# An insightful approach to bearings-only tracking in log-polar coordinates

Athena Helena Xiourouppa  0000-0003-0033-744X Dmitry Mikhin Melissa Humphries
0009-0002-6228-2958 0000-0002-0473-7611 and John Maclean  0000-0002-5533-0838

###### Abstract

The choice of coordinate system in a bearings-only (BO) tracking problem
influences the methods used to observe and predict the state of a moving
target. Modified Polar Coordinates (MPC) and Log-Polar Coordinates (LPC) have
some advantages over Cartesian coordinates. In this paper, we derive closed-
form expressions for the target state prior distribution after ownship
manoeuvre: the mean, covariance, and higher-order moments in LPC. We explore
the use of these closed-form expressions in simulation by modifying an
existing BO tracker that uses the UKF. Rather than propagating sigma points,
we directly substitute current values of the mean and covariance into the time
update equations at the ownship turn. This modified UKF, the CFE-UKF, performs
similarly to the pure UKF, verifying the closed-form expressions. The closed-
form third and fourth central moments indicate non-Gaussianity of the target
state when the ownship turns. By monitoring these metrics and appropriately
initialising relative range error, we can achieve a desired output mean
estimated range error (MRE). The availability of these higher-order moments
facilitates other extensions of the tracker not possible with a standard UKF.

###### Index Terms:

Statistical analysis, bearings-only tracking, target tracking, Modified Polar
Coordinates, Log-Polar Coordinates, Unscented Kalman Filter

##  I Introduction

Consider a sensing platform (ownship) tracking a vessel of interest (target)
using a passive acoustic sensor.111For simplicity, we ignore the potential
differences between the ownship coordinates / orientation and the sensor
coordinates / orientation. Assume energy propagates in simple, horizontal
straight lines, so that there is only one energy path (ray) from target to
sensor. The sensor obtains the target bearing by measuring the direction of
target noise arrival. We aim to recover the target’s full state, i.e.,
position and speed, from a sequence of this data.

Modified Polar Coordinates (MPC) have been established as a stable approach
for bearings-only (BO) target tracking. State update equations in MPC are non-
linear; to implement estimation for non-linear systems, existing literature
suggests the use of the Extended Kalman Filter (EKF) [1, 2] or Unscented
Kalman Filter (UKF) [3].

Log-Polar Coordinates (LPC) are another viable candidate [4, 5], especially
for theoretical analysis; using LPC, one can obtain the posterior Cramér-Rao
bound (PCRB) for the target state distribution [4]. This paper further
explores the theoretical properties of the state distribution in LPC.

We do so in the context of a simple explicit ownship manoeuvre, described in
Section II; broader scenarios can be approximated as combinations of simple
manoeuvres.

The main contribution of this paper is proving that the target state
distribution after an ownship manoeuvre can be described in LPC with closed-
form expressions, derived in Section III. Closed-form expressions allow us to
better understand the relationships between target state variables. We
demonstrate the advantage of this for tracking simulation in Section IV by
implementing a Kalman Filter that incorporates the closed-form expressions,
CFE-UKF, and comparing its capabilities to a pure UKF. Conclusions are
presented in Section V.

##  II Passive Bearings-Only Tracking

###  II-A Straight Line Movement

The target’s state in MPC [2] is described as a vector of

  * •

absolute bearing $\beta$ that is the angle from the receiver to the target
measured from True North,

  * •

bearing rate $\dot{\beta}$,

  * •

scaled range rate $\dot{\rho}=\dot{r}/r$, where $r$ is range, and

  * •

inverse range $s=1/r$.

Here and below, the upper dot denotes the derivative with respect to time.

We define the target state at time $t$ as
$\bm{x}=[\beta,\dot{\beta},\dot{\rho},s]$. If we assume that the ownship and
target do not manoeuvre, the vessel trajectory in MPC is:

$\displaystyle\beta^{+}$
$\displaystyle=\beta+\arctan{\frac{t_{\beta}}{1+t_{\rho}}},$ $\displaystyle
s^{+}$ $\displaystyle=\frac{s}{\sqrt{{(1+t_{\rho})}^{2}+{t_{\beta}}^{2}}},$
(1) $\displaystyle\dot{\beta}^{+}$
$\displaystyle=\frac{\dot{\beta}}{{(1+t_{\rho})}^{2}+{t_{\beta}}^{2}},$
$\displaystyle\dot{\rho}^{+}$ $\displaystyle=\frac{\dot{\rho}+\Delta
t(\dot{\rho}^{2}+\dot{\beta}^{2})}{{(1+t_{\rho})}^{2}+{t_{\beta}}^{2}}.$

Here $t^{+}$ is some later time, $\Delta t=t^{+}-t$,
$t_{\beta}=\dot{\beta}\Delta t$ and $t_{\rho}=\dot{\rho}\Delta t$. The update
equations in Eq. 1 are non-linear. To use these equations for target state
updates in a sequential estimator, we must apply the EKF, UKF, or other
advanced methods of non-linear estimation (e.g., [6]). By taking $\Delta t$ as
the update interval of the EKF or UKF, Eq. 1 would define the state transition
function for the filter.

The bearing transition equation $\beta^{+}$ in Eq. 1 does not contain the
initial value of inverse range $s$. Hence, we cannot determine the range from
BO measurements when the ownship and target move straight.

###  II-B Equations of Motion for Manoeuvring Platforms

Assume that at the initial time $t$ the ownship position and speed in a fixed
Cartesian coordinate system are $x_{\text{O}}$, $y_{\text{O}}$,
$v_{x\text{O}}$, and $v_{y\text{O}}$, and the corresponding position and speed
at a later time $t^{+}$ are $x^{+}_{\text{O}}$, $y^{+}_{\text{O}}$,
$v^{+}_{x\text{O}}$, and $v^{+}_{y\text{O}}$. Define the ownship displacement
vector and speed change vector as

$\displaystyle\bm{w}_{\text{O}}$
$\displaystyle=\left[x^{+}_{\text{O}}-x_{\text{O}}-v_{x\text{O}}\Delta
t,y^{+}_{\text{O}}-y_{\text{O}}-v_{y\text{O}}\Delta t\right],$ (2)
$\displaystyle\dot{\bm{w}}_{\text{O}}$
$\displaystyle=\left[v^{+}_{x\text{O}}-v_{x\text{O}},v^{+}_{y\text{O}}-v_{y\text{O}}\right].$

Similarly, define $\bm{w}_{\text{T}}$ and $\dot{\bm{w}}_{\text{T}}$ for the
target displacement. The overall acceleration-related displacement and speed
change are

$\displaystyle\bm{a}$ $\displaystyle=\bm{w}_{\text{T}}-\bm{w}_{\text{O}},$ (3)
$\displaystyle\dot{\bm{a}}$
$\displaystyle=\dot{\bm{w}}_{\text{T}}-\dot{\bm{w}}_{\text{O}}.$

Using these definitions, target movement in MPC in the presence of target and
ownship manoeuvres follows the equations

$\displaystyle\beta^{+}$
$\displaystyle=\beta+\arctan{\frac{D_{\beta}}{D_{\rho}}},$ $\displaystyle
s^{+}$
$\displaystyle=\frac{s}{{\left(D^{2}_{\beta}+D^{2}_{\rho}\right)}^{1/2}},$ (4)
$\displaystyle\dot{\beta}^{+}$
$\displaystyle=\frac{\dot{D}_{\beta}D_{\rho}-\dot{D}_{\rho}D_{\beta}}{D^{2}_{\beta}+D^{2}_{\rho}},$
$\displaystyle\dot{\rho}^{+}$
$\displaystyle=\frac{\dot{D}_{\rho}D_{\rho}+\dot{D}_{\beta}D_{\beta}}{D^{2}_{\beta}+D^{2}_{\rho}},$

where

$\displaystyle D_{\beta}$ $\displaystyle=t_{\beta}+s(\bm{a}\cdot\bm{n}),$
$\displaystyle\dot{D}_{\beta}$
$\displaystyle=\dot{\beta}+s(\dot{\bm{a}}\cdot\bm{n}),$ (5) $\displaystyle
D_{\rho}$ $\displaystyle=1+t_{\rho}+s(\bm{a}\cdot\bm{q}),$
$\displaystyle\dot{D}_{\rho}$
$\displaystyle=\dot{\rho}+s(\dot{\bm{a}}\cdot\bm{q}),$

and the unit vectors $\bm{q}$ and $\bm{n}$ point from the ownship to the
target and orthogonally at time $t$:

$\displaystyle\bm{q}$ $\displaystyle=\left[\sin{\beta},\cos{\beta}\right],$
(6) $\displaystyle\bm{n}$
$\displaystyle=\left[\cos{\beta},-\sin{\beta}\right].$

The bearing update equation 4 depends on the initial target range $s$. If we
know the ownship manoeuvres, we can estimate $s$, assuming that the target
moves straight. We often generalise this conclusion as “the ownship must
outmanoeuvre the target to determine its range” [2]. For the non-manoeuvring
target $\bm{w}_{\text{T}}=\dot{\bm{w}}_{\text{T}}=0$, so the subscript “O” can
be omitted in Eqs. 2 and 3 without causing ambiguity. From the Kalman Filter
viewpoint, the acceleration-related terms $(\bm{a}\cdot\bm{n})$,
$(\dot{\bm{a}}\cdot\bm{n})$, $(\bm{a}\cdot\bm{q})$, and
$(\dot{\bm{a}}\cdot\bm{q})$ now represent control inputs to the system.

###  II-C Modified Polar Coordinates Versus Log-Polar Coordinates

BO trackers commonly use bearing, bearing rate, and scaled range rate as state
variables because these values are directly observable on a straight leg of
the ownship [2]. The choice of the fourth state component is not fixed. One
conventional choice is MPC, in which we use inverse range $s$ as the fourth
coordinate, as in Sections II-A and II-B. An alternative is to select the
logarithm of range $\rho=\ln r$ as the fourth state component; this gives LPC
with the state vector $\bm{x}=[\beta,\dot{\beta},\dot{\rho},\rho]$.

The choice of LPC has several theoretical advantages over MPC. Firstly, it
allows the derivation of closed-form expressions for the posterior Cramér-Rao
bound (PCRB) in a BO tracking problem [4]. Compared to MPC, the LPC state
vector appears more intuitive because it contains two values (bearing and log-
range) and their time derivatives, while MPC mixes $s$ and $\dot{\rho}$. Also,
a Gaussian distribution of the fourth state vector component $\rho$ never
results in zero or negative range values, which theoretically is always
possible in MPC at the tails of the distribution. A corollary is that sigma
points of the UKF algorithm [7] would never end up at infeasible negative
ranges.

The transition from MPC to LPC is simple: we describe the target motion in LPC
by the system of differential equations

$\frac{d}{dt}\begin{bmatrix}\beta\\\ \dot{\beta}\\\ \dot{\rho}\\\
\rho\end{bmatrix}=\begin{bmatrix}\dot{\beta}\\\ -2\dot{\beta}\dot{\rho}\\\
\dot{\beta}^{2}-\dot{\rho}^{2}\\\ \dot{\rho}\end{bmatrix},$ (7)

obtained from Eqs. (11) and (18) of [2] by substituting $s=e^{-\rho}$. The
same substitution into Eqs. 1, 4 and 5 provides the solutions of Eq. 7 for
straight and manoeuvring vessels, respectively. For use in the UKF, one should
also adapt the range-related elements of the process noise matrix $Q$.

In accordance with Kalman Filter theory assumptions, we model the target state
vector in LPC, $\bm{x}=[\beta,\dot{\beta},\dot{\rho},\rho]$ as a Gaussian
random variable:

$\bm{x}\sim\mathcal{N}(\bm{x};\bm{\mu},\Sigma).$ (8)

With LPC in mind, we return to the equations of motion and make a key
simplification.

###  II-D Instant Manoeuvres

Assume that the ownship makes an elementary manoeuvre: an instant turn. The
state space variables after the manoeuvre,
$[\beta^{+},\dot{\beta}^{+},\dot{\rho}^{+},\rho^{+}]$, are related to the
corresponding values before the turn, $[\beta,\dot{\beta},\dot{\rho},\rho]$ by
Eqs. 4 and 5 with $\Delta t=0$. The target coordinates and speed do not
change, and neither do the ownship coordinates, as the turn is instant.
Therefore, the only non-zero acceleration-related term is
$\dot{\bm{a}}=-\dot{\bm{w}}=-\left[\Delta v_{x},\Delta v_{y}\right]$, and Eq.
4 simplifies to

$\displaystyle\beta^{+}$ $\displaystyle=\beta,$ $\displaystyle\rho^{+}$
$\displaystyle=\rho,$ (9) $\displaystyle\dot{\beta}^{+}$
$\displaystyle=\dot{\beta}-\frac{(\dot{\bm{w}}\cdot\bm{n})}{r},$
$\displaystyle\dot{\rho}^{+}$
$\displaystyle=\dot{\rho}-\frac{(\dot{\bm{w}}\cdot\bm{q})}{r}.$

The properties of instant manoeuvres can be used to model non-instant, gradual
manoeuvres by discretisation. Any ownship trajectory can be approximated as a
series of instant manoeuvres interwoven with straight leg motion. The total
cumulative duration of the straight legs should equal the duration of the
original complete manoeuvre. Using sufficiently short legs, we can achieve the
desired accuracy of approximation.

With all the preliminaries in place, we now begin to explore the state space
distribution from a statistical perspective.

##  III Closed-Form Expressions for the Post-Manoeuvre Distribution

###  III-A The Post-Manoeuvre Distribution

Gaussian approximations of the distribution of state space variables are
ubiquitous in filtering, for example, when using UKF [7] or ensemble methods
[8]. Assume the pre-manoeuvre distribution of the state space variables is
Gaussian. What about the post-manoeuvre distribution?

######  Proposition 1 (Post-manoeuvre distribution).

Let $f_{x}$ refer to the pre-manoeuvre probability distribution function for
the target in coordinates $\bm{x}$ (either MPC or LPC). Then, the post-
manoeuvre distribution is given by

$f_{x^{+}}(\bm{x}^{+})=f_{x}(\bm{h}^{-1}(\bm{x}^{+})),$

where

$\bm{h}^{-1}(\bm{x}^{+})=\begin{bmatrix}\beta^{+}\\\
\dot{\beta}^{+}+f_{1}(\beta^{+},\rho^{+})\\\
\dot{\rho}^{+}+f_{2}(\beta^{+},\rho^{+})\\\ \rho^{+}\end{bmatrix},$ (10)

and

$\displaystyle f_{1}(\beta,\rho)$ $\displaystyle=e^{-\rho}(\Delta
v_{x}\cos\beta-\Delta v_{y}\sin\beta),$ (11a) $\displaystyle
f_{2}(\beta,\rho)$ $\displaystyle=e^{-\rho}(\Delta v_{x}\sin\beta+\Delta
v_{y}\cos\beta).$ (11b)

If working in MPC, then the fourth coordinate of $\bm{x}^{+}$ is $s^{+}$ and
factors $e^{-\rho}$ in $f_{1}$ and $f_{2}$ are rewritten as $s$.

###### Proof.

Define the coordinate transformation by Eq. 9 from the pre-manoeuvre
distribution in $\bm{x}$ to the post-manoeuvre distribution in $\bm{x}^{+}$,
as the mapping $\bm{h}(\bm{x})$:

$\bm{h}(\bm{x})=\begin{bmatrix}\beta\\\ \dot{\beta}-f_{1}(\beta,\rho)\\\
\dot{\rho}-f_{2}(\beta,\rho)\\\ \rho\end{bmatrix}.$ (12)

By observation, $\bm{h}^{-1}$ defined in the proposition is the inverse of
$\bm{h}$. The multivariate change of variable theorem then expresses the post-
manoeuvre distribution:

$f_{x^{+}}(\bm{x}^{+})=f_{x}(\bm{h}^{-1}(\bm{x}^{+}))\lvert\det\mathcal{J}(\bm{x}^{+})\rvert,$
(13)

where $\mathcal{J}$ is the Jacobian matrix

$\mathcal{J}(\bm{x}^{+})=\begin{bmatrix}1&0&0&0\\\
-f_{2}(\beta,\rho)&1&0&-f_{1}(\beta,\rho)\\\
f_{1}(\beta,\rho)&0&1&-f_{2}(\beta,\rho)\\\ 0&0&0&1\\\ \end{bmatrix}.$

It is easy to check that $\mathcal{J}$ has determinant $1$, so Eq. 13 reduces
to the appropriate form. ∎

As the transformation by Eq. 12 does not alter the bearing $\beta^{+}$ or the
range coordinate ($\rho^{+}$ or $s^{+}$), we anticipate that the marginal
post-manoeuvre distribution of these two variables is unchanged from the pre-
manoeuvre distribution. Information about the target range comes from the
entanglement of $\rho^{+}$ or $s^{+}$ with $\dot{\beta}^{+}$ and
$\dot{\rho}^{+}$. We now lay out some special cases in which the post-
manoeuvre distribution can be found exactly.

######  Proposition 2.

Assume that the pre-manoeuvre distribution (in either LPC or MPC) is well
approximated by the Gaussian distribution
$\bm{x}\sim\mathcal{N}(\bm{\mu},\Sigma)$, so Proposition 1 provides

$f_{x^{+}}(\bm{x}^{+})\propto\text{exp}\left\\{-\frac{1}{2}\left[\bm{y}^{\text{T}}\Sigma^{-1}\bm{y}\right]\right\\},$
(14)

where $\bm{y}=h^{-1}(\bm{x}^{+})-\bm{\mu}$.

  1. (1)

If the coordinate system is MPC, $x_{4}=s$, then

    1. (i)

the post-manoeuvre distribution is non-Gaussian, but

    2. (ii)

the post-manoeuvre distribution is Gaussian if conditioned on the bearing rate
$\beta$. In this case, the post-manoeuvre distribution in MPC is

$\begin{pmatrix}\dot{\beta}\\\ \dot{\rho}\\\
s\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}\mu_{\dot{\beta}}-c_{1}\mu_{s}\\\
\mu_{\dot{\rho}}-c_{2}\mu_{s}\\\
\mu_{s}\end{pmatrix},\mathbf{P}\Sigma_{-\beta}\mathbf{P}^{\text{T}}\right).$

In this distribution, $c_{1}=\Delta v_{x}\cos\beta-\Delta v_{y}\sin\beta$ and
$c_{2}=\Delta v_{x}\sin\beta+\Delta v_{y}\cos\beta$ denote the constant
factors in Eq. 11, $\Sigma_{-\beta}$ means the covariance matrix $\Sigma$ with
the first row and column removed, and

$\mathbf{P}=\begin{pmatrix}1&0&-c_{1}\\\ 0&1&-c_{2}\\\ 0&0&1\end{pmatrix}.$

It is important to recall that $s$ is strictly positive and any significant
probability mass for $s<0$ should formally be treated as follows:
$s\rightarrow-s,\,\beta\rightarrow\beta\pm\pi$.

  2. (2)

If the coordinate system is LPC, $x_{4}=\rho$, then

    1. (i)

the post-manoeuvre distribution is non-Gaussian, but

    2. (ii)

the post-manoeuvre distribution is Gaussian if conditioned on $\beta$ and
$\rho$, and in this case

$\begin{pmatrix}\dot{\beta}\\\
\dot{\rho}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}\mu_{\dot{\beta}}+f_{1}(\beta,\,\rho)\\\
\mu_{\dot{\rho}}+f_{2}(\beta,\,\rho)\end{pmatrix},\Sigma_{-\beta\rho}\right),$

where $\Sigma_{-\beta\rho}$ refers to the $2\times 2$ covariance matrix formed
by omitting the first and last row and column of $\Sigma$.

The proof of Proposition 2 is in Appendix A-1. In order to understand the
Proposition clearly, let us consider an illustrative example. Consider the
situation of a nearby, poorly estimated target. In Cartesian coordinates, the
pre-manoeuvre distribution estimates that the target is at $(100,150)$ m from
the ownship, but with a standard deviation of $400$ m. Assume that $\beta$ is
fairly accurately known from data. Then Fig. 1 shows how unusual the MPC and
LPC distributions appear in practice. The LPC distribution, which is the topic
of this paper, has many samples with great deviation from the sample mean;
that is, the LPC distribution has large kurtosis.

Figure 1: Visualisation by samples of the post-manoeuvre distribution in MPC
and LPC. The colour of each sample indicates the estimated range of the target
in $s$ or $\rho$. As described below Proposition 2, the unusual situation in
play is one in which the actual distance to the target is less than the
_uncertainty_ in the target position. The MPC distribution consists of two
roughly Gaussian blobs, which are separated by $\pi$ in $\beta$ and represent
the probability mass on each side of the ownship. The LPC distribution has
clear skew and kurtosis; that is, there are rare, large deviations from the
sample mean. Notice in both plots that there is a clear correlation between
$\dot{\beta}$ and the range, shown by the colour banding; as explained in
Proposition 2, this correlation has been created by the manoeuvre.

Altering the initial position of the target to $(1000,1500)$ m, and preserving
all other parameters, we obtain roughly identical Gaussian distributions for
MPC and LPC in Fig. 2. Full details for both Figs. 2 and 1 are given in
Appendix A-2.

Figure 2: Visualisation by samples of the post-manoeuvre distribution in MPC
and LPC. The colour of each sample indicates the estimated range. As described
below Proposition 2, this target is located far from the ownship. The MPC and
LPC distribution are approximately identical.

Our goal in investigating the post-manoeuvre distribution is to assess the
viability of Kalman Filter approaches to target tracking. Such approaches work
by estimating the target mean and covariance. Our concern is that those
estimates can be inadequate or misleading if the underlying distribution has,
for example, significant kurtosis. Proposition 2 establishes some special
cases in which we do understand the distribution, but Figs. 1 and 2
demonstrate that, in the general case, the LPC post-manoeuvre distribution
both can, and can not, resemble a Gaussian distribution. In order to assuredly
use a Kalman Filter in LPC in the general case, we need to know whether a
Gaussian approximation is defensible. For this reason, the main body of the
paper derives the moments of the post-manoeuvre distribution. The skew and
kurtosis of the post-manoeuvre distribution allow monitoring whether a Kalman
Filter estimate can be trusted.

###  III-B Mean and Covariance of the Post-Manoeuvre Distribution

Define the pre-manoeuvre mean and covariance of the target state vector as
$\bm{\mu}_{x}$ and $\Sigma_{x}$; we seek the corresponding post-manoeuvre mean
and covariance, $\bm{\mu}^{+}_{x}$ and $\Sigma^{+}_{x}$.

From Eq. 12, observe that any post-manoeuvre state space variable $x_{k}^{+}$,
where $k\in\\{0,1,2,3\\}$, is defined in terms of its pre-manoeuvre value,
bearing, and range as

$x_{k}^{+}(x_{k})=x_{k}+a_{k}\frac{\sin\beta}{r}+b_{k}\frac{\cos\beta}{r},$
(15)

where $a_{0}=a_{3}=b_{0}=b_{3}=0$, $a_{1}=\Delta v_{y}$ and $b_{1}=-\Delta
v_{x}$, $a_{2}=-\Delta v_{x}$ and $b_{2}=-\Delta v_{y}$.

The following results present the exact mean and covariance for the post-
manoeuvre distribution.

######  Theorem 1 (Mean of the post-manoeuvre distribution).

The expected value of a post-manoeuvre variable $x_{k}^{+}$ defined in Eq. 15
is:

$\displaystyle\mathbb{E}\left[x_{k}^{+}\right]=\;$
$\displaystyle\mathbb{E}\left[x_{k}\right]+e^{-\mu_{\rho}-\frac{1}{2}(\sigma_{\beta\beta}-\sigma_{\rho\rho})}(a_{k}\sin(\mu_{\beta}-\sigma_{\beta\rho})$
$\displaystyle+b_{k}\cos(\mu_{\beta}-\sigma_{\beta\rho})).$ (16)

###### Proof.

The integrals for the mean values take the form

$\mathbb{E}\left[x_{k}^{+}\right]=\int
x_{k}^{+}(\bm{x})\mathcal{N}(\bm{x};\bm{\mu},\Sigma)d\bm{x}.$ (17)

For $k\in\\{0,3\\}$ (bearing and log-range), $x_{k}^{+}=x_{k}$, hence
$\mathbb{E}\left[x^{+}_{k}\right]=\mu_{k}$.

Now consider the mean for bearing rate, $k=1$. Substituting Eq. 9 for
$\dot{\beta}^{+}$ into Eq. 17, we get

$\displaystyle\mathbb{E}\left[\dot{\beta}^{+}\right]$
$\displaystyle=\int\dot{\beta}^{+}\mathcal{N}\\!\left(\bm{x};\bm{\mu},\Sigma\right)d\bm{x}$
$\displaystyle=\int\left(\dot{\beta}-\frac{\dot{\bm{w}}\cdot\bm{n}}{r}\right)\mathcal{N}\\!\left(\bm{x};\bm{\mu},\Sigma\right)d\bm{x}$
$\displaystyle=\mathbb{E}\left[\dot{\beta}\right]-\dot{\bm{w}}\cdot\mathbb{E}\left[\frac{\bm{n}}{r}\right].$
(18)

If we substitute the inverse range $r^{-1}$ with $e^{-\rho}$ and the sine and
cosine components of $\bm{n}$ with real and imaginary parts of $e^{i\beta}$,
the last term becomes a linear combination of the real and imaginary parts of
$\mathbb{E}\left[e^{-\rho+i\beta}\right]$.

Note how this last expected value is similar to the moment-generating function
(of the multivariate normal distribution), just for a complex-valued argument.
As the case of a complex argument appears unnamed in the literature, we use
the phrase ‘Generalised Gaussian MGF’.

######  Proposition 3 (The ‘Generalised Gaussian MGF’ (GGMGF)).

Let $\bm{h}\in\mathbb{C}^{n}$ be fixed and $\bm{x}\in\mathbb{R}^{n}$ be a
Gaussian random variable. Then

$\mathbb{E}\left[e^{\bm{h}^{\text{T}}\bm{x}}\right]=e^{\bm{\mu}^{\text{T}}\bm{h}+\frac{1}{2}\bm{h}^{\text{T}}\Sigma\bm{h}}\text{.}$
(19)

This generalises the Gaussian moment-generating function (MGF) to complex
arguments. The proof is described in §1.2 of [9] and §2 of [10].

######  Corollary 1 (GGMGF for specific $\bm{h}$).

For any $\bm{h}=[in,0,0,-m]$, where $m,n\in\mathbb{N}\cup\\{0\\}$, the GGMGF
has the form

$\\!\\!\mathbb{E}\\!\left[r^{-m}e^{in\beta}\right]=e^{-m\mu_{\rho}-\frac{1}{2}\left(n^{2}\sigma_{\beta\beta}-m^{2}\sigma_{\rho\rho}\right)+i(n\mu_{\beta}-mn\sigma_{\beta\rho})}.$
(20)

The real and imaginary parts of Eq. 20 give
$\mathbb{E}\left[r^{-m}\cos\\!{(n\beta)}\right]$ and
$\mathbb{E}\left[r^{-m}\sin\\!{(n\beta)}\right]$, respectively.

Now, continuing the proof, to obtain the expected value of $e^{-\rho+i\beta}$,
we use Corollary 1 with $\bm{h}=[i,0,0,-1]$ to find

$\\!\mathbb{E}\\!\left[e^{-\rho+i\beta}\right]=\exp{\\!\big{(}\\!-\mu_{\rho}-\frac{\sigma_{\beta\beta}-\sigma_{\rho\rho}}{2}+i(\mu_{\beta}-\sigma_{\beta\rho})\big{)}}.$
(21)

The respective real and imaginary parts are
$\mathbb{E}\left[r^{-1}\cos\beta\right]$ and
$\mathbb{E}\left[r^{-1}\sin\beta\right]$. Substitution of these expressions
into Section III-B produces the mean of the post-manoeuvre bearing rate:

$\mathbb{E}\\!\left[\dot{\beta}^{+}\right]=\mathbb{E}\\!\left[\dot{\beta}\right]-\left(\Delta
v_{x}\cos(\mu_{\beta}-\sigma_{\beta\rho})\right.-\\\ \left.\Delta
v_{y}\sin(\mu_{\beta}-\sigma_{\beta\rho})\right)e^{-\mu_{\rho}-\frac{1}{2}(\sigma_{\beta\beta}-\sigma_{\rho\rho})}.$
(22)

We can further simplify Eq. 22 by describing the ownship change in velocity
through the turning angle $\alpha$. Then, $\Delta v_{x}=\Delta v\cos\alpha$
and $\Delta v_{y}=\Delta v\sin\alpha$, so that

$\\!\\!\mathbb{E}\\!\left[\dot{\beta}^{+}\right]\\!=\mathbb{E}\\!\left[\dot{\beta}\right]-\Delta
ve^{-\mu_{\rho}-\frac{\sigma_{\beta\beta}-\sigma_{\rho\rho}}{2}}\cos{(\alpha+\mu_{\beta}-\sigma_{\beta\rho})}.$
(23)

Finally, consider scaled range rate, $k=2$. Using Eq. 9 for $\dot{\rho}^{+}$
in Eq. 17, and again employing Proposition 3, we obtain

$\displaystyle\mathbb{E}\\!\left[\dot{\rho}^{+}\right]$
$\displaystyle=\int\dot{\rho}^{+}\mathcal{N}\\!\left(\bm{x};\bm{\mu},\Sigma\right)d\bm{x}$
$\displaystyle=\int\left(\dot{\rho}-s(\dot{\bm{w}}\cdot\bm{q})\right)\mathcal{N}\\!\left(\bm{x};\bm{\mu},\Sigma\right)d\bm{x}$
$\displaystyle=\mathbb{E}\left[\dot{\rho}\right]-\dot{\bm{w}}\cdot\mathbb{E}\left[e^{-\rho}\bm{q}\right]$
$\displaystyle=\mathbb{E}\left[\dot{\rho}\right]-\left(\Delta
v_{x}\sin(\mu_{\beta}-\sigma_{\beta\rho})\right.$
$\displaystyle\quad\quad\left.+\Delta
v_{y}\cos(\mu_{\beta}-\sigma_{\beta\rho})\right)e^{-\mu_{\rho}-\frac{1}{2}(\sigma_{\beta\beta}-\sigma_{\rho\rho})}$
(24) $\displaystyle=\mathbb{E}\left[\dot{\rho}\right]-\Delta
ve^{-\mu_{\rho}-\frac{1}{2}(\sigma_{\beta\beta}-\sigma_{\rho\rho})}\sin{(\alpha+\mu_{\beta}-\sigma_{\beta\rho})}.$

Equations 22 and 24 match Theorem 1 with the coefficients $a_{k}$ and $b_{k}$
from Eq. 15, thus completing the proof. ∎

Now derive the second raw moments for the post-manoeuvre distribution. By
combining these second raw moments with the first raw moments 1, we will also
obtain the post-manoeuvre covariance matrix $\Sigma^{+}$.

######  Theorem 2 (Second raw moment of the post-manoeuvre distribution).

For any two post-manoeuvre state space variables $x_{j}^{+}$ and $x_{k}^{+}$,
their second raw moment is:

$\displaystyle\mathbb{E}\\!\left[x_{j}^{+}x_{k}^{+}\right]=\;$
$\displaystyle\mathbb{E}\left[x_{j}x_{k}\right]+a_{k}\mathbb{E}\\!\left[x_{j}\frac{\sin{\beta}}{r}\right]+b_{k}\mathbb{E}\\!\left[x_{j}\frac{\cos{\beta}}{r}\right]$
$\displaystyle+a_{j}\mathbb{E}\\!\left[x_{k}\frac{\sin{\beta}}{r}\right]+b_{j}\mathbb{E}\\!\left[x_{j}\frac{\cos{\beta}}{r}\right]$
$\displaystyle+\frac{a_{j}a_{k}+b_{j}b_{k}}{2}\mathbb{E}\\!\left[\frac{1}{r^{2}}\right]$
$\displaystyle+\frac{b_{j}b_{k}-a_{j}a_{k}}{2}\mathbb{E}\\!\left[\frac{\cos{\\!(2\beta)}}{r^{2}}\right]$
$\displaystyle+\frac{a_{j}b_{k}+a_{k}b_{j}}{2}\mathbb{E}\\!\left[\frac{\sin{\\!(2\beta)}}{r^{2}}\right].$
(25)

All the expected values in this expression can be obtained in closed form
using the GGMGF 19.

######  Corollary 2 ($k=j$ case of second raw moment).

If $k=j$ for the two state space variables in Theorem 2, then

$\displaystyle\mathbb{E}\\!\left[\left(x_{j}^{+}\right)^{2}\right]=\;$
$\displaystyle\mathbb{E}\\!\left[x_{j}^{2}\right]+2a_{j}\mathbb{E}\\!\left[x_{j}\frac{\sin{\beta}}{r}\right]+2b_{j}\mathbb{E}\\!\left[x_{j}\frac{\cos{\beta}}{r}\right]$
$\displaystyle+\frac{a_{j}^{2}+b_{j}^{2}}{2}\mathbb{E}\\!\left[\frac{1}{r^{2}}\right]+\frac{b_{j}^{2}-a_{j}^{2}}{2}\mathbb{E}\\!\left[\frac{\cos{\\!(2\beta)}}{r^{2}}\right]$
$\displaystyle+a_{j}b_{j}\mathbb{E}\\!\left[\frac{\sin{\\!(2\beta)}}{r^{2}}\right].$
(26)

Before presenting the proof of Theorem 2, we need one more auxiliary result.
The theorem proof follows.

######  Lemma 1 (First derivative of GGMGF for specific $\bm{h}$).

For any $\bm{h}=[in,0,0,-m]$, where $m,n\in\mathbb{N}\cup\\{0\\}$, the first
derivative of the GGMGF with respect to $x_{j}$ has the form

$\mathbb{E}\left[x_{j}r^{-m}e^{in\beta}\right]=(\mu_{j}-m\sigma_{\rho
j}+in\sigma_{\beta j})\mathbb{E}\left[r^{-m}e^{in\beta}\right].$ (27)

###### Proof.

Consider the partial derivative of the GGMGF with respect to $h_{j}$. Swapping
the order of expectation and differentiation, we have

$\displaystyle\mathbb{E}\left[x_{j}e^{\bm{h}^{\text{T}}\bm{x}}\right]$
$\displaystyle=\frac{\partial}{\partial
h_{j}}\left(\mathbb{E}\left[e^{\bm{h}^{\text{T}}\bm{x}}\right]\right)$
$\displaystyle=\frac{\partial}{\partial
h_{j}}e^{\bm{\mu}^{\text{T}}\bm{h}+\frac{1}{2}\bm{h}^{\text{T}}\Sigma\bm{h}}$
$\displaystyle=\left(\mu_{j}+\Sigma_{j}\bm{h}\right)e^{\bm{\mu}^{\text{T}}\bm{h}+\frac{1}{2}\bm{h}^{\text{T}}\Sigma\bm{h}},$

where $\Sigma_{j}$ denotes the $j^{\text{th}}$ row of the matrix $\Sigma$.
Without loss of generality, substitute $\bm{h}=[in,0,0,-m]$ into this
expression to obtain Eq. 27. ∎

###### Proof of Theorem 2.

In the integral form, the second-order raw moments of the post-manoeuvre
distribution are

$\mathbb{E}\\!\left[x_{j}^{+}x_{k}^{+}\right]=\int
x_{j}^{+}x_{k}^{+}\mathcal{N}\\!\left(\bm{x};\bm{\mu},\Sigma\right)d\bm{x}.$
(28)

Substituting Eq. 15 for $x_{j}^{+}$, $x_{k}^{+}$, expanding the product, then
interchanging the order of integration and expectation, and grouping the terms
with the same expected values we obtain the decomposition 2. All the expected
values in Theorem 2 are given in closed form by the real and imaginary parts
of Eqs. 20 and 27 for the respective $m$ and $n$ indices. ∎

Corollary 2 immediately follows from Theorem 2 for $j=k$. Therefore, we have
derived closed-form expressions for all the post-manoeuvre first and second
(raw and central) moments.

###  III-C Higher-Order Moments of the Post-Manoeuvre Distribution

In Theorems 1 and 2, we see that the first two moments of the post-manoeuvre
distribution are linear combinations of pre-manoeuvre moments. We generalise
this for the arbitrary $N^{\text{th}}$ raw moment of the post-manoeuvre
distribution.

######  Proposition 4 ($N^{\text{th}}$ raw moment of the post-manoeuvre
distribution).

For post-manoeuvre state space variables $x_{k}^{+}$ each raised to powers
$q_{k}$, $k=0,\dots,3$, such that $\sum_{j=0}^{3}q_{k}=N$, the $N^{\text{th}}$
raw moment is

$\displaystyle\\!\\!\\!\mathbb{E}\\!\left[\prod_{k=0}^{3}(x_{k}^{+})^{q_{k}}\right]\\!=\mathbb{E}\\!\left[\prod_{k=0}^{3}\\!\left(x_{k}+a_{k}\frac{\sin\beta}{r}+b_{k}\frac{\cos\beta}{r}\\!\right)^{q_{k}}\right]\\!\\!,$
(29)

which, when expanded, yields a linear combination of

$\displaystyle E_{p_{0},p_{1},p_{2},p_{3},m,n}^{(s)}$
$\displaystyle=\mathbb{E}\left[\prod_{k=0}^{3}x_{k}^{p_{k}}r^{-m}\sin(n\beta)\right]\text{,
and}$ $\displaystyle E_{p_{0},p_{1},p_{2},p_{3},m,n}^{(c)}$
$\displaystyle=\mathbb{E}\left[\prod_{k=0}^{3}x_{k}^{p_{k}}r^{-m}\cos(n\beta)\right],$
(30)

where $\sum_{k=0}^{3}p_{k}+m=N$, for $0\leq n\leq m$.

###### Proof.

Consider the product

$\prod_{k=0}^{3}\left(x_{k}+a_{k}\frac{\sin\beta}{r}+b_{k}\frac{\cos\beta}{r}\right)^{q_{k}}$

from the right-hand side of Eq. 29. Expansion via the binomial theorem yields
a linear combination of terms

$\prod_{k=0}^{3}x_{k}^{p_{k}}r^{-(c+d)}\sin^{c}{\beta}\cos^{d}{\beta},$ (31)

where $\sum_{k=0}^{3}p_{k}+c+d=N$.

Using De Moivre’s formula, Euler’s formula, and the binomial theorem, we
convert the powers $\sin^{c}{\beta}$ and $\cos^{d}{\beta}$ to sums of
$\sin(c^{*}\beta)$ and $\cos(d^{*}\beta)$, where $c^{*}=0,\dots,c$ and
$d^{*}=0,\dots,d$. Expanding again the product of these sums, and using the
trigonometric addition identities, each element 31 becomes a linear
combination of terms

$\prod_{k=0}^{3}x_{k}^{p_{k}}r^{-m}\sin\\!{(n\beta)},\quad\prod_{k=0}^{3}x_{k}^{p_{k}}r^{-m}\cos\\!{(n\beta)},$
(32)

where $m=c+d$, $\sum_{k=0}^{3}p_{k}+m=N$, and $0\leq n\leq m$.

Therefore, using the linearity of expectation, the right-hand side of Eq. 29
is a linear combination of $E_{p_{0},p_{1},p_{2},p_{3},m,n}^{(s)}$ and
$E_{p_{0},p_{1},p_{2},p_{3},m,n}^{(c)}$ defined in Proposition 4, which are
the expected values of expressions 32. ∎

The expected values in Proposition 4 can be found in closed form by recursive
application of the following Lemma.

######  Lemma 2 ($N^{\text{th}}$ partial derivative of the GMGF).

The $N^{\text{th}}$ partial derivative of the GMGF, conditioned by
$\sum_{k=0}^{3}p_{k}+m=N$, $p_{j}\geq 0$, and $m\geq 0$, is

$\displaystyle\mathbb{E}\left[e^{\bm{h}^{\text{T}}\bm{x}}\prod_{k=0}^{3}x_{k}^{p_{k}}\right]=$
$\displaystyle\frac{\partial^{N}\Phi_{\bm{x}}(\bm{h})}{\partial
h_{j}^{p_{j}}\partial h_{k}^{p_{k}}\dots}$ $\displaystyle=$
$\displaystyle\sum_{j\neq
k}p_{j}\sigma_{jk}\frac{\partial^{N-2}\Phi_{\bm{x}}(\bm{h})}{\partial
h_{j}^{p_{j}-1}\partial h_{k}^{p_{k}-1}\dots}$ (33a)
$\displaystyle+(p_{k}-1)\sigma_{kk}\frac{\partial^{N-2}\Phi_{\bm{x}}(\bm{h})}{\partial
h_{j}^{p_{j}}\partial h_{k}^{p_{k}-2}\dots}$ (33b)
$\displaystyle+\left(\mu_{k}+\Sigma_{k}\bm{h}\right)\frac{\partial^{N-1}\Phi_{\bm{x}}(\bm{h})}{\partial
h_{j}^{p_{j}}\partial h_{k}^{p_{k}-1}\dots},$ (33c)

where $k$ is selected such that $p_{k}>1$, which is always possible for $N\geq
1$. The index $k$ is not specific, any index can be selected due to the
symmetry of derivatives.

For the sake of brevity, the notation in Eq. 33 technically allows negative
derivatives of order $-1$ if $p_{j}=0$ for some elements in Eq. 33a, or if
$p_{k}=1$ in Eq. 33b. However, such terms are eliminated by the corresponding
$p_{j}=0$ and $p_{k}-1$ multipliers in front.

###### Proof.

For $N=1$, we re-write Eq. 27 as

$\frac{\partial}{\partial
h_{k}}\Phi_{\bm{x}}(\bm{h})=\left(\mu_{k}+\Sigma_{k}\bm{h}\right)\Phi_{\bm{x}}(\bm{h}).$
(34)

For $N=2$, we take the second partial derivative to obtain

$\displaystyle\frac{\partial^{2}}{\partial h_{k}\partial
h_{j}}\Phi_{\bm{x}}(\bm{h})$ $\displaystyle=\frac{\partial}{\partial
h_{k}}\left(\mu_{j}+\Sigma_{j}\bm{h}\right)\Phi_{\bm{x}}(\bm{h})$
$\displaystyle=\bigl{(}\sigma_{jk}+\left(\mu_{j}+\Sigma_{j}\bm{h}\right)\left(\mu_{k}+\Sigma_{k}\bm{h}\right)\bigr{)}\Phi_{\bm{x}}(\bm{h})$
$\displaystyle=\leavevmode\resizebox{258.36667pt}{}{$\sigma_{jk}\Phi_{\bm{x}}(\bm{h})+\left(\mu_{k}+\Sigma_{k}\bm{h}\right)\frac{\partial}{\partial
h_{j}}\Phi_{\bm{x}}(\bm{h})$}.$ (35)

These expressions start the recursion of Eq. 33: for $j\neq k$, the first term
of Section III-C yields Eq. 33a, and for $j=k$, it yields Eq. 33b. Eq. 34 and
the last term of Section III-C produce Eq. 33c for the respective $N=1,2$.

The proof for a general order $N$ is obtained through direct differentiation
of Eq. 33 by $h_{m}$ for some arbitrary $m$: the partial derivatives in Eqs.
33a to 33c increase the order of their $h_{m}$-th derivatives by one. Then the
derivative of the $\mu_{k}+\Sigma_{k}\bm{h}$ term in Eq. 33c yields one more
summand with a $\sigma_{km}$ multiplier. If $k\neq m$, it is combined with the
corresponding $j=m$ term of Eq. 33a to increase its $p_{m}$ multiplier in
front to $p^{\prime}_{m}=p_{m}+1$. Alternatively, for $k=m$ the new summand is
combined with Eq. 33b to increase the $p_{k}-1=p_{m}-1$ multiplier to
$p_{m}=p^{\prime}_{m}-1$. In both cases, we obtain Eq. 33 with substitutions
$N\Rightarrow N^{\prime}=N+1$ and $p_{m}\Rightarrow p^{\prime}_{m}=p_{m}+1$,
which is the next order of recursion. ∎

Substituting $\bm{h}=[in,0,0,-m]$ into Eq. 33 recursively yields the expected
values from Proposition 4.

By Proposition 4, any $N^{\text{th}}$ order raw moment is

$\displaystyle\mathbb{E}\left[\prod_{k=0}^{3}(x_{k}^{+})^{q_{k}}\right]$
$\displaystyle=\sum_{\\{q,m\\}}\sum_{n=0}^{m}\left(S_{p_{0},\dots,p_{3},m,n}^{(p_{0},\dots,p_{3})}E_{p_{0},\dots,p_{3},m,n}^{(s)}\right.$
$\displaystyle\left.+C_{p_{0},\dots,p_{3},m,n}^{(p_{0},\dots,p_{3})}E_{p_{0},\dots,p_{3},m,n}^{(c)}\right).$
(36)

The first sum considers all possible combinations of $p_{k}$, $k=0,\dots,3$
and $m$ subject to the conditions $\sum_{k=0}^{3}p_{k}+m=N$, $p_{k}\geq 0$,
$m\geq 0$. The coefficients $S_{p_{0},\dots,p_{3},m,n}^{(p_{0},\dots,p_{3})}$
and $C_{p_{0},\dots,p_{3},m,n}^{(p_{0},\dots,p_{3})}$ are products of $a_{k}$
and $b_{k}$ and can be found from Eqs. 29 and 15.

We demonstrated that any higher-order moment of the post-manoeuvre
distribution can be found in closed form by recursive application of Eqs. 33
and III-C. However, as the order of the moment increases, so does the
complexity of its closed-form expression. Explicit calculation of the
coefficients $S_{p_{0},\dots,p_{3},m,n}^{(p_{0}^{+},\dots,p_{3}^{+})}$ and
$C_{p_{0},\dots,p_{3},m,n}^{(p_{0}^{+},\dots,p_{3}^{+})}$ quickly becomes
prohibitive. Combined with the non-Gaussianity proven in Section III-A, such
that we have no existing statistical results to draw on, we turn to computer
algebra for deriving specific cases of Section III-C. We have developed a
package in SageMath [11] to generate these moments and export them as LaTeX or
Python code. A demonstration of this package is shown in Appendix B.

The first two moments are explicitly given by Theorems 1 and 2. In the next
section, we apply these two moments in a modified UKF estimator, while using
the higher-order moments to monitor its performance.

##  IV Applications of the Closed-Form Expressions in Tracking Simulation

###  IV-A UKF using Closed-Form Expressions for Mean and Covariance Prediction

The Kalman Filter provides a minimum square error estimate for the mean and
covariance of the target location [12]. For our non-linear dynamical system,
the UKF is appropriate. It uses the UT to propagate deterministic sigma points
that accurately capture the target state mean and covariance [7].

In order to test our closed-form expressions in tracking simulation, we will
implement a pure UKF and a modified UKF that incorporates our closed-form
expressions. We assume the ownship alternates between straight legs and
instant turns. The modified UKF employs the first two moments of the post-
manoeuvre distribution to calculate the target state and covariance after each
ownship turn.

We define the ownship manoeuvre at time step $i$ as a tuple of three values:

  * •

$\Delta\bm{r}_{i}$, the accelerated displacement during the step illustrated
in Fig. 3.

  * •

$\Delta\bm{v}_{i}$, the speed change during the step; the speed changes
instantly at the point of the turn, at unknown time within the step.

  * •

$\Delta t_{i}$, the time duration of the step.

As the turn may happen at a time instance within the step, we seek to split
the manoeuvre into elementary movements, or sub-manoeuvres: a straight leg, an
instant turn, and another straight leg after the turn. The duration of the two
straight legs adds up to the total duration of the time step $\Delta t_{i}$.
Computation of these sub-manoeuvres is described in Theorem 3.

$\bm{r}_{1}$$\bm{r}_{1}^{\prime}$$\bm{r}_{3}$$\bm{v}_{1}$$\Delta t_{1}$$\Delta
t_{3}$$\Delta t_{3}$$\bm{v}_{3}$ Figure 3: A diagram of the predicted ownship
position, $\bm{r}_{1}^{\prime}$, versus the true position, $\bm{r}_{3}$, where
the black node indicates the point of the instant turn. Note the jump from
index $1$ to $3$ as the second sub-manoeuvre, the instant turn, has zero
duration, zero accelerated displacement (hence, zero length on the plot), but
non-zero change in velocity: $\Delta\bm{v}=\bm{v}_{3}-\bm{v}_{1}$. The
accelerated displacement is $\Delta\bm{r}=\bm{r}_{3}-\bm{r}_{1}^{\prime}$.

######  Theorem 3 (Manoeuvre splitting).

The manoeuvre containing a turn $M(\Delta\bm{r},\Delta\bm{v},\Delta t)$ is
equivalently described as a list of three sub-manoeuvres:

  1. 1.

$M_{1}(0,0,\Delta t_{1})$,

  2. 2.

$M_{2}(0,\Delta\bm{v},0)$, and

  3. 3.

$M_{3}(0,0,\Delta t_{3})$

where

$\Delta t_{3}=\frac{\Delta\bm{r}\cdot\Delta\bm{v}}{|\Delta\bm{v}|^{2}}$ (37)

and $\Delta t_{1}=\Delta t-\Delta t_{3}$.

###### Proof.

We write the final position in terms of the initial position and two
velocities.

$\bm{r}_{3}=\bm{r}_{1}+\Delta t_{1}\bm{v}_{1}+\Delta t_{3}\bm{v}_{3}.$

Substituting $\Delta t_{3}=\Delta t-\Delta t_{1}$ and regrouping, we get

$\bm{r}_{3}-(\bm{r}_{1}+\Delta t\bm{v}_{1})=\Delta
t_{3}(\bm{v}_{3}-\bm{v}_{1}).$

As illustrated in Fig. 3, the left-hand side is the accelerated displacement
$\Delta\bm{r}$; the term in brackets in the right-hand side is the speed
change $\Delta\bm{v}$. Therefore,

$\Delta\bm{r}=\Delta t_{3}\Delta\bm{v}.$

By assumption of a non-zero turn, $|\Delta\bm{v}|\neq 0$. Thus, multiplying
both sides by $\Delta\bm{v}$ we obtain Eq. 37. ∎

In cases where the turn is at the start or at the end of the total manoeuvre,
without loss of generality, we split the total manoeuvre into two: a turn and
a straight leg.

Having established the approach for isolating instant turns, we present the
modified filter, CFE-UKF, as Algorithm 1.

Algorithm 1 The modified UKF algorithm, CFE-UKF, that uses the closed-form
expressions 1 and 2 to update the mean and covariance over instant turns.

1:function CFE-UKF

2: Initialise at time $t_{0}$, iteration $0$

3: $N_{T}$ is the total count of time steps

4: for $i\in[1,N_{T}]$ do

5: Advance platforms forward by time step $\Delta t_{i}$

6: Compute the manoeuvre $M_{i}(\Delta\bm{r}_{i},\Delta\bm{v}_{i},\Delta
t_{i})$

7: if $|\Delta\bm{v}_{i}|<\epsilon$ then $\triangleright$ no turn

8: Time update over $\Delta t_{i}$ using UT

9: else

10: Split $M_{i}$ into sub-manoeuvres $[M_{j}]$

11: for all $M_{j}\in[M_{j}]\ \textbf{do}$

12: if $\Delta\bm{v}_{j}\neq\bm{0}$ then $\triangleright$ ownship turned

13: State update by Theorems 1 and 2

14: else$\triangleright$ straight leg

15: Time update over $\Delta t_{j}$ using UT

16: end if

17: end for

18: end if

19: Measurement update of the UKF as usual

20: end for

21:end function

Figure 4: Motion of the ownship and target in the simulated scenario. In
Cartesian coordinates, the ownship velocity along straight legs is $[\pm
10,10]$ m/s and the initial target velocity is $[0,12.5]$ m/s. Each straight
leg is 2 hours long. The short cyan segment shows the stretch of time used for
mean range error computations in Section IV-B.

To compare the CFE-UKF with a pure UKF, we use a scenario presented in Fig. 4.
The ownship moves deterministically along straight lines, and makes exact
turns, while the target movement is subject to the process noise provided by
the continuous-time white noise model with nearly constant velocity [13]. The
time increment $\Delta t$ is 10 seconds.

For the purposes of verification, we first run the CFE-UKF and UKF with
artificially precise parameters. The process noise intensity is set to
$10^{-8}$ and the sensor standard deviation of bearing measurements to 0.001°.
We set the initial track range to $150$ km (the true range from ownship to
target) and initialise the range variance to $10$% of the true range. In this
idealised case, we find both trackers run very similarly: the averaged
absolute difference between their means is only $[1.3\cdot 10^{-9},1.8\cdot
10^{-11},3.9\cdot 10^{-10},1.8\cdot 10^{-5}]$.

The CFE-UKF is slightly less efficient than a pure UKF; for each turning
point, we perform up to two UTs, depending on whether there is a preceding and
succeeding straight leg, and then additionally compute the closed-form
expressions to update the mean and covariance over the turn. However, due to
our small state space dimension $n_{x}=4$, and the small proportion of ownship
turns relative to the total scenario, this increase in complexity has minimal
effect on total run-time.

###  IV-B Monitoring Gaussianity Using the Third and Fourth Central Moments of
the Post-Manoeuvre Distribution

In Section III-A we have shown that the post-manoeuvre distribution is not
Gaussian. Now we can quantify this statement. When the ownship turns, we know
the higher-order moments of the post-manoeuvre state distribution, and can use
them to monitor how close is this distribution to Gaussian. The turning points
are critical for passive BO tracking because only there the range becomes
observable, and therefore, it is vital for the estimator assumptions, e.g.
Gaussianity, to be satisfied.

We consider two metrics for use as a monitoring tool. The first it is the
Frobenius norm of the third central moment tensor

$M_{3}=\big{\lVert}\,\mathbb{E}\left[(\bm{x}^{+}-\bm{\mu}^{+})^{3}\right]\\!\big{\rVert}.$
(38)

For a Gaussian distribution $M_{3}\equiv 0$, hence a non-zero value is a
measure of non-Gaussianity.

The second metric is the norm of the deviation of the post-manoeuvre fourth
central moment, a $4^{\text{th}}$-order tensor, from the expected fourth
central moment of a Gaussian distribution:

$M_{4}=\big{\lVert}\,\mathbb{E}\\!\left[(\bm{x}^{+}-\bm{\mu}^{+})^{4}\right]-\mathbb{E}\\!\left[(\bm{y}-\bm{\mu}^{+})^{4}\right]\\!\big{\rVert},$
(39)

where $y\sim\mathcal{N}(\bm{\mu}^{+},\Sigma^{+})$. Again, $M_{4}\equiv 0$ for
a Gaussian distribution; a non-zero value quantifies non-Gaussianity.

Figure 5 shows the time series of the proposed metrics for the scenario of
Fig. 4. To make the simulations somewhat less idealised, from now on the
process noise intensity is increased to $10^{-6}$ and the bearing measurement
error to 0.1°. Clearly, the distribution is not Gaussian when the ownship
turns. On the positive side, we observe that with every subsequent turn the
metrics decrease, implying that the distribution becomes more Gaussian with
time as the estimators converge. Consequently, it may imply that the observed
non-Gaussianity is related to the estimator initialisation, and that is what
we examine next.

Figure 5: Time series of the two non-Gaussianity metrics. The three peaks
correspond to the three ownship turns (bars have been offset for visibility).

###  IV-C Effect of Range Error Initialisation on the Post-Manoeuvre
Distribution

To start the estimator, we need to provide the initial state $\bm{x}_{0}$ and
covariance $\Sigma_{0}$. In the examples presented in this paper, we set
$\bm{x}_{0}$ using the first meaured bearing for $\beta_{0}$, zeros for the
derivatives $\dot{\beta}_{0}$ and $\dot{\rho}_{0}$, and the true value for the
initial range $r_{0}$. The covariance $\Sigma_{0}$ is set to a diagonal matrix
of

  * •

the sensor measurement variance for bearing,

  * •

${(v_{\text{max}}/r_{0})}^{2}$ for the bearing and scaled range rates, where
$v_{\text{max}}$ is some “large” speed value, and

  * •

$\sigma_{rr}=\log{(1+{(\delta r)}^{2})}$ for log-range, were we introduce
$\delta r$ as the initial relative range error.

We express the log-range error in terms of $\delta r$, because in the limit of
small errors, the range variance of ${(r_{0}\delta r)}^{2}$ indeed corresponds
to the log-range variance of $\sigma_{rr}$. For larger errors the relationship
breaks down, but $\delta r$ still provides a convenient description of the
error in human terms, as opposed to the less intuitive variance of the range
logarithm.

To examine the impact of initialisation on the tracking accuracy and on the
properties of the post-manoeuvre distribution, we consider the estimator
output after the first manoeuvre of our scenario, where the non-Gaussianity is
most pronounced. As a measure of accuracy, we use the mean estimated range
error (MRE) calculated over ten minutes of the scenario at the end of the
second leg (highlighted in Fig. 4 with cyan).

Figure 6: Effect of the initial range variance on MRE (a, top), third central
moment (b, middle), and fourth central moment difference (c, bottom). The
dashed and dotted lines respectively show the thresholds to achieve the MRE of
$5\%$ and $10\%$ of the initial range.

The dependence of the MRE on tracker initialisation is presented in Fig. 6.
The corresponding two metrics of target state Gaussianity after the first
ownship turn are shown in Figs. 6 and 6. As expected, estimation error grows
with initialisation ambiguity. Correspondingly, the post-manoeuvre state
distribution becomes less Gaussian, thus violating the UKF assumptions and
reducing estimator accuracy.

A corollary of the presented result is that monitoring non-Gaussianity enables
a measure of control of the estimator performance. This is the motivation for
the CFE-UKF. While it may not perform that differently from the pure UKF, the
new estimator provides the user with a capability to monitor whether the
underlying assumptions of the Gaussian estimator are being met.

Whenever the non-Gaussianity metrics exceed certain thresholds222 The specific
thresholds for $M_{3}$ and $M_{4}$ metrics can be established by Monte-Carlo
modelling, but are beyond the scope of this paper., the user would know that
the target range estimate after the manoeuvre cannot be trusted, even if the
estimated range variance is low. In such a case, the ownship may, for example,
attempt the next manoeuvre to improve the solution. Figure 5 shows how
Gaussianity of the post-manoeuvre distribution improves on each turn, thus
validating the estimator convergence. Alternatively, the same Gaussianity
metrics can guide the choice of the initial range variance $\delta r$.

The focus of this section has been the use of our results in a Kalman Filter
framework. One may consider more powerful filtering algorithms that do not
assume Gaussianity of the target state; these are discussed in the next
section.

##  V Conclusions and Future Work

The paper extends the theoretical understanding of object tracking in LPC. In
order to make the theory tractable, we assume an instantaneous manoeuvre and a
target moving in a straight line. Under these simplifications, we obtain some
strong results. The main contribution of the paper is Proposition 4: we derive
all moments of the post-manoeuvre distribution in closed form. Additionally,
Proposition 2 derives, for both LPC and MPC, special cases in which the post-
manoeuvre distribution is Gaussian, with known parameters. Our results, while
formally obtained for instantaneous manoeuvres, may be applied in the general,
non-instantaneous case through discretisation.

As the contributions of the paper are primarily theoretical, let us now
discuss possible applications of the theory. The most common class of tracking
algorithms, consisting of extensions of the Kalman Filter for non-linear
system dynamics, work best when the state distribution of the target is
approximately unimodal and without large outliers. Our contribution allows a
user to track the third and fourth moments of the LPC distribution, and
thereby assess when these conditions are met.

We demonstrate the utility of the obtained results in filtering algorithms.
For state and covariance prediction, we define a new algorithm, the CFE-UKF,
that uses the closed-form expressions for mean and covariance in place of the
UT-based time update over an instant ownship turn. By comparing with the well-
understood UKF in a simple tracking scenario, we confirm our results for the
first two moments of the post-manoeuvre distribution are correct. However, in
addition to the mean and covariance, our estimator also computes the third and
fourth post-manoeuvre moments to allow monitoring of the quality of the
Gaussian approximation, an ability that the pure UKF lacks. This insight gives
the CFE-UKF an edge over the UT-based estimator.

With this understanding, we conclude that there is a need to evaluate
algorithms that do not blindly assume Gaussianity of the underlying target
state distribution. Similar to the approaches presented in [14], we could use
our higher-order moment metrics as splitting criteria for a Gaussian sum
filter; a threshold established by Monte-Carlo modelling can determine when
the estimator would benefit from a split of the prior when the ownship turns.

Alternatively, the conditional Gaussianity properties proved in Proposition 2
empower the use of a Rao-Blackwell-type particle filter; a particle filter
would track the bearing rate and scaled range rate, while bearing and log-
range are conditionally Gaussian and can be tracked with a Kalman Filter.

## Acknowledgements

We thank Acacia Systems for their support and funding.

##  Appendix A Supplemental details for Section III-A

### 1\. Proof of Proposition 2

###### Proof.

Claims (1)(i) and (2)(i), that the post-manoeuvre distributions of MPC and LPC
are non-Gaussian, are clearly visible from the terms in $\cos\beta$ and
$\sin\beta$, which are entangled with $\dot{\beta}$ and $\dot{\rho}$ in the
exponent of the post-manoeuvre distribution. Let $P_{i,j}$ refer to the
$i,j^{\text{th}}$ entry of the precision matrix $\Sigma^{-1}$; then the
exponent of Eq. 14 contains a term

$(\dot{\rho}-\mu_{\rho}+s\Delta v_{x}\sin\beta+s\Delta
v_{y}\cos\beta)^{2}P_{2,2}\;.$

Expanding this quadratic reveals terms like, for example,
$\dot{\rho}sP_{2,2}\sin\beta$, that render the distribution non-Gaussian. One
might wonder whether these terms can cancel with other terms in the
distribution — but cancellation is impossible without requiring special
conditions on the precision matrix entries.

Claim (1)(ii), that the MPC distribution is known exactly if we condition on
the bearing $\beta$, is proven as follows. As in the Proposition, let $c_{1}$
and $c_{2}$ refer to the constant factors in $f_{1}$ and $f_{2}$. Accordingly,
Eq. 10 in MPC becomes

$\bm{h}^{-1}(\bm{x}^{+})=\begin{bmatrix}\beta^{+}\\\
\dot{\beta}^{+}+c_{1}s^{+}\\\ \dot{\rho}^{+}+c_{2}s^{+}\\\
s^{+}\end{bmatrix}\;,$

and on substitution into Eq. 14 and conditioning on $\beta$, the exponent of
the post-manoeuvre distribution is

$-\frac{1}{2}\left[\left(\begin{bmatrix}\dot{\beta}^{+}+c_{1}s^{+}\\\
\dot{\rho}^{+}+c_{2}s^{+}\\\
s^{+}\end{bmatrix}-\bm{\mu}\right)^{{}^{\text{T}}}\Sigma^{-1}_{-\beta}\left(\begin{bmatrix}\dot{\beta}^{+}+c_{1}s^{+}\\\
\dot{\rho}^{+}+c_{2}s^{+}\\\ s^{+}\end{bmatrix}-\bm{\mu}\right)\right].$

We now write the vectors that pre- and post-multiply $\Sigma$ as linear
transformations of Gaussian innovations. Let $d_{3}=s^{+}-\mu_{s}$, then

$\dot{\beta}^{+}+c_{1}s^{+}-\mu_{\dot{\beta}}=\dot{\beta}-(\mu_{\dot{\beta}}-c_{1}\mu_{s})+c_{1}d_{3},$
$\dot{\rho}^{+}+c_{2}s^{+}-\mu_{\dot{\rho}}=\dot{\rho}-(\mu_{\dot{\rho}}-c_{2}\mu_{s})+c_{2}d_{3}.$

Accordingly, let $d_{1}=\dot{\beta}-(\mu_{\dot{\beta}}-c_{1}\mu_{s})$ and
$d_{2}=\dot{\rho}-(\mu_{\dot{\rho}}-c_{2}\mu_{s})$, then the exponent
simplifies to

$-\frac{1}{2}\left[\begin{pmatrix}d_{1}+c_{1}d_{3}\\\ d_{2}+c_{2}d_{3}\\\
d_{3}\end{pmatrix}^{\text{T}}\Sigma^{-1}\begin{pmatrix}d_{1}+c_{1}d_{3}\\\
d_{2}+c_{2}d_{3}\\\ d_{3}\end{pmatrix}\right].$

Clearly, the vectors are a linear combination of $d_{1},\,d_{2},\,d_{3}$.
Recalling the definition of $\mathbf{P}$ from the Proposition, we get

$\begin{pmatrix}d_{1}+c_{1}d_{3}\\\ d_{2}+c_{2}d_{3}\\\
d_{3}\end{pmatrix}=\begin{pmatrix}1&0&c_{1}\\\ 0&1&c_{2}\\\
0&0&1\end{pmatrix}\begin{pmatrix}d_{1}\\\ d_{2}\\\
d_{3}\end{pmatrix}=\mathbf{P}^{-1}\begin{pmatrix}d_{1}\\\ d_{2}\\\
d_{3}\end{pmatrix}$

and simplify the exponent to

$-\frac{1}{2}\left[\begin{pmatrix}d_{1}\\\ d_{2}\\\
d_{3}\end{pmatrix}^{\text{T}}\left(\mathbf{P}^{-1}\right)^{\text{T}}\Sigma^{-1}\mathbf{P}^{-1}\begin{pmatrix}d_{1}\\\
d_{2}\\\ d_{3}\end{pmatrix}\right]\;.$

Recognising that
$\left(\mathbf{P}^{-1}\right)^{\text{T}}\Sigma^{-1}\mathbf{P}^{-1}=\left(\mathbf{P}\Sigma\mathbf{P}^{\text{T}}\right)^{-1}$,
we conclude the stated result.

Claim (2)(ii), on the Gaussian distribution for LPC if conditioned on $\beta$
and $\rho$, is trivial: simply recognise that the terms $f_{1}(\beta,\rho)$
and $f_{2}(\beta,\rho)$ are fixed and can be thought of as modifications to
the means of $\dot{\beta}$ and $\dot{\rho}$. ∎

### 2\. Details of Figs. 1 and 2

Each Figure considers a Gaussian prior in MPC / LPC coordinates using a
diagonal $\Sigma$. The main challenge is to select the mean and variance for
$s$ and $\rho$ so that the priors are roughly comparable between the two
coordinate systems. In order to do so, we select an initial location for the
target in Cartesian coordinates — $(0.1,0.15)$ km in Fig. 1 and $(1,1.5)$ km
in Fig. 2 — and an initial standard deviation for target location uncertainty
of $0.4$ km in Cartesian space. We sample 10,000 times from this Gaussian
prior in Cartesian coordinates, transform each sample into $s$ and $\rho$, and
use the variance of the transformed samples as the variance of the prior in
LPC / MPC. The resulting standard deviations, and all other parameters for the
figures, are shown in Table I. The figures are then obtained by sampling
10,000 times from each prior and transforming those samples: label the
$i^{\text{th}}$ sample $\bm{x}_{i}$, then the figures plot
$\bm{x}_{i}^{+}=\bm{h}(\bm{x}_{i})$ for all $i$.

|  | $\mu_{\beta}$ | $\mu_{\dot{\beta}}$ | $\mu_{\dot{\rho}}$ | $\mu_{s/\rho}$ | $\sigma^{2}_{\beta}$ | $\sigma^{2}_{\dot{\beta}}$ | $\sigma^{2}_{\dot{\rho}}$ | $\sigma^{2}_{s/\rho}$
---|---|---|---|---|---|---|---|---|---
Fig. 1 | MPC | 0.58 | 1.1 | 0.1 | 5.5 | 0.05 | 0.2 | 0.2 | 4.0
LPC | 0.58 | 1.1 | 0.1 | -1.7 | 0.05 | 0.2 | 0.2 | 0.63
Fig. 2 | MPC | 0.58 | 1.1 | 0.1 | 0.55 | 0.05 | 0.2 | 0.2 | 0.14
LPC | 0.58 | 1.1 | 0.1 | 0.59 | 0.05 | 0.2 | 0.2 | 0.23
TABLE I: Means and variances for Gaussian priors, obtained as described in
Appendix A.

##  Appendix B Demonstration of Higher-Order Moment Generation via Computer
Algebra

Our SageMath package, seaweed , exports any moment of the post-manoeuvre
target state distribution as Python or LaTeX code. For example, we present the
third raw moment of bearing rate in LaTeX in Appendix B. We have manually
written the expected value on the left-hand side, but we directly obtain the
right-hand side by calling the function moment_0300/latex_0300 in the pre-
generated post_manoeuvre module.

There is a minor stylistic difference between the generated moments and the
main body of this paper; the two components of ownship speed change, $\Delta
v_{x}$ and $\Delta v_{y}$, are respectively written as $\Delta v_{0}$ and
$\Delta v_{1}$. We have also manually added line breaks and alignment markers
to fit the equation to the page.

$\displaystyle\mathbb{E}\\!\left[x^{3}_{1}\right]=$
$\displaystyle-\frac{3}{4}\,$ $\displaystyle{\Delta
v_{0}}^{3}\cos\left(\mu_{0}-3\,\sigma_{30}\right)e^{\left(-3\,\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}-\frac{3}{4}\,{\Delta
v_{0}}{\Delta
v_{1}}^{2}\cos\left(\mu_{0}-3\,\sigma_{30}\right)e^{\left(-3\,\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}$
$\displaystyle-\frac{1}{4}\,$ $\displaystyle{\Delta
v_{0}}^{3}\cos\left(3\,\mu_{0}-9\,\sigma_{30}\right)e^{\left(-3\,\mu_{3}-\frac{9}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}+\frac{3}{4}\,{\Delta
v_{0}}{\Delta
v_{1}}^{2}\cos\left(3\,\mu_{0}-9\,\sigma_{30}\right)e^{\left(-3\,\mu_{3}-\frac{9}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}$
$\displaystyle+\frac{3}{4}\,$ $\displaystyle{\Delta v_{0}}^{2}{\Delta
v_{1}}e^{\left(-3\,\mu_{3}-\frac{9}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}\sin\left(3\,\mu_{0}-9\,\sigma_{30}\right)-\frac{1}{4}\,{\Delta
v_{1}}^{3}e^{\left(-3\,\mu_{3}-\frac{9}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}\sin\left(3\,\mu_{0}-9\,\sigma_{30}\right)$
$\displaystyle+\frac{3}{4}\,$ $\displaystyle{\Delta v_{0}}^{2}{\Delta
v_{1}}e^{\left(-3\,\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}\sin\left(\mu_{0}-3\,\sigma_{30}\right)+\frac{3}{4}\,{\Delta
v_{1}}^{3}e^{\left(-3\,\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{9}{2}\,\sigma_{33}\right)}\sin\left(\mu_{0}-3\,\sigma_{30}\right)$
$\displaystyle+\frac{3}{2}\,$ $\displaystyle{\Delta
v_{0}}^{2}{\left(\mu_{1}-2\,\sigma_{31}\right)}e^{\left(-2\,\mu_{3}+2\,\sigma_{33}\right)}+\frac{3}{2}\,{\Delta
v_{1}}^{2}{\left(\mu_{1}-2\,\sigma_{31}\right)}e^{\left(-2\,\mu_{3}+2\,\sigma_{33}\right)}$
$\displaystyle+\frac{3}{2}\,$
$\displaystyle{}{\left({\left(\mu_{1}-2\,\sigma_{31}\right)}\cos\left(2\,\mu_{0}-4\,\sigma_{30}\right)e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}-2\,\sigma_{10}e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}\sin\left(2\,\mu_{0}-4\,\sigma_{30}\right)\right)}{\Delta
v_{0}}^{2}$ $\displaystyle-3\,$
$\displaystyle{}{\left(2\,\sigma_{10}\cos\left(2\,\mu_{0}-4\,\sigma_{30}\right)e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}+{\left(\mu_{1}-2\,\sigma_{31}\right)}e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}\sin\left(2\,\mu_{0}-4\,\sigma_{30}\right)\right)}{\Delta
v_{0}}{\Delta v_{1}}$ $\displaystyle-\frac{3}{2}\,$
$\displaystyle{}{\left({\left(\mu_{1}-2\,\sigma_{31}\right)}\cos\left(2\,\mu_{0}-4\,\sigma_{30}\right)e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}-2\,\sigma_{10}e^{\left(-2\,\mu_{3}-2\,\sigma_{00}+2\,\sigma_{33}\right)}\sin\left(2\,\mu_{0}-4\,\sigma_{30}\right)\right)}{\Delta
v_{1}}^{2}$ $\displaystyle-3\,$
$\displaystyle{}{\left(\sigma_{11}\cos\left(\mu_{0}-\sigma_{30}\right)e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}+{\left({\left(\mu_{1}-\sigma_{31}\right)}\cos\left(\mu_{0}-\sigma_{30}\right)e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}\right.}\right.}$
$\displaystyle-{\left.{\left.\sigma_{10}e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}\sin\left(\mu_{0}-\sigma_{30}\right)\right)}{\left(\mu_{1}+\sigma_{10}+\sigma_{31}\right)}\right)}{\Delta
v_{0}}$ $\displaystyle+3\,$
$\displaystyle{}{\left(\sigma_{11}e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}\sin\left(\mu_{0}-\sigma_{30}\right)+{\left(\sigma_{10}\cos\left(\mu_{0}-\sigma_{30}\right)e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}\right.}\right.}$
$\displaystyle+{\left.{\left.{\left(\mu_{1}-\sigma_{31}\right)}e^{\left(-\mu_{3}-\frac{1}{2}\,\sigma_{00}+\frac{1}{2}\,\sigma_{33}\right)}\sin\left(\mu_{0}-\sigma_{30}\right)\right)}{\left(\mu_{1}-\sigma_{10}+\sigma_{31}\right)}\right)}{\Delta
v_{1}}+{\left(\mu_{1}^{2}+\sigma_{11}\right)}\mu_{1}+2\,\mu_{1}\sigma_{11}\text{.}$
(40)

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## Biography Section

Athena Helena Xiourouppa Having completed a Bachelor of Mathematical Sciences
(Advanced) at the University of Adelaide, Athena is pursuing postgraduate
research in Statistics and Applied Mathematics via a Master of Philosophy. Her
studies are supported by Acacia Systems. Athena has proficiency in statistical
modelling, simulation, and dynamical systems. Outside of her work and study,
Athena holds a position on the Women in Science, Technology, Engineering, and
Mathematics Society (WISTEMS) committee at the University of Adelaide. This
allows her to meaningfully apply her experience in academia and industry to
inspire women and other minorities in STEM.
---
Dmitry Mikhin Dmitry started his research career in early 90’ working on sound
propagation modelling in the ocean using ray acoustics, and later, parabolic
equation methods. Since then he worked in areas as varied as acoustical
modelling, infrared propagation in the atmosphere, flight planning and route
search, asset optimisation and planning in mining, agricultural robotics, and
lately, estimation, tracking, and data fusion. When wearing his second hat,
Dmitry is a professional software developer combining his research and coding
skills to bridge the vast badlands separating the academic science and
industry applications.
---
Melissa Humphries Dr. Melissa Humphries is a statistician — she creates, and
applies, analytical tools that make sense of our world. Working in areas like
forensic science, health and psychology, the goal is always to increase
efficiency, accuracy, and transparency. Building bridges between machines and
experts, Melissa’s work aims to support experts in making decisions in an
explainable way. Working at the interface between statistics and AI, Melissa’s
research draws meaning out of complex systems by leveraging the explainable
components of statistical and physical models and elevating them using novel
techniques. She strongly advocates the integration of the end-user from the
outset of all research and is committed to finding impactful ways to implement
science. Melissa is also a passionate advocate of equity across all areas of
her work and life.
---
John Maclean I use Bayesian methods to attack inference problems in which one
has access to forecasts from a noisy, incorrect, chaotic model, and also
access to noisy, incomplete data. The key question is how to use forecasts and
data together in order to infer model states or parameters. The methods I use
work very well in low dimensional problems, and my research attacks the curse
of dimensionality found when employing high dimensional data.
---