Statistical synthesis of phase alignment algorithms for localization of wave field sources

Abstract

This article is devoted to the problem of determining the coordinates of wave field sources using phases of complex Fourier transforms of signals recorded by the sensor array. Phase-based source localization algorithms have an important reliability property: they provide the quality of determining the source coordinates, which weakly depends on the statistical characteristics of random noise affecting the array sensors. For this reason, phase algorithms are widely used to localize wave radiation sources in radio and acoustic applications and, more recently, in seismology. Naturally, preference should be given to phase-based source localization algorithms that provide the smallest average errors in estimating source coordinates. Such algorithms can be constructed and analyzed using methods of mathematical statistics. In this article, we synthesized a phase-based statistically optimal (PSO) algorithm, which is a modification of the asymptotically efficient (ASE) statistical algorithm for estimating the coordinates of microseismic sources proposed in Kushnir et al. (Int J Geomath 4(2):201–225, 2013. https://doi.org/10.1007/s13137-013-0049-6). Unlike the ASE algorithm, the PSO algorithm does not require additional observations of “pure” noise and is resistant to the statistical properties of random noise. The article shows that the PSO algorithm is an extended version of the well-known phase-based SRP-PHAT algorithm (Brandstein and Ward in Microphone arrays signal processing techniques and applications, Springer, Berlin, 2001, Chapter 8; Zhang et al. in IEEE Trans Multimedia 10(3):538–548, 2008), which is widely used in acoustic applications. In this article, we describe Monte-Carlo simulation, the purpose of which is to compare the accuracy of determining the coordinates of micro-seismic sources provided by various phase-based algorithms. This simulation showed that the proposed PSO algorithm provides much better source positioning accuracy than the traditional SRP-PHAT algorithm. It is also shown that the PSO algorithm provides the source positioning accuracy, which is almost the same as for the phase-based source location algorithm recently synthesized by the maximum likelihood (ML) method in Varypaev and Kushnir (Int J Geomath 9(2):335–358, 2018. https://doi.org/10.1007/s13137-018-0108-0). However, the PSO algorithm has much higher computational efficiency than the ML algorithm, which allows it to be used for online processing of array data.

Appendices

Appendix 1

1. Consider the first term $\tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right)$ in Eq. (14) for the functional of the ASE estimate:

$$ \tilde{\varLambda }_{\text{ALF}} \left( {{\dot{\mathbf{X}}}_{mn} |{\mathbf{r}}} \right) = \tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right) + \tilde{\varPsi }\left( {{\dot{\mathbf{X}}}_{mn} |{\mathbf{r}}} \right), $$

(37)

where $\tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right) = \frac{1}{n}\sum\limits_{j = 1}^{n} \,lndet\,{\dot{\mathbf{F}}}_{{{\varvec{\upxi}},j}} + \frac{1}{n}\sum\limits_{j = 1}^{n} lndet{\mathbf{Q}}_{j} \left( {\mathbf{r}} \right),{\mathbf{Q}}_{j} \left( {\mathbf{r}} \right) = \left( {{\mathbf{I}} + {\dot{\mathbf{F}}}_{{{\varvec{\upxi}},j}}^{ - 1} d_{j} {\dot{\mathbf{h}}}_{j} \left( {\mathbf{r}} \right){\dot{\mathbf{h}}}_{j}^{*} \left( {\mathbf{r}} \right)} \right).$

Let’s evaluate the dependence of variations of a function ${\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)$ on variations of its argument ${\mathbf{r}}$. In accordance with Eq. (37), we can write:

$$ \begin{aligned} det{\mathbf{Q}}_{\text{j}} & = det\left( {{\mathbf{I}} + \left[ {\begin{array}{*{20}l} {\rho_{1} } \hfill & 0 \hfill & \ldots \hfill & 0 \hfill \\ 0 \hfill & {\rho_{2} } \hfill & \ldots \hfill & 0 \hfill \\ \ldots \hfill & \ldots \hfill & {} \hfill & \ldots \hfill \\ 0 \hfill & 0 \hfill & \ldots \hfill & {\rho_{m} } \hfill \\ \end{array} } \right]_{j} .\left[ {\begin{array}{*{20}l} {a_{1}^{2} } \hfill & {a_{1} a_{2} \dot{\beta }_{1,2} } \hfill & \ldots \hfill & {a_{1} a_{m} \dot{\beta }_{1,m} } \hfill \\ {a_{2} a_{1} \dot{\beta }_{2,1} } \hfill & {a_{2}^{2} } \hfill & \ldots \hfill & {a_{2} a_{m} \dot{\beta }_{2,m} } \hfill \\ \ldots \hfill & \ldots \hfill & {} \hfill & \ldots \hfill \\ {a_{m} a_{1} \dot{\beta }_{m,1} } \hfill & {a_{m} a_{2} \dot{\beta }_{m,2} } \hfill & \ldots \hfill & {a_{m}^{2} } \hfill \\ \end{array} } \right]_{j} } \right) \\ & = det\left[ {\begin{array}{*{20}l} {1 + a_{1}^{2} \rho_{1} } \hfill & {a_{1} a_{2} \dot{\beta }_{1,2} \rho_{1} } \hfill & \ldots \hfill & {a_{1} a_{m} \dot{\beta }_{1,m} \rho_{1} } \hfill \\ {a_{1} a_{2} \dot{\beta }_{1,2}^{*} \rho_{2} } \hfill & {1 + a_{2}^{2} \,\rho_{2} } \hfill & \ldots \hfill & {a_{2} a_{m} \dot{\beta }_{2,m} \rho_{2} } \hfill \\ \ldots \hfill & \ldots \hfill & {} \hfill & \ldots \hfill \\ {a_{1} a_{m} \dot{\beta }_{1,m}^{*} \rho_{m} } \hfill & {a_{m} a_{2} \dot{\beta }_{2,m}^{*} \rho_{m} } \hfill & \ldots \hfill & {1 + a_{m}^{2} \rho_{m} } \hfill \\ \end{array} } \right]_{j} , \\ \end{aligned} $$

(38)

where $\dot{\beta }_{k,l,j} = { \exp }\left\{ {i\phi_{k,j} - \phi_{l,j} } \right\}$; $\rho_{k,j} = {{d_{j} }}{{g_{k,j} ^{ - 1}}}$ is the ratio of signal power to noise power for spectral component $f_{j}$. Since $a_{k} > 0,\,\,k \in \overline{1,m}$, it is easy to show that

$$ \left| {det{\mathbf{Q}}_{j} } \right| \le det\left[ {\begin{array}{*{20}l} {a_{1} \rho_{1} a_{1} + 1} \hfill & {a_{1} \rho_{1} a_{2} } \hfill & { \cdot \cdot \cdot } \hfill & {a_{1} \rho_{1} a_{m} } \hfill \\ {a_{2} \rho_{2} a_{1} } \hfill & {a_{2} \rho_{2} a_{2} + 1} \hfill & { \cdot \cdot \cdot } \hfill & {a_{2} \rho_{2} a_{m} } \hfill \\ { \cdot \cdot \cdot } \hfill & { \cdot \cdot \cdot } \hfill & {} \hfill & { \cdot \cdot \cdot } \hfill \\ {a_{m} \rho_{m} a_{1} } \hfill & {a_{m} \rho_{m} a_{2} } \hfill & { \cdot \cdot \cdot } \hfill & {a_{m} \rho_{m} a_{m} + 1} \hfill \\ \end{array} } \right]_{j} = \left[ {q_{k,l,j} ,\,\,k \in \overline{1,m} } \right]. $$

(39)

By the Hadamard inequality (Bellman 1997) (see¹) we have:

$$ ln\left| {det{\mathbf{Q}}_{j} } \right| = lndet\left[ {q_{k,l,j} ,j,k \in \overline{1,m} } \right] \le ln\prod\limits_{k = 1}^{m} {\sqrt {\sum\limits_{l = 1}^{m} {q_{k,l,j}^{2} } } } = \frac{1}{2}\sum\limits_{k = 1}^{m} {\ln \sum\limits_{l = 1}^{m} {q_{k,l,j}^{2} } } . $$

(40)

$$ ln\left| {det{\mathbf{Q}}_{j} } \right| \le \frac{1}{2}\sum\limits_{k} {ln\left[ {\left( {1 + a_{k}^{2} \rho_{k,j} } \right)^{2} + \left( {a_{k} \rho_{k,j} } \right)^{2} \left( {\left( {\sum\limits_{l} {a_{l}^{2} } } \right) - a_{k}^{2} } \right)} \right]} , $$

(41)

The coefficients $a_{l}^{2} \left( {\mathbf{r}} \right)$ characterize the attenuation of the source signal during its propagation from the source location to the sensors of the array. They are always less than unity due to the absorption of seismic waves by the medium and the transformation of P-waves into other types of waves:

$$ 0 \le A_{{\mathbf{r}}}^{2} \le a_{{k,{\mathbf{r}}}}^{2} \le B_{{\mathbf{r}}}^{2} \le 1. . $$

(42)

where $A_{{\mathbf{r}}}^{2} = min_{k} a_{{k,{\mathbf{r}}}}^{2}$, $B_{{\mathbf{r}}}^{2} = max_{k} a_{{k,{\mathbf{r}}}}^{2}$.

In real seismic monitoring, the PSD of the source signal and the PSD of interferences have lower and upper bounds (see Eq. (9)):

$$ \begin{aligned} 0 < D_{\text{L}} & = \mathop {min}\limits_{f} d\left( f \right) \le d\left( f \right) \le \mathop {max}\limits_{f} d\left( f \right) = D_{\text{U}} ; \\ 0 < G_{\text{L}} & = \mathop {min}\limits_{f,k} g_{k} \left( f \right) \le g_{k} \left( f \right) \le \mathop {max}\limits_{f,k} g_{k} \left( f \right) = G_{\text{U}} ,f \in \left[ {0,f_{s} /2} \right],k \in \overline{1,m} . \\ \end{aligned} $$

(43)

The ratio of signal power to noise power in frequency components of the sensor signals are equal: $a_{{k,{\mathbf{r}}}}^{2} \rho_{k,j}$, where $\rho_{k,j} = d_{j} g_{k,j}^{ - 1}$. Therefore, they satisfy the following inequalities (see Eqs. (10), (11)):

$$ A_{{\mathbf{r}}} \gamma \le a_{{k,{\mathbf{r}}}}^{2} \rho_{k,j} \le B_{{\mathbf{r}}} \delta , $$

(44)

where $\gamma = {{min_{j} d_{j} }}\,{\left[{{max_{k,j} g_{k,j} }}\right]^{-1}}$; $\delta = {{K_{\hbox{max} } }}{{\left(C_{\hbox{min} }\right)^{-1} }}\,{{max_{j} d_{j} }}\,{{\left[min_{k,j} g_{k,j} \left( {f_{j} } \right)\right]^{-1}}}$.

Taking into account the restriction B (Eqs. (9)–(11)) we get the following inequality:

$$ \frac{1}{2n}\sum\limits_{j = 1}^{n} {\sum\limits_{k = 1}^{m} {\ln \left\{ {1 + \rho_{k,j} \left( {2a_{{k,{\mathbf{r}}}}^{2} + a_{{k,{\mathbf{r}}}}^{4} \rho_{k,j}^{{}} + a_{{k,{\mathbf{r}}}}^{2} \rho_{k,j}^{{}} \left[ {\left( {\sum\nolimits_{l} {a_{{l,{\mathbf{r}}}}^{2} } } \right)\, - a_{k}^{2} } \right]\,} \right)} \right\}} } . . $$

(45)

If $\delta = max_{{j \in \overline{1,n} ,{\kern 1pt} k \in \overline{1,m} }} \,\,\rho_{k,j} \ll 1$, then the following approximate equality is correct:

$$ \begin{aligned} \frac{1}{n}\sum\limits_{j = 1}^{n} {\ln \left| {\det {\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)} \right|} & \le \ln \left\{ {1 + \rho_{k,j} \left( {2a_{{k,{\mathbf{r}}}}^{2} + a_{{k,{\mathbf{r}}}}^{4} \rho_{k,j}^{{}} + a_{{k,{\mathbf{r}}}}^{2} \rho_{k,j}^{{}} \left[ {\left( {\sum\limits_{l = 1}^{m} {a_{{l,{\mathbf{r}}}}^{2} } } \right)\, - a_{k}^{2} } \right]\,} \right)} \right\} \\ &\quad \sim 1 - 2a_{{k,{\mathbf{r}}}}^{2} \rho_{k,j} - \varepsilon_{{k,j,{\mathbf{r}}}} , \end{aligned} $$

(46)

where $\varepsilon_{{k,j,{\mathbf{r}}}} = \rho_{k,j}^{2} a_{{k,{\mathbf{r}}}}^{4} + \rho_{k,j}^{2} a_{{k,{\mathbf{r}}}}^{2} \left[ {\left( {\sum\nolimits_{l = 1}^{m} {a_{{l,{\mathbf{r}}}}^{2} } } \right) - a_{{k,{\mathbf{r}}}}^{2} } \right]$.

That is, $\,\varepsilon_{{k,j,{\mathbf{r}}}} \to 0,\,\,\,{\text{if}}\,\,\delta \to 0$, and this convergence is uniform on ${\mathbf{r}} \in Q$.

From Eqs. (45), (46) we get:

$$ \begin{aligned} & \frac{1}{n}\sum\limits_{j = 1}^{n} {\ln \left| {\det {\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)} \right|} < \frac{1}{2n}\sum\limits_{j = 1}^{n} {\sum\limits_{k = 1}^{m} {\left( {1 - 2a_{k}^{2} \left( {\mathbf{r}} \right)\rho_{k,j} - \varepsilon_{{k,j,{\mathbf{r}}}} } \right)} } \\ & \quad = \left[ {\frac{m}{2} - \sum\limits_{k = 1}^{m} {\left( {2a_{k}^{2} \left( {\mathbf{r}} \right)f_{s}^{ - 1} \sum\limits_{j = 1}^{n} {\rho_{k} \left( {\frac{{j\,f_{\text{s}} }}{n}} \right)\frac{{f_{s} }}{n}} + f_{s}^{ - 1} \sum\limits_{j = 1}^{n} {\varepsilon_{{k,{\mathbf{r}}}} \left( {\frac{{j\,f_{\text{s}} }}{n}} \right)\frac{{f_{s} }}{n}} } \right)} } \right]. \\ \end{aligned} $$

(47)

Since the sums over j in the right-hand side of Eq. (47) are integral sums, we can write the following inequality:

$$ \begin{aligned} \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{j = 1}^{n} {\ln \left| {\det {\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)} \right|} &< \frac{m}{2} + \sum\limits_{k = 1}^{m} {\left( {a_{k}^{2} \left( {\mathbf{r}} \right)f_{s}^{ - 1} \int\limits_{0}^{{f_{s} }} {\rho_{k} \left( f \right)df + f_{s}^{ - 1} \int\limits_{0}^{{f_{s} }} {\varepsilon_{{k,{\mathbf{r}}}} \left( f \right)df} } } \right)} \\ & = \frac{m}{2} + f_{s}^{ - 1} \sum\limits_{k} {\left( {\int\limits_{0}^{{f_{s} }} {a_{k}^{2} \left( {\mathbf{r}} \right)\rho_{k} \left( f \right)\left( {1 + \varepsilon_{{k,{\mathbf{r}}}} \left( f \right)} \right)df} } \right)} . \\ \end{aligned} $$

(48)

Consequently,

$$ \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{j = 1}^{n} {\ln \left| {\det {\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)} \right|} < \frac{m}{2} + \delta f_{s}^{ - 1} \sum\limits_{k = 1}^{m} {a_{{k,{\mathbf{r}}}}^{2} } + m \cdot o\left( \delta \right) $$

(49)

Similarly, if $n \to \infty$ we can get the following equality:

$$ \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{j = 1}^{n} \,ln det \,{\dot{\mathbf{F}}}_{{{\varvec{\upxi}},j}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^{m} \,f_{s}^{ - 1} \sum\limits_{j = 1}^{n} {\ln g_{k} \left( {\frac{{j\,f_{s} }}{n}} \right)\frac{{f_{s} }}{n}} = f_{s}^{ - 1} \sum\limits_{k = 1}^{m} \,\int\limits_{0}^{{f_{s} }} {\ln g_{k} \left( f \right)df} . $$

(50)

Therefore, if number $n$ of the array observations is large enough and a power of the source signal is small enough:

$n \gg 1$ and $d\left( f \right) \ll g_{k} \left( f \right)$ for all $f \in \left[ {0,f_{s} } \right],\,\,k \in \overline{1,m}$, then the term $\tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right)$ of Eq. (37) satisfies the following inequality:

$$ \begin{aligned} \tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right) &= \frac{1}{n}\sum\limits_{j = 1}^{n} \,ln det \,{\dot{\mathbf{F}}}_{{{\varvec{\upxi}},j}} + \frac{1}{n}\sum\limits_{j = 1}^{n} {\ln \left| {\det {\mathbf{Q}}_{j} \left( {\mathbf{r}} \right)} \right|} \\ & \le f_{s}^{ - 1} \sum\limits_{k = 1}^{m} \,\int\limits_{0}^{{f_{s} }} {\ln g_{k} \left( f \right) + a_{k}^{2} \left( {\mathbf{r}} \right)\rho_{k} \left( f \right)\left( {1 + \varepsilon_{{k,{\mathbf{r}}}} \left( f \right)} \right)df} + \frac{m}{2}. \\ \end{aligned} $$

(51)

That is, if $\delta \to 0$, then

$$ \begin{aligned} \mathop {\lim }\limits_{\delta \to 0} \tilde{\varOmega }_{mn} \left( {{\mathbf{r}} ;\delta } \right) &< \left( {\frac{m}{2} + f_{s}^{ - 1} \sum\limits_{k = 1}^{m} \,\int\limits_{0}^{{f_{s} }} {\ln g_{k} \left( f \right)} \,df\,} \right)\, + \mathop {\lim }\limits_{\delta \to 0} \left[ {\delta \left( {1 + o\left( \delta \right)} \right)\sum\limits_{k = 1}^{m} {a_{k}^{2} \left( {\mathbf{r}} \right)} } \right] \\ & = \left( {\frac{m}{2} + f_{s}^{ - 1} \sum\limits_{k = 1}^{m} \,\int\limits_{0}^{{f_{s} }} {\ln g_{k} \left( f \right)} \,df\,} \right). \\ \end{aligned} $$

(52)

As follows from Eqs. (13), (14) and limit inequality (52), in the case when the power of the source signal is small compared with the power of the interferences acting on the sensors of the array, and the number of the array observations is quite large, the influence of the term $\tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right)$ on the quality of the ASE-estimate of the source coordinates becomes very small. This result allows us to ignore the term $\tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right)$ when calculating the ASE-estimate by maximizing the functional $\tilde{\varLambda }_{\text{ALF}} \left( {{\dot{\mathbf{X}}}_{mn} |{\mathbf{r}}} \right)$$= \tilde{\varOmega }_{mn} \left( {\mathbf{r}} \right) + \tilde{\varPsi }\left( {{\dot{\mathbf{X}}}_{mn} ;{\mathbf{r}}} \right)$. That is, under conditions (51), (52), the ASE-estimate of the source coordinates can be found in the form:

$$ {\hat{\mathbf{r}}}_{\text{ASE}} \left( {{\dot{\mathbf{X}}}_{mn} } \right) = \mathop {\hbox{max} }\limits_{{{\mathbf{r}} \in Q}} \tilde{\varPsi }\left( {{\dot{\mathbf{X}}}_{mn} ;{\mathbf{r}}} \right). $$

(53)

2. The coefficients $v_{j} \left( {\mathbf{r}} \right) = \sum\nolimits_{k = 1}^{m} {a_{k}^{2} \left( {\mathbf{r}} \right)d_{j} g_{k,j}^{ - 1} }$ in Eq. (19) are the sums of ratios of the source signal power to the noise power for frequency components of the array data. In the case when the seismic source has a spherical radiation pattern of the P-wave and the region $\left\{ {{\mathbf{r}} \in Q} \right\}$ is located inside the array aperture, the fraction of power of the source radiation entering the array weakly depends on the position ${\mathbf{r}}$ of the source in the region $Q$. That is, variations of coefficients $v_{j} \left( {\mathbf{r}} \right)$ with changes of coordinates ${\mathbf{r}} \in Q$ can be neglected.

Appendix 2

A significant difference in the accuracy of estimating the source coordinates between the traditional SRP-PHAT algorithm and statistically optimal algorithms can be explained from the point of view of statistical estimation theory. The SRP-PHAT, PML and PSO algorithms are different estimators of the unknown parameter r in the statistical observation model defined by the Eq. (2). As follows from Eq. (32), the SRP-PHAT algorithm minimizes phase discrepancies using the method recommended in statistical regression analysis in the special case of uniform distributions of regression errors. But in the case of different probability distributions of these errors, the regression analysis recommends a weighted summation of the discrepancies using different factors for each discrepancy.

The PSO-algorithm follows the general recommendations of the statistical regression analysis in case of different distributions of regression errors. It calculates the weighted sum of cosines of the phase discrepancies for sensor observations using the coherence estimates of the observations as the weights. This provides a significant increase in the accuracy of estimating the parameter r.

The maximum likelihood PML-algorithm also takes into account the different distributions of phase discrepancies for pairs of sensor observations, but instead of the sum of the weighted cosines of the phase discrepancies, it calculates the likelihood functions of the discrepancies, which depend on the coherences of the pairs of sensor observations.

Appendix 3

3.1 Comparison of the computational complexity of PSO and PML algorithms for estimating coordinates of wave-field sources

The algorithms being compared have the structure

$$ {\tilde{\mathbf{r}}}\left( {S_{\text{M}} } \right) = \mathop {argmax}\limits_{{{\mathbf{r}} \in R}} \sum\limits_{j = 1}^{n} {\sum\limits_{k = 2}^{m} {\sum\limits_{l = k + 1}^{m} {S_{\text{M}} \left( {u_{j,k,l} ,\,v_{j,k,l} ;{\mathbf{r}}} \right)} } } , $$

(54)

where $u = \left( {\psi_{k,l,j} - \theta_{k,l,j} \left( {\mathbf{r}} \right)} \right)$ is the phase discrepancy for observations of the k-th and l-th sensors, $\psi_{k,l,j}$ is the phase difference of these observations at DFFT frequency $f_{j}$, $v = \left| {\hat{\dot{\gamma }}_{k,l,j} } \right|$ is value of the coherence function of these observations at DFFT-frequency $f_{j}$, M is the symbol of estimation method (PSO or PML), m is the number of array sensors, n is the number of multichannel observations; R is the three-dimensional region of the source coordinates in which the maximum is sought.

For the PSO-algorithm the function $S_{\text{M}} = S_{\text{PSO}} = v\cos \left( u \right)$.

For the PML-algorithm the function $S_{\text{M}} = S_{\text{PML}}$, where

$$ S_{\text{PML}} = \ln \left\{ {\frac{{1 - v^{2} }}{2\pi }\left( {\frac{1}{{1 - v^{2} \cos^{2} \left( u \right)}} + v\cos \left( u \right)\frac{{\pi - \arccos \left[ {v\cos \left( u \right)} \right]}}{{\left[ {1 - v^{2} \cos^{2} \left( u \right)} \right]^{\,3/2} }}} \right)} \right\}. $$

Let us denote $M_{q}$ the number of elementary computational operations for multiplying two q-digit numbers u and v. Table 3 shows the number of operations (complexity) for calculating the following functions of the arguments u, v (Borwein and Borwein 1987):

Table 3 allows evaluate the complexity $Q\left[ {S_{\text{PSO}} \left( {u ;v} \right)} \right]$ and $Q\left[ {S_{\text{PML}} \left( {u ;v} \right)} \right]$ of calculating the terms on the right side of Eq. (54):

Table 3 Computational complexity of simple operations

Complexity of calculating the function $S_{\text{PSO}} = v\cos \left( u \right)$ is:

$$ Q\left[ {S_{\text{PSO}} \left( {u ;v} \right)} \right] = U_{q} + V_{q} + M_{q} \left( {1 + q^{1/2} } \right) $$

Complexity of calculating the function $S_{\text{PML}} \left( {u ;v} \right)$ is composed by complexities of operations A–D presented in Table 4.

Table 4 Computational complexity of some expressions of S_PML function

The difference in computational complexity in calculating the functions $S_{\text{PML}} \left( {u ;v} \right)$ and $S_{\text{PSO}} \left( {u ;v} \right)$ is equal to:

$$ Q\left[ {S_{\text{PML}} \left( {u ;v} \right)} \right] - Q\left[ {S_{\text{PML}} \left( {u ;v} \right)} \right] = A + B + C + D + \ln D = M_{q} \left( {7 + 3\log q + 2q^{1/2} } \right). $$

The difference in computational complexity in calculating the estimates ${\tilde{\mathbf{r}}}\left( {S_{\text{M}} } \right)$ given by the Eq. (30) using functions $S_{\text{M}} = S_{\text{PML}} \left( {u_{j,k,l} ;v_{j,k,l} } \right)$ and $S_{\text{M}} = S_{\text{PSO}} \left( {u_{j,k,l} ;v_{j,k,l} } \right)$ is equal to:

$$ Q\left[ {{\tilde{\mathbf{r}}}_{n} \left( {S_{\text{PML}} } \right)} \right] - Q\left[ {{\tilde{\mathbf{r}}}_{n} \left( {S_{\text{PSO}} } \right)} \right] = N_{R} nm^{2} M_{q} \left( {7 + 3\log q + 2q^{1/2} } \right) \approx N_{R} nm^{2} M_{q} \left( {7 + 2q^{1/2} } \right), $$

where $N_{R}$ is the number of points of three dimensional area R, which are analyzed when searching for the maximum in the Eq. (54).

Thus, the number of multiplication operations required to calculate one value of the term $S_{\text{PML}} \left( {u_{j,k,l} ,\,v_{j,k,l} ;{\mathbf{r}}} \right)$ is more than seven times greater than for calculation of one value of the term $S_{\text{PSO}} \left( {u_{j,k,l} ,\,v_{j,k,l} ;{\mathbf{r}}} \right)$. To calculate each of the estimates ${\tilde{\mathbf{r}}}_{n} \left( {S_{\text{PML}} } \right)$ and ${\tilde{\mathbf{r}}}_{n} \left( {S_{\text{PSO}} } \right)$, it is necessary recalculate the term $S_{\text{PML}}$ or the term $S_{\text{PSO}}$$N_{R} nm^{2}$ times. If the number $N_{R} nm^{2}$ is significant (which is typical for the problem considered in this paper) then the computational complexity of the estimate ${\tilde{\mathbf{r}}}\left( {S_{\text{PML}} } \right)$ becomes much larger than for the estimate ${\tilde{\mathbf{r}}}\left( {S_{\text{PSO}} } \right)$.