# Inferring rheology from free-surface observations

Edward M. Hinton1<EMAIL_ADDRESS>1School of Mathematics and
Statistics, The University of Melbourne, Parkville, VIC 3010, Australia

###### Abstract

We develop direct inversion methods for inferring the rheology of a fluid from
observations of its shallow flow. First, the evolution equation for the free-
surface flow of an inertia-less current with general constitutive law is
derived. The relationship between the volume flux of fluid and the basal
stress, $\tau_{b}$ is encapsulated by a single function $F(\tau_{b})$, which
depends only on the constitutive law. The inversion method consists of (i)
determining the flux and basal stress from the free-surface evolution, (ii)
comparing the flux to the basal stress to constrain $F$ and (iii) inferring
the constitutive law from $F$. Examples are presented for both steady and
transient free-surface flows demonstrating that a wide range of constitutive
laws can be directly obtained. For flows in which the free-surface velocity is
known, we derive a different method, which circumvents the need to calculate
the flux.

##  1 Introduction

In many environmental and industrial settings, it is important to determine
the rheology of a fluid from observations of its free surface (Sellier, 2016).
A major example is the simple slump test used in the mining, concrete and food
industries to characterize particular features such as the yield stress of the
fluid. In other contexts, such as for lava flows and ice sheets, deploying a
rheometer may not be practical and instead free-surface observations provide
an alternative constraint on the rheology (Roussel & Coussot, 2005; Balmforth
et al., 2007; Martin & Monnier, 2015).

Previous methods for inferring the rheology of a fluid from its free surface
typically involve comparing observations to simulated predictions (Piau &
Debiane, 2005; Sayag & Worster, 2013; Sellier, 2016; Al-Behadili et al.,
2019). Simulated predictions are obtained from a forward model, which
calculates the free-surface shape for given fluid properties. A particular
form of the constitutive equation that has one or more free parameters, such
as the Herschel-Bulkley law, is assumed. The free-surface evolution is
calculated from this model over a sample of the parameter space and the
mismatch between the simulated predictions and the observed data is minimised
to constrain the rheological parameters.

These ‘best-fit’ methods rely upon a particular pre-supposed form of the
constitutive law, which may be inaccurate or even unphysical at some values of
the stress (Matsuhisa & Bird, 1965; Barnes & Walters, 1985; Myers, 2005). They
also require a large amount of forward computation. Finally, since the entire
form of the constitutive law is determined by a global comparison to the
observed flow, the parameters may be non-unique. The latter two of these
concerns are exacerbated as the number of free parameters is increased.

In this paper, direct inversion methods to infer general constitutive laws
from free-surface data are described for inertia-less steady and transient
flows. Our direct inversion methods provide conceptual insight into how
predictions of the stress at the base of the fluid can be obtained from the
free-surface elevation. The constitutive law is reconstructed by comparing
these stresses to the observed flux or velocity of the flow. In contrast to
the indirect, global ‘best-fit’ methods, the rheology is constrained in a
pointwise and direct fashion. This reduces the computation required and
enables a wide range of constitutive laws to be inferred.

The first step in developing direct inversion methods is to derive a model for
the flow thickness of a fluid with a given general constitutive law. We
restrict our attention to inelastic, isothermal fluids with time-independent
rheology (known as ‘generalized Newtonian fluids’). For two-dimensional flows,
Pritchard et al. (2015) presented a governing equation for the flow thickness
that is applicable to such rheologies. They exploited the fact that in shallow
flows, the stress distribution is a known linear function of the vertical
coordinate. The flux is written in terms of an integral over the stress that
depends only on the constitutive law. In §2, we derive a forward model for a
generalised Newtonian fluid in three dimensions by observing that the velocity
and pressure gradient of shallow flows are parallel and their direction is
independent of vertical position. This model suggests direct inversion methods
for steady and transient flows (§3 and §4). Concluding remarks are made in §5.

##  2 Forward model

In this section, we develop a three-dimensional forward model for the flow of
a generalised Newtonian fluid. The two-dimensional version was presented by
Pritchard et al. (2015). We analyse the flow of a generalized Newtonian fluid
on topography with elevation given by $z=E(x,y)$. The fluid has thickness
$h(x,y,t)$ and the free surface is at $z=E(x,y)+h(x,y,t)$. The effects of
surface tension and inertia are neglected. The flow is assumed to be shallow;
the thickness scale is much smaller than the lengthscales in the $x$ and $y$
direction. The vertical velocity, $w$, is negligible relative to the
velocities ($u$, $v$) in the $x$ and $y$ directions. Using this lubrication
approximation, the strain rate simplifies to

$\dot{\gamma}=\Bigg{\rvert}\frac{\partial\boldsymbol{u}}{\partial
z}\Bigg{\rvert}=\sqrt{(\partial u/\partial z)^{2}+(\partial v/\partial
z)^{2}},$ (1)

where $\boldsymbol{u}=(u,v)$. As a constitutive law, we use a generalised
Newtonian model with

$\dot{\gamma}=\phi(\tau),$ (2)

for a prescribed function $\phi(\tau)$, where $\tau=|\boldsymbol{\tau}|$ is
the magnitude of the shear stress. This generalised form of the constitutive
law is applicable to a wide range of inelastic fluids. Well-known constitutive
laws such as the Newtonian, Bingham and Herschel-Bulkley models are special
cases of this representation.

Under the shallow flow assumption, the pressure $p$ within the fluid is
hydrostatic,

$p=\rho g(h+E-z),$ (3)

where $\rho$ is the fluid density. The momentum equation in the $x$ and $y$
directions is given by

$\frac{\partial\boldsymbol{\tau}}{\partial
z}=-\boldsymbol{G},\qquad\text{where}\qquad\boldsymbol{G}=-\nabla p=-\rho
g\nabla(h+E).$ (4)

The vector $\boldsymbol{G}$ is the (negative) hydrostatic pressure gradient
and $\nabla=(\partial/\partial x,\partial/\partial y)$. Upon integrating with
respect to $z$ we obtain

$\boldsymbol{\tau}=(h+E-z)\boldsymbol{G},$ (5)

since there is zero stress at the free surface. The stress is a linear
function of the vertical position within the flowing layer. The volume flux is
rearranged as follows (Schwartz & Eley, 2002; Pritchard et al., 2015)

$\boldsymbol{Q}=\int_{E}^{E+h}\boldsymbol{u}\,\mathrm{d}z=\int_{E}^{E+h}(h+E-z)\frac{\partial\boldsymbol{u}}{\partial
z}\,\mathrm{d}z,$ (6)

where we have used no slip at the base, $z=E$. Note that the vectors,
$\boldsymbol{G}$, $\boldsymbol{\tau}$, $\boldsymbol{u}$ and $\boldsymbol{Q}$
are parallel with the same direction (in the $(x,y)$ plane) as the flow. Since
the pressure gradient, $\boldsymbol{G}$ is independent of $z$, the direction
of $\boldsymbol{\tau}$ and $\boldsymbol{u}$ is also independent of $z$ (as is
the direction of $\partial\boldsymbol{u}/\partial z$ and
$\partial\boldsymbol{\tau}/\partial z$). We take the magnitude of both sides
of (6) and these observations allow us to take the magnitude through the
integral (the direction of the integrand is independent of $z$). Next, we
substitute for $h+E-z$ from (5) in (6). Also, equation (5) furnishes the
relation $-|\boldsymbol{G}|\mathrm{d}z=\mathrm{d}\tau$ and the magnitude of
the flux becomes an integral over the stress within the layer,

$|\boldsymbol{Q}|=|\boldsymbol{G}|^{-2}\int_{0}^{|\boldsymbol{G}|h}\tau\phi(\tau)\,\mathrm{d}\tau.$
(7)

The quantity, $|\boldsymbol{G}|h$ is the magnitude of the stress at $z=E(x,y)$
(the basal stress). In this formulation, the flux has been written in terms of
an integral of the stress, $\tau$, and the shear rate, $\phi(\tau)$.

Since the flux and pressure gradient are parallel (in the $(x,y)$ plane), we
write

$\boldsymbol{Q}=\frac{F(|\boldsymbol{G}|h)}{|\boldsymbol{G}|^{3}}\boldsymbol{G},\qquad\text{where}\quad
F(\tau_{0})=\int_{0}^{\tau_{0}}\tau\phi(\tau)\,\mathrm{d}\tau.$ (8)

A similar equation was derived by Schwartz & Eley (2002) with the integrand
written in terms of a stress-dependent viscosity. The function $F(\tau_{0})$
depends only on the rheology of the fluid. It accounts for how the volume flux
varies with the basal stress. Schwartz & Eley (2002) describe this variation
as the ‘fluidity’ of the flow. Mass continuity furnishes the governing
equation for the flow thickness,

$\frac{\partial h}{\partial
t}+\nabla\cdot\Big{[}|\boldsymbol{G}|^{-3}F(|\boldsymbol{G}|h)\boldsymbol{G}\Big{]}=0,$
(9)

which completes the forward model for $h(x,y,t)$ since the pressure gradient
$\boldsymbol{G}$ depends linearly on the gradient of the flow thickness. In
the case that $\partial h/\partial y=\partial E/\partial y=0$, the two-
dimensional governing equation of Pritchard et al. (2015) is recovered. Since
we are interested in general constitutive laws, there is no intrinsic scaling
for the stress and strain rate so we keep the problem dimensional throughout
this paper. For steady flows ($\partial h/\partial t=0$), the forward problem
is solved with boundary conditions for $h(x,y)$ at the edges of a particular
domain. The system is integrated numerically and details of the method are
given in Appendix A. The solution for steady flow over a topographical feature
is shown in figure 1a,b for a specific constitutive law, given in figure 1d.
Corresponding plots for the steady flow around a cylinder on an inclined plane
for a different constitutive law are shown in figure 2.

##  3 Inversion methods for steady flows

The steady inverse problem involves inferring the rheology of the fluid (i.e.
the function, $\phi(\tau)$), given the elevation of the topography, the fluid
free surface and its density. To obtain the constitutive law, the function
$F(\tau)$ is reconstructed. We rewrite (8) as

$F(|\boldsymbol{G}|h)=|\boldsymbol{Q}||\boldsymbol{G}|^{2}.$ (10)

The quantities $|\boldsymbol{Q}|$, $|\boldsymbol{G}|$ and $h$ (the flux,
pressure gradient and flow thickness) are determined at various points in the
flow, each of which provides a prediction of $F(\tau)$ at
$\tau=|\boldsymbol{G}|h$. We calculate these three quantities as follows.
First, we measure the flow thickness, $h$, from the difference between the
free-surface elevation and the (known) topographic elevation. The pressure
gradient, $|\boldsymbol{G}|$, is calculated from the gradients of the free
surface using (4)b. Finally, to determine the flux, $|\boldsymbol{Q}|$, the
steady version of the governing equation (9) is rewritten as

$\nabla\cdot\Big{(}|\boldsymbol{Q}|\boldsymbol{\hat{G}}\Big{)}=0,\qquad\text{where}\quad\boldsymbol{\hat{G}}=\boldsymbol{G}/|\boldsymbol{G}|,$
(11)

and $\boldsymbol{\hat{G}}$ is the unit vector in the $(x,y)$ plane in the
direction of the flow. This equation is solved for $|\boldsymbol{Q}|$ rather
than for $h$ as in the forward problem. The method of characteristics
furnishes

$\frac{\mathrm{d}\boldsymbol{x}}{\mathrm{d}s}=\hat{\boldsymbol{G}},\qquad\frac{\mathrm{d}|\boldsymbol{Q}|}{\mathrm{d}s}=-|\boldsymbol{Q}|\nabla\cdot\hat{\boldsymbol{G}},$
(12)

where $\boldsymbol{x}=(x,y)$, and $s$ parameterises the characteristic curves
in the $(x,y)$ plane, which are the streamlines of the flow. A similar method
was used by Sellier & Panda (2010) for reconstructing bottom topography from
the free-surface flow of thin films of Newtonian fluid (see also Heining et
al., 2012). The direction of the streamlines is given by the (known) direction
of $\boldsymbol{G}$. Then by integrating (12), the magnitude of the flux
$|\boldsymbol{Q}|$ is obtained along the streamlines.

The integration along characteristics requires the value of $|\boldsymbol{Q}|$
at some upstream location. If we only have the steady free-surface elevation
and we have no data concerning the timescale of the flow (such as the flux or
free-surface velocity), then the inverse problem is incomplete. In its
simplest form, this incompleteness arises in the problem of determining the
viscosity of a Newtonian flow from its steady free-surface elevation; one must
know something of the timescale to constrain the shear rate. In general,
without knowing the flow timescale, the constitutive law can be determined up
to a multiplicative constant (essentially a reference viscosity).

In some cases, the flux far upstream may be known or could at least be
estimated. In the case where the far upstream flux is not known, but there is
some other data concerning the timescale of the flow, we make an arbitrary
choice of $|\boldsymbol{Q}|=1$\mathrm{m}^{2}\mathrm{/}\mathrm{s}$$ far
upstream, which will subsequently be corrected. We then determine
$|\boldsymbol{Q}|$ along the streamlines, from which we calculate $F$ and
$\phi$ as described below. Owing to the linearity of (12)b and the linear
relationship between $|\boldsymbol{Q}|$, $\phi$ and $F$, the incorrect
upstream flux merely leads to a constant multiplicative error in these
quantities. This multiplicative constant may be constrained by comparison with
observed data of the flux or free-surface velocity from at least one point
along the streamline (e.g. by using the formulation in §4.2).

Figure 1: Forward and inversion method for steady flow over topography. (a)
Topographic and free-surface elevation. (b) Flow thickness and locations of
the points used for the inversion method. (c) Predictions of $F(\tau)$ (dotted
lines) and its true form (continuous line). (d) Reconstructed constitutive law
for the three sets of points in (b) is compared to its true shape (14, solid
line). (e,f) Absolute error between the inferred and true quantities in panels
(c) and (d).

We use the data (of $|\boldsymbol{Q}|$, $|\boldsymbol{G}|$ and $h$) along the
streamlines to infer the relationship between $|\boldsymbol{G}|h$ and
$|\boldsymbol{Q}||\boldsymbol{G}|^{2}$, which provides an approximation for
$F(\tau)$ using (10). For a given stress, $\tau^{\prime}$, we determine
$F(\tau^{\prime})$ from observations of the flow at a point where the basal
stress equals $\tau^{\prime}$. With a prediction for $F(\tau)$ in hand, the
constitutive relation is reconstructed via

$\dot{\gamma}=\phi(\tau)=\frac{1}{\tau}\frac{\mathrm{d}F}{\mathrm{d}\tau}.$
(13)

This requires a prediction of the derivative of $F$ from scattered data; we
either fit a polynomial or a piecewise linear function (e.g. figure 1c). We
can only predict the function $F(\tau)$, and hence the constitutive law
$\phi(\tau)$, over the range of values that the basal shear stress,
$|\boldsymbol{G}|h$, takes within the flow. The more the steady flow is
diverted by the topography or obstruction, the greater the range of $\tau$
over which the constitutive law can be inferred. Further details of the method
are provided in the following subsections by studying two examples; flow over
topography (§3.1) and flow around a cylinder (§3.2). In each case, we assume
that the upstream flux is known.

###  3.1 Rheology from steady flow over topography

We deploy the inversion method to show how the constitutive law can be
inferred from the shape of the free surface that arises in steady flow over a
topographic feature. The topography consists of a slope at an angle
$\beta=0.6$ to the horizontal with a Gaussian mound added at the origin (shown
in figure 1a). The feature extends to infinity in the positive and negative
$y$ directions and hence the problem is two-dimensional. The flow is in the
$x$ direction and the volume flux, $|\boldsymbol{Q}|$, is constant along the
$x$ axis. The forward problem for the flow thickness is solved numerically as
described in Appendix A with the following constitutive relation (black line
in figure 1d)

$\dot{\gamma}=\phi(\tau)=\Bigg{(}1+\frac{M-1}{\sqrt{1+a^{2}\tau^{2}}}\Bigg{)}\frac{\tau}{\mu},$
(14)

where $M=0.2$, $\mu=400\,$\mathrm{Pa}\,\mathrm{s}$$ and $a=1.1\,\times
10^{-4}\,$\mathrm{Pa}^{-1}$$. The flux far upstream is
$1.67\,$\mathrm{m}^{2}\mathrm{/}\mathrm{s}$$ per unit width and the fluid
density is $\rho=1200\,$\mathrm{k}\mathrm{g}\,\mathrm{m}^{-3}$$. The flow
thickness far upstream is thus 1 metre. The calculated free-surface elevation
is shown relative to the topography in figure 1a and the flow thickness is
plotted in figure 1b. The flow deepens (and slows) upstream of the mound,
where the basal stress is small (c.f. Hinton et al., 2019). The flow is
thinner and faster on the downstream side of the mound and the basal stress is
greater.

The inversion method is carried out for three sets of observations of the
synthetic forward prediction. The first inversion uses the free-surface
gradient, free-surface elevation and topographic elevation at 80 points along
the $x$ axis, shown by red crosses in figure 1b. The function $F$ is predicted
from this data (see equation 10). At each of the 80 points, we determine the
flow thickness, $h$ and the pressure gradient, $|\boldsymbol{G}|$ from the
gradient of the free surface (calculated via a finite difference approximation
using the 80 points). The flux is a known constant along the $x$ axis. From
these measurements, we calculate the basal shear stress, $|\boldsymbol{G}|h$
and $|\boldsymbol{G}|^{2}|\boldsymbol{Q}|$ at the 80 points. This data is
sorted from the smallest to the largest basal stress and plotted using linear
interpolation. This prediction is shown as a red dashed line in figure 1c,
which shows close agreement with the true $F(\tau)$ (continuous black line).
The prediction of $F$ furnishes an estimate of $\phi(\tau)$ via (13). This is
plotted as a red dashed line in figure 1d and shows excellent agreement with
the true constitutive law (continuous black line). The same method is then
carried out with 15 sample points (blue circles and blue dotted lines) and 6
sample points (green stars and green dashed lines) demonstrating that good
predictions of the constitutive law may be obtained with fifteen observations
focused in the region where the flow thickness varies significantly. Figure 1
illustrates that when there are fewer data points (e.g. the green curves),
small errors in $F$ can lead to more significant errors in the constitutive
law because the derivative of $F$ is not well-approximated. The error is
larger at smaller values of $\tau$ owing to the $1/\tau$ pre-factor in (13).

Finally, it should be noted that this method requires the topographic
elevation. In the context of ice sheets, this is frequently obtained via
either airborne or ground-based radio-echo sounding. For lava flows, the pre-
eruption topography is often known. The other examples in this paper do not
require such detailed topographic data (e.g. slump tests on a horizontal
plane).

###  3.2 Rheology from steady flow around cylinders

Figure 2: Steady flow around a cylinder. (a) Free-surface elevation (blue)
relative to the cylinder and plane. (b) Flow thickness and three streamlines
(dashed lines). (c) Predictions of $F(\tau)$ from the streamlines (circles)
and its true form (continuous line). (d) Prediction of the constitutive law
(red dashed line) and its true form (15, continuous line).

In this subsection, we infer the rheology of a fluid from observations of its
steady flow around a circular cylinder on an inclined plane with the axis of
the cylinder in the $z$ direction (see figure 2). The fluid has the following
constitutive law (black line in figure 2d),

$\dot{\gamma}=\phi(\tau)=\frac{\tau^{2}/\mu_{2}}{1+a^{2}\tau^{2}},$ (15)

with $\mu_{2}=10^{7}\,$\mathrm{Pa}^{2}\,\mathrm{s}$$ and $a=2\,\times
10^{-4}\,$\mathrm{Pa}^{-1}$$. The plane is inclined with gradient
$\tan\beta=0.2$. Far upstream of the cylinder, the steady thickness is $H=4$
metres and the flux per unit width is
$9.74\,$\mathrm{m}^{2}\mathrm{/}\mathrm{s}$$. The cylinder radius is 3 metres
and the fluid has density
$\rho=1200\,$\mathrm{k}\mathrm{g}\,\mathrm{m}^{-3}$$. The forward problem is
solved numerically as described in Appendix A with no-flux boundary condition
on the cylinder. The flow is symmetric about the centreline (which is the $x$
axis). Figure 2a shows the free-surface elevation and figure 2b shows the flow
thickness, which increases by approximately 15% upstream of the cylinder and
decreases by a similar amount downstream (c.f. Hinton et al., 2020).

The solution for the free-surface elevation is used as the input for the
inversion method. The free-surface elevation and its gradient are known on a
discrete grid with $10\,$\mathrm{c}\mathrm{m}$$ spacing in the $x$ and $y$
directions. The pressure gradient can be obtained from the free-surface
gradient and the flux is calculated along three streamlines using (12) (dashed
lines in figure 2b). The prediction for $F(\tau)$ from each streamline is
shown as a different set of coloured circles in figure 2c. There is good
agreement with the true form (black line). A fourth order polynomial is fitted
to the three sets of data from which we obtain an inference of the
constitutive law via (13). Figure 13d compares the predicted constitutive law
(red dashed line) to its true form (black line). The range of strain rates
over which the constitutive law is inferred is fairly small because the
cylinder is relatively thin leading to only small variations in the basal
stress. Observations of flow around a wider cylinder would provide predictions
of $\dot{\gamma}=\phi(\tau)$ over a larger range.

##  4 Inversion methods for transient flows

For the forward transient problem, the governing equation (9) is solved with
the relevant initial and boundary conditions and a source term may be included
on the right-hand side in the case of a supplied input flux. Details of the
transient forward numerical method are given in Appendix A.

In the following two subsections (§4.1 and §4.2), two inversion methods for
obtaining the rheology from transient free-surface flow are described and
examples are given in each case. The method of §4.1 is relevant to two-
dimensional and axisymmetric flows. We determine the flux at a given location
and compare this to the pressure gradient to predict $F(\tau)$ in an analogous
fashion to the steady method. §4.2 introduces a different approach, which
utilises the free-surface velocity of the flow.

###  4.1 Flux-based method

Figure 3: (a) Slump shape at 12.5, 50 and 200 seconds (black lines) and the
edges of a control rectangle, $x=\pm 1.5$ metres (blue dashed lines). (b) Flow
thickness (black line, left axis) and basal stress (dashed line, right axis)
at the edge location. (c) Prediction of $F$ (circles) versus true $F$ (solid
line). (d) Inferred constitutive law (red crosses) and the true constitutive
law, (17) (solid line).

For a two-dimensional flow, the flux across any given station, $x=L$, can be
obtained from the rate of change of the volume in $x<L$, which in turn can be
calculated from the evolution of the free surface. The flux at $x=L$ can be
written as

$Q(x=L,t)=q_{s}-\frac{\partial}{\partial
t}\Bigg{(}\int_{-\infty}^{L}h\,\mathrm{d}x\Bigg{)},$ (16)

where $q_{s}$ is any source flux in $-\infty<x<L$ if it exists. Given the
free-surface elevation at various times, the flux, the flow thickness and the
pressure gradient can be obtained at $x=L$ at those times. The function
$F(\tau)$ is reconstructed by using equation (10). Unlike the steady method,
additional data about the upstream flux or free-surface velocity is not
required to fully constrain $F(\tau)$ (and the constitutive law) because the
transient evolution of the free surface provides sufficient data to determine
the timescale of the flow.

As an example, we consider a two-dimensional slump of material, with volume
per unit length of $1\,$\mathrm{m}^{2}$$, released as a square block on a
horizontal plane. Slumps of Herschel-Bulkley fluids on a horizontal plane were
described in detail by Balmforth et al. (2000). We use the following
constitutive law

$\dot{\gamma}=\phi(\tau)=\frac{\tau}{\mu_{0}}+\alpha\sin(\tau/\tau^{*}),$ (17)

with $\tau^{*}=10\,$\mathrm{Pa}$$, $\alpha=0.08\,$\mathrm{s}^{-1}$$ and
$\mu_{0}=83\,$\mathrm{Pa}\,\mathrm{s}$$. Figure 3a shows the free-surface
evolution, calculated numerically using the forward model (9). The flow is
symmetric about $x=0$.

For the inversion method, we use the free-surface elevation data at 10
$\mathrm{c}\mathrm{m}$ intervals along the $x$ axis and half second intervals
in time. Equation (16) is used to estimate the flux at $x=\pm 1.5$ metres
(blue dashed line in figure 3a). Figure 3b shows the flow thickness and the
basal stress, $|G|h$, calculated from the gradient of the free surface, at
$x=L=1.5$ metres as a function of time. The undulations arise from the
nonlinearities in the constitutive law. The basal stress, flow thickness and
flux provide a prediction for $F(\tau)$, which is plotted with blue circles in
figure 3c and compared to the true shape of $F$ (black line). The derivative
of $F$ is obtained via linear interpolation and this furnishes a prediction of
the constitutive law (red crosses in figure 3d). Provided the flow thickness
in $-L<x<L$ and its gradient at $x=L$ can be obtained to sufficient accuracy
at multiple times, the inversion method is able to capture the detailed
nonlinearities in the constitutive law. The constitutive law is determined for
larger stresses at earlier times and smaller stresses at late times as the
fluid slumps (see figure 3b).

Figure 4: (a) Free-surface shape for a yield-stress fluid at $10$, $10^{2}$,
$10^{3}$ and $10^{4}$ seconds. (b) The constitutive law (black line) and
inference from the inversion method (red crosses).

The inversion method can also be applied to a material with a yield stress. We
repeat the forward solution from above but the fluid now has a yield stress of
$25\,$\mathrm{Pa}$$. The free surface is shown at four times in figure 4a. At
late times, the stationary slumped shape is approached. The inversion method
is applied to predict the constitutive law and we include the additional
constraint that $\phi(0)=0$ (see figure 4b). The basal stress at $x=L$ reduces
in time and slowly converges to the yield stress (algebraically with respect
to time (c.f Hogg & Matson, 2009)). Hence, to capture the constitutive law
close to the yield stress, the slumping fluid must be observed for a long
time. The yield stress is accurately captured at $t=10^{4}$ seconds (see
figure 4b). In previous research, the final slump shape has been used to
obtain the yield stress and our method can reproduce those results as well as
capturing the constitutive law at other strain rates from the transient
evolution (Roussel & Coussot, 2005; Balmforth et al., 2007). Finally, it
should be noted that the method of this section is easily adapted to
axisymmetric flows.

###  4.2 Utilising the free-surface velocity

Figure 5: Inferring the constitutive law from the free-surface velocity data
of Al-Behadili et al. (2019). (a) $\Phi(\tau)$ predicted using our inversion
method applied to their data (circles and squares) with a polynomial best fit
(red dashed line) and the correct shape (black continuous line). (b) Inferred
(red dashed line) and true (black line) constitutive law.

For unsteady three-dimensional flows, a different inversion method is
required. In this section, we show how the free-surface velocity can be
written in terms of the rheology of the fluid. This formulation provides a
basis for an inversion method that uses the free-surface elevation and
velocity to determine the fluid rheology. Such a method is ideal for
exploiting experimental Particle Image Velocimetry (PIV) data.

We write the magnitude of the free-surface velocity in the form

$|\boldsymbol{u}(z=E+h)|=\int_{E}^{E+h}\bigg{\rvert}\frac{\partial\boldsymbol{u}}{\partial
z}\bigg{\rvert}\,\mathrm{d}z,$ (18)

where the magnitude is taken through the integral because the direction of the
integrand is independent of $z$ (see §2). We apply the change of variables
$-|\boldsymbol{G}|\mathrm{d}z=\mathrm{d}\tau$ to obtain the following
relationship between the free-surface velocity and the fluid rheology,

$|\boldsymbol{u}(z=E+h)|=|\boldsymbol{G}|^{-1}\Phi(|\boldsymbol{G}|h),\qquad\Phi(\tau_{0})=\int_{0}^{\tau_{0}}\phi(\tau)\,\mathrm{d}\tau.$
(19)

A similar formulation can be applied to obtain the velocity at each height
within the layer. The inversion method is similar to that of previous
sections. We constrain $\Phi(\tau)$ by comparing the observed free-surface
velocity of the flow, $|\boldsymbol{u}(z=E+h)|$, to the pressure gradient,
$|\boldsymbol{G}|$, and basal stress, $|\boldsymbol{G}|h$ at various locations
and times. The constitutive law is simply recovered from
$\phi(\tau)=\mathrm{d}\Phi/\mathrm{d}\tau$.

By way of an example, we apply this inverse approach to the data of Al-
Behadili et al. (2019). Their figures 5 and 6 show the flow thickness and
free-surface velocity at three times for a dam break of an Ellis fluid
calculated from a full Navier-Stokes simulation. First, we calculate the basal
stress and pressure gradient at two times and thirty-five spatial points from
the flow thickness and its gradient. The data for the basal stress,
$|\boldsymbol{G}|h$, is plotted against the observed free-surface velocity
multiplied by the magnitude of the pressure gradient, $|\boldsymbol{G}|$, at
each point to obtain a prediction for $\Phi(\tau)$, shown as scattered squares
and circles in figure 5a. The true form of $\Phi(\tau)$ is shown as a solid
black line. A fourth order polynomial (red dashed line in figure 5a) is fitted
to the scattered data with the condition that its derivative vanishes at the
origin, which ensures that $\phi(0)=0$. This polynomial is differentiated to
obtain a prediction for the constitutive law (red dashed line in figure 5b),
which shows reasonable agreement with the constitutive law used in the
simulations of Al-Behadili et al. (2019). The simulations included relatively
small amounts of inertia and surface tension, which are not accounted for in
our inversion method, although the latter could be included in the pressure
gradient. Despite this, our simple direct inversion method reproduces the
rheology to within a 15% error over most shear rates.

##  5 Conclusion

This contribution has presented direct inversion methods for determining the
constitutive law for a fluid from observations of its free surface. For steady
flows, the flux along streamlines is compared to the pressure gradient and
flow thickness to constrain a function, $F$, that encapsulates the rheology.
We have also presented a technique for inferring the constitutive law from the
free-surface velocity. The methods could be extended to incorporate an
undetermined sliding law at the bed as is relevant to the migration of ice
sheets. In other settings, the flow may be compressible or inertia may be
important. In these cases, the relationship between $|Q|$, $F$ and $\phi$ may
be nonlinear, which would lead to a more complicated inversion method. Further
work could also study optimum methods for taking the derivative of $F$ from
noisy data.

Declaration of Interests: The author reports no conflict of interest.

## Acknowledgements

The author is grateful to the School of Mathematics and Statistics at the
University of Melbourne for the award of a Harcourt-Doig research fellowship.

##  Appendix A Numerical methods

In this appendix, the numerical methods for the forward problems are
described. The flow thickness for two-dimensional steady flow over topography
(figure 1) is calculated numerically using equation (8). The flux is set to a
constant value, which furnishes a first order ordinary differential equation
for $h$. This is integrated numerically in the upstream direction using a
fixed value of the flow thickness far downstream; a numerical instability
arises from integration in the downstream direction (see Hinton et al., 2019).
The steady three-dimensional flow around a cylinder (figure 2) is solved
numerically using MATLAB’s Partial Differential Equation (PDE) ToolboxTM. The
method is given in section III of Hinton et al. (2020) with the function
inside the divergence operator adjusted to account for the general rheology.
For this flow, $h>0$ and $|\boldsymbol{G}|>0$ and no regularisation is
required.

For transient flows (figures 3 and 4), the forward problem (9) is solved using
the toolbox’s transient solver. In the case that the fluid has a yield stress,
a regularisation of the constitutive law is required. If the constitutive law
without a yield stress is $\dot{\gamma}=\phi(\tau)$ then for a fluid with
yield stress, $\tau_{Y}$, we use the following regularisation

$\dot{\gamma}=\phi\Big{\\{}\Big{[}\tau-\tau_{Y}+\sqrt{(\tau-\tau_{Y})^{2}+\epsilon^{2}\tau_{Y}^{2}}\Big{]}/2\Big{\\}},$
(20)

where $\epsilon=10^{-5}$. We also replace $|\boldsymbol{G}|$ with
$\sqrt{|\boldsymbol{G}|^{2}+(\delta\rho g)^{2}}$ where $\delta=10^{-8}$.
Checks confirm that volume is conserved. The inversion methods provide an
additional check on the forward computations.

## References

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  * Balmforth et al. (2000) Balmforth, N. J., Burbidge, A. S., Craster, R. V., Salzig, J. & Shen, A. 2000 Visco-plastic models of isothermal lava domes. Journal of Fluid Mechanics 403, 37–65.
  * Balmforth et al. (2007) Balmforth, N. J., Craster, R. V., Perona, P., Rust, A. C. & Sassi, R. 2007 Viscoplastic dam breaks and the Bostwick consistometer. Journal of Non-Newtonian Fluid Mechanics 142 (1-3), 63–78.
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  * Martin & Monnier (2015) Martin, N. & Monnier, J. 2015 Inverse rheometry and basal properties inference for pseudoplastic geophysical flows. European Journal of Mechanics-B/Fluids 50, 110–126.
  * Matsuhisa & Bird (1965) Matsuhisa, S. & Bird, R. B. 1965 Analytical and numerical solutions for laminar flow of the non-Newtonian Ellis fluid. AIChE Journal 11 (4), 588–595.
  * Myers (2005) Myers, T. G. 2005 Application of non-Newtonian models to thin film flow. Physical Review E 72 (6), 066302.
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# Can Balloons Produce Li-Fi?
A Disaster Management Perspective

Atchutananda Surampudi1, Sankalp Sirish Chapalgaonkar1 and Paventhan Arumugam2
1Department of Electrical Engineering, Indian Institute of Technology Madras,
2ERNET, IIT Madras Research Park.
1{ee16s003<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Natural calamities and disasters disrupt the conventional communication setups
and the wireless bandwidth becomes constrained. A safe and cost effective
solution for communication and data access in such scenarios is long needed.
Light-Fidelity (Li-Fi) which promises wireless access of data at high speeds
using visible light can be a good option. Visible light being safe to use for
wireless access in such affected environments, also provides illumination.
Importantly, when a Li-Fi unit is attached to an air balloon and a network of
such Li-Fi balloons are coordinated to form a Li-Fi balloon network, data can
be accessed anytime and anywhere required and hence many lives can be tracked
and saved. We propose this idea of a Li-Fi balloon and give an overview of
it’s design using the Philips Li-Fi hardware. Further, we propose the concept
of a balloon network and coin it with an acronym, the LiBNet. We consider the
balloons to be arranged as a homogenous Poisson point process in the LiBNet
and we derive the mean co-channel interference for such an arrangement.

Index Terms\- Attocell dimension, interference, Li-Fi, light emitting diode,
photodiode, rate, time division multiple access.

##  I Introduction

Visible-light-communications (VLC) or Light-Fidelity (Li-Fi) has gained a lot
of prominence among the research groups around the world for it’s secure,
safe, energy efficient and high speed data transfer characteristics over a
wireless medium. The use of visible light, which has given life to the planet,
is now paving the path for a new form of wireless communications [1]. This
technology has both the characteristics of providing data access and
illumination at the same time over linear-time-invariant optical wireless
channels [2], [3].

Natural disasters like earthquakes or floods, disrupt the available wireless
communication infrastructure and as a result the wireless bandwidth becomes
scarce. But, in such scenarios, communication becomes an important aspect of
the emergency response. A safe, high speed, energy efficient and spectrum
efficient wireless access technology is the need of the hour. There have been
several technologies in use for emergency communications, like the ham radio
[4], cognitive radio [5], [6] etc. But, when the demand becomes large or large
number of people have to be saved, these technologies remain sub-optimal due
to the limited radio frequency (RF) spectrum available in such a situation.
Also, if floods are considered, the people trapped below flood waters have to
be saved immediately. Communicating with them, or to at least receive their
location using some uplink signatures becomes a priority. RF waves cannot
penetrate through flood waters, whereas the light wave can. So, Li-Fi can be a
reliable and energy efficient option to improve the communication capacity
along with illuminating the affected area. Lately, for practical purposes and
outdoor applications, Li-Fi has been implemented using fixed light-emitting-
diode (LED) sources [7]. The question that arises is, can these Li-Fi LEDs fly
in the air, without any constraints, at a given height near the ground at
around $10$ to $15$ metres, to provide the same data access whenever and
wherever required? This idea of a mobile Li-Fi unit can be a good solution
during such natural calamities. This can be achieved by using a bunch of
centrally monitored balloons, which can carry the Li-Fi LEDs operated by low
power batteries to illuminate as well as provide data access (or broadcast
services) to a given region for a given period of time.

###  I-A Our approach and contributions

Our contributions are as follows.

  * •

The Li-Fi Balloon -  We present an overview of the design for the balloon
using the Philips Li-Fi hardware.

  * •

The concept of LiBNet and the overall functioning.

  * •

The mean co-channel interference is derived for homogenous Poisson arrangement
of balloons in one and two dimensions.

###  I-B Context of application:

####  I-B1 Earthquakes

During earthquakes, every other communication infrastructure above or below
the ground gets disrupted. Also, the problem deepens when any gas leaks happen
and become susceptible to combustion. So, using the LiBNet, both illumination
and high speed data services can be provided, in a safe way, without
interfering with the RF frequencies.

####  I-B2 Floods

In a flood affected region, LiBNet will be able to send a broadcast inside the
water surface and receive any uplink messages from the people or pets trapped
below. The balloons can coordinate to provide tracking services as well.

The further organisation of the paper is as follows. In section II, we give a
brief overview of the design of the Li-Fi balloon using the hardware from
Philips. In section III, we present the concept of LiBNet and assuming a
poisson point process arrangement, we derive the mean co-channel interference
in such a scenario. The paper concludes with section IV.

##  II The Li-Fi Balloon

The Li-Fi balloon can be designed with the outline presented in Fig. 1.

$h_{b}$$h_{w}$$h_{l}$$R$FRONT VIEW$D_{l}$$D_{b}$$l_{w}$$w_{w}$TOP VIEW

Figure 1: This figure is to a scale of 1:30. The front and top views of the Li-Fi balloon have been shown. All dimensions are in centimetres (cm) as given in Table I. The balloon’s aerial structure can be of any shape depending on the aerodynamical aspects. In the top view, the dashed circular boundary depicts the base of the balloon. There are four shaded regions on the base, on each of which a Li-Fi LED and an uplink receiver are placed. The space in the centre of the base is reserved for the LED drivers, constant power source and signal processing modem. For inter-balloon communication, at the corners of the base, four wireless nodes are placed (filled regions), with beamforming capability. Table I: dimensions of the balloon Part of the Balloon |  Symbol used |  Dimension
---|---|---
Diameter of the base |  $D_{b}$ |  $90$cm
Height of the basket |  $h_{b}$ |  $45$cm
Radius of the balloon sheet |  $R$ |  $70$cm
Diameter of the area for downlink and uplink components (Shaded area) |  $D_{l}$ |  $30$cm
Height of the Downlink LED |  $h_{l}$ |  $15$cm
Height of the wireless node(fixed near the roof of the basket) |  $h_{w}$ |  $10$cm
Length of the wireless node |  $l_{w}$ |  $10$cm
Width of the wireless node |  $w_{w}$ |  $10$cm

###  II-A The Base

A Li-Fi balloon will contain all the necessary communication equipment and the
power source. The construction of the balloon is as follows. It consists of a
base (closed basket) onto which four downlink LEDs and uplink receivers are
placed (shaded area). The shaded areas surround the empty area at the centre,
where the power and signal processing modem is placed. Four wireless nodes
(with beam forming) are placed in alternate corners of the base, for inter-
balloon communication. The material used for the basket can be any strong
waterproof and non-conducting fiber [8].

Distribution of weight should be uniform and homogenous to stabilize the
balloon at a given height. So, a symmetric arrangement of components is
desired.

###  II-B The balloon sheet

The frame of the balloon’s arial sheet, can be of circular or hexagonal shape,
appropriate to the aerodynamic balancing aspects [9]. The material used can be
a thin rigid polymer as proposed in [10] and [11] or water resistant polymers
as in [12]. Light weight composites are preferred.

###  II-C Components and Hardware

The Li-Fi components used are shown in Table II.

Table II: components used per balloon. Component |  Name |  Quantity |  Payload (max. value)
---|---|---|---
Luxspace DN561B |  Philips downlink Li-Fi transmitter |  $4$ |  $0.4\times 4=1.6$kg
Philips modem board |  Philips modem board |  $1$ |  $0.2$kg
LBRD1514-1 |  Xitanium $20$W LED power driver |  $1$ |  $0.3$kg
LBRD14016-3 |  Uplink-IR Receiver |  $4$ |  $0.1\times 4=0.4$kg
DC Power supply |  Rechargeable DC power supply |  $1$ |  $2$kg
D-Link wireless access points |  Wireless nodes A |  $4$ |  $4\times 0.5=2$kg
Motion and Position sensor |  Sensor |  $1$ (each) |  $0.1$kg

From Table II, we have the total payload per balloon as $6.6$kg. The
dimensions in Table I suggest that the balloon will be closely packed. We now
discuss the use of each component in Table II. The Philips Luxspace DN561B is
a Li-Fi downlink transmitter and LBRD14016-3 is an uplink infrared receiver.
These are used for illumination and data access with various emission
properties [13]. In [14], these components have been used to characterize
outage in an indoor Li-Fi environment. The LBRD1514-1 is a set of LED power
drivers which are an integral part for intensity modulation (IM) purposes. The
modem board acts as a central switch for all the LEDs and for IM. The modem
board, drivers and the DC power supply can be appropriately fitted in the
central space provided. The sensors will help in stabilizing the position and
the inclination of the balloon. The wireless nodes will be useful to form an
interconnected wireless balloon network.

##  III The LiBNet

A network of Li-Fi balloons is called LiBNet. Each balloon, using the wireless
nodes, forms an interconnected wireless network. The balloons can be assumed
to be arranged as a Poisson point process111This assumption gives the worst
case co-channel interference in the network., as in [15], in both one and two
dimensions which is shown in Fig. 2 and Fig. 3 respectively. For any
arrangement, we consider all the balloons to be stabilized at a uniform height
$h$ and have an effective222Effective, because each balloon has four downlink
LEDs. Lambertian emission order $m=\frac{-\ln(2)}{\ln(\cos(\theta_{h}))}$.
$\theta_{h}$ is the effective half-power-semi-angle (HPSA) of the balloon. The
trapped victim is assumed to be at a distance $z$ from the tagged balloon,
which is nearest to the victim.

$h$$h$$z$The balloon network in one dimension $(0,0)$$(z,0)$xattocelltagged-
balloonvictim / trapped pet

Figure 2: (One dimension model) This figure shows the one dimension balloon
network. The balloons can be assumed to be arranged deterministically or as a
Poisson point process $\psi$, stabilized at a height $h$ (black circular
dots). The rectangular dotted regions on ground depict the attocells
corresponding to each balloon above. The trapped victim (small cuboid) at
$(z,0)$ (inside one of the attocell), receives data wirelessly from the
tagged-balloon corresponding to the attocell in which he/she is located. Here,
that attocell is highlighted as dash-dot. All other balloons become the co-
channel interferers. Here, we assume that the victim can be trapped only along
the thick line on ground. Every balloon is interconnected with each other
wirelessly.

attocelltagged-balloonx$(0,0,0)$$h$victim at $(d_{x},d_{y},0)$$z$The balloon
network in two dimensions

Figure 3: (Two dimension model) This figure shows the two dimension balloon
network. The balloons can be assumed to be arranged deterministically or as a
Poisson point process $\psi$, stabilized at a height $h$ (black circular
dots). The rectangular dotted regions on ground depict the attocells
corresponding to each balloon above. The trapped victim (small cuboid) at
$(d_{x},d_{y},0)$ (inside one of the attocell), receives data wirelessly from
the tagged-balloon corresponding to the attocell in which it is located. Here,
that attocell is highlighted as dash-dot. All other balloons become the co-
channel interferers. Here we assume that the victim can be trapped anywhere on
the ground plane. Every balloon is interconnected with each other wirelessly.

###  III-A How does the LiBNet work?

Each balloon with the necessary Li-Fi equipment will cover a certain affected
region via illumination. Uplink signatures from the trapped lives, can be
received and downlink broadcast can be done. The sensing of uplink signatures
can be done efficiently by using machine learning as in [16] for cognitive
radio networks. All the balloons will be interconnected wirelessly and linked
to a fusion centre (FC) which is in-turn connected to the Internet. The uplink
and downlink data will be routed through the network and through the FC as a
gateway to the Internet. This model can be assumed similar to a wireless
sensor network (WSN) in a cooperative sensing environment [17]. Now, various
routing protocols can be used or devised [18], [19] to route the data
efficiently. So, such a network of Li-Fi balloons can be used anywhere in the
affected region. Their interconnected network can be used to track the lives
of many and save them immediately.

###  III-B The mean co-channel interference

Let all the points in the given balloon network be arranged as a Poisson point
process $\psi$, in both one and two dimensions. Given an origin, let the
balloons be at a distance $x$ from the origin. The user photodiode (PD) is at
a distance of $z$ from the origin. Let the tagged balloon be at origin i.e.
$x=0$. Now, assuming a wavelength reuse factor of unity, all other balloons
become co-channel interferers. We now derive the mean co-channel interference
in the Poisson network of balloons with the assumptions in [20]. For a
distance $D_{x}$ between the PD and the balloon, the Signal-to-Interference-
plus-Noise-Ratio (SINR) for such a network is given as

$\gamma(z)=\frac{(z^{2}+h^{2})^{-m-3}\rho(D_{0})}{\sum_{x\in\psi\setminus
0}((x+z)^{2}+h^{2})^{-m-3}\rho(D_{x})+\Omega},$ (1)

where, $\rho(D_{x})$ is the field-of-view (FOV) $\theta_{f}$ constraint
function of the PD used by the trapped victim, which is defined as

$\rho(D_{x})=\left\\{\begin{array}[]{ll}1,&|D_{x}|\leq h\tan(\theta_{f}),\\\
0,&|D_{x}|>h\tan(\theta_{f}).\end{array}\right.$

The co-channel interference $\mathcal{I}_{x}$ is

$\mathcal{I}_{x}=\sum_{x\in\psi\setminus 0}f(x),$

where $f(x)$ is given as

$((x+z)^{2}+h^{2})^{-m-3}\rho(D_{x}).$

Hence, we state the theorem for mean interference below.

######  Theorem 1.

Consider a photodiode, with FOV $\theta_{f}$ radians, situated at a distance
$z$ (inside an attocell) from the origin, in a Poisson distributed balloons of
Li-Fi LEDs, emitting light with an effective Lambertian emission order $m$,
installed at a height $h$. Then, for a wavelength reuse factor of unity, the
mean co-channel interference caused by the Poisson distributed co-channel
interferers of intensity $\lambda(x)$ at the photodiode is

$\displaystyle\mathbb{E}(\mathcal{I}_{x})=\int_{\mathbb{S}}\lambda(x)f(x)\mathrm{d}x,$

where, $\mathbb{E}(.)$ is the expectation operator over all possible random
arrangements in the Poisson point process and $\mathbb{S}$ is the support of
the integration.

###### Proof.

The proof is provided in Appendix Proof of Mean Interference. ∎

We now extend Thm. 1 to one and two dimension LiBNets in the following Lem. 1
and 2 respectively.

######  Lemma 1.

The mean co-channel interference in a one dimension LiBNet with homogenously
distributed Poisson point balloons is given as

$\displaystyle\mathbb{E}(\mathcal{I}_{x})$
$\displaystyle=\lambda\bigg{(}h^{1-2\beta}\tan(\theta_{f}){}_{2}F_{1}\bigg{(}0.5,\beta;1.5;-\tan^{2}(\theta_{f})\bigg{)}$
$\displaystyle\ \ \ \ \
-zh^{-2\beta}{}_{2}F_{1}\bigg{(}0.5,\beta;1.5;-\frac{z^{2}}{h^{2}}\bigg{)}\bigg{)},$

where, ${}_{2}F_{1}(.;.;.)$ is the ordinary hypergeometric function and
$\beta=m+3$.

###### Proof.

From Thm. 1, with the support $\mathbb{S}=[z,h\tan(\theta_{f})]$ we have

$\displaystyle\mathbb{E}(\mathcal{I}_{x})$
$\displaystyle=\int_{z}^{h\tan(\theta_{f})}\lambda(x)f(x)\mathrm{d}x,$ (2)

because, the interferers are present only after a distance $z$ limited by the
FOV of the PD. Now, for a one dimension network

$f(x)=\frac{1}{(x^{2}+h^{2})^{\beta}}$

and $\lambda(x)=\lambda$ for a homogenous point process. Hence, integrating
the same in (2) derives the result. ∎

######  Lemma 2.

The mean co-channel interference in a two dimension LiBNet homogenously
distributed Poisson point balloons is given as

$\displaystyle\mathbb{E}(\mathcal{I}_{x})$
$\displaystyle=\lambda\bigg{(}\frac{\pi(h^{2}+z^{2})^{1-\beta}}{\beta-1}-\frac{\pi
h^{2-2\beta}\cos^{2\beta-2}(\theta_{f})}{\beta-1}\bigg{)}.$

where, $\beta=m+3$.

###### Proof.

For a two dimension homogenous Poisson point process of balloons we have
$\lambda(x)=\lambda$ and $f(x,y)=\frac{1}{(x^{2}+y^{2}+h^{2})}$. So, from Thm.
1, we can write

$\mathbb{E}(\mathcal{I}_{x})=\int\int_{\mathbb{S}}\lambda\frac{1}{(x^{2}+y^{2}+h^{2})}\mathrm{d}x\mathrm{d}y.$

Converting to polar coordinates $(r,\theta)$, we have

$\mathbb{E}(\mathcal{I}_{x})=2\pi\int_{z}^{h\tan(\theta_{f})}\frac{1}{(r^{2}+h^{2})}\mathrm{d}r.$

Integrating further, derives the result. ∎

##  IV Conclusion

In this paper, a novel idea of using balloons for Li-Fi has been proposed. A
disaster or a natural calamity has been taken as a use case. An overview of
the physical design of the balloon has been given. Given the payload, the
installation of the communication components has been proposed using the
Philips Li-Fi equipments. Symmetry of installation has been kept in mind.
Further, the functioning of a network of such Li-Fi balloons has been
discussed and coined an acronym of LiBNet. The mean co-channel interference in
both one and two dimension LiBNets has been derived, assuming the balloons are
arranged as a homogenous Poisson point process. Given the problems of
contemporary interest, the possibilities of future work are extensive. Li-Fi,
being a safe and high speed alternative to RF emergency communication set-ups,
this concept, if implemented can save the lives of many. Tracking and
positioning of people / pets becomes easier. But, Li-Fi, requiring too much
line of sight, will have to co-exist with blockages. Hence an extension to a
blockage model is desired. Also, as a future work, this concept can be further
standardised and practically implemented.

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## Appendix A

## Proof of Mean Interference

###### Proof.

The Laplace transform of the Interference $\mathcal{I}_{x}$ is given as

$\displaystyle\mathcal{L}_{\mathcal{I}_{x}}(s)$
$\displaystyle=\mathbb{E}_{\psi}(e^{-s\mathcal{I}_{x}}),$
$\displaystyle=\mathbb{E}_{\psi}(e^{-s\sum_{x\in\psi\setminus 0}f(x)}),$
$\displaystyle=\mathbb{E}_{\psi}\bigg{(}\prod_{x\in\psi\setminus
0}e^{-sf(x)}\bigg{)},$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}e^{-\lambda(x)\int_{\mathbb{R}}(1-e^{-sf(x)})\mathrm{d}x},$
(3)

where, (a) follows from the definition of expectation operator. Now, taking
the $n^{th}$ order derivative on L.H.S of (3) w.r.t $s$, we have

$(-1)^{n}\frac{\partial^{n}}{\partial
s^{n}}(\mathbb{E}(e^{-s\mathcal{I}_{x}}))=\mathbb{E}(\mathcal{I}^{n}_{x}e^{-s\mathcal{I}_{x}}).$

Using the result in (3), we can write for $n=1$ as

$\displaystyle\mathbb{E}(\mathcal{I}_{x}e^{-s\mathcal{I}_{x}})$
$\displaystyle=(-1)e^{-\lambda(x)\int_{\mathbb{R}}(1-e^{-sf(x)})\mathrm{d}x}$
$\displaystyle\ \ \ \ \ \
(-1)\big{(}\lambda(x)\int_{\mathbb{R}}(-1)(-f(x))\mathrm{d}x\big{)}.$ (4)

Substituting $s=0$ in (4) we have

$\mathbb{E}(\mathcal{I}_{x})=\int_{\mathbb{R}}\lambda(x)f(x)\mathrm{d}x,$

proving the theorem. ∎


Experiments sec:experiments

In this section, we first introduce the datasets used to evaluate our proposed
approach. The results are then compared to the state of the art, and an
ablation study is performed.

Datasets and Evaluation

We consider two popular benchmarks for 3D HPE following recent works
DBLP:journals/corr/abs-2103-16385.

Human3.6M DBLP:journals/pami/IonescuPOS14: This is the most widely-used
dataset for 3D human pose estimation. It uses a motion capture system and 4
video cameras to capture the 3D pose information and accompanying images,
respectively. The camera calibration parameters are used to project the 3D
joint positions to the 2D image plane of each camera. The dataset contains 3.6
million images of 7 professional actors performing 15 generic movements such
as walking, eating, sitting, making a phone call and engaging in a discussion.
We follow the standard protocol, using subjects 1, 5, 6, 7 and 8 for training,
and subjects 9 and 11 for evaluation.

MPI-INF-3DHP 8374605: This dataset contains images of three different
scenarios: (i) studio with a green screen (GS), (ii) studio without green
screen (noGS) and (iii) outdoor scene (Outdoor). We follow the common skeleton
representation for both of these datasets, which contains $17$ joints.

Following common practice, for Human3.6M we evaluate by using the mean per-
joint pixel error (MPJPE) in $mm$ between the ground truth and our predictions
across all joints and cameras, after alignment of the root joint (Protocol
#1). It is also common to align the prediction with the ground truth using a
rigid transformation moreno20173d (Protocol #2). For the MPI-INF-3DHP test set
we use 3D-PCK (Percentage of Correct Keypoints) and AUC (Area Under the Curve)
as evaluation metrics following previous works DBLP:conf/bmvc/LuoCY18,
DBLP:journals/corr/abs-2103-16385, ci2019optimizing.

Comparison with State-of-the-Art Methods

0.8!  (a) CPN keypoints Direct Discuss Eating Greet Phone Photo Pose Purch.
Sitting SittingD. Smoke Wait WalkD. Walk WalkT. Avg. Martinez
[martinez2017simple] ICCV’17 51.8 56.2 58.1 59.0 69.5 78.4 55.2 58.1 74.0 94.6
62.3 59.1 65.1 49.5 52.4 62.9 Tekin [DBLP:conf/iccv/TekinMSF17] ICCV’17 54.2
61.4 60.2 61.2 79.4 78.3 63.1 81.6 70.1 107.3 69.3 70.3 74.3 51.8 63.2 69.7
Sun [DBLP:conf/iccv/0001SLW17] ICCV’17 (+) 52.8 54.8 54.2 54.3 61.8 67.2 53.1
53.6 71.7 86.7 61.5 53.4 61.6 47.1 53.4 59.1 Yang [yang20183d] CVPR’18 (+)
51.5 58.9 50.4 57.0 62.1 65.4 49.8 52.7 69.2 85.2 57.4 58.4 43.6 60.1 47.7
58.6 Fang [DBLP:conf/aaai/FangXWLZ18] AAAI’18 50.1 54.3 57.0 57.1 66.6 73.3
53.4 55.7 72.8 88.6 60.3 57.7 62.7 47.5 50.6 60.4 Pavlakos
[DBLP:conf/cvpr/PavlakosZD18] CVPR’18 (+) 48.5 54.4 54.5 52.0 59.4 65.3 49.9
52.9 65.8 71.1 56.6 52.9 60.9 44.7 47.8 56.2 Zhao [zhao2019semantic] CVPR’19
48.2 60.8 51.8 64.0 64.6 53.6 51.1 67.4 88.7 57.7 73.2 65.6 48.9 64.8 51.9
60.8 Sharma [DBLP:conf/iccv/SharmaVBSJ19] ICCV’19 48.6 54.5 54.2 55.7 62.2
72.0 50.5 54.3 70.0 78.3 58.1 55.4 61.4 45.2 49.7 58.0 Ci [ci2019optimizing]
ICCV’19 (+) 46.8 52.3 44.7 50.4 52.9 68.9 49.6 46.4 60.2 78.9 51.2 50.0 54.8
40.4 43.3 52.7 Liu [DBLP:conf/eccv/LiuDZ0T20] ECCV’20 46.3 52.2 47.3 50.7 55.5
67.1 49.2 46.0 60.4 71.1 51.5 50.1 54.5 40.3 43.7 52.4 Lin [lin2020end]
CVPR’21 - - - - - - - - - - - - - - - 54.0 Xu
[DBLP:journals/corr/abs-2103-16385] CVPR’21 45.2 49.9 47.5 50.9 54.9 66.1 48.5
46.3 59.7 71.5 51.4 48.6 53.9 39.9 44.1 51.9 Base Transformer 50.3 53.7 49.7
53.0 56.7 62.8 52.2 49.0 61.8 69.0 54.5 51.0 56.1 42.1 44.7 53.8 Ours 45.0
48.8 46.6 49.4 53.2 60.1 47.0 46.7 59.6 67.1 51.2 47.1 53.8 39.4 42.4 50.5
Ours + Refinement 45.6 49.7 46.0 49.3 52.2 58.8 47.5 46.1 58.2 66.1 50.7 47.5
52.6 39.2 41.6 50.1
0.8!  (b) GT keypoints Direct Discuss Eating Greet Phone Photo Pose Purch.
Sitting SittingD. Smoke Wait WalkD. Walk WalkT. Avg. Martinez
[martinez2017simple] ICCV’17 45.2 46.7 43.3 45.6 48.1 55.1 44.6 44.3 57.3 65.8
47.1 44.0 49.0 32.8 33.9 46.8 Zhou [DBLP:conf/iccv/ZhouHJJL19] ICCV’19 (+)
34.4 42.4 36.6 42.1 38.2 39.8 34.7 40.2 45.6 60.8 39.0 42.6 42.0 29.8 31.7
39.9 Ci [ci2019optimizing] ICCV’19 (+) 36.3 38.8 29.7 37.8 34.6 42.5 39.8 32.5
36.2 39.5 34.4 38.4 38.2 31.3 34.2 36.3 Zhao [zhao2019semantic] CVPR’19 37.8
49.4 37.6 40.9 45.1 41.4 40.1 48.3 50.1 42.2 53.5 44.3 40.5 47.3 39.0 43.8 Liu
[DBLP:conf/eccv/LiuDZ0T20] ECCV’20 36.8 40.3 33.0 36.3 37.5 45.0 39.7 34.9
40.3 47.7 37.4 38.5 38.6 29.6 32.0 37.8 Xu [DBLP:journals/corr/abs-2103-16385]
CVPR’21 35.8 38.1 31.0 35.3 35.8 43.2 37.3 31.7 38.4 45.5 35.4 36.7 36.8 27.9
30.7 35.8 Ours 31.0 36.6 30.2 33.4 33.5 39.0 37.1 31.3 37.1 40.1 33.8 33.5
35.0 28.7 29.1 34.0 Ours + Refinement 32.2 37.4 29.7 33.4 33.4 38.1 38.1 32.1
37.1 40.1 33.9 34.1 35.2 28.6 28.8 34.2

Table : Quantitative results on the Human3.6M
[DBLP:journals/pami/IonescuPOS14] dataset for single-frame 3D HPE with (a) CPN
[DBLP:conf/cvpr/ChenWPZYS18] predictions and (b) GT keypoints used as input.
The evaluation is performed using MPJPE in $mm$ under Protocol #1. Methods
denoted by (+) use extra data from MPII [DBLP:conf/cvpr/AndrilukaPGS14]. Best
results are shown in bold.

We compare the performance of our proposed model with recent state-of-the-art
methods for single-frame 3D human pose estimation across two popular datasets:
Human3.6M and MPI-INF-3DHP. We include both direct estimation and 2D-3D
lifting methods in our analysis. Existing lifting methods use either
multilayer perceptron or graph convolutional models to map from 2D to 3D. In
contrast, the Joint Transformer uses a transformer architecture with
intermediate supervision, error prediction and refinement. To highlight the
significance of our improvements, we also compare to a ”Base Transformer”
architecture that consists of a linear embedding, spatial positional encoding
[zheng20213d] and transformer encoder blocks. We first investigate the in-the-
wild performance of our network where it takes the 2D predictions of a 2D pose
detector as input. Table (a) shows the results on the Human3.6M dataset on 2D
predictions of the cascaded pyramid network (CPN). Our Joint Transformer
outperforms the state-of-the-art approaches by more than $1.4$ $mm$. The
addition of the Refinement Transformer decreases our error a further $0.4$
$mm$ for a total improvement of $1.8$ $mm$. Our Joint Transformer also
outperforms the Baseline Transformer without intermediate supervision, error
prediction and with linear pose embedding by more than $3$ $mm$. Since the
above results are dependent on the quality of the output of the 2D pose
detector, it is useful to remove the effect of the 2D backbone to characterise
the theoretical performance of our network. Table (b) presents results on the
Human3.6M dataset with ground truth 2D joint positions passed as input. Here,
we find that our Joint Transformer surpasses the state of the art by $1.8$
$mm$. In this case, we find that our Refinement Transformer does not further
decrease the prediction error. In fact, the performance of the Refinement
Transformer seems correlated to the magnitude of the error prediction of the
Joint Transformer. In the case of the ground-truth 2D inputs the predicted
error is quite small, whilst the larger magnitude of the error prediction for
the CPN keypoints can be effectively used by the Refinement Transformer to
achieve a better prediction. In comparison to the state of the art, our method
only uses $1.48$M parameters, compared to $3.70$M and $4.38$M parameters for
the second- [DBLP:journals/corr/abs-2103-16385] and third-
[DBLP:conf/eccv/LiuDZ0T20] placed methods respectively. To show the
generalisation ability of our method, we evaluate our Joint Transformer
trained on Human3.6M directly on the MPI-INF-3DHP test dataset without any
additional training. As shown in Table , we outperform the previous state of
the art, even though some previous methods use additional training data from
MPI-INF-3DHP. We also present the performance of our Joint Transformer
followed by the Refinement Transformer. As can be seen, the Refinement
Transformer further enhances our results by an additional small margin. This
shows the ability of our method to generalise to unseen datasets.

0.8!  Training data GS (PCK) noGS (PCK) Outdoor (PCK) All (PCK) All (AUC)
Martinez [martinez2017simple] ICCV’17 H3.6M 49.8 42.5 31.2 42.5 17.0 Mehta
[8374605] 3DV’17 H3.6M 70.8 62.3 58.8 64.7 31.7 Yang [yang20183d] CVPR’18
H3.6M+MPII - - - 69.0 32.0 Zhou [Zhou_2017_ICCV] ICCV’17 H3.6M+MPII 71.1 64.7
72.7 69.2 32.5 Luo [DBLP:conf/bmvc/LuoCY18] BMVC’18 H3.6M 71.3 59.4 65.7 65.6
33.2 Ci [ci2019optimizing] ICCV’19 H3.6M 74.8 70.8 77.3 74.0 36.7 Zhou
[DBLP:conf/iccv/ZhouHJJL19] ICCV’19 H3.6M+MPII 75.6 71.3 80.3 75.3 38.0 Xu
[DBLP:journals/corr/abs-2103-16385] CVPR’21 H3.6M 81.5 81.7 75.2 80.1 45.8
Base Transformer H3.6M 83.3 81.6 81.3 82.1 49.2 Ours H3.6M 83.8 83.8 81.4 83.0
51.2 Ours + Refinement H3.6M 84.2 85.4 81.4 83.7 52.4

Table : Quantitative results for PCK and AUC on the MPI-INF-3DHP [8374605]
test dataset with GT keypoints used as input. Best results are shown in bold.
The Jointformer is not trained on this dataset, showing the generalisation
ability of the proposed method to unseen data.

##  Ablation

0.8!  $T_{layers}$ $D_{h}$ $Embed_{Pose}$ 2*Int Sup. 2*Err Pred. Pos Enc.
Scores 4 5 6 64 128 256 Conv1D Linear Spatial Frequency MPJPE P-MPJPE
$\checkmark$ $\checkmark$ $\checkmark$ 44.2 33.7 $\checkmark$ $\checkmark$
$\checkmark$ $\checkmark$ 34.5 26.5 $\checkmark$ $\checkmark$ $\checkmark$
$\checkmark$ $\checkmark$ 34.0 26.5 $\checkmark$ $\checkmark$ $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ 35.5 27.1 $\checkmark$ $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 35.2 27.0 $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 34.6 26.7 $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 34.8 26.9 $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 34.4 26.9 $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 34.3 26.5 $\checkmark$
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ 38.1 28.7

Table : Ablation study with quantitative results in $mm$ on the Human3.6M
dataset [DBLP:journals/pami/IonescuPOS14]. As input, GT keypoints are used. We
start with a basic Jointformer with $T_{layers}=4$, $D_{h}=64$ and
$Embed_{Pose}=Conv1D$ (first row) and next add intermediate supervision
(second row), error prediction (third row) and positional encoding (fourth and
fifth row). We notice improvements with Int Sup. and Error Pred. but Pos Enc.
negatively impacts results. Next, with Int Sup. and Error Pred. added to the
basic Jointformer, we experiment with $T_{layers}$ (fifth and sixth rows),
$D_{h}$ (seventh and eighth rows) and $Embed_{Pose}=Linear$ (ninth row). We
obtain the best results at $T_{layers}=4$, $D_{h}=64$ , $Embed_{Pose}=Conv1D$,
without position encoding and with Int Sup. and Err Pred.

In this section, we evaluate the effectiveness of our modifications to the
standard transformer architecture on the Human3.6M dataset
[DBLP:journals/pami/IonescuPOS14]. The results are reported in Table . To
remove the influence of the 2D pose detector, the results are reported using
$17$ 2D ground truth joints as input. The purpose of the ablation study is to
investigate the effects of the number of transformer layers ($T_{layers}$),
size of the encoder ($D_{h}$), type of pose embedding used ($Embed_{Pose}$),
intermediate supervision (Int Sup.), error prediction (Err Pred.) and
positional encoding (Pos Enc.). All reported errors are the average of $3$
training runs. In the upper part of the table we observe the impact of the
main contributions of this paper. Just by adding the intermediate supervision,
we drastically improve the performance. Adding the error prediction and
training the network in a multi-task framework further improves the results.
We also observe that explicitly adding positional encoding to the transformer
tokens reduces the performance. Since the order of the joints never changes,
their positions are already implicitly encoded and further addition of an
explicit encoding seems to confuse the network somewhat. This is the case for
both the frequency based positional encoding, as well as the spatial encoding
[zheng20213d]. In the lower part of the table, we show the effect on a
different number of transformer encoders and size of the hidden dimension. We
show that the highest performance and most stable training can be achieved by
using $4$ layers and a hidden dimension of $64$. While a larger number in
either of these parameters increases the capacity of the network, the larger
capacity seems not to be needed for best results. Finally, we can also observe
that using a 1D convolution to embed the 2D inputs to the hidden dimension
instead of a linear layer not only saves some network parameters, but also
leads to better performance.