2.1. The Degeneration Mechanism of the RS Spectrum Similarity and the Local Similarity Characteristics
Monitoring the RS spectrum of the sensing fiber is commonly utilized to sense variations of the distributed strain along the sensing fiber in conventional OFDR methods. That is based on the characteristics of the sensing fiber that RS spectrum remains stable and unique once the fiber was manufactured and is only influenced by variations of the temperature or strain [
7]. RS spectrum is induced by random fluctuation of the refractive index in the fiber, and its properties are similar to the fiber Bragg grating (FBG). Therefore, the RS can be considered as the weak random Bragg grating [
3]. As the FBG’s spectrum varies with the loaded strain, the external strain also causes a shift of the RS spectrum. The ratio of spectrum shifts to strain variations is usually constant for a specific fiber. Thus, distributed strain sensing can be accomplished by monitoring the RS spectrum shift of the measured gauge.
In the OFDR system, a tunable laser source working in linear-frequency-sweeping mode is utilized to demodulate the position information and distributed strain of every gauge. The beat signal consists of the RS light and local oscillator light. Based on properties of the linear frequency swept light source, the relationship between the position and the beat frequency is directly proportional. Therefore, position information can be achieved through transforming signals from time-domain into frequency-domain using fast Fourier transform (FFT). Accordingly, the RS spectrum located at different positions can be extracted by a short-time Fourier transform (STFT). The spatial resolution and the measured gauge position are controlled by setting the length and center position of the window of STFT. Then, the cross-correlation is used to calculate spectrum shift between the MeS (with unknown distributed strain) and the ReS (with known distributed strain). The achieved corresponding spectrum shift is proportional to the distributed strain. Theoretically, the distributed strain of the whole sensing fiber can be obtained by performing the STFT and the cross-correlation on every measured gauge along the fiber. The conventional OFDR method is illustrated in
Figure 1. The cross-correlation between WRS
i and WMS
i can be expressed as:
where,
n = −
N, −
N + 1, …,
N − 1,
N is the length of the WRS
i and WMS
i sequences, and the elements WRS and WMS sequences are zeros when the index is not within [0, N].
The swept spectrum range should be expanded to enhance the strain measurable range which would induce a long sweep time and large amount of data. However, a broad swept spectrum range would limit the data processing and occupy a big memory when a high spectrum or strain/temperature resolution is required. When the swept spectrum range is
B1 and the spectrum shift induced by suffered strain is
S1, the different spectrum percentage between ReS and MeS is written as:
The positive and negative shift of the reference spectrum is equivalent. So the positive is taken as an example to investigate the influence of the new segments. The reference spectrum is
R(
n) (wherein,
n = 1, 2, 3, …,
K) and the measurement spectrum
Mi can be expressed as:
where,
Ni(
n) is the new spectrum segment,
n = 1, 2, 3, …,
K −
m,
m is the spectrum shift, and
m ≤ 0.5
K.
Then, simulation is run to investigate the influence of the parameter
p on finding the true peak representing the spectrum shift. In the simulation, experimental results of 180 reference spectra are statistically analyzed and new spectrum segments is randomly selected in the reference spectra (except the current reference spectrum).
Figure 2 demonstrates the statistical results of the amplitude ratio of the peak to highest peak when the parameter
p varies from 0 to 50% with a 10% step and the simulation is run 180 times under a same condition.
Figure 2 demonstrates the results of 180 times and every curve represents an experiment. It can be concluded that the cross-correlation method is valid when the ratio is greater than 1 because the true peak is the highest and it can be determined by this method.
Figure 2 indicates that the cross-correlation method is invalid when the different spectrum percentage between ReS and MeS is over ~13%. So the measurable strain is less than ~1100 με when the swept spectrum range is 10 nm.
This physical description is demonstrated in
Figure 3 and it indicates that the MeS (0 με) has a high similarity with the ReS (0 με), however, the original MeS shifts relative to the ReS when the strain is loaded on the sensing fiber. New spectrum segment appears in the MeS spectrum and it has no corresponding similar spectrum segment in the ReS. The spectrum shift under a small strain load is relatively small, so that the proportion of new spectrum is not large enough to cause significant similarity degeneration. Therefore, it can be deduced that the low similarity between the MeS and the ReS is owing to the large spectrum shift induced by the large strain.
The experimental data in
Figure 4 are derived from distributed strain experiments in which the strain loads are 700 με, 1300 με, 2100 με and 2300 με when the swept spectrum range is 10 nm. In practice, however, this method only works well when the distributed strain is small. Experimental results indicate that SNR of the cross-correlation result degenerates significantly and multi or fake peaks begin to appear in the cross-correlation results when the loaded distributed strain exceeds 1000 µε.
In conventional OFDR methods, spectrum shifts are calculated by true peaks’ position in cross-correlation results. Under a small strain load, the true peak is usually unique and keeps highest amplitude.
Figure 4a shows the ideal cross-correlation result and the true peak is unique and highest.
Figure 4b,c shows that there are multiple peaks in the cross-correlation result and this will make the determination of the true peak difficult.
Figure 4d shows that fake peaks refer to that there are some peaks which are higher than the true peak and will lead to wrong determination of the spectrum shifts. Based on the experimental data, peaks in cross-correlation results are relatively lower when multi or fake peaks begin to appear. As illustrated in
Figure 3, the highest peak in the ideal cross-correlation results exceeds 0.8, while the highest peaks of multi and fake peaks are lower than 0.2. Therefore, it can be concluded that the appearance of multi and fake peaks is mainly caused by the degeneration of similarity between the MeS and ReS. With the increase of the loaded strain, the similarity between the MeS and ReS decreases significantly, and multi and fake peaks begin to appear in cross-correlation results. Finally, the determination of the spectrum shift is no more robust and becomes inaccurate. The strain measurable range of OFDR is thus limited.
During the experiments, the sweep wavelength range is 10 nm (~1200 GHz) and the spectrum shift caused by 3000 με is ~3.8nm. Thus, the new spectrum without similarity accounts for ~38% of the ReS under a strain of 3000 με, which will result in a significant decrease of similarity between the MeS and the ReS and the appearance of multiple or fake peaks. In summary, avoiding the appearance of multiple and fake peaks is mainly to rule out the influence of the new spectrum segment. On the other hand, there exists a section of MeS spectrum always located in the ReS. As illustrated in
Figure 3, the local spectrum is defined as a part of the original ReS, which will remain in the latter MeS (after loading strain) and have high similarity to a segment of the MeS. So the improvement of the similarity can be achieved through the utilization of the local spectrum of high similarity. The appearance of multiple and fake peaks can be effectively avoided under a large strain load.
Figure 5 illustrates the MeS and ReS of a distributed sensing gauge in 3 mm length under a strain of 1000 με and it indicates that a segment of the MeS (marked in solid purple line) has high similarity with the shifted ReS segment (marked in solid blue line), even though it is hard to identify the similarity and spectrum shift between the whole MeS (marked in dotted purple line) and ReS (marked in dotted purple line). Therefore, the high local spectrum similarity characteristics are experimentally verified.
During this experiment, a segment of ReS is selected as the local ReS (LRS) and the corresponding local MeS (LMS) with the highest similarity is found by comparing the similarity among the LRS and each LMS. The Pearson correlation coefficient is utilized to evaluate the similarity. Then, the corresponding spectrum shift can be achieved by the wavelength difference between the LRS and the matched LMS.
Figure 6 shows the cross-correlation results of the whole spectra and local spectra. The comparison results indicate that the cross-correlation result of whole spectra has fake peaks and the similarity is lower than 0.25. However, a single peak of a high SNR is achieved through the cross-correlation of the local spectra and the similarity is ~0.8 which is raised by 4-fold. It is experimentally verified that utilization of the local spectrum similarity characteristics can effectively suppress the appearance of multi and fake peaks, and improve the similarity and the SNR of the cross-correlation results, so this method can achieve a higher strain measurable range using a narrow swept spectrum range.
2.2. The Similarity Evaluation Function Based on the Least-Square Method
Based on the local similarity characteristics of the RS spectrum, the spectrum shift between the LRS and LMS can be achieved through matching the LRS of the highest similarity in the MeS. Therefore, similarity evaluating function determines the accuracy of the achieved spectrum shift and data processing rate. In conventional OFDR methods, the evaluation criterion of similarity is on the basis of the Pearson correlation coefficient. The Pearson correlation coefficient cannot fully express the trend between the LRS and LMS because it is only a measure of the linear correlation [
8]. Besides, the Pearson correlation coefficient cannot maintain a high performance when the distributed strain is over 2000 με, even if the similarity is improved by local spectrum characteristics. Multiple and fake peaks will appear in the similarity evaluation results and affect matching the LRS on the MeS, so the similarity evaluating function must be researched and improved. The cross-correlation is invalid in measuring a larger strain, so similarity between reference and measurement spectra is used to find their shift. In the conventional method, the Pearson correlation coefficient can reflect the correlation or similarity between two signals. The sequence shift represents spectrum shift. The Pearson correlation coefficient is written as:
where,
is the covariance of spectra sequences
and
;
and
are the variance of spectra sequences
and
.
The least squares function is usually utilized to evaluate the approximating degree of two curves and achieve the fitting function [
9,
10]. On the other hand, the least squares function can also represent the Euclidean distance between two curves. In the match of the LRS in the LMS, the trend of these two spectra is more considerable and the least squares function is therefore more suitable for evaluating the spectrum similarity. It can be concluded that when the residual sum of squares reaches the lowest, the trend bias between spectra is smallest. A similarity evaluation function based on the least-square method is thus proposed. Furthermore, there are two factors need to be taken into consideration for the practical application: (1) the similarity between each LMS and a fixed LRS is needed to be evaluated and compared; (2) two local spectrum segments with a large amplitude difference may influence similarity evaluation badly. Therefore, normalization operation is needed before each evaluation function. The similarity evaluation function is constructed by residual sum of squares between normalized LMS and LRS:
wherein,
and
are the local spectrum sequences and the sequence length is
n (note: the data is discrete and variables of the above function are also discrete).
The sum of the squares of the residuals between the fixed LRS and each LMS can be considered as a similarity index in the matching procedure. The value of reaches its minimum when the most similar spectrum is matched.
Next, the performances of the similarity evaluation function based on the Pearson correlation coefficient and the least squares function are compared experimentally. Experiments on processing rate are conducted firstly. The time consumption of the two evaluation functions in MATLAB is compared when processing the data amount ranges from 1000 to 10,000.
Figure 7 shows the comparison results and it indicates that the processing rate of the similarity evaluation function based on the least squares is at least 10 times faster. Furthermore, the evaluation method based on the least squares takes more advantage in terms of the growth rate of time consumption to data amount, which has benefits on processing the distributed sensing of a high density and resolution.
Subsequently, the performance of the two evaluation functions on improving the similarity is compared experimentally. SNR is taken as the index for comparing the similarity improvement achieved by the different methods. The definition of SNR for the conventional evaluation function is the amplitude ratio of the highest peak to the secondary peak in the cross-correlation results wherein the highest peak represents the available spectrum shift signal and the secondary peak represents the noise signal. According to this definition, SNR of the whole and local spectrum cross-correlation result in
Figure 6 is respectively ~1.1 and ~3.1. By employing the local spectrum similarity characteristics, the SNR is raised by 3-fold. Considering the true peak value is the minimum sum of squares of residuals, the definition of SNR of the latter method is the ratio of the secondary lowest peak amplitude to the lowest peak amplitude.
Figure 8 illustrates the result of the method based on the least squares function and the SNR is over than 7. The similarity is further improved by 2-fold through combining the local spectrum similarity characteristics with the evaluation method based on the least squares function, so the improvement of the proposed method on the similarity can be experimentally verified.