Table 1 The abbreviation of used symbols in network model

UMSN

Underwater mobile sensor networks

CSSP

CH–SH Selection and Sink Path-planning scheme

CH

Cluster head node in the network

SH

Sub-cluster head node in the network

AUV

Autonomous underwater vehicle

UA

Underwater acoustic communication

RF

Radio frequency communication

Upper-layer network

Multi-hop network consisting of CHs, SHs and base station, transmitting small amount of message and command among CHs, SHs and base station

Lower-layer network

Multi-hop network consisting of CHs, SHs and sensor nodes, forwarding large amount of data from nodes to CHs, and from nodes to SHs

v

Set of total sensor nodes in the network

\(v_{{{\text{CH}}}}^{j}\)

j-th Cluster head in the network

\(v_{{{\text{SH}}_{i} }}^{j}\)

i-th Sub-cluster head of the j-th cluster in the network

d(v_CH)

Density of CH nodes in the network, which is CH number versus sensor nodes number

D_Sink

Maximum traveling distance of Sink moving path in a round trip

\(D(v_{i}^{{}} ,v_{j}^{{}} )\)

Distance between node v_i and v_j

\(P(v_{i} )\)

Position coordinate of node v_i

N(v)

Number of sensor nodes in the network

N(v_CH)

Number of assumed cluster head nodes in the network

N^H(v_i)

Number of one-hop neighbor nodes of v_i

N^In(v_i)

Number of inner-cluster (or inner-sub-cluster) nodes of v_i (v_i is CH or SH)

E^H(v_i)

Energy of one-hop neighbor nodes of v_i

E^In(v_i)

Energy of inner-cluster (or inner-sub-cluster) nodes of v_i (v_i is CH or SH)

E_R(v_i)

Remaining energy of sensor node v_i

E_C(v_i)

Consumed energy of sensor node v_i

E₀(v_i)

Initial energy of sensor node v_i