## 1 Introduction  
Wireless sensor networks (WSNs) are used in a variety of applications, such as environmental data collection, dangerous event monitoring, and disaster prevention [1,2]. However, they have some performance limits (e.g., energy consumption). Conventional WSNs protocols, like location-based protocols [3] and hierarchical protocols [4], are helpful in reducing bandwidth requirements. The theory of compressive sensing (CS) [5,6], a novel sensing/sampling paradigm that goes against common wisdom in data acquisition, can further reduce the bandwidth requirements and save more energy. [PERSON] and [PERSON] [5] provided an introduction to compressive sampling, which is usually used in the field of efficient digital image compression. If a signal is known to be compressible, [PERSON] [6] found that the number of measurements \(M\) for this signal is dramatically smaller than the size of signal \(N\) (\(M<<N\)). Note that a signal being compressible means that it can be exploited to hold the correlation of the data both temporally and spatially based on some sparsification bases. A number of papers focus on the combination of CS theory and WSNs [1,7,8]. [PERSON] _et al_. [1] proposed a distributed energy-efficient sensor network scheme based on random access compressed sensing. However, the authors in [1] do not address the scenario of multi-hop wireless networks. A recent survey of CS theory as applied in WSNs is given in [7], which recovers sparse data in WSNs by solving a convex optimization via \(L1\) norm. The authors in [8] discussed EW-CS scheme that is better overall recovery quality for non-uniform compressible signals than ordinary CS schemes. However, it is not feasible to simply combine CS theory with WSNs since the recovery procedure would fail due to the coherence between measurement and sparsity matrices in a real WSN scenario as described later and reference [9]. The research group of [PERSON], [PERSON], [PERSON] and [PERSON] has done great work in the field of data collection in wireless sensor networks [9, 10, 11, 12]. Reference [9] illustrated two crucial insights: firstly, applying CS naively may not bring any improvement, which coincides with our viewpoint in this paper, such as the bad performance of \(L1\) described in Table 1; and secondly, they put forward the idea that the hybrid-CS can achieve significant throughput improvements. Based on their previous hybrid-CS, reference [10] further proposed two solutions for data collection, _i.e._, the optimal solution and the near-optimal solution. Reference [11] proposed the Dual-I-Evel Compressed Aggregation framework to recover the physical signal from incomplete data from WSNs, which is also able to be done from our HRM_CS. The amazing work of their group is the CS-based aggregation scheme, which achieves both recovery fidelity and energy efficiency in WSNs with arbitrary topology according to the reference [12]. They employed diffusion wavelets to design the sparse basis while we focus on the recovery method based on Bayesian Compressed Sensing and the problem of coherence in the model of HRM_CS, which is more practical under the environment with real world signals. [PERSON] _et al_. [13] proposed a sparsity model that allows the use of CS theory for the online recovery of large data sets. While some of our work was inspired by the study reported in [13], it not onlyextends the results given in [13] by using the hierarchical routing method, but also provides a mathematical analysis of sparsity and coherence. In our previous paper [2], we proposed a data fusion method based on CS, which was used to monitor the cyanobacteria bloom-forming in a lake using a single hop network. In the past years, considering the energy consumption in the procedure of route, reference [14] put forward a recurrent neural network to realize the range-free localization of WSNs while we decrease the energy consumption by making use of compressive sampling. These are two different methods to solve a similar problem.