with fast-speed traffic produces low-resolution results for low-speed
vehicles causing inaccurate speed tracking.

To address the challenge, we first characterize the vehicle impulse response.
Based on our observation that the vehicle’s spatial wavelet is robust to speed
change, we propose a new deconvolution U-Net model (space-domain DAE model)
using a space-domain vehicle wavelet. The U-Net model takes quasi-static
signals as inputs and outputs sharp deconvolution results reflecting the
vehicle trajectory. It is a weak-supervised approach not requiring any
groundtruth labels: The data-fitting term in the loss function only involves
network input and a reconstructed input resulted from a spatial convolution
between the network output and the assumed wavelet. The benefits of our method
over the time-domain DAE model is that it is effective and robust to compress
signals of speed-varying cars into compact pulses leading to more accurate
vehicle tracking and better resolving complicated patterns in heavy traffic
condition.

The rest of the paper is organized as follows: Section II demonstrates the
resolution reduction of traffic-induced DAS signals due to a combination
effect of physics and the limitation of the sensor. Through the numerical
characterization of vehicle impulse response and observations from field
experiments, we design a spatial DAE model aiming to improve the signal
resolution for speed-varying vehicles. Section III describes the method of our
spatial DAE model in detail. Section IV describes the training procedure and
baseline approaches. Section V benchmarks the proposed DAE model with
baselines for car motion and speed tracking, and resolving closely traveling
vehicles in heavy traffic. Furthermore, we demonstrate the use of the space-
domain DAE model for bus axle number counting and length estimation. Besides,
we discuss a potential limitation of the spatial DAE model for a case with a
spatially non-stationary vehicle response. Section VI concludes our work.

##  II Vehicle-Induced DAS Response

This section aims to better understand the vehicle-induced DAS response to
guide the design of the proposed DAE model. We discuss two types of vehicle-
induced DAS responses as background. We then conduct theoretical study and
numerical simulation to demonstrate a resolution reduction of the quasi-static
signals due to physics and sensor limitation, which motivates resolution
improvement through deconvolution. Lastly, we observe the speed-invariant
nature of the spatial wavelet leading to the design of the space-domain DAE
model.

###  II-A Quasi-static and dynamic response

Permanently deployed fibre-optic cables are often buried (trenched) or placed
within underground conduits. We assume here that the DAS fiber is deployed
alongside a road at some depth below the surface. When a vehicle passes near-
by the virtual sensors of the roadside telecom fiber cable, the interaction
between the vehicle and the road structure induces the deformation of the
telecom fiber cable. The signal pattern of vehicle-induced telecom fiber
deformation is a function of the vehicle characteristics, fiber conduit
properties, ambient conditions, etc. There are mainly two components of
signals produced by moving vehicles: quasi-static signals ($<$ 1 Hz) resulting
from the ground deformation due to the vehicle’s weight, and surface waves (2
to 20 Hz) caused by the dynamic vehicle-road interaction resulting from the
roughness of the road (e.g., bumps). Previous studies [17, 16] have found that
the quasi-static component dominates the energy of vehicle-induced telecom
fiber vibration and is theoretically described by the Flamant-Boussinesq
approximation [23, 24]. As a vehicle approaches the virtual sensor, ground
deformation above the sensor increases, and the fiber coupled to the earth is
stretched, resulting in increased tension in the fiber. As the vehicle moves
away, ground deformation near the virtual sensor and the fiber tension
decreases. Due to the relatively strong energy, simplicity and compactness
compared to the surface-wave component, quasi-static signals have been used
for car tracking and detection tasks [22]. For the same reason, our car
tracking method is also based on the quasi-static response.

###  II-B Sensing resolution limited by physics and sensor limitation

We conduct synthetic experiments to demonstrate the resolution degradation of
quasi-static signals due to physics and sensor limitations. We model vehicular
seismic sources using a collection of vertical point forces located at the
vehicle’s wheel-road contacts [24, 25, 16, 17]. Define $x$ as the horizontal
distance along the fiber. The quasi-static or geodetic strain ($E_{x}$) DAS
signal from a vehicle centered at $x=0$ is equal to the change in displacement
over the DAS gauge length ($L$). Displacement in the direction of the fiber
($U_{x}$) can be modeled using the Flamant-Boussinesq equation for a point
load applied to a half space with basic knowledge of the fiber and vehicle
locations, the vehicle load ($F_{z}$), the soil shear modulus ($\mu$), and
Poisson’s ratio ($\nu$):

$U_{x}(x)=\frac{F_{z}}{4\pi\mu}(\frac{zx}{r^{3}}-\frac{(1-2\nu)}{r+z}(\frac{x}{r})),$
(1)

$E_{x}(x)=\frac{U(x+\frac{L}{2})-U(x-\frac{L}{2})}{L},$ (2)

where $z$ is fiber depth, $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is the 3-D distance
from the fiber position to the car, $y$ is the lateral fiber-road offset. From
the equation, we can see that for fixed soil properties, the shape of spatial
strain response depends on the gauge length and the relative position between
the load and the fiber. To compute the modeled synthetic horizontal strain
signal, we assumed a fiber depth of 2 m, appropriate values for sandy/clayey
soils, such as a Poisson’s ratio of 0.4. Fig. 1 shows the simulated strain
along the fiber for a point load at $x=0$ m and $y=0,15,25$ m. We investigate
the gauge length effect by simulating wavelets with $L$ of 0 (equivalent to
point sensor), 8, and 16 m in Fig. 1 (a), (b), and (c), respectively. From the
point sensor wavelets shown in Fig. 1 (a), we can see that the wavelet is
compact and sharp when the fiber is right beneath the load. As $y$ increases,
we can observe an increasing smoothing effect leading to wider wavelets and
lower resolution. By comparing Fig. 1 (a) with (b) and (c), we can see that
increasing gauge length leads to a wider wavelet for 0 offset, but has a
limited impact on larger $y$ (15 and 25 m). Fig. 2 (a) and (b) show the
simulation of a two-axle car with 2.8 m axle spacing and a three-axle vehicle
with 9 m axle spacing, respectively. We compare the point senor recording to
that obtained with $L=16$ m and the fiber-road offset of 15 m and 25 m. We can
see that the axles are resolved from the point sensor recordings with zero
lateral offset, but become indistinguishable in recording with a larger offset
due to the smoothing effect.

Figure 1: Simulation of quasi-static signals of a DAS array subjected to a
point load with various lateral offsets, $y$, to the fiber. (a), (b), and (c)
shows the simulated wavelets for gauge lengths, $L$, of 0, 8, and 16 m,
respectively. Figure 2: Simulation of quasi-static signals of a DAS array with
zero and 16 m gauge length, $L$, loaded with (a) a two-axle car with an axle
spacing of 2.8 m; (b) a three-axle vehicle with an axle spacing of 9 m with
various lateral offsets, $y$, to the fiber.

###  II-C Wavelet dependency on vehicle speed

This subsection studies the effect of vehicle speed on the DAS response in the
temporal and spatial domain, which is important to guide the design of the DAE
model. For a vehicle traveling from an initial position, $x_{0}$, with a
constant speed of $c$, the measured strain response at fiber location $x$ is,

$\varepsilon_{x}(x,t)=E_{x}(x-ct-x_{0}).$ (3)

The temporal wavelet at a constant time $x_{0}$ and the spatial wavelet at a
constant location $t_{0}$ can be written as
$\phi(t)=\varepsilon_{x}(x=x_{0},t)$ and $\psi(x)=\varepsilon_{x}(x,t=t_{0})$,
respectively. From equations (1) to (3), we can see that the car speed, $c$,
acts as a scaling factor of $\phi(t)$, determining the compactness of the
temporal wavelet. This can be confirmed through numerical experiments. Fig. 3
shows the simulation of the temporal wavelet of a two-axle car traveling at
10, 20, 30, and 40 mph. We can see that faster car speed leads to sharper
temporal wavelet. On the other hand, we can see from the equations that the
shape of the spatial wavelet is speed-independent. The speed term only
introduces a space shift accounting for the different vehicle positions at
time $t_{0}$ for different speeds.

Figure 3: Numerical simulation of the time-domain quasi-static signals of a
two-axle car with constant speeds of 10, 20, 30, and 40 mph.

Our observation that the shape of the space-domain rather than the time-domain
wavelet is speed-invariant is confirmed using field experiments at Sand Hill
road monitoring by the Stanford DAS2 array [16]. Fig. 4 shows the map of a
subsection of the roadside fiber labeled with distances along the fiber. The
horizontal distances from the centers of the south- and north-bound traffic
lanes to the fiber are around 15 m and 25 m, respectively. We drove a test car
southward to understand the car impulse responses for various driving speeds,
including 10, 20, 30, and 40 mph. Fig. 5 shows the quasi-static signals of our
car indicated from DAS recordings using orange arrows. The bottom panels show
the corresponding Frequency-Wavenumber spectra. It can be seen that the
frequency components vary with speeds, i.e., a higher car speed leads to a
broader bandwidth (narrower temporal wavelet). We can also observe that the
wavenumber components remain relatively invariant to car speeds, implying that
the spatial wavelet for different speeds is stationary.

Figure 4: Map view of a roadside section of the Stanford DAS‐2 Array.
Distances along the fiber are labeled on the map Figure 5: (Top) DAS
recordings of the test car driving with constant speeds of 10, 20, 30 and 40
mph. The time axes are scaled differently. (Bottom) The corresponding F-K
spectra of the DAS recordings above. We can see that the frequency range
becomes broader with increasing speed, whereas the wavenumber components are
invariant to car speeds.

##  III Method

As discussed in Section II B., the resolution of the quasi-static signals of
vehicles is degraded due to the smoothing effect of large fiber-road offset
and gauge length. In order to improve the resolution, we introduce a space-
domain DAE model to perform deconvolution along the spatial axis with a space-
domain wavelet taking the offset and gauge length effect into account.

The space-domain DAE model is based on the idea of the original time-domain
DAE model that performs deconvolution along the time axis with a stationary
wavelet in time. The time-domain model is effective when car speed is
approximately constant in time. However, as was shown in the previous section,
the car’s time-domain impulse response changes with speed, indicating the
method could be suboptimal when the speed varies (e.g. stopping or
accelerating). In contrast, the impulse response in the space domain is speed-
invariant and can be tied to wheelbase and axle numbers.

###  III-A Network Architecture

Our space-domain DAE model shown in Fig. 6 is a 2-D fully convolutional U-Net
adapted from [22]. The inputs are quasi-static traffic recordings, a set of
$N_{x}=256$ (256 meters) consecutive waveforms of $N_{t}=1024$ time samples
(20.48 s) in duration, organized in an $N_{x}\times N_{t}$ matrix. The outputs
are sharp deconvolution results with the same shape as the inputs. The U-Net
model comprises 3 convolutional layers, followed by 3 encoder blocks
containing a downsampling (max pooling) layer and 3 convolutional layers. The
kernel sizes for the convolution layers are 3 × 5. The number of convolutional
filters is initialized at 8 and gets doubled after each downsampling
operation. The maxpooling operation downsamples the data by a factor of 2
along the DAS sensor axis and by a factor of 4 along the time axis (i.e, the
maxpooling kernel and strides are of size 2 × 4). The decoder reverses the
encoding operations with 3 blocks of bilinear upsampling. The U-Net contains
skip-connections, which directly connect the output of one encoder block with
the corresponding decoder block. Lastly, the output layer is a single
convolutional layer with 1 output channel and ReLU activation, which enforces
positivity in the model output.

Our model is a semi-supervised algorithm, in the sense that no ground truth
deconvolution is required as labels to train the model. Weak supervision comes
from the spatial car impulse response kernel shown as the red bell-shaped
curve in Fig. 6. The key difference between our DAE model and the original
time-domain DAE model is that we obtain the reconstructed input by spatial
convolution using the impulse response and the network output. For a batch of
quasi-static inputs, $\\{y_{1},y_{2},\dots,y_{N_{b}}\\}$ ($N_{b}=128$), the
loss function is defined as a combination of the L-2 norm of the difference
between the reconstructed input and the original input, and the regularization
term (L-1 or L-2 norm) of the outputs:

$\mathcal{L}=\frac{1}{N_{b}}\sum_{i=1}^{N_{b}}(||[k*x_{i}]_{d}-y_{i}||^{2}_{2}+\rho||x_{i}||_{1,2}),$
(4)

where $x_{i}$ and $y_{i}$ denote the $i$-th deconvolution output and quasi-
static input of the U-Net model, respectively. $\rho$ is a weighting term that
promotes sparseness in the deconvolved results. $[*]_{d}$ refers to
convolution along the sensor axis. As we will see in the following sections,
unlike conventional channel-independent linear filters, the 2-D deconvolution
operations incorporate spatial-temporal features in the DAS recordings. The
non-linear nature of the U-Net introduces high frequencies that are not
present in the inputs, producing sharp and localized outputs.

###  III-B Spatial kernel estimation

In practice, the spatial kernel can be estimated either through numerical
simulation described in the previous section or by performing statistical
averaging of responses of multiple passing cars assuming that the spatial
impulse response is constant in time at each fiber location. The statistical
averaging approach requires detecting several passing cars in a subsection of
the fiber. The detection can be achieved through manual inspection. Herein, we
apply a find-local-maximum algorithm from the SciPy library [26] to the
recordings at quiet midnight to detect an average waveform from isolated cars.

Figure 6: Conceptual overview of the spatial DAE model. The input is the
quasi-static response of DAS to cars, which can be viewed as a matrix, 256
channels (256 meters) x 1024 time steps (20.48 seconds). The output is the
deconvolution results. The loss is computed with the input and the
reconstructed input is obtained through a spatial convolution of the output
with a known car impulse response in the spatial domain.

##  IV Training and evaluation baselines

We describe the training procedure of the DAE models and the baseline methods
for evaluation.

###  IV-A Training procedure

We trained our proposed space-domain DAE models with the simulated spatial
wavelets of a point load at $y=15$ and $y=25$ m (targeting traffic at the
south- and north-bound lanes, respectively) with a DAS gauge length of 16 m
shown as the orange and green curves in Fig 1 (c). The choice of $y$ and gauge
length match the real settings of the Sand Hill road DAS experiments. The
training dataset we used is 2-hours’ worth of traffic recordings of the Sand
Hill DAS fiber. We split the 2-hour recordings into a training and evaluation
set with a ratio of $80\%$ to $20\%$.

###  IV-B Evaluation baselines

As a baseline, we trained a time-domain DAE model with the same dataset. The
temporal wavelet used for training is simulated with a car speed of 30 mph
(shown in Fig. 3) matching the speed limit of the Sand Hill road.

We also benchmark our spatial DAE model using a conventional spatial
deconvolution algorithm with an objective function:

$\hat{x}_{q}=\operatorname*{argmin}_{x_{q}}\\{\frac{1}{2}||[k*x_{q}]_{d}-y_{q}||_{2}^{2}+\rho||x_{q}||_{1}\\}$
(5)

Notations have the same meaning as in equation (4). Note that $[k*x]_{d}$
stands for convolution in space between a known spatial impulse response
kernel and the underlying impulse model, $x$. One commonly used algorithm to
solve this optimization problem is the Iterative Shrinkage Thresholding
Algorithm (ISTA; [27, 28]). For this study, we adopt an accelerated version of
ISTA (Fast-ISTA or FISTA) due to the reason described in [29], which exhibits
faster convergence guarantees. With FISTA performing spatial deconvolution,
signals at each time index are processed independently.

##  V Results

We show the evaluation results of the proposed DAE model for car tracking,
resolving closely traveling vehicles, large-vehicle axle counting, and a
potential limitation of the approach with our solution.

###  V-A Car tracking

To test the performance of the proposed spatial DAE model, we conducted
controlled driving experiments where we drove a test car equipped with a speed
sensor and a GPS receiver southward along the subsection of the Sand Hill road
shown in Fig. 4. In Fig. 7, we focus on a case where our car is first speeding
up and then slowing down. Fig. 7 (a) shows the quasi-static signal of our car.
The car speed is inversely proportional to the slope in the time-space
coordinate. We can see in (a) that when the car speed is low ($<10$ s and
$>30$ s), the wavelet is “stretched” in time. When the car speed is relatively
higher, the time-domain wavelet is “compressed”, agreeing with our observation
in Section II. Fig. 7 (b), (c), and (d) show the deconvolution results from
the proposed space-domain DAE model, the original time-domain DAE model and
the space-domain FISTA algorithm. Both the space- and time-domain DAE models
are trained with L-1 norm regularization loss to promote sparsity in the
outputs. We can see that the proposed space-domain DAE model yields the
sharpest and the most localized results regardless of car-speed variation.
Meanwhile, we can see that the background noise is suppressed by the DAE
model. In contrast, the time-domain DAE model yields results that are
dependent on the speed, e.g. the output is more compact in space with a higher
speed but becomes stretched out when the speed is lower. The spatial
deconvolution via the FISTA algorithm yields result that is speed-invariant
but is not as compact as the spatial-DAE results. Signals $<5$ seconds are
oversuppressed by the L-1 norm term in the equation 5. We can also observe
artifacts shown in black in the FISTA outputs.

Figure 7: (a) DAS recording of the quasi-static signal of a controlled driving
experiment where we first speed up and then slow down; Deconvolution results:
(b) the proposed space-domain DAE model, (c) the time-domain DAE model; (d)
spatial deconvolution via the FISTA algorithm. The amplitude picks in each
panel are shown as a light blue curve reflecting the car trajectory.

Using the signals in Fig. 7 (a) to (d), we can track the car movement by
picking the amplitude at each time step. The estimated trajectory for each
panel is shown as a thin blue curve in Fig. 7. Car speed estimates can be
obtained efficiently through a local beamforming algorithm applied to either
the quasi-static signals or the deconvolution results (Fig. 8): The
beamforming spectrum at each time step is computed using a local 2-D window (3
s x 30 m) following the estimated trajectories. In Fig. 8, a brighter color
indicates higher stack energy in each panel. The red curves represent speed
measurements from an onboard speed sensor. Our speed estimates, indicated with
the black curves, are obtained by picking the maximum amplitude of the
beamforming spectrum at each time step. We can see that our beamforming
estimates using the space-domain DAE output match the red curve the best.
Using the Controller Area Network (CAN) bus reading from the vehicle as the
ground truth, the root-mean-squared errors (RMSE) for the original recording,
space-domain DAE, time-domain DAE, and FISTA are 2.56, 2.14, 4.55, and 2.48
mph, respectively. Our space-domain DAE model achieves the lowest RMSE thanks
to the sharp deconvolution outputs and its highest noise suppression
performance. The large error of the time-domain DAE model is mainly due to the
falsely high-speed event around 33 s as shown in Fig.7 (c) and the wiggled
trajectory pickings when the car speed is low ($t>$33 s).

Figure 8: Local beamforming spectra is performed following the trajectory
picks in Fig. 7 using (a) the quasi-static signal, (b) space-domain DAE
results, (c) time-domain DAE results, and (d) FISTA results. The red curve in
each panel shows the speed measurements of an onboard speed sensor. The black
curves indicate the speed estimates picked from the amplitude of the spectra
at each time step.

###  V-B Traffic pattern monitoring

This section investigates the performance of the space-domain DAE model under
dense traffic conditions. Fig. 9 (a) shows a 200-second quasi-static recording
in heavy traffic. Vehicles transiting southward are closer to the fiber and
thus generate stronger quasi-static signals than the northbound traffic. In
the plot, we label several interesting events. We can see that when multiple
cars trailing closely, their quasi-static signals interfere with each other
causing complicated patterns that pose ambiguity for individual car
identification. Fig. 9 (b) and (c) show the FISTA and time-domain DAE (L-1
norm regularization) results, respectively. We can see that FISTA generates
poorly resolved results and contains evident sidelobe artifacts shown in
black. The time-domain DAE model yields sharp results with fast car speed.
However, as can be seen from the boxes in Fig. 9 (c), the results get blurred
when the car speed is low, e.g. cars stopping for a red light and restarting.
Fig. 9 (d) shows the results of the proposed space-domain DAE model trained
with the near-lane wavelet ($y=15$ m) and L-1 norm regularization loss. We can
see that the model yields sharp results regardless of speed changes. However,
signals of the northbound traffic in a further lane are oversuppressed. To
recover the further lane traffic, we experiment with a spatial DAE model
trained with the wider far-lane wavelet (simulated with $y=25$ m matching the
fiber-northbound lane offset). Fig. 9 (e) shows the far-lane DAE results. We
observe less oversuppression issues for the northbound traffic and sharper
signals for both near and far lanes. Nonetheless, artifacts that could be
misinterpreted as vehicles appear as circled out in the plot. In Fig. 9 (f),
we experiment with space-domain DAE models trained with L-2 norm
regularization loss using the near-lane wavelet. We can see that with L-2 norm
regularization, weak signals of the far-lane traffic are better preserved,
which benefits the tracking of the northbound vehicles. Fig. 9 (g) shows the
results of the L-2 norm space-domain DAE model trained with the far-lane
wavelet. Although the signals are slightly sharper than that shown in (f), the
background noise level gets increased.

Figure 9: (a) Quasi-static signals of passing vehicles in heavy traffic. (b)
spatial deconvolution via FISTA. (c) temporal deconvolution via the time-
domain DAE model. Boxes indicate signals in lower resolution due to slower car
speeds. (d) and (e) show the results from the proposed space-domain DAE model
with L-1 norm regularization using near and far lane wavelets, respectively.
Artifacts that could be misinterpreted as transiting cars are circled out. (f)
and (g) show the results of the space-domain DAE model with L2 regularization
using the near and far wavelets, respectively.

###  V-C Large-size vehicle monitoring

Heavier and longer vehicles, e.g. buses, trucks, and trains, generate quasi-
static signals with much larger amplitudes and wider spatial wavelets than
cars, e.g., sedans and SUVs. The wider spatial wavelet can be viewed as the
superposition of quasi-static signals of wheels at axles that are farther
apart. Fig. 10 (a) shows a quasi-static recording of an 18-meter three-axle
bus transiting southward (video captured by our camera). Due to the smoothing
effect of the road-fiber offset and gauge length, the axles are
indistinguishable from the quasi-static signals. Fig. 10 (b) shows the
deconvolution results of the space-domain DAE model trained with L-1 norm
regularization loss using the simulated near-lane impulse response wavelet. We
can see the three axles are recovered as three strong-energy trajectories
corresponding to the three axles. The amplitude difference between the three
trajectories could imply a non-uniform weight distribution on different wheel
groups. The distance between the two most substantial peaks is about 17 m
agreeing with the bus length, indicating the usage of our L-1 norm model for
counting long-size vehicle axles and length characterization. Fig. 10 (b)
shows the results of the space-domain DAE model trained with L-2 norm
regularization. We can see that axles are smoothed out by the L-2 norm
regularization. The L-2 norm model could be used to track the bus motion.

Monitoring large-size vehicles benefits not only traffic management but also
non-intrusive shear-wave velocity imaging, which can support city
sustainability on applications including sinkhole detection, excavation
monitoring, and earthquake hazard analysis. A shear-wave profile can be
estimated through an optimization algorithm using traffic-induced surface
waves as inputs. Fig. 11 (a) and (b) show the dynamic surface wave components
($>1$ Hz) excited by a moving bus and a car, respectively. We can see that the
bus excites much stronger surface waves than the car. With the surface waves
within the two dashed lines, we compute the dispersion spectrum reflecting the
relationship between the phase velocity and frequency as shown in Fig. 11 (c)
and (d) for the bus and car, respectively. We can see that large-size
vehicles, such as a bus, generates surface waves with frequency as low as
$2.5$ Hz, which is absent from the car-induced surface waves. Low frequencies
at $2.5$ Hz have a wavelength of $\sim 200$ m. The maximum investigation depth
of the shear-wave velocity structure using surface waves is about half of the
longest wavelength [30]. Therefore, with buses, we can retrieve depths down to
100 m, whereas using only the car, results at depths below approximately 50 m
are unreliable.

Figure 10: (a) Signals of an 18-meter bus and two regular-sized cars.
Deconvolution results of (b) space-domain DAE model with L-1 regularization,
and (c) space-domain DAE model with L-2 regularization, respectively. Figure
11: (a) Surface-wave components of (a) a passing bus and (b) a car. (c) and
(d) show the dispersion image estimated from the surface waves in (a) and (b),
respectively

###  V-D Spatial non-stationarity

Our spatial model assumes that the car impulse response is stationary in
space. However, the impulse response is a function of the recording system and
the near-surface conditions surrounding the fiber optic cable. Thus, the
deconvolution of a DAS array covering heterogeneous near-surface conditions
using a single stationary spatial impulse response could be suboptimal. Fig.
12 (a) shows DAS recordings of traffic in downtown San Jose City (SJC). (b)
shows deconvolution results of the proposed space-domain DAE model with a
simulated car impulse response kernel. The deconvolution results look
encouraging for most of the fiber, which can be contributed to the consistency
of the assumed wavelet to the real wavelet. However, we can see that the
results in the red box are relatively poorly resolved. The low resolution is
more noticeable from the zoomed-in view of data and deconvolution results in
the red box in Fig. 13 (a) and (b). The under-performance could be explained
as the wavelet inconsistency due to the non-stationarity of the near-surface
properties. Assuming that the near-surface properties, and in turn the car
response, at each channel location is stationary in time. A possible solution
would be to estimate the spatial car response at different parts of the fiber
by averaging responses of multiple passing cars around each location. To
verify this, we employ a local maximum finding algorithm as a simple car
detector for the quasi-static signals. We average the responses of six
identified cars passing the area in the red box of Fig. 12 as the impulse
response input to the U-net model. We retrain the U-net using this estimated
kernel, which produces a sharper result in Fig. 13, indicating the
effectiveness of our approach.

Figure 12: Deconvolution of traffic recording in downtown SJC via a space-
domain DAE model with a simulated impulse response stationary in space. (a)
Input data; (b) deconvolution results. The red box indicates results that are
poorly resolved due to the spatial non-stationarity of the car wavelets.
Figure 13: (a) and (b) are respectively zoomed-in views of the data and
deconvolution results in the red box shown in Fig. 12. We can see artifacts in
(b) due to the wavelet mismatch. (c) shows the better-resolved deconvolution
results with a statistical estimate of spatial wavelet for this zoomed-in
location.

##  VI Discussion and conclusions

This paper focuses on the application of traffic monitoring with car-induced
quasi-static signals recorded by an urban DAS array. To denoise the data and
to reduce the interference among closely traveling cars, we propose a self-
supervised convolutional U-Net model (space-domain DAE model) that can
compress the quasi-static signals into sharp pulses and remove the background
noises. The goal is achieved through spatial deconvolution with an assumed
spatial wavelet of the quasi-static signals, which is a major difference from
the previously proposed time-domain DAE model.

This paper shows that using the spatial instead of the temporal kernel is
advantageous because it is invariant to car speed variation. This leads to an
improved precision robustness of tracking cars with varying speeds, which is
essential for driving behavior identification and accident detection. The
benefits of our DAE model are more obvious for heavy-traffic conditions where
car signals interfere. With our space-domain DAE model, compressing signals of
individual cars to sharp pulses, traffic patterns are much better revealed.
Furthermore, we show that our space-domain DAE model is robust to deconvolve
signals of large-size and heavy vehicles, such as a bus. Being able to
identify large vehicles can be helpful in extracting low-frequency surface
waves to image the velocity structure down to hundreds of meters beneath the
fiber. Lastly, we point out that a limitation of our spatial-domain DAE model
arises when car impulse responses at different locations along the fiber vary
due to coupling heterogeneity and/or fiber properties. We address the issue by
extracting car wavelets using statistical averaging, and train separated
networks. Future work would be adapting the DAE model to different parts of
the fiber using location-dependent spatial wavelets.

## Acknowledgments

This research was supported financially by the affiliates of the Stanford
Exploration Project and the Stanford Sustainability Initiative and the UPS
Foundation Endowment Fund. Jingxiao Liu is supported by Leavell Fellowship on
Sustainable Built Environment at Stanford University. Martijn van den Ende was
supported by the French government through the 3IA Côte d’Azur Investments in
the Future project managed by the National Research Agency (ANR) with the
reference number ANR-19-P3IA-0002. The interrogator unit was loaned to us by
OptaSense Inc. We thank Martin Karrenbach, Victor Yartsev, and Lisa LaFlame
from Optasense, as well as the Stanford ITS fiber team, and in particular
Erich Snow, for crucial help with the Stanford DAS-2 experiment. We also thank
the Stanford School of Earth IT team for hosting the interrogator in the
Scholl computer room.

## References

  * [1] K. Robert, “Video-based traffic monitoring at day and night vehicle features detection tracking,” in _2009 12th International IEEE Conference on Intelligent Transportation Systems_ , 2009, pp. 1–6.
  * [2] S. R. E. Datondji, Y. Dupuis, P. Subirats, and P. Vasseur, “A survey of vision-based traffic monitoring of road intersections,” _IEEE Transactions on Intelligent Transportation Systems_ , vol. 17, no. 10, pp. 2681–2698, 2016.
  * [3] P. Reinartz, M. Lachaise, E. Schmeer, T. Krauss, and H. Runge, “Traffic monitoring with serial images from airborne cameras,” _ISPRS Journal of Photogrammetry and Remote Sensing_ , vol. 61, no. 3, pp. 149–158, 2006, theme Issue: Airborne and Spaceborne Traffic Monitoring. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0924271606001146
  * [4] N. K. Jain, R. Saini, and P. Mittal, “A review on traffic monitoring system techniques,” _Soft Computing: Theories and Applications_ , pp. 569–577, 2019\.
  * [5] S.-T. Jeng and L. Chu, “A high-definition traffic performance monitoring system with the inductive loop detector signature technology,” in _17th International IEEE Conference on Intelligent Transportation Systems (ITSC)_ , 2014, pp. 1820–1825.
  * [6] T. Cherrett, H. Bell, and M. McDonald, “Traffic management parameters from single inductive loop detectors,” _Transportation Research Record_ , vol. 1719, no. 1, pp. 112–120, 2000. [Online]. Available: https://doi.org/10.3141/1719-14
  * [7] J. Zhang, Y. Lu, Z. Lu, C. Liu, G. Sun, and Z. Li, “A new smart traffic monitoring method using embedded cement-based piezoelectric sensors,” _Smart Materials and Structures_ , vol. 24, no. 2, p. 025023, 2015.
  * [8] Z.-X. Li, X.-M. Yang, and Z. Li, “Application of cement-based piezoelectric sensors for monitoring traffic flows,” _Journal of transportation engineering_ , vol. 132, no. 7, pp. 565–573, 2006.
  * [9] R. Posey, “Rayleigh scattering based distributed sensing system for structural monitoring,” in _Fourteenth International Conference on Optical Fiber Sensors_ , A. G. Mignani and H. C. Lefèvre, Eds., vol. 4185, International Society for Optics and Photonics. SPIE, 2000, p. 41850E. [Online]. Available: https://doi.org/10.1117/12.2302157
  * [10] A. Masoudi and T. P. Newson, “Contributed review: Distributed optical fibre dynamic strain sensing,” _Review of Scientific Instruments_ , vol. 87, no. 1, p. 011501, 2016. [Online]. Available: https://doi.org/10.1063/1.4939482
  * [11] B. L. Biondi, S. Yuan, E. R. Martin, F. Huot, and R. G. Clapp, _Using Telecommunication Fiber Infrastructure for Earthquake Monitoring and Near-Surface Characterization_. American Geophysical Union (AGU), 2021, ch. 10, pp. 131–148. [Online]. Available: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/9781119521808.ch10
  * [12] E. R. Martin, C. M. Castillo, S. Cole, P. S. Sawasdee, S. Yuan, R. Clapp, M. Karrenbach, and B. L. Biondi, “Seismic monitoring leveraging existing telecom infrastructure at the SDASA: Active, passive, and ambient-noise analysis,” _The Leading Edge_ , vol. 36, no. 12, pp. 1025–1031, 12 2017\. [Online]. Available: https://doi.org/10.1190/tle36121025.1
  * [13] G. Fang, Y. E. Li, Y. Zhao, and E. R. Martin, “Urban near-surface seismic monitoring using distributed acoustic sensing,” _Geophysical Research Letters_ , vol. 47, no. 6, p. e2019GL086115, 2020, e2019GL086115 10.1029/2019GL086115. [Online]. Available: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2019GL086115
  * [14] Optasense, a Luna company, “Traffic Monitoring — Road & Highway Incident Detection,” 7 2022. [Online]. Available: https://www.optasense.com/transportation/traffic-monitoring/
  * [15] ——, “Quantx das interrogator: Optasense,” 2022. [Online]. Available: https://www.optasense.com/technology/quantx/
  * [16] S. Yuan, A. Lellouch, R. G. Clapp, and B. Biondi, “Near-surface characterization using a roadside distributed acoustic sensing array,” _The Leading Edge_ , vol. 39, no. 9, pp. 646–653, 09 2020. [Online]. Available: https://doi.org/10.1190/tle39090646.1
  * [17] N. J. Lindsey, S. Yuan, A. Lellouch, L. Gualtieri, T. Lecocq, and B. Biondi, “City-scale dark fiber das measurements of infrastructure use during the covid-19 pandemic,” _Geophysical Research Letters_ , vol. 47, no. 16, p. e2020GL089931, 2020, e2020GL089931 2020GL089931. [Online]. Available: https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2020GL089931
  * [18] S. Yuan, J. Liu, H. Young Noh, and B. Biondi, “Urban system monitoring using combined vehicle onboard sensing and roadside distributed acoustic sensing,” in _SEG International Exposition and Annual Meeting_ , vol. Day 1 Sun, September 26, 2021, 09 2021, d011S137R003. [Online]. Available: https://doi.org/10.1190/segam2021-3584136.1
  * [19] X. Wang, Z. Zhan, E. F. Williams, M. G. Herráez, H. F. Martins, and M. Karrenbach, “Ground vibrations recorded by fiber-optic cables reveal traffic response to covid-19 lockdown measures in pasadena, california,” _Communications Earth & Environment_, vol. 2, 2021.
  * [20] M. van den Ende, A. Ferrari, A. Sladen, and C. Richard, “Next-generation traffic monitoring with distributed acoustic sensing arrays and optimum array processing,” in _2021 55th Asilomar Conference on Signals, Systems, and Computers_ , 2021, pp. 1104–1108.
  * [21] F. Huot, E. R. Martin, and B. Biondi, _Automated ambient-noise processing applied to fiber-optic seismic acquisition (DAS)_ , 2018, pp. 4688–4692. [Online]. Available: https://library.seg.org/doi/abs/10.1190/segam2018-2997880.1
  * [22] M. v. d. Ende, A. Ferrari, A. Sladen, and C. Richard, “Deep deconvolution for traffic analysis with distributed acoustic sensing data,” _IEEE Transactions on Intelligent Transportation Systems_ , pp. 1–0, 2022.
  * [23] Y. Fung, “Foundations of solid mechanics prentice-hall,” _Inc, New Jersey_ , 1965.
  * [24] P. Jousset, T. Reinsch, T. Ryberg, H. Blanck, A. Clarke, R. Aghayev, G. P. Hersir, J. Henninges, M. Weber, and C. M. Krawczyk, “Dynamic strain determination using fibre-optic cables allows imaging of seismological and structural features,” _Nature communications_ , vol. 9, no. 1, pp. 1–11, 2018.
  * [25] L. Li, S. Nimbalkar, and R. Zhong, “Finite element model of ballasted railway with infinite boundaries considering effects of moving train loads and rayleigh waves,” _Soil Dynamics and Earthquake Engineering_ , vol. 114, pp. 147–153, 2018. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0267726118303968
  * [26] P. Virtanen, R. Gommers, T. E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, İ. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” _Nature Methods_ , vol. 17, pp. 261–272, 2020.
  * [27] A. Chambolle, R. De Vore, N.-Y. Lee, and B. Lucier, “Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage,” _IEEE Transactions on Image Processing_ , vol. 7, no. 3, pp. 319–335, 1998.
  * [28] I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” _Communications on Pure and Applied Mathematics_ , vol. 57, no. 11, pp. 1413–1457, 2004. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.20042
  * [29] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm with application to wavelet-based image deblurring,” in _2009 IEEE International Conference on Acoustics, Speech and Signal Processing_ , 2009, pp. 693–696.
  * [30] C. B. Park, R. D. Miller, and H. Miura, “Optimum field parameters of an masw survey,” _Japanese Society of Exploration Geophysics Extended Abstracts_ , vol. 36, 2002.