The proposed technique named “geodesic search algorithm” is described in this section. It is a technique which separates ripple region from anchor node geometry, as per proposition 4.1 mentioned below.
Proposition 4.1
Let \(\overline{f}\) be the curve representing the scope of the ripple region in a uniform phasespace. Assume that \(\overline{f}\) has an anchor node position \(\overline{p}\) such that \(\overline{f}^{\prime}\left( {\overline{p}} \right)\) represents the slope of the ripple region boundary and \(\overline{f}^{\prime\prime}\left( {\overline{p}} \right)\) is the rate of change of ripple region boundary. If \(\lambda_{1}\) and \(\lambda_{2}\) are the eigenvectors corresponding to true anchor node and malicious anchor node, then the anchor node geometry becomes independent of the Ripple Region, provided that the Ripple Region contains \(\overline{p}\).
Let us introduce the concept of “Geodesic norm” by equation (2)
$$\left\ {u,v} \right\: = \sqrt {\left\langle {u,u} \right\rangle + \varepsilon \left\langle {v,v} \right\rangle }$$
(2)
where \(u,v\) belong to manifolds \(T_{p} ,M\) and some function \(\varepsilon < \frac{1}{K}\), where \(K\) is the connectivity of UWSNs.
\(\frac{{\left\langle {Y,\dot{Y}} \right\rangle }}{{\left\ {Y,\dot{Y}} \right\^{2} }} \ge \delta\) defines the propagation of acoustic signal traversing in underwater domain.
Let us define a lemma that restricts the extent of signal variation in an underwater scenario:
Lemma 4.2
For Metastasis \(\delta\) which is limited to \(\delta < \frac{K}{{1 + K^{{{\raise0.5ex\hbox{$\scriptstyle { – 3}$} \kern0.1em/\kern0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} }}\), the family of acoustic trajectory \(C_{\delta }\) given by \(\frac{{\left\langle {Y,\dot{Y}} \right\rangle }}{{\left\ {Y,\dot{Y}} \right\^{2} }} \ge \delta\) is strictly immune to malicious effects.
It would be easier to prove this only for No Metastasis of anchor node, but we need some positive metastasis \(\delta\) later to dwell upon the acoustic localization in presence of malicious anchor nodes.
Proof
For some \(\varepsilon\) and \(k \in K\),
$$\frac{{\sqrt {\left\langle {Y,Y} \right\rangle \left\langle {\dot{Y},\dot{Y}} \right\rangle } }}{{\left\ {Y,\dot{Y}} \right\^{2} }} \le \frac{1}{2\sqrt \varepsilon }$$
(3)
$${\text{and}}\quad \frac{{\left\langle {\dot{Y},\dot{Y}} \right\rangle + k\left\langle {Y,Y} \right\rangle }}{{\left\ {Y,\dot{Y}} \right\^{2} }} \ge k$$
(4)
From Cauchy–Schwarz Inequality, and by limiting the metastasis \(\delta\) to less than 0.05, we get
$$\frac{d}{dt}\left( {\frac{{\left\langle {Y,\dot{Y}} \right\rangle }}{{\left\ {Y,\dot{Y}} \right\^{2} }}} \right) = \frac{{\left( {\left\langle {\dot{Y},\dot{Y}} \right\rangle + \left\langle {\ddot{Y},Y} \right\rangle } \right)\left\ {Y,\dot{Y}} \right\^{2} – 2\left\langle {Y,\dot{Y}} \right\rangle \left( {\left\langle {Y,\ddot{Y}} \right\rangle + \varepsilon \left\langle {\ddot{Y},\dot{Y}} \right\rangle } \right)}}{{\left\ {Y,\dot{Y}} \right\}}$$
(5)
Therefore, the first order derivative of the unit norm of \(Y\) and \(\dot{Y}\) is positive when Metastasis \(\delta\) equals the unit norm of \(Y\) and \(\dot{Y}\). □
Lemma 4.2 shall be useful when proving that the proposed UWSN localization technique separates the dependence of ripple region on anchor node topology.
Geodesic representation of the UWSN topology
To delve into this section using the symplectic geometry developed for underwater sensor networks, a function called “Geodesic formalism” representing the phasespace distribution is introduced here.

(A)
Geodesic Formalism and stretch Ripple Region
The Geodesic concept is widely used in the domain of image processing to map a curved surface for specific display of perspective. Unlike L2 norm which maps the distance between two points as a straight line, the underwater positioning system requires a more sophisticated form of acoustic trajectory measurement, especially when dealing with stratified sound speed profile.
Let \(A\) denote the set of anchor nodes amongst a subset of all the deployed sensor nodes \({\mathbb{R}}^{p}\). It is assumed throughout this work that for all anchor nodes \(\overline{a} \in A\), the noise profile is independent of their physical location, that is, all anchor nodes face equal amount of noise irrespective of their actual position in the topology. Let \(\left\ x \right\_{G}^{{\overline{a}}}\) denote the geodesic norm of anchor node \(\overline{a}\) from the target. The geodesic norm \(\left\ x \right\_{G}^{{\overline{a}}}\) is, thereby, given as
$$\left\ x \right\_{G}^{{\overline{a}}} = \inf \left\{ {t > 0:x \in \sum\limits_{count\left( A \right)} {\left( {\frac{{\partial {\rm X}}}{\partial A}} \right)} } \right\}$$
(6)
where \(t\) is the time instance of measurement, \(count\left( A \right)\) is the number of anchor node hops taken by the acoustic signal to reach the target, and \({\rm X}\) is the total distance traversed by the acoustic signal when its path is considered as a continuous signal.
The issues of Malicious Node on anchor node information are a topic of analysis of their own. For now, let us restrict ourselves to the penalty incurred due to the stretch Ripple Region under the framework of Geodesic Formalism. Let a linear measurement model of intensity of stretch \(\Phi\) for a simple Ripple Region be represented by \(x^{ * }\), which is the dual of geodesic norm for the set of anchor nodes \(A\). If the signal model is
$$\overline{y} = \Phi \overline{x}^{ * }$$
(7)
Then, the estimated geodesic norm \(\widehat{{\overline{x}}}\) is computed from \(\overline{x}\) such that
$$\widehat{{\overline{x}}} = \arg \min \left\ x \right\_{G}^{{}}$$
(8)
$${\text{Subject}}\,{\text{to}}\quad \overline{y} = \Phi \overline{x}$$
(9)
A convex formulation may be easily obtained by relaxing constraint in (9) to \(\left {\overline{y} – \Phi \overline{x}} \right < \Delta_{G}\), where \(\Delta_{G}\) is the tolerable Geodesic noise floor. Subsequently, we state and prove the condition under which the Geodesic Formalism provides a unique and optimal solution to the anchor node topology under stretch ripple region.
Proposition 4.3
The dual of the Geodesic norm \(x^{ * }\) equals the estimated geodesic norm \(\hat{\overline{x}}\) and this estimated geodesic norm \(\hat{\overline{x}}\) is the matched acoustic trajectory from the source anchor node to the target if and only if
$$E\left( {Y\left {T^{ * } = t} \right.} \right) = E\left( {h\left( {X_{1} \left( t \right), \ldots ,X_{n – 1} \left( t \right),\Phi \left( {t, \cup } \right)} \right)} \right)$$
(10)
where \(X_{i} \left( t \right)\) are independent random variables uniformly distributed and independent of connectivity region \(\cup\), and \(\cup\) has density \(g\left( {u\left t \right.} \right)\).
Proof
$$\begin{aligned} & E\left( {Y\left {T^{ * } = t} \right.} \right) \\ & \quad = E\left( {h\left( {X_{1} , \ldots ,X_{n} } \right)\left {T^{ * } = t} \right.} \right) \\ & \quad = E\left( {h\left( {X_{1} , \ldots ,X_{{\left( {n – 1} \right)}} ,X_{n} } \right)\left {T^{ * } = t} \right.} \right) \\ \end{aligned}$$
where \(E\left( \cdot \right)\) is the expectation operator. By Crofton’s Theorem^{30}, conditional on \(T^{ * } = t\), the terms \(X_{1} , \ldots ,X_{{\left( {n – 1} \right)}}\) have the same distribution as the generalized order statistics of \(X_{1} \left( t \right), \ldots ,X_{{\left( {n – 1} \right)}} \left( t \right)\) as in the proposition 4.3 and \(X_{\left( n \right)}\) is distributed as \(\Phi \left( {t, \cup } \right)\). By symmetry of \(h\), the result follows that the geodesic norm of the true measurement shall equal the geodesic norm of the estimated measurement. In other words, localization using geodesic formulation is feasible in case of UWSN as well.□
In order to model the behavior of Ripple Region stretching outwards, we first have to relate the support lines of the anchor nodes forming a convex space. Let \(O_{1}\) denote the proximity of anchor node \(a_{1}\) while \(O_{2}\) be the proximity of \(a_{2}\). Let the anchor nodes have support lines \(A_{1} P\) and \(A_{2} P\) such that \(P\) denotes the point of intersection of the two supports, and \(P\) be the location of the malicious node. Let \(\tau_{1}\) and \(\tau_{2}\) denote the vectors of support lines; change of angle \(\omega\) indicates Ripple Region stretching (\(\omega\) increasing) or folding (\(\omega\) decreasing). In either case, the relationship between nonmalicious and malicious anchor nodes would be critical to locating the target accurately. The Geodesic Formalism is explained below:

a.
Knowledge of \(\left( {x,y} \right)\), \(\sin \phi\),\(\cos \phi\), distance \(p\).

b.
To express \(x\) and \(y\) in terms of geometric coordinates.

c.
To express \(dx\) and \(dy\), and subsequently estimate the extent of perturbation caused by the malicious anchor node.

d.
To estimate the loss of information.

e.
To discretize (curved) acoustic measurement and determine the Geodesic norm to remove the Malicious effects.

f.
Optimal underwater localization.
According to Crofton’s Formula^{31}, the position \(\left( {x,y} \right)\) of the malicious node is given by
$$x\cos \phi_{1} + y\sin \phi_{1} – A_{1} P = 0$$
(11)
$${\text{and}}\quad x\cos \phi_{2} + y\sin \phi_{2} – A_{2} P = 0$$
(12)
Differentiating (11) with respect to \(\Phi\), we get
$$\left( {A_{1} P} \right)^{\prime } = y\cos \phi – x\sin \phi$$
(13)
Then, (11) is rewritten as
$$\Rightarrow x\cos \phi_{1} = \left( {A_{1} P} \right) – y\sin \phi_{1}$$
(14)
$$\begin{aligned} & \Rightarrow x\cos^{2} \phi_{1} = \left( {A_{1} P} \right)\cos \phi_{1} – y\sin \phi_{1} \cos \phi_{1} \\ & \Rightarrow x\left( {1 – \sin^{2} \phi_{1} } \right) = \left( {A_{1} P} \right)\cos \phi_{1} – y\sin \phi_{1} \cos \phi_{1} \\ & \Rightarrow x = \left( {A_{1} P} \right)\cos \phi_{1} – y\sin \phi_{1} \cos \phi_{1} + x\sin^{2} \phi_{1} \\ & \Rightarrow x = \left( {A_{1} P} \right)\cos \phi_{1} – \left( {y\cos \phi_{1} – x\sin \phi_{1} } \right)\sin \phi_{1} \\ \end{aligned}$$
$$\Rightarrow x = (A_{1} P)\cos \phi_{1} – \left( {A_{1} P} \right)^{{\prime }} \sin \phi_{1}$$
(15)
Similarly, for (12)
$$\Rightarrow x = \left( {A_{1} P} \right)\cos \phi_{2} – \left( {A_{1} P} \right)^{{\prime }} \sin \phi_{2}$$
(16)
y coordinate is similarly computed as
$$y = \left( {A_{1} P} \right)\sin \phi_{1} + \left( {A_{1} P} \right)^{{\prime }} \cos \phi_{1} \quad {\text{for}}\,{\text{anchor}}\,{\text{node}}\,A_{1}$$
(17)
$$\& \quad y = \left( {A_{2} P} \right)\sin \phi_{2} + \left( {A_{2} P} \right)^{{\prime }} \cos \phi_{2} \quad {\text{for}}\,{\text{anchor}}\,{\text{node}}\,A_{2}$$
(18)
Next, the extent of perturbation of malicious node maybe parametrically expressed for xcoordinate as
$$x = A_{1} P\cos \phi_{1} – \left( {A_{1} P} \right)^{\prime } \sin \phi_{1}$$
(19)
$$\Rightarrow dx = – A_{1} P\sin \phi_{1} d\phi_{1} – \left( {A_{1} P} \right)^{{\prime }} \cos \phi_{1} d\phi_{1} + \left( {A_{1} P} \right)^{{\prime }} \cos \phi_{1} d\phi_{1} – \left( {A_{1} P} \right)^{{\prime \prime }} \sin \phi_{1} d\phi_{1}$$
$$\Rightarrow dx = – \left( {\left( {A_{1} P} \right) + \left( {A_{1} P} \right)^{\prime \prime } } \right)\sin \phi_{1} d\phi_{1} \quad {\text{according}}\,{\text{to}}\,{\text{anchor}}\,{\text{node}}\,A_{1}$$
$${\text{And}} \Rightarrow dx = – \left( {\left( {A_{2} P} \right) + \left( {A_{2} P} \right)^{{\prime \prime }} } \right)\sin \phi_{2} d\phi_{2} \quad {\text{according}}\,{\text{to}}\,{\text{anchor}}\,{\text{node}}\,A_{2}$$
Similarly, the extent for perturbation for ycoordinate is given from derivative of (17) as
$$\Rightarrow dy = A_{1} P\cos \phi_{1} d\phi_{1} – \left( {A_{1} P} \right)^{\prime } \sin \phi_{1} d\phi_{1} + \left( {A_{1} P} \right)^{\prime } \sin \phi_{1} d\phi_{1} + \left( {A_{1} P} \right)^{\prime \prime } \cos \phi_{1} d\phi_{1}$$
(20)
$$= \left( {A_{1} P} \right)\cos \phi_{1} d\phi_{1} + \left( {A_{1} P} \right)^{\prime \prime } \cos \phi_{1} d\phi_{1}$$
$$\Rightarrow dy = \left( {\left( {A_{1} P} \right) + \left( {A_{1} P} \right)^{\prime \prime } } \right)\cos \phi_{1} d\phi_{1} \, {\text{from anchor node}}\,A_{1} \,{\text{perspective}},$$
$${\text{and}},\quad \Rightarrow dy = \left( {\left( {A_{2} P} \right) + \left( {A_{2} P} \right)^{{\prime \prime }} } \right)\cos \phi_{2} d\phi_{2} \, {\text{from anchor node}}\,A_{2} \,{\text{perspective}}.$$
\(dx\) and \(dy\) respectively indicate the consensus in malicious anchor node positioning. Assuming that the movement of anchor node topology is smooth, the Phasespace of the geometry is approximated by the proposed Geodesic Formalism.