Geometric phase seismic sensing using green's functions obtained from cross correlation

Abstract

Embodiments of a system and methods for utilizing ambient seismic noise to implement a modality based on the geometric phase for sending the effect of environmental conditions on ground properties are disclosed. The geometric phase is a global measure of the geometry of the seismic field which can be reconstructed via seismic interferometry. The change in geometric phase is the defined as the rotation angle of the complex state vector.

Description

Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215GEOMETRIC PHASE SEISMIC SENSING USING GREEN’S FUNCTIONS OBTAINED FROM CROSS CORRELATIONGOVERNMENT SUPPORT

[0001] This invention was made with government support under Grant No. 2242925 awarded by National Science Foundation. The government has certain rights in the invention.CROSS REFERENCE TO RELATED APPLICATIONS

[0002] This is a PCT application that claims benefit to U.S. provisional application serial number 63 / 689,471 filed on August 30, 2024 which is incorporated by reference in its entirety.FIELD

[0003] The present disclosure generally relates to acoustic technologies and environmental monitoring; and in particular to geometric phase seismic sensing using Green’s functions obtained from cross correlation.BACKGROUND

[0004] Global warming is causing a variety of environmental perturbations such as permafrost thawing which has a significant impact on the landscapes and infrastructure worldwide. For example, the Arctic region has been warming four times faster than the global average since 1979 (Rantanen et al., 2022). It is therefore crucial to develop and have at hand effective methods for monitoring in a continuous manner changes in ground properties.

[0005] Seismic techniques can be useful in monitoring changes in the physical properties of the ground by measuring sensitive changes in seismic wave velocity over time. However, in most cases, these velocity changes are small and the resolution is limited to relatively low frequency (e.g., 0.1 to 20Hz), long wavelength, and high depth. Therefore, it is desirable to monitor continuously the physical properties of the shallow ground over long periods of time.Attorney Docket No.: 085067-851342 Client’s Ref.: UA24-215

[0006] It is with these observations in mind, among others, that various aspects of the present disclosure were conceived and developed.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215BRIEF DESCRIPTION OF THE DRAWINGS

[0007] FIG. 1 A is an image of a distribution of the seismic stations in southwestern Iceland.

[0008] FIG. 1 B is a report of noise correlation functions (unfiltered) between vertical components of all stations pairs.

[0009] FIG. 2A is a time-frequency feature map of Ar] measurements using the vertical component and setting the reference station as MA1 - the white bands presenting data gaps.

[0010] FIG. 2B is a report showing correlation between Li and surface temperature (the color of each grid denotes the Pearson CC value between temperature and Aq at a certain frequency (horizontal axis) and with a reference station (vertical axis).

[0011] FIG. 2C is a series of reports showing examples of measured Aq time series in panel of FIG. 2A.

[0012] FIG. 3A is a report of enhanced correlation (in comparison to FIG. 2B) after correcting the estimated delay times (st).

[0013] FIG. 3B presents Li estimations in two typical Aq time series.

[0014] FIG. 3C presents Li estimations in two typical Aq time series.

[0015] FIG. 4 is a report illustrating comparison between variations in geometric phase (A / ) and relative seismic velocity (Av / v).

[0016] FIG. 5 is a series of graphs showing some environmental factors: all records were collected from ERA5 program (Hersbach et al. 2018) and localized to the present study location.

[0017] FIG. 6 is a series of Aq measurements by using two horizontal components: (a) and (b) are a correlation matrix between temperature and Aq that are measured from North-South and West-East components, respectively; and (c) and (d) are some typical Aq time series that are extracted from (a) and (b), respectively.

[0018] FIG. 7 is a series of reports showing correlation between Aq and surface pressure (P): (a) and (b) are correlation matrix and enhancement after delay correction, respectively, and (c) and (d) are time-lag analysis and delay estimationsAttorney Docket No.: 085067-851342Client’s Ref.: UA24-215 from two Ar| examples. The Aq are measured from vertical components, all settings are the same with FIGS. 2 and 3 above.

[0019] FIG. 8 is A v / v measurements in four frequency ranges. All time series are smooth with a 10-day running mean. The measurement from 0.1 -1 Hz is displayed in FIG. 4.

[0020] FIG. 9 is a series of measurements showing travel time shifts (df) in MWCS (a) and noise correlation functions (b) in Pair MA1-MA4 of four different frequency ranges. Gray bands denote the measurement windows that were set in the NCFs. Circles in Panel (b) are color-coded by the measured shift times of the example day from Panel (a).

[0021] Corresponding reference characters indicate corresponding elements among the view of the drawings. The headings used in the figures do not limit the scope of the claims.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215DETAILED DESCRIPTION

[0022] The present inventive concept relates to methods, systems, and devices / apparatuses for geometric phase seismic sensing using Green’s functions obtained from cross correlation. More specifically, Green's functions can be obtained from cross correlation techniques when either the location of an acoustic source is known or when it is unknown. The geometric phase as a function of frequency is a sensitive indicator of the structure of surface and subsurface features in the geological environment. The sensitivity of geometric phase sensing provides unique advantages in the detection of features (e.g., oil and gas deposits, geological structures) compared to current active or passive source techniques that depend only of the amplitude of reflected and transmitted waves.

[0023] This inventive concept opens new opportunities for the detection of surface and subsurface features in the geological environment which will allow for accurate and robust detection of oil and gas deposits, among other subsurface and surface resources. The information available from the geometric phase as a function of frequency will enable industrial application in sensing and discovery that are unavailable from current active source methods. As one example test application, using seismic data recorded in Iceland over a two-year period, the measured changes in geometric phase were shown to correlate with the seasonal fluctuations of surface air temperatures. An average delay time of 12 days is estimated between geometric phase sensing and surface temperature. It is demonstrated herein that the response of a geometric phase to temperature changes might be more sensitive than changes in seismic velocity. The high sensitivity and short delay times in the response of the geometric phase suggest that it may play a significant role in future environmental seismology.Introduction

[0024] Topological acoustic sensing, which focuses on the geometric attribute of mechanical waves, offers a new approach to monitor the state of wavesupporting media (Deymier & Runge, 2017). Theoretical models and experiments suggest that the changes in the geometric phase of seismic waves in the ground due to environmental perturbations or acoustic waves in materials due to defects andAttorney Docket No.: 085067-851342Client’s Ref.: UA24-215 imperfections are large (Lata, Deymier, Runge, Ferriere, & Huettmann, 2022; Lata, Deymier, Runge, & Clark, 2022; Lata et al., 2023; G. Zhang et al., 2024). The geometric phase is a global measure of the spatial geometry of an acoustic or seismic field, making it a highly sensitive metric to changes in the wave supporting medium (Lata, Deymier, Runge, Ferriere, & Huettmann, 2022). In this paper, we demonstrate the capability of the geometric phase for sensing changes in ground properties as a result of perturbations in environmental conditions. For this demonstration, we use realistic seismic recordings from a network of six sensors in southwest Iceland. These sensors are located at an elevation of "300 meters, where there is a deep-to-shallow seasonal freezing of the ground (Van Vliet-Lanoe & Gudmundsson, 2018). Iceland which is located on an active plate boundary, covered with volcanoes resulting from upwelling magma from the Mid-Atlantic Range spreading center and a deep-seated man tie plume (Einarsson, 2008; Weisenberger & Selbekk, 2009). In addition to geothermal fields, Iceland also has glaciers and permafrost above ~ 800 meters above sealevel, based on geomorphic features (Van Vliet-Lanoe & Gudmundsson, 2018). We reconstructed the Green’s function by cross-correlating seismograms between pairs of sensors among the six stations. The complex frequency-dependent Green’s functions are used to construct a discrete representation of the seismic field. This representation is then employed to calculate the geometric phase change with respect to a reference as a function of time. We observe seasonal changes of the geometric phase which correlate with the freezethaw cycle of the ground, which in turn is influenced by environmental factors, such as surface air temperature.Data and MethodsSeismic Noise Recordings

[0025] Seismic array and noise recordings were collected from six stations that are deployed near the Hengill geothermal region, in the southwestern Iceland (FIG. 1A). FIG. 1 A shows a distribution of the seismic stations - all stations belong to the CO-SEISMIQ project (Oberman & Swiss Seismological Service (SED) at ETH Zurich, 2018; Dahm at al., 2021 ). FIG. 1 B shows noise correlation functions (unfiltered) between vertical components of all station pairs on January 1 , 2020.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215

[0026] All of these seismic stations are temporary broadband stations and are part of the Control Seismicity and Manage Induced Earthquakes (COSEISMIQ) project, operated by GFZ German Research Centre for Geosciences and the Swiss Federal Institute of Technology (Dahm et al., 2021 ; Obermann & Swiss Seismological Service (SED) at ETH Zurich, 2018). These stations were originally deployed to monitor and forecast induced seismicity, which can be generated by the two largest geothermal power plants in Iceland (Grigoli et al., 2022). Approximately two years of seismic recordings were collected from August 2019 to July 2021 . Each station consists of two horizontal components (HHN and HHE: North-South and East-West) and one vertical component (HHZ), all sampled at 200 samples per second.Geometric phase change measurements

[0027] First, it was desired to reconstruct Green’s functions by crosscorrelating seismograms from two stations. The following preprocessing steps can be applied to raw seismograms: demeaning, detrending, and a highpass filter of 0.1 Hz. All seismograms can be divided into 1800-second slices with a 50% overlap to perform cross-correlation. To eliminate short perturbations from possible seismicity and improve the quality of signal data, the threshold was set as three times the root mean square and conduct spectral whitening for each slice (Lecocq et al., 2014). All daily noise correlation functions (NCFs) were saved and stacked every 10 days to retrieve robust Green’s functions. An example of daily NCFs of all station pairs is displayed in FIG. 1 B. The Green’s functions are Fourier transformed in the spectral domain. For each frequency, the spectral Green’s functions are complex quantities with amplitude and phase. We can define one station as the reference. For instance, setting station MA4 as the reference, we only cross-correlate each individual station that is linked to MA4. The NCFs of other station pairs (e.g., the pair of stations MA1 -MA7) are not considered. So the NCFs of five pairs of stations, are used to construct a five-component complex state vector. Such a state vector can be constructed for every station serving as the reference.

[0028] We build the complex state vector by involving NCFs as follows:Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215

[0029] where the complex state vector at a certain day t, ( >), is a function of frequency OJ and consists of N components. We apply the Fast Fourier Transform to each component of CNei0n, which includes its both real (Re) and imaginary (Im) parts. The denominator is to normalize all magnitudes to scale all components at the same level. So, the change in geometric phase (A„) of two state vectors can be represented by their angle difference:

[0030] We measure the angle difference by taking the areas function of the real part of the dot product between these two complex vectors. * denotes the complex conjugate. Here, we define the state vectors that are averaged from 2020-01-01 to 2020-01 -31 as the reference vector Cref. We suggest that the selection of the reference vector is important to Anmeasurements, and additional information is provided in the “Discussion” section.

[0031] We measure Anusing all daily NCFs throughout the frequency range in the Fourier domain. The final measured An(o>, t) is a time-frequency matrix, where each value represents the rotated angle with respect to the reference vector CrefThe temporal interval of Anis one day. All values of An(<o, t) will fall in the range from 0 to 7i(180°).Quantitative Analysis and Time-lags between Independent Variables

[0032] We calculate the linear Pearson correlation coefficient values between Anand environmental perturbations to evaluate their corresponding changes:Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215

[0033] where cov denotes the covariance between two variables X and Y, a denotes their standard deviation. Before calculating the correlation, we smooth both variables with a 30-day running mean to remove high-frequency signals. The calculated correlation coefficient (CC) value ranges from -1 to 1 , representing the regression slope between these two variables. Positive or negative values indicate that X and Y are in a correlation or anti-correlation relationship, respectively.

[0034] Furthermore, we utilize two methods to measure the time lag of Anwith respect to environmental perturbations. The first method is a simple movingwindow correlation. We calculate the Pearson CC values by shifting A?lwith a range of lag times, and we determine the time delay (At) with the highest CC value. Another method is cross-wavelet transform, which enables us to evaluate each local phase lag in periodogram (Torrence & Compo, 1998; Grinsted et al., 2004). We stretch and translate a wavelet function (i ~) to filter the variable X or Y into a continuous wavelet domain ( ^.):

[0035] where s and n are the stretch scale and translate index of i . 5t - 1day is an uniform time step. We then compute the cross-wavelet transform (XWT) between X and Y :WXY= WXW*,

[0036] where * denotes complex conjugation. Thus, we can measure the local relative phase lags between X and Y in the WXYargument, and we determine the time delay At as the average of all local phase lags within a significant area of 99%.ResultsSeasonal Fluctuations of Geometric Phase Changes

[0037] We display a measured time-frequency feature map of Anusing vertical components with MA1 as the reference station (FIG. 2A). FIG. 2 showsAttorney Docket No.: 085067-851342Client’s Ref.: UA24-215 measurements of geometric phase changes (An). We observe seasonal fluctuations of Anduring these two years, including low values in winters and high values in summers. These seasonal fluctuations seem to be more significant in the frequency range from 20 to 60 Hz (FIG. 2A). We then collect environmental factors that may have effects on the variations in ground properties, including surface temperature, air pressure, snow depth, rainfall, and soil moisture (FIG. 5). All these environmental datasets are collected from the ERA5 global reanalysis, which uses data assimilation and combines model forecast with new observations to produce the best-fitted estimates of the parameters (Hersbach et al., 2018). The spatial resolution of the gridded ERA5 datasets are 0.25°x 0.25° (around 27 km> \2km in the study site), and we select datasets at 64°N, 21 ,5°W that are closest to the seismic array. Surface temperature and pressure (combined air and snowstatic measurements) are two major factors that involve distinct seasonal fluctuations (FIG. 5). Therefore, we consider surface temperature and pressure as two most related environmental factors, and then analyze the measured Anwith them. We calculate a correlation matrix between temperature and measured Anat different frequency ranges and different reference stations (FIG. 2B). In different reference stations, the major correlations consistently occur from 20 to 40 Hz, and other strong correlations occur at higher frequencies, such as 60 to 80 Hz with reference stations MA2, MA4 and MEI05). The frequencies with high CC values are overall consistent with the significant seasonal fluctuations as shown in FIG. 2A. We display four typical Antime series that have high CC values with temperature (FIG. 2C).

[0038] All geometric phase changes displayed in FIG. 2 are measured using vertical components of six seismic stations. To validate the robustness of Anmeasurements and analysis, we provide two additional tests using horizontal components (HHN and HHE) of these seismic stations. The correlation between temperature and Anmeasured from the horizontal components is generally consistent with the vertical component results in FIG. 2, especially high CC values in the range of 20 to 40 Hz (FIG. 6A and 6B). Some Antime series achieve even higher correlations (FIGS. 6C and 6D). Therefore, we suggest that the measured geometric phase are highly consistent among all three components of seismic stations.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215Time-lag analysis of geometric phase changes

[0039] The delayed responses of seismic observables with respect to environmental factors have been widely identified in various monitoring cases (Tsai, 2011 ; Mordret et al. , 2016; Luo et al. , 2023; S. Zhang et al. , 2023). This delay commonly occurs when factors such as temperature or pressure diffuse from the surface to the investigated depth. The delay time may be controlled by the diffusion rate and investigated depth (Tsai, 2011 ; Mordret et al., 2016). We utilize two methods to conduct a time-lag analysis between the major periods of temperature and An, providing similar delay time (At) evaluations. Hence, we re-calculated the correlation matrix (FIG. 3A) by shifting Anwith the evaluated At from the sophisticated cross-wavelet transform method (bottom panels in FIGS. 3B-3C). FIGS. 3A-3C show time-lag analysis of geometric phase changes (An). We suggest that for the most correlated Antime series, which have high CC values, their correlations can be further enhanced after correcting At. Furthermore, we note that the similar At values are evaluated from reference stations of MA4, MA6, MA7, and MEI05 at around 30 Hz (FIG. 3A), which possibly agree with the significant An, measurements from 20 to 40 Hz in FIGS. 2A and 2B. We provide an example of the reference station of MA4 at 32 Hz in FIG. 3B. In a few cases, the At could be too small for us to consider, such as the example in FIG. 3C. In summary, that the evaluated At in FIG. 3A has an average and maximum value of 12 and 47 days, respectively, with respect to temperature. Therefore, the delay time we evaluate here serves as a factor when evaluating the geometric phase as an effective monitoring tool. We filter out the high-frequency perturbations and only consider the major seasonal fluctuations in the time-lag analysis. Therefore, we claim that the delay might be overestimated when the temperature experiences strong perturbations, such as in March, 2021 .

[0040] In addition to temperature perturbations, we also analyzed the correlation between variations in At and the local surface pressure field, combining air and snow static pressure in winters (FIG. 7). Overall, the correlation between A?1and pressure is not as high as that with temperature. We evaluate the At values with respect to pressure, and overall they have average and maximum values of 19 and 50 days, respectively. The relatively blurred seasonal fluctuations of pressure perturbations mayAttorney Docket No.: 085067-851342 Client’s Ref.: UA24-215 hinder the higher their correlations with An. However, we observed that some measuredAnhave consistent fluctuations with pressure, such as cases in FIGS. 7C and 7D.Therefore, we still consider the possible contribution of changes in the pressure field to the geometric phase.Comparison of changes in geometric phase and seismic wave velocity

[0041] As mentioned, the relative seismic velocity changes (Av / v) have also been widely used to monitor environmental factors (Sens-Schonfelder & Wegler, 2006; Mordret et al., 2016; Lecocq et al., 2017; James et al., 2019; Lindner et al., 2021 ; Albaric et al., 2021 ; Luo et al., 2023; S. Zhang et al., 2023). Here, we measure the regional averaged (Av / v) by using the same noise records from all stations. We set the same reference level by stacking NCFs within January 2020 and measure (Av / v) using a moving-window cross-spectrum (MWCS) method (Clarke et al., 2011 ; Lecocq et al., 2014). More parameters in the MWCS can be found in Table 1.

[0042] Table 1 (Parameters for Av / v measurements in the Moving-window cross-spectrum (MWCS) technique):Control parameters MWCS filter

[0043] Among the four test frequency ranges, only the (Av / v) from 0.1 -1 Hz shows likely seasonal fluctuations (FIG. 8). From the MWCS analysis, we observe that the measured traveltime shifts (dt in FIG. 9A) in high frequencies are relatively small, possibly resulting from the low-coherent dynamic phases in NCFs (FIG. 9B).

[0044] FIG. 4 shows a comparison between variations in geometric phase (An) and relative seismic velocity. We display the An(in black) measured from the reference station of MA6 at 22 Hz. The Av / v (in blue) is measured using 01 .-1 Hz and is smoothed with a 5-day running mean. The blue error bars denote the misfits in travel time shift measurements (FIG. 9). In FIG. 4, we display the measured (Av / v) from 0.1 -1Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215Hz and one of the measured Antogether with the temperature, including overall corresponding seasonal fluctuations. We note that some small perturbations in (Av / v) do not respond as well as those in An, such as in March, 2021 (FIG. 4). The mismatch of (Av / v) may be due to strong temperature attenuation in the deeper subsurface, as we use a low-frequency (0.1 to 1 Hz) band for investigation. However, the low coherence of phases in high frequencies (FIG. 9) may limit the investigation of (Av / v) in the shallower near-surface. Furthermore, different measurement windows, which may correspond to velocity changes of different wave types, can introduce additional discrepancies in (Av / v) . On the other hand, the frequency-dependent Ancan be continuously measured across the entire frequency range. This may assist us in obtaining robust measurements at high frequencies and reliable monitoring of the near- surface. The measured angle difference of state vectors also helps alleviate potential uncertainties resulting from window selection in NCFs. Consequently, this comparison indicates that the response of geometric phase to temperature might be more effective than seismic velocity in our monitoring case. However, we do not exclude the possibility that other optimal parameters or workflows can contribute to better (Av / v) measurements. The responses of geometric phase to other environmental conditions are expected to be investigated by future studies.Discussion

[0045] Based on observation and analysis of the measured geometric phase changes, we suggest that the changes of geometric phase probably respond to physical properties of the ground, which further result from environmental perturbations over time. In the following section, we aim to explain the mechanism and quality control involved in geometric phase measurement.Illustration of changes in phase space based on a harmonic oscillator model

[0046] We consider the equation of motion of an externally driven damped harmonic oscillator as a metaphor for a ground wave:Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215

[0047] where / J is a damping coefficient of the ground,is a characteristic frequency-dependent on mass density and stiffness of the medium. When the magnitude of the driving froce, F, is unity, then g is the temporal Green’s function of the oscillator. Conventionally, one assumes that g(t) will follow the driving force in time, that is, g(t) = Ge10'^ where G is the spectral amplitude of the Green’s function to be determined. Inserting this ansatz into Equation 6 and simplifying by e1^:

[0048] To calculate the change in geometric phase with frequency )d, we first normalize the Green’s function amplitude

[0049] Where G* denotes the complex conjugate of G. Let us assume that the physical properties of the oscillator are a slowly varying function of time, / z(t) or c (t) . If the variation of the physical properties with time is much slower than the period of the oscillator, we can assume that the oscillator will be in its ground state at all times and the change in geometric phase is then given by (Deymier & Runge, 2017):

[0050] For the sake of illustration, we limit the time dependency to the damping coefficient and assume a>Qto be a constant. After extensive mathematical manipulations, inserting Equations 7 and 8 into Equation 9 yields:Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215

[0051] The change in geometric phase as a function of time is directly related to the change in damping coefficient as a function of time. Let us consider that darning is related to time indirectly through temperature, / z(T(t)), then Equation 10 may be rewritten as:

[0052] Equation 11 relates the change in geometric phase with temperature to the change in damping with temperature.Sensitivity evaluation of the change in geometric phase

[0053] Based on the demonstration above, the sensitivity of the change in geometric phase varies with different frequencies, and our correlation and time-lag analysis (FIGS. 2B and 3A) reinforce this variability. The optimal sensitive frequency may be related to the local resonance of a noise source (Lata, Deymier, Runge, Ferriere, & Huettmann, 2022). Colombi et al. (2016) suggested that in the case of field measurements, the local resonant structure is possibly determined by the arrangement of scatters such as trees, which interact with seismic surface waves during propagation. Theoretical sensitivity tests also support that the change in geometric phase tends to be more sensitive near the resonance frequency of local tree distributions (Lata, Deymier, Runge, Ferriere, & Huettmann, 2022). Therefore, in our case, the significant phase changes from 20 to 40 Hz possibly indicate a sensitive frequency range that is close to the local resonant structure (Colombi et al., 2016). This possible resonance frequency might be determined by some high-frequency noise sources, such as geothermal power plants or near-surface scatters, since there are no forests covered in our study area. We expect that the unique local resonant structure can be further investigated to enhance the monitoring of the change in geometric phase in different environmental conditions.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215In addition to the frequency range, we note that the sensitivity of phase changes may also vary with different reference stations (FIG. 2B). Effectively, the Green’s function G(r, r') spans an infinite space, since r and r' represent two continuous location variables. However, in our field measurement, we reduce the dimension of the space by setting a certain reference station and including NCFs associated with it within a state vector (Equation 1 ). Therefore, here we are exploring discrete subspaces by setting different reference stations, which may result in different sensitivities of geometric phase sensing.

[0054] In contrast with velocity changes, the change of geometric phase has a given range from 0 to n, indicating that a proper reference vector (Crefin Equation 2) is important to measure the change in geometric phase. For example, if we choose the state vector of a certain day to as the reference, the phase change at t0will be at the lower limit, An(t0) = 0, due to other values are referenced to An(t0). However, this defined A„(t0) = 0 may be spurious when we choose t0with its true An(t0) value is close to the upper limit n. To mitigate the uncertainty given by the reference vector selection, we define it as an average level of all daily vectors within January, 2020. Therefore, in our measurements, no An(t) reaches the boundary values and all An(t) time series are within a consistent range (from 1 .2 to 1 .8). We suggest that the selection of the reference vector is not exclusive, and some blind tests are helpful for us to select an appropriate reference. For example, we can start by setting the first day’s vector as the reference, which will give the overall variations of the time series without too many outliers. Then it may be a good idea to average vectors of a few days near the minimum level as the reference and re-calculate throughout the workflow.

[0055] The measured changes in geometric phase over time for the Iceland array are likely related to the changes in physical properties of the ground. We suggest that these temporal changes are most likely caused by temperature, but they could also be influenced by surface pressure, water saturation or other factors, depending on the season. Our study demonstrates a first but strong correlation between geometric phase and seasonal variations in the shallow ground layers, and this method holds great potential for monitoring changes in the shallow layers due to environmental conditions. We are currently working on developing a more comprehensiveAttorney Docket No.: 085067-851342Client’s Ref.: UA24-215 mathematical and physical framework. We are aiming to provide an analytical solution, along with an uncertainty analysis, for measuring the changes in geometric phase.Conclusion

[0056] A novel geometric phase sensing approach from topological acoustics is introduced to monitor changes in the environmental conditions and ground properties. We define the geometry phase of a seismic field by reconstructing the ground Green’s function through seismic noise recordings. The changes in geometric phase can be measured from the rotation of a complex state vector which dimension is associated with the number of sensors in the seismic network. In the southwestern Iceland case study, we have observed highly consistent seasonal fluctuations between geometric phase and surface air temperature. A generalized harmonic oscillator model helps us understand how changes in the phase space are influenced by the ground property changes, which in turn result from environmental conditions. The technique has several advantages, including high sensitivity and a short delay in responding to changes in ground property. Moreover, since it does not rely on seismic phases, such as body or surface waves, this technique helps to mitigate any additional biases associated with wave selection. We propose that the benefits of geometric phase sensing can be further investigated using high-frequency, longterm seismic recordings. We expect that this novel technique can be potentially applied to investigate complex environmental hazards.

[0057] It should be understood from the foregoing that, while particular embodiments have been illustrated and described, various modifications can be made thereto without departing from the spirit and scope of the invention as will be apparent to those skilled in the art. Such changes and modifications are within the scope and teachings of this invention as defined in the claims appended hereto.

Claims

Attorney Docket No.: 085067-851342Client’s Ref.: UA24-215CLAIMSWhat is claimed is:1 . A method for geometric phase sending of environmental changes based on seismic noise, comprising: accessing data defining seismic recordings, the seismic recordings including ambient seismic noise; and implementing, by at least one processor, a modality based on geometric phase for sensing environmental conditions associated with the seismic recordings, the modality including a correlation between the geometric phase and the environmental conditions.

2. The method of claim 1 , wherein the geometric phase is a global measure of the geometry of the seismic field which can be reconstructed via seismic interferometry.

3. The method of claim 1 , wherein the change a change in the geometric phase is defined as the rotation angle of the complex state vector.

4. The method of claim 1 , further comprising: defining the geometry phase of a seismic field by reconstructing the ground Green’s function through seismic noise recordings; and measuring changes in geometric phase from the rotation of a complex state vector which dimension is associated with a number of sensors in a seismic network associated with the seismic recordings.

5. The method of claim 1 , wherein the modality characterizes the geometry of the seismic field in the ground and is strongly correlated to seasonal fluctuations in air temperature.Attorney Docket No.: 085067-851342Client’s Ref.: UA24-2156. The method of claim 1 , further comprising: reconstructing the Green’s function by cross-correlating seismograms of the data between pairs of sensors generating the seismic recordings.

7. the method of claim 6, further comprising: constructing, using the complex frequency-dependent Green’s functions, a discrete representation of a seismic field defined by the seismic recordings, the representation configured to calculate the geometric phase change with respect to a reference as a function of time.

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