  For a distribution function $F$ on $\mathbb{R}^d$ and a point $q\in
\mathbb{R}^d$, the \emph{spherical depth} $\SphD(q;F)$ is defined to be the
probability that a point $q$ is contained inside a random closed hyper-ball
obtained from a pair of points from $F$. The spherical depth $\SphD(q;S)$ is
also defined for an arbitrary data set $S\subseteq \mathbb{R}^d$ and $q\in
\mathbb{R}^d$. This definition is based on counting all of the closed
hyper-balls, obtained from pairs of points in $S$, that contain $q$. The
significant advantage of using the spherical depth in multivariate data
analysis is related to its complexity of computation. Unlike most other data
depths, the time complexity of the spherical depth grows linearly rather than
exponentially in the dimension $d$. The straightforward algorithm for computing
the spherical depth in dimension $d$ takes $O(dn^2)$. The main result of this
paper is an optimal algorithm that we present for computing the bivariate
spherical depth. The algorithm takes $O(n \log n)$ time. By reducing the
problem of \textit{Element Uniqueness}, we prove that computing the spherical
depth requires $\Omega(n \log n)$ time. Some geometric properties of spherical
depth are also investigated in this paper. These properties indicate that
\emph{simplicial depth} ($\SD$) (Liu, 1990) is linearly bounded by spherical
depth (in particular, $\SphD\geq \frac{2}{3}SD$). To illustrate this
relationship between the spherical depth and the simplicial depth, some
experimental results are provided. The obtained experimental bound ($\SphD\geq
2\SD$) indicates that, perhaps, a stronger theoretical bound can be achieved.
