Sum-rate maximization and data delivery for wireless seismic acquisition

Abstract

Traditional seismic acquisition systems suffer from a number of difficulties related to telemetry cables that are used as a means of data transmission. Transforming the traditional seismic acquisition system to a wireless system has been considered as a potential solution to most of these difficulties . Wireless seismic acquisition systems have to serve a huge aggregate data rate requirement, as is usually the case in a large wireless sensor network. This paper considers a novel wireless acquisition system with a generalized tree topology, and studies the maximum achievable transmission data rates from the geophones to the wireless gateways, where successive interference cancellation decoding is used at the gateway nodes. We consider the problem of sum-rate maximization by optimizing the decoding set at each gateway, i.e. which group of geophones will be decoded at each gateway. Various integer programming algorithms are proposed for solving this maximization problem. These optimization algorithms are simulated and compared, where it is shown that the ant system algorithm achieves the highest sum-rate, with up to $82\%$ increase compared to the case of no optimization, and a relatively low computational complexity compared to other algorithms, e.g., it may take the ant system algorithm only $19\%$ of the time consumed by the angle-modulated particle swarm optimization. Furthermore, the data delivery from the gateways to the data center is also considered. In this stage, two cases, with different buffer sizes at the gateways, are studied. For small-size buffers, two optimization problems are identified and solved. The first problem considers the minimization of the total power of the gateways, and the second problem considers power fairness between the gateways. It is demonstrated that the first problem reduces the total power of the gateways by $43\%$ compared to the second problem. For large-size buffers, the problem of maximizing the weighted sum rate of the gateways is shown to be convex, and the globally optimal solution is obtained efficiently.

Appendix

Consider the last constraint on the sum of rates in (4), i.e. $\sum _{i=1}^{N}Q_i \le \log _2 \left( 1+\frac{\sum _{i=1}^{N}P_i |g_i|^2}{N_0}\right)$. It is obvious that, at optimality, this constraint must hold with equality for minimum total power. Assume the channel gains $g_i,\forall i\in \{1,\ldots ,N\}$ are ordered in a descending order, and assume $P_{N,{\text {min}}}$ is the power required to decode the signal of GP_N at the end without interference. Now, if the DC decides not to decode GP_N at the end and exchanges its order with another GP; assume without loss of generalization GP₁, then GP_N will suffer from more interference, while GP₁ will suffer from less interference. Therefore, we will have to increase the power of GP_N and we can decrease the power of GP₁ in order to keep the constraint satisfied for each individual GP. However, in order for the sum rate constraint to hold with equality, the following summation must remain constant

$$\begin{aligned} A = P_1 |g_1|^2 + \cdots + P_{N-1} |g_{N-1}|^2 + P_{N,{\text {min}}} |g_{N-1}|^2, \end{aligned}$$

(15)

and is equal to $A =N_0(2^{\sum _{i=1}^{N}Q_i}-1)$. Now, assume that we will increase $P_{N,{\text {min}}}$ by $\Delta _N$ and decrease $P_1$ by $\Delta _1$. Then in order to keep A constant, then we must have: $\Delta _N |g_N|^2 = \Delta _1 |g_1|^2$, and hence, $\Delta _1 = \Delta _N \frac{|g_N|^2}{|g_1|^2}<\Delta _N$. The total power in this case is,

$$\begin{aligned} P_{\text {Total}}&= P_1-\Delta _1 +\cdots +P_{N-1}+P_{N,{\text {min}}}+\Delta _N \nonumber \\&= P_1 + \cdots +P_{N-1}+P_{N,{\text {min}}}+ (\Delta _N - \Delta _1) \nonumber \\&> P_1+\cdots +P_{N-1}+P_{N,{\text {min}}}. \end{aligned}$$

(16)

Since $\Delta _N - \Delta _1$ is a positive term, then the total power will be larger than the case where the DC decodes GP_N last. By repeating the same procedure for $P_{N-1}$ up to $P_{1}$, it is found that the decoding order should follow the decreasing order of the channel gain values in order to have the minimum total power.