## 1 Introduction  
Ocean covers more than 70% of the surface area of Earth. There are various applications related to the ocean environment such as scientific exploration and surveillance of the sea [1]. Underwater sensor networks (U-WSNs) have emerged to study undersea circumstances for these and other applications [2; 3]. In U-WSNs, there are many sensor nodes available in the network [4]. The nodes have the capability to sense the information and forward towards the upper surface sink node. The information packets of the sink node are processed further for various applications. The received data packet accuracy is mostly dependent on the errorless communication. The ocean has time-varying link properties and severe noise [5], which make the underwater networks more challenging to have an optimal data transmission. Water has absorbing properties for radio waves so U-WSNs use acoustic signals [6], which provide optimal performance [7; 8]. Reducing the energy consumption is one of the major concerns in U-WSNs [9]. The replacement of the node batteries in the sea is quite a difficult job. However, for better network lifetime and error-free communication, researchers are focused on introducing innovative routing schemes for U-WSNs as one of the potent remedies. Furthermore, to design the routing protocols for U-WSNs, it is necessary to identify the challenges of this channel: low battery power, time-varying channel, and low allowable bandwidth [10; 11]. Another important issue faced by underwater communication is network reliability. Cooperative routing (CR) is one of the optimal solutions to ensure a reliable network by traveling the information over multi-paths [12]. Cooperative routing also decreases the contrary channel effects and holds a reliable information transmission by considering the source node (SRN) towards the sink. CR uses two techniques for packet advancement: in one technique the data is first amplified and then forwarded (AF) and in the second technique the data is first decoded and forwarded (DF). CR makes the SRN able to advance the packet over multi-cast mode, where the SRN transmits the information towards the multi-nodes at the same time, instead of sending to a single node. The existing optimized depth base routing (ODBR) is basically a single-path routing scheme, which considers only local depth information for the selection of data advancer node [13]. This information is not adequate to specify the position of the nodes. In addition, the situation becomes worse when two nodes at the same depth forward the redundant information towards the sink node. The lowest depth nodes in ODBR become dead quickly due to continuous transmission of packets towards the sink node. Likewise, the cooperative depth base routing (CoDBR) is a multi-path routing scheme [14]. Like ODBR scheme, the CoDBR scheme also considers the lowest depth criteria for the selection of best forwarder nodes. Moreover, the CoDBR scheme selects two relay nodes (RLNs) with the destination (DSN), which increases the contribution of these nodes during the routing but consumes high energy. To solve the aforementioned issues, this paper introduces two routing schemes: EERD and CoEERD. The EERD scheme is a non-cooperative scheme, which reduces the net energy cost by traveling the information over a single path. In addition, during the selection of the best forwarder node, the robust weight function parameters make it more efficient to consume minimum energy. In EERD, the SRN transmits a control packet (CP) to collect the nodes information in terms of node IDs, residual energy, distance, and bit error rate (BER). The node with maximum residual energy, lowest BER, and lowest distance is considered to be DSN, which is the highest priority node for information packet forwarding in the network. The second proposed scheme, CoEERD, also defines the same criteria for the selection of DSN. In this scheme, the DSN is selected with the same criterion as in EERD. The RLN is also selected by the SRN which holds the second highest weight function value.