A Multi-sensor Data Fusion Method for Nondestructive Testing of Oil Pipelines

During nondestructive testing of oil pipelines, the quality of data collection and processing directly hinges on the sensor, the collection method and the data source [1]. In general, single-sensor data collection cannot obtain complete and precise data. The collected data are often highly uncertain, because each sensor has a unique level of quality, performance and noise [2, 3]. By contrast, multi-sensor data fusion reduces the data uncertainty through integration of data from multiple signals, making it easier to make robust decisions [4]. For example, multiple ultrasonic sensors and magnetic flux leakage (MFL) sensors can be coupled to make up for their respective defects and limitations, creating an effective nondestructive method to capture the abnormal parameters of oil pipelines [5, 6]. The combination between the two types of sensors can provide more accurate data for pipeline management and maintenance.

This paper designs a multi-sensor data fusion method capable of detecting wall thickness and pipe damage at any position, with a few number of ultrasonic sensors [7]. The fused data can approximate the actual size of damages, and thus improve the safety of oil pipelines [8, 9]. Example analysis shows that the proposed method did not mistake excessive damage for non-excessive damage, eliminating the need for costly and risky error checking.

3.1 DST-based fuzzy linear regression

(1) According to the DST, the confidence function of sensor i relative to the measure function of damage j can be expressed as:

$m_{i}(j)=\frac{C_{i}(j)}{\sum_{j} C_{i}(j)+N_{c}\left(1-R_{i}\right)\left(1-\omega \alpha_{i} \beta_{i}\right)}$      (1)

The measure function of the uncertainty θ of sensor i can be described as:

$m_{i}(\theta)=\frac{N_{c}\left(1-R_{i}\right)\left(1-\omega \alpha_{i} \beta_{i}\right)}{\sum_{j} C_{i}(j)+N_{c}\left(1-R_{i}\right)\left(1-\omega \alpha_{i} \beta_{i}\right)}$   (2)

where, ω is the performance coefficient of sensor i (e.g. geometry and surface roughness); α_i is the maximum correlation coefficient between sensor i and each damage; β_i is the distribution coefficient of sensor i; R_i is the reliability coefficient of sensor i; N_c is the type of damage.

Two confidence functions were cited to explain the fusion rule. In essence, the rule combines the probability assignments of two evidences under the same recognition framework into an overall probability assignment. Let m_i(1) and m_i(2) be the confidence functions relative to the measure functions of the two evidences under the same recognition framework, with the focal elements being A₁, A₂, ⋯, A_k and B₁, B₂, ⋯, B_r, respectively. Then, we have:

$K=\sum_{i, j \atop A_{i} \cap B_{j}=\varphi} m_{1}\left(A_{i}\right) m_{2}\left(B_{j}\right)<1$      (3)

Then, the overall probability assignment can be expressed as:

$m(c)=\left[\begin{array}{l}{\Sigma_{i, j} \quad m_{1}\left(A_{i}\right) m_{2}\left(B_{j}\right)} \\ {\frac{A_{i} \cap B_{j}=c}{1-K}, \forall c \subset U} \\ {0, c=\varphi}\end{array}\right]$       (4)

(2) Damage membership, fuzzy interval and membership estimation. The values and probabilities of the same physical quantity measured by multiple sensors generally obey the normal distribution [15]. Hence, this paper takes the standard normal distribution function as the membership function of fuzzy set.

Through the analysis on the data collected by ultrasonic and MFL sensors, the estimated size z and the measured size z' of a damage has the following correlation:

$\phi(z)=\lambda_{0}+\lambda_{1} \phi\left(z^{\prime}\right)+\lambda_{2} \phi^{2}\left(z^{\prime}\right)+\cdots+\lambda_{m} \phi^{m}\left(z^{\prime}\right)+\varepsilon, \varepsilon \sim N\left(0, \sigma^{2}\right)$    (5)

where, the polynomial about ϕ(•)is the estimated size of the damage; ϕ(•)is the real function; ε is a normally-distributed error term; λ_i, i=1, 2, ⋯, m and σ² are the parameters to be regressed.

In most cases, $\varphi(z)=\ln z, \varphi\left(z^{\prime}\right)=\ln z^{\prime}$ and $m=1$. Then, formula 5 can be converted into:

$\ln z=\lambda_{0}+\lambda_{1} \ln z^{\prime}+\varepsilon, \varepsilon \sim N\left(0, \sigma^{2}\right)$      (6)

The next step is to determine parameters λ₀, λ₁ and σ². Let z₁, z₂, ⋯, z_n and $Z_{1}^{\prime}, Z_{2}^{\prime}, \cdots Z_{n}^{\prime}$ be the estimated sizes and measured sizes of n damages, respectively. The n damages were assumed to subjected to an independent nondestructive test. It can be seen from regression analysis that the values of λ₀ and λ₁ can be estimated as:

$\hat{\lambda}_{0}=\frac{1}{n} \sum_{i=1}^{n} \ln z_{i}-\hat{\lambda}_{1} \frac{1}{n} \sum_{i=1}^{n} \ln z_{i}^{\prime}$       (7)

$\hat{\lambda}_{1}=\frac{\sum_{i=1}^{n}\left(\ln z_{i}-\frac{1}{n} \sum_{j=1}^{n} \ln z_{j}\right)\left(\ln z_{i}^{\prime}-\frac{1}{n} \sum_{j=1}^{n} \ln z_{j}^{\prime}\right)}{\sum_{i=1}^{n}\left(\ln z_{i}^{\prime}-\frac{1}{n} \sum_{j=1}^{n} \ln z_{j}^{\prime}\right)^{2}}$      (8)

The regression equation of the above formulas can be expressed as:

$\ln \tilde{z}=\hat{\lambda}_{0}+\hat{\lambda}_{1} \ln z^{\prime}$     (9)

where, $\tilde{Z}$ is the median size of the damage.

Then, the value of variance σ² can be obtained by:

$\hat{\sigma}^{2}=\frac{1}{n-2} \sum_{i=1}^{n}\left(\ln z_{i}-\hat{\lambda}_{0}-\hat{\lambda}_{1} \frac{1}{n} \sum_{j=1}^{n} \ln z_{j}^{\prime}\right)^{2}$     (10)

Since the fuzzy membership function is assumed to obey standard normal distribution, the membership of the damage in the interval (0, z) can be derived from formulas (9) and (10) as:

$F(z | \hat{z})=\Phi\left[\left(\ln z-\hat{\lambda}_{0}-\hat{\lambda}_{1} \ln \hat{z}\right) / \hat{\sigma}\right]$         (11)

similarly, the membership of the damage in the interval [Z,+∞) can be expressed as:

$P(z | \hat{z})=1-F(z | \hat{z})$      (12)

where, Φ(•)is the standard normal distribution function; $F(z | \hat{z})$is the fuzzy distribution of the damage size in the detection range.

Substituting the measured size $\hat{z}$ of the said damage into formulas (9) and (10), the logarithm mean and the logarithm standard deviation can estimate as $\ln \tilde{z}=\hat{\lambda}_{0}+\hat{\lambda}_{1} \ln \hat{z}$. Hence, the lower and upper limits, $F_{L}\left(z_{0} | \hat{z}\right)=\Phi\left(u_{P L}\right)$ and $F_{U}\left(z_{0} | \hat{z}\right)=\Phi\left(u_{P U}\right)$, of the confidence γ of the damage relative to the membership interval (0,z₀) can be obtained as:

$u_{P L}=\frac{\ln z_{0}-\ln \hat{z}}{\hat{\sigma}} \sqrt{\frac{2 v-2}{2 v-1}}-u_{\gamma} \sqrt{\frac{1}{n^{\prime}}+\frac{\left(\ln z_{0}-\ln \hat{z}\right)^{2}(2 v-2)}{\sigma^{2} \omega(2 v-1)}}$   (13)

$u_{P U}=\frac{\ln z_{0}-\ln \hat{z}}{\hat{\sigma}} \sqrt{\frac{2 v-2}{2 v-1}}+u_{\gamma} \sqrt{\frac{1}{n^{\prime}}+\frac{\left(\ln z_{0}-\ln \hat{z}\right)^{2}(2 v-2)}{\sigma^{2} \omega(2 v-1)}}$       (14)

$\omega=2\left|v+u_{\gamma}-c-\frac{1}{\sqrt{v+u_{\gamma}-c}}\right|$       (15)

similarly, the unilateral confidence of lower and upper limits of confidence γ relative to the membership interval [z₀,+∞) can be derived as:

$P_{U}\left(z_{0} | \hat{z}\right)=1-\Phi\left(u_{P L}\right)$       (16)

$P_{L}\left(z_{0} | \hat{z}\right)=1-\Phi\left(u_{P U}\right)$        (17)

In nondestructive testing, the use of $F_{L}\left(z_{0} | \hat{z}\right)$ and $P_{U}\left(z_{0} | \hat{z}\right)$ can prevent the mistake of excessive damage for non-excessive damage, but cannot prevent the mistake of non-excessive damage for excessive damage. If the damage falls between the excessive level and the non-excessive level, more independent nondestructive tests should be conducted and the test results should be regressed repeatedly to increase the sample size n. Then, the confidence intervals $F_{U}\left(z_{0} | \hat{z}\right)-F_{L}\left(z_{0} | \hat{z}\right)$ and $P_{U}\left(z_{0} | \hat{z}\right)-P_{L}\left(z_{0} | \hat{z}\right)$ can be greatly shortened.

(3) Fuzzy data fusion of multiple data sources. Let $\hat{z}_{1}$ and $\hat{z}_{2}$ be the sizes of the same damage measured in two independent nondestructive tests. Substituting them into formula (11), the fuzzy distributions $F\left(z | \hat{z}_{1}\right)$ and $F\left(z | \hat{z}_{2}\right)$ of the damage size can be obtained as:

$F\left(z | \hat{z}_{1}\right)=\Phi\left[\left(\ln z-\hat{\lambda}_{0}-\hat{\lambda}_{1} \ln \hat{z}_{1}\right) / \hat{\sigma}\right]$              (18)

$F\left(z | \hat{z}_{2}\right)=\Phi\left[\left(\ln z-\hat{\lambda}_{0}-\hat{\lambda}_{1} \ln \hat{z}_{2}\right) / \hat{\sigma}\right]$              (19)

Then, the DST was applied to deduce the unilateral confidence lower limit of the membership in the interval (0, z₀):

$F_{L}\left(z_{0} | \hat{z}_{1}, \hat{z}_{2}\right)=\frac{F_{L}\left(z_{0} | \hat{z}_{1}\right) F_{L}\left(z_{0} | \hat{z}_{2}\right)}{1-F_{L}\left(z_{0} | \hat{z}_{1}\right)-F_{L}\left(z_{0} | \hat{z}_{2}\right)+2 F_{L}\left(z_{0} | \hat{z}_{1}\right) F_{L}\left(z_{0} | \hat{z}_{2}\right)}$ (20)

similarly, $P_{U}\left(z_{0} | \hat{z}_{1}\right), P_{U}\left(z_{0} | \hat{z}_{2}\right)$, and the unilateral confidence limit of the membership interval [z₀,+∞) can be derived by formula (14) as:

$P_{U}\left(z_{0} | \hat{z}_{1}, \hat{z}_{2}\right)=\frac{P_{U}\left(z_{0} | \hat{z}_{1}\right) P_{U}\left(z_{0} | \hat{z}_{2}\right)}{1-P_{U}\left(z_{0} | \hat{z}_{1}\right)-P_{U}\left(z_{0} | \hat{z}_{2}\right)+2 P_{U}\left(z_{0} | \hat{z}_{1}\right) P_{U}\left(z_{0} | \hat{z}_{2}\right)}$ (21)

Obviously, the number of independent testing sources is positively correlated with the number of iterations and the computing load.

3.2 Fuzzy representation of the size of the same damage measured by one type of sensors

Let $\hat{z}$ be the size of a damage on an oil pipeline measured by one type of sensors in a nondestructive test [16]. Then, the fuzzy distribution of the damage size can be expressed as: $F\left(z_{0}\right)=\Phi\left[\left(\ln z_{0}-\hat{\lambda}_{0}-\hat{\lambda}_{1} \ln \hat{z}\right) / \hat{\sigma}\right]$.

For the damage size, the upper limit and lower limit of reliability can be obtained respectively by:

$\ln z_{U}=\hat{\lambda}_{0}+\hat{\lambda}_{1} \ln \hat{z}+k_{R}\left(n^{\prime}, \nu\right) \hat{\sigma}$        (22)

$\ln z_{U}=\hat{\lambda}_{0}+\hat{\lambda}_{1} \ln \hat{z}-k_{R}\left(n^{\prime}, v\right) \hat{\sigma}$      (23)

where,

$k_{R}\left(n^{\prime}, v\right)=\frac{u_{R}+u_{\gamma} \sqrt{\frac{1}{n^{\prime}}\left(1-\frac{u_{\gamma}^{2}}{w}\right)+\frac{u_{R}^{2}}{w}}}{1-\frac{u_{\gamma}^{2}}{w}} \sqrt{\frac{2 v-1}{2 v-2}}$   (24)

$w=2\left(v+u_{\gamma}-0.64-\frac{1}{\sqrt{v+u_{\gamma}-0.64}}\right)$       (25)

where, ν=n-2.

3.3 Fuzzy representation of the sizes of the same damage measured by two type of sensors

In this paper, the MFL sensors and ultrasonic sensors are applied to measure the same damage interpedently. The measured data must satisfy the following formulas:

$\ln z=\sum_{i=1}^{2} r_{i} \lambda_{0 i}+\sum_{i=1}^{2} r_{i} \lambda_{1 i} \overline{(\ln \hat{z})}_{i}+\varepsilon$      (26)

$\varepsilon \sim N\left[0,\left(\frac{1}{\sigma_{1}^{2}}+\frac{1}{\sigma_{2}^{2}}\right)^{-1}\right]$       (27)

Compared with the measurement method in 3.2, the variance $\left(\frac{1}{\sigma_{1}^{2}}+\frac{1}{\sigma_{2}^{2}}\right)^{-1}$ of ln Z in this section is relatively small, indicating that different sensors can detect the same damage independently and accurately. The logarithm means of the measured sizes of ultrasonic sensors and the MFL sensors can be denoted as $\overline{(\ln \hat{z})_{1}}$ and $\overline{(\ln \hat{z})_{2}}$, respectively. If the sample space is sufficiently large, the actual statistics can be replaced by their respective statistical estimators:

$\hat{r}_{\mathrm{i}}=\frac{1 / \hat{\sigma}_{1}^{2}}{1 / \hat{\sigma}_{1}^{2}+1 / \hat{\sigma}_{2}^{2}}=\frac{\hat{\sigma}_{2}^{2}}{\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2}}$           (28)

$\hat{r}_{2}=\frac{1 / \hat{\sigma}_{2}^{2}}{1 / \hat{\sigma}_{1}^{2}+1 / \hat{\sigma}_{2}^{2}}=\frac{\hat{\sigma}_{1}^{2}}{\hat{\sigma}_{1}^{2}+\hat{\sigma}_{2}^{2}}$          (29)

$\hat{\sigma}^{2}=\left(\frac{1}{\hat{\sigma}_{1}^{2}}+\frac{1}{\hat{\sigma}_{2}^{2}}\right)^{-1}$       (30)

$F(z | \hat{z})=\Phi\left[\frac{\ln z-\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{0 i}-\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{1 i} \overline{(\ln \hat{z})}_{i}}{\left(\frac{1}{\hat{\sigma}_{1}^{2}}+\frac{1}{\hat{\sigma}_{2}^{2}}\right)^{-\frac{1}{2}}}\right]$      (31)

Then, the upper and lower limits of the reliability can be expressed as:

$\ln z_{U}=\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{0 i}+\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{1 i} \overline{(\ln \hat{z})}_{i}+k_{R}\left(n^{\prime}, \nu\right)\left(\frac{1}{\hat{\sigma}_{1}^{2}}+\frac{1}{\hat{\sigma}_{2}^{2}}\right)^{-\frac{1}{2}}$ (32)

$\ln z_{L}=\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{0 i}+\sum_{i=1}^{2} \hat{r}_{i} \hat{\lambda}_{1 i} \overline{(\ln \hat{z})}_{i}-k_{R}\left(n^{\prime}, v\right)\left(\frac{1}{\hat{\sigma}_{1}^{2}}+\frac{1}{\hat{\sigma}_{2}^{2}}\right)^{-\frac{1}{2}}$ (33)

where,

$k_{R}\left(n^{\prime}, v\right)=\frac{u_{R}+u_{\gamma} \sqrt{\frac{1}{n^{\prime}}\left(1-\frac{u_{\gamma}^{2}}{w}\right)+\frac{u_{R}^{2}}{w}}}{1-\frac{u_{\gamma}^{2}}{w}} \sqrt{\frac{2 v-1}{2 v-2}}$         (34)

$w=2\left(v+u_{\gamma}-0.64-\frac{1}{\sqrt{v+u_{\gamma}-0.64}}\right)$           (35)

where, ν=ν₁+ν₂, ν₁=n₁-2, ν₂=n₂-2.

The data collected by ultrasonic and the MFL sensors exhibit heavy noises and high redundancy. To solve the problems, this paper designs a multi-stage data fusion method, which integrates fuzzy algorithm with linear regression, and relies on membership to describe damage size and judge if the damage is excessive. The application in several cases shows that our method can represent any test data in a form closer to the actual damage size, and display the fused data in an intuitive manner. Coupled with visual inspection and expert database, this algorithm can determine the exact size and type of damage, judge if the pipeline needs repairing or replacement, and disclose other important information. The research findings have great application potential in nondestructive testing for oil and gas transport and mining.