Passive seismic monitoring with nonstationary noise sources

Evan Delaney1, Laura Ermert2, Korbinian Sager2, Alexander Kritski3, Sascha Bussat4, andAndreas Fichtner2

ABSTRACT

Heterogeneous, nonstationary noise sources can cause trav-eltime errors in noise-based seismic monitoring. The effectworsens with increasing temporal resolution. This may leadto costly false alarms in response to safety concerns and limitour confidence in the results when these systems are used forquasi-real-time monitoring of subsurface changes. Therefore,we have developed a new method to quantify and correct thesetraveltime errors to more accurately monitor subsurface con-ditions at daily or even hourly timescales. It is based on theinversion of noise correlation asymmetries for the time-depen-dent distribution of noise sources. The source model is thenused to simulate time-dependent ambient noise correlations.The comparison with correlations computed for homogeneousnoise sources yields traveltime errors that translate intospurious changes of the subsurface. The application of

our method to data acquired at Statoil’s SWIM array, a perma-nent seismic installation at the Oseberg field, demonstrates thatfluctuations in the noise source distribution may induce appar-ent velocity changes of 0.25% within one day. Such biasesthereby likely mask realistic subsurface variations expectedon these timescales. These errors are systematic, being pri-marily dependent on the noise source location and strength,and not on the interstation distance. Our method can thenbe used to correct for source-induced traveltime errors by sub-tracting these quantified biases in either the data or modelspace. It can furthermore establish a minimum threshold forwhich we may reliably attribute traveltime changes to actualsubsurface changes, should we not correct for these errors.In addition to the aforementioned real data scenario, we applyour method to a synthetic case for a daily passive monitoringoverburden feasibility study. Synthetics and field experimentsvalidated the method’s theory and application.

INTRODUCTION

Passive monitoring using ambient noise

Though its first successful application begins on the sun (e.g.,Duvall Jr. et al., 1993; Woodard, 1997; Rickett and Claerbout,1999), ambient noise tomography has become a standard tool forimaging earth structure on local, regional, and global scales (e.g.,Sabra et al., 2005; Shapiro et al., 2005; Larose et al., 2006; Lin et al.,2008; Ruigrok et al., 2008; Nishida and Montagner, 2009; Bussatand Kugler, 2011; Verbeke et al., 2012; Boué et al., 2013; Nakataet al., 2015; Haned et al., 2016). The omnipresence of the ambient

noise field makes this method well-suited for the monitoring oftime-lapse subsurface changes beneath volcanoes (Brenguier et al.,2008b; Obermann et al., 2013), along active seismic faults ( Bren-guier et al., 2008a; Obermann et al., 2014), in hydrocarbon reser-voirs (Bakulin and Calvert, 2004; de Ridder and Biondi, 2013;Mordret et al., 2014b; de Ridder and Biondi, 2015), and withinthe vicinity of geothermal stimulation sites (Hillers et al., 2015;Obermann et al., 2015).Most ambient noise tomographies rest on the approximation of

an interstation Green’s function by the crosscorrelation of noise re-corded at two stations. Theoretical requirements for Green’s func-tion retrieval include the homogeneous distribution of uncorrelated

Manuscript received by the Editor 21 June 2016; revised manuscript received 8 February 2017; published online 19 May 2017.1Formerly TU Delft, Delft, The Netherlands; Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland; and Rheinisch-Westfälische Tech-

nische Hochschule (RWTH) Achen, Aachen, Germany; presently Statoil, Stavanger, Norway. E-mail: evde@statoil.com.2Institute of Geophysics, Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland. E-mail: laura.ermert@erdw.ethz.ch; korbinian.sager@

erdw.ethz.ch; andreas.fichtner@erdw.ethz.ch.3Statoil, Trondheim, Norway. E-mail: akr@statoil.com.4Statoil, Bergen, Norway. E-mail: sbus@statoil.com.© 2017 Society of Exploration Geophysicists. All rights reserved.

KS57

GEOPHYSICS, VOL. 82, NO. 4 (JULY-AUGUST 2017); P. KS57–KS70, 13 FIGS.10.1190/GEO2016-0330.1

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noise sources, the equipartitioning of the ambient noise field, andthe absence of attenuation (Lobkis and Weaver, 2001; Weaver andLobkis, 2001; Malcolm et al., 2004; Weaver and Lobkis, 2004; Cur-tis et al., 2006; Wapenaar and Fokkema, 2006; Wapenaar et al.,2006). Even though Lawrence and Prieto (2011) and Lawrence et al.(2013) suggest that attenuation may be included when azimuthallywell-distributed noise sources are present, their result has been con-tested by the analytical work of Tsai (2011).In the earth, the conditions for perfect Green’s function retrieval

are not satisfied. The earth has spatially variable scattering proper-ties and locally strong attenuation, especially in the near-surface,which prevents equipartitioning on a wide range of scales (Mulga-ria, 2012; Sens-Schoenfelder et al., 2015). Furthermore, noisesources at any frequency are heterogeneously distributed and non-stationary (Peterson, 1993; Schulte-Pelkum et al., 2004; Bonnefoy-Claudet et al., 2006; Stehly et al., 2006; Tanimoto et al., 2006;Stutzmann et al., 2012; Latorre et al., 2014; Ermert et al., 2016).The detrimental effects on Green’s function retrieval include incor-rect traveltimes and amplitudes (Tsai, 2009; Cupillard and Capde-ville, 2010; Froment et al., 2010; Tsai, 2011; Fichtner, 2014) andthe appearance of spurious arrivals that largely preclude the use ofhigher mode surface waves in noise correlations (Halliday and Cur-tis, 2008; Kimman and Trampert, 2010). To some extent, Green’sfunction retrieval may be improved by processing (Bensen et al.,2007; Schimmel et al., 2011; Groos et al., 2012), but the proceduresare subjective and lead to hardly predictable effects on the sensitiv-ity of noise correlations to earth structure (Fichtner, 2014).

Motivation: Limitations caused by heterogeneous,nonstationary noise sources

Although traveltime errors induced by the heterogeneity of noisesources are often small enough for time-independent tomography(Froment et al., 2010), the nonstationarity of noise sources maytranslate into spurious time-lapse changes of subsurface structure.Indeed, Bussat (2015) presents an example where 5 Hz Scholtewaves undergo a 2% shear-wave phase velocity change in fourhours, unrelated to subsurface production activity, but instead asso-ciated to a passing storm.To reduce such artifacts, two different approaches are typically

followed: Either the coda of the correlations is used under theassumption that multiple scattering reduces the influence of thenoise source direction (Brenguier et al., 2008a, 2008b; Obermannet al., 2013, 2014) or the averaging interval of the noise correlationsis increased until noise source fluctuations are believed to averageout to potentially recover surface and body waves (Claerbout, 1968;Bakulin and Calvert, 2004; Bensen et al., 2007; de Ridder et al.,2014; Mordret et al., 2014b). Although coda waves have been usedsuccessfully in monitoring, their use is often not an option eitherdue to the lack of sufficiently strong scatterers or to attenuation thatmay prevent their clear observation. This aspect is particularly im-portant in near-surface applications in which Q can be in the orderof 10, thus preventing the emergence of coda.In the context of hydrocarbon and geothermal reservoirs, the

required time resolution is dictated by the timescales over whichserious incidents may evolve, i.e., a few hours or days. Concrete ap-plications include detecting — and preferably preventing — out-of-zone injections (Verdon et al., 2011), monitoring slop (producedwater) and cuttings injectors, overseeing carbon sequestration sites(Eiken et al., 2011; Raknes et al., 2015), and optimizing future pro-

duction targets by corroborating changes in pressure and productionhistory profiles when current wells undergo shut-in tests and whennew producers and injectors are brought online.Because the extent to which source-related limitations are practi-

cally relevant is likely to be application-dependent, we consider thespecific case of Statoil’s Seismic Waste Injection Monitoring(SWIM) array, stationed above the Oseberg field off the Norwegiancoast (Figure 1), and we aim to answer the following questions:(1) Within the frequency band 0.55–1.55 Hz, do noise source dis-

tributions remain the same on daily timescales — in terms of azi-muthal coverage, strength, and frequency? (2) If not, how much cannoise sources fluctuate from day to day? (3) What traveltime errorsdo we produce when we wrongly assume that correlations equalGreen’s functions? (4) What is the potential magnitude of the imagedchanges in apparent velocity structure from day to day due to time-variable noise sources? (5) Are these artifacts sufficiently significantas to mask true variations of the subsurface? (6) If they are, how canwe remove them and further maximize our time resolution?

Outline

This paper is the first real data application of the theory estab-lished by Hanasoge (2014) and extends upon this via methodsfound by Ermert et al. (2016). It is organized as follows: First,we compute daily stacks of noise correlations. Using measurementsof correlation asymmetry, we invert for the power-spectral densitydistribution of the noise sources on a daily basis. With this realisticpicture of noise sources and their temporal variability, we perform aseries of synthetic inversions for 2D velocity structure: As artificialdata, we use correlation functions computed for a homogeneousearth model and the heterogeneous noise source distribution of aspecific day. For the computation of synthetic data, we use the samehomogeneous earth model and a noise source distribution that ishomogeneous as well. Although the earth model used to computeartificial and synthetic data is identical, the different distributions ofnoise sources induce systematic traveltime differences. Wave-equa-tion traveltime inversion then allows us to translate these traveltimedifferences into 2D maps of apparent velocity. The temporal vari-ability of these maps provides a measure of apparent subsurfacechanges caused by the nonstationarity of the heterogeneous noisesource distribution. Finally, instead of blindly performing waveforminversion with these biased traveltimes, we formulate how to re-move these errors so that our velocities are unaffected by noisesource nonstationarity. Preferably, this is done in the data space be-fore proceeding to the tomography. We now discuss these steps infurther detail.

DATA: AMBIENT NOISE CORRELATIONSAT THE SWIM ARRAY

Acquisition layout, data characteristics, and processing

The SWIM array, shown in Figure 1, consists of 172 4C receivers(micro electro-mechanical systems (MEMS) accelerometer and hy-drophone) linked through a single ocean-bottom cable. The con-figuration lies along the ocean floor, roughly 108 m below meansea level, approximately 150 m south of the Oseberg C platform.Its purpose is to monitor cuttings injected into the overburden.Receiver spacing is approximately 50 m for receivers on the outsideboundaries of the array and reduces to approximately 25 m for

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receivers in its inner portion. The spread of the array’s outer rightside is approximately 1.74 km and the horizontal separation of theright side’s inner and outside segments halfway down the array isapproximately 300 m. Similar properties apply to its left side. Fur-ther information about its layout can be found in Bussat (2015).The frequency response ranges from direct current, used to mea-

sure the gravity vector, to beyond 200 Hz. All receivers operate con-tinuously, sampling at 500 Hz. The accessible data range from April30, 2014 to May 18, 2015 totals approximately 40 TB.Above 1.55 Hz, we observe strong noise originating from the

platform. Because noise at higher frequencies is most likelyanthropogenic (Bonnefoy-Claudet et al., 2006) and — in our case— predominantly spatially stationary, we limit our study tofrequencies of 1.55 Hz and below. We also do not study frequenciesless than 0.55 Hz because our method requires that causal andacausal arrivals be clearly distinguishable.To compute interstation correlation functions, we first divide the

data for one day into 225 s segments. For each segment, we rotatethe data to correct for misorientation, extract the vertical compo-nent, low-pass filter to 10 Hz, downsample to 20 Hz, and computegeometrically normalized crosscorrelations for all possible receiverpairs (Schimmel, 1999; Seats et al., 2012). We then stack all seg-ments to obtain the averaged correlations for one day. Even thoughwe do not expect to observe waves taking more than 10 s to propa-gate across an array with <1.8 km maximum offset, we lag the cor-relations up to 30 s. This allows us to estimate the signal-to-noiseratio (S/N) and reject correlations without clear signals. We delib-erately do not apply nonlinear processing techniques such as spec-tral whitening or one-bit normalization (Bensen et al., 2007; Grooset al., 2012) even though the latter has been shown to not add asystematic bias to the measurement in the case of Gaussian wave-field statistics and for a variety of other distributions (Hanasoge andBranicki, 2013; Fichtner et al., 2016). This is intended to avoid non-physical biases in the correlations (Fichtner, 2014) and to preservevaluable information on the distribution of the noise sources.

Spatiotemporal variability

Figure 2 shows a selection of correlation functions for differentreceiver-receiver azimuths and for two different days: July 21 andJuly 22, 2014. The correlation functions are dominated by large-am-plitude Scholte wave arrivals with a group velocity of approxi-mately 300 m∕s. All examples exhibit a pronounced asymmetrythat varies with the azimuth and time, thus attesting to the spatio-temporal variability of the underlying noise sources.

THE POWER-SPECTRAL DENSITY DISTRIBUTIONOF NOISE SOURCES

Modeling interstation correlations

To estimate the distribution of noise sources quantitatively in spaceand time, we start with the formulation of a forward model that allowsus to compute synthetic correlation functions for a predefined noisesource. Because our focus is on horizontally propagating Scholtewaves in vertical-component correlations, we adopt the acousticapproximation of the frequency-domain representation theorem:

uðx;ωÞ ¼Z

Gðx; ξ;ωÞNðξ;ωÞd2ξ; (1)

where uðx;ωÞ is the seismic wavefield at position x, ω denotes thecircular frequency, Nðξ;ωÞ is the noise source force density at posi-tion ξ, and Gðx; ξ;ωÞ is the Green’s function of the medium thatrelates the noise source to the wavefield. Based on equation 1, theinterstation correlation Cðxi; xkÞ ¼ uðxiÞu�ðxkÞ takes the form,

Cðxi; xkÞ ¼ZZ

Gðxi; ξÞG�ðxk; ξ 0ÞNðξÞN�ðξ 0Þd2ξd2ξ 0;

(2)

where we omit the dependence on ω in the interest of a condensednotation. Ambient noise in the oceans at frequencies between 0.5–2.0 Hz is most likely caused by the interaction of wind waves withwavelengths less than 25 m (Gimbert and Tsai, 2015). We can, there-fore, assume that the temporal correlation of neighboring noise sourcesNðξÞN�ðξ 0Þ in equation 2 is a δ-function compared with the seismicwavelength that exceeds several hundred meters. This widely usedapproximation (Woodard, 1997; Hanasoge, 2013, 2014; Fichtner,2014; Nishida, 2014) simplifies the interstation correlation to

Cðxi; xkÞ ¼Z

Gðxi; ξÞG�ðxk; ξÞSðξÞd2ξ; (3)

where SðξÞ ¼ NðξÞN�ðξÞ is the power-spectral density (psd) distribu-tion of the noise sources as a function of space and frequency. Equa-tion 3 constitutes a forward modeling equation that allows usto directly compute synthetic interstation correlations from the deter-ministic quantity SðξÞ, without the need to model a quasi-randomwavefield excited by stochastic sources.

Figure 1. SWIM array layout with its location given in the inset.The receiver spacing is approximately 50 m for the array’s outerborder, and it decreases to approximately 25 m for the inner section.The array’s outer right-side offset (from receiver 137 to 172) is1737.2 m and the horizontal separation of the right side’s innerand outer segments roughly halfway down the array (from receiver96 to 155) is 298.4 m. Due to the array’s symmetry, similar proper-ties apply to its left side.

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Measuring causal-to-acausal energy ratios

To extract robust information on the psd of the noise sources, wefollow the approach of Ermert et al. (2016) and measure causal-to-acausal energy ratios in the daily correlations, defined as

A ¼ ln

R ½wþðtÞC̄ðtÞ�2dtR ½w−ðtÞC̄ðtÞ�2dt¼ ln

EþE−

; (4)

where A is the correlation asymmetry, C̄ðtÞ is the time-domainversion of CðωÞ; wþðtÞ represents the window centered aroundthe causal wave of C̄ðtÞ, and w−ðtÞ signifies the window centeredaround the acausal wave, Eþ and E− denote the energy of the causaland acausal wave packets under wþðtÞ and w−ðtÞ, respectively. Themeasurement process is illustrated in Figure 3 for receiver pair 36-145 that shows clear causal and acausal Scholte wave arrivals.The amplitude ratio equals 0 in the case of a symmetric corre-

lation function, thus indicating that noise source energy firstrecorded by an interstation pair’s first receiver, which later propa-gates onward to the second receiver is the same as the energy firstarriving at the second receiver which then propagates toward thefirst receiver. However, when A < 0, the acausal amplitudes arelarger than those on the causal side, as observed for instance in Fig-ure 3. This signifies that the source strength on the side of the firstreceiver (receiver 36 in our example) is stronger than that on the sideof the second receiver.The measurement of amplitude ratios has the advantage of being

primarily sensitive to the noise source distribution, while beingpractically insensitive to viscoelastic attenuation (Lawrence andPrieto, 2011) and focusing induced by unknown earth structure.

Sensitivity kernels for the noise source psd

To solve an inverse problem for the psd of the noise sources, wecompute synthetic correlation functions using the forward modeling

Figure 2. Correlation functions for July 21 and July 22 for selectedreceiver-receiver azimuths in the frequency band 1.15–1.55 Hz. Thelogarithmic causal-to-acausal signal energy ratio, explained in moredetail in the following section, is given to the right of the plottedcorrelations. For a homogeneous noise source distribution, the log-arithmic energy ratios would be equal to zero. Instead, they varysignificantly between −2.64 and 1.88, with dependence on azimuthand time. This indicates a pronounced spatiotemporal variability ofthe noise sources that can be observed robustly. For interstation pair134–162 (black), its azimuth points slightly west of north, as indi-cated by the azimuth compass. Receiver 134 thus lies at the nock,whereas receiver 162 is at the arrow tip — in terms of the receivers’relative orientation to each other, and not absolute position in thearray. More energy comes from the north than from the south forthis receiver pair on July 22 (i.e., causal amplitudes are larger thanacausal). This is also supported by receiver pair 21-130 (white), inwhich more energy arrives from a northwesterly direction than fromthe southeast on this same day (i.e., acausal amplitudes are largerthan causal with this pair’s orientation flipped to that of pair 134-162). The fact that we observe spurious arrivals of nonzero ampli-tude information and an elongated wave train for receiver pair 36-111 (gray) on July 22 indicates that noise sources are movingthroughout the day and that even shorter time intervals would benecessary to better constrain their azimuthal position.

Figure 3. Correlation function for receiver pair 36-145. Blackdashed lines mark the measurement windows w−ðtÞ and wþðtÞ fromequation 4, centered around the acausal and causal Scholte waves,respectively. Gray dashed lines indicate windows used to estimateS/N, needed to eliminate correlation functions of insufficient qual-ity. These windows are centered at lag times in which no directarrivals are expected. In this example, the amplitude ratio is−0.55, and the S/N is 234.4 and 464.1 on the causal and acausalbranch, respectively. The S/N is defined as Eþ∕∫ ½wnþðtÞC̄ðtÞ�2dtfor the causal side and E−∕∫ ½wn−ðtÞC̄ðtÞ�2dt for the acausal side,where wnþðtÞ and wn−ðtÞ, respectively, denote the causal andacausal windows lacking any expected meaningful signal.

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equation 3. As an initial noise source model, we use an annulus ofconstant psd SðξÞ, as shown in the background of Figure 4. We thenquantify the difference between the synthetic correlation asymmetryAik and the observed asymmetry Ao

ik using the l2 misfit functional

χ ¼ 1

2

Xik

ðAik − AoikÞ2; (5)

where the subscripts i and k, respectively, denote the receivers in areceiver pair. Using adjoint techniques (Tarantola, 1988; Fichtner et al.,2006; Plessix, 2006; Fichtner, 2010), we can compute sensitivity ker-nels of the misfit χ with respect to the noise source psd (Tromp et al.,2010; Hanasoge, 2013; Fichtner, 2014; Ermert et al., 2016).Figure 4a shows the finite-frequency noise source kernel for sta-

tion pair 36-145, computed from the asymmetry measurement fea-tured in Figure 3. The kernel indicates where the noise source psdshould be increased (darker colors) and where it should be de-creased (lighter colors), relative to the azimuthally homogeneousinitial psd (background annulus).To reduce the computational costs of noise source inversion, we

adopt the ray-theoretical simplification of the noise source kernelsproposed by Ermert et al. (2016). The ray-theoretical kernel corre-sponding to the finite-frequency kernel in Figure 4a is shown belowin Figure 4b. The simplified kernels allow us to obtain images ofnoise sources with negligible computational effort, which is particu-larly beneficial for noise source imaging on a daily basis or evenshorter time intervals.As shown by Ermert et al. (2016), kernel-based noise source in-

version leads to results that are qualitatively similar to beamform-ing, in the sense that nearly identical source regions are mapped byboth methods. This is to be expected because sensitivity kernelsroughly act as beams pointing in the direction of possible noisesources, and with strength proportional to the noise source ampli-tudes. The outstanding advantage of our inversion over beamform-ing is that it provides the actual psd of the noise sources with properphysical units (m−4N4s2), which are needed to solve the forwardmodeling equation 3.

Noise sources on July 21 and July 22, 2014

Although we analyze data for a complete year, we limit ourselvesto the presentation of results for July 21 and July 22, 2014, in thefrequency band 1.15–1.55 Hz. This is sufficient to demonstrate theconcept. However, we have also generated noise source sensitivitykernels for all other days from April 30, 2014 to May 18, 2015, inthe frequency bands 0.55–0.95 Hz, 0.75–1.15 Hz, 0.95–1.35 Hz,and 1.15–1.55 Hz. These can be found in the supplementarymaterial at the following links: s1.pdf, 001.mp4, 002.mp4. Thesetwo days are chosen because July 21 and 22 are good representa-tives of the complete data set because there are days when noisesources change less, and other days when they change more.Though 172 receivers yield 14706 correlation functions, we limit

our analysis to station pairs with an offset that is sufficiently large toclearly distinguish causal and acausal arrivals. For the frequencyband 1.15–1.55 Hz, we therefore use offsets ⩾855.2 m, whichleaves 4190 station pairs.For July 21 and July 22, 2014, we reject a few station pairs with

poor S/N. This further reduces the number of correlations to 4144and 4054, respectively. The final azimuthal coverage, shown in Fig-ure 5, is very similar for both days. Nonetheless, we bin and weightthe sensitivity kernels appropriately (i.e., by dividing each bin’s cu-

mulative kernel sum by its fold to produce its kernel average) suchthat all azimuths are treated equally. This ensures that any changesobserved in the noise source kernels are not biased by data selection.The cumulative noise source kernels of all receiver pairs for July

21 and July 22, 2014, are displayed in Figure 6. Prior to an actualinversion for the noise source psd, these kernels provide an initialimage of the noise sources and their temporal variability. On July21, 2014, noise sources west of the SWIM array were stronger thanaverage, and they were weaker than average to the east. On July 22,2014, the distribution has changed, and stronger than average noisesources are now located to the north.The sensitivity kernels indicate that only azimuthal variations of

the noise source psd can be constrained. This justifies the inversion

Figure 4. Sensitivity kernels for the noise source psd for receiverpair 36-145, as indicated by the black triangles. Plotted in the back-ground is the annulus of the azimuthally homogeneous noise sourcepsd used as the initial model in the noise source inversion. (a) Finite-frequency kernel containing the first and higher Fresnel zones.Darker areas indicate regions where the psd (within the annulus)should be increased, and vice versa. (b) Ray-theoretical simplifica-tion of the finite-frequency kernel (Ermert et al., 2016) used to con-strain noise sources for low computational costs on a daily basis.

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for the psd distribution along an annulus surrounding the array. Forthis inversion, we use the previously computed kernels in a steepest-descent minimization of the misfit defined in equation 5. Althoughthe convexity of a misfit functional, required to ensure convergencetowards the global optimum, is generally difficult to assess, thereare at least plausibility arguments: (1) Because we measure wave-form energies, our misfit does not suffer from cycle skipping. Thismajor source of nonlinearity and nonconvexity is therefore auto-matically eliminated. (2) The step-length tests performed in thisstudy show that misfit generally increases away from the optimalstep length. This indicates that the misfit is convex at least alongselected directions. The optimized psd distributions for July 21and July 22, 2014, are shown in Figure 7.

APPARENT TRAVELTIME DIFFERENCESAND EARTH STRUCTURE INDUCED

BY HETEROGENEOUS NOISE SOURCES

Systematic traveltime biases

The SWIM array data provide us with realistic distributions of thenoise source psd in the North Sea surrounding the array on a daily

basis. In the following, we will use these field measurements innumerical experiments intended to quantify systematic traveltimebiases and apparent subsurface structure.Our modeling is based on the time-domain finite-difference sol-

ution of the acoustic wave equation (Virieux, 1984) with absorbingboundary conditions (Cerjan et al., 1985). Using an acousticapproximation as an analog for Scholte wave propagation is com-putationally efficient and appropriate for narrow frequency bands inwhich dispersion can be ignored. We choose a homogeneous earthmodel m that permits waves to propagate at 300 m∕s, which cor-responds to the average group velocity of Scholte waves recordedby the SWIM array at frequencies between 1.15 and 1.55 Hz.Using the psd distributions of July 21 and July 22, 2014 (Fig-

ure 7), we compute time-domain interstation correlationsC̄oðxi; xk; tÞ by solving the forward modeling equation 3. We treatthese correlation functions as artificial data. Emulating the commonpractice in ambient noise interferometry that assumes that noisesources are favorably distributed to enable Green’s function recov-ery (Sabra et al., 2005; Shapiro et al., 2005; Mordret et al., 2014a;de Ridder and Biondi, 2015), we then compute synthetic correlationfunctions C̄ðxi; xk; tÞ using an isotropic psd distribution.To quantify the difference between traveltimes for the hetero-

geneous psd T oikðmÞ and the homogeneous psd T ikðmÞ we measure

the time in which the crosscorrelation between the synthetics

Figure 5. Binning map highlighting the azimuthal coverage for allreceiver pairs used for July 21 (top) and July 22 (bottom), 2014, inthe frequency band 1.15–1.55 Hz. The bin hits run along great circlesegments extending 1000 km radially from the center of every in-terstation pair used. Latitudes range from 50.6°N to 70.6°N and lon-gitudinally from 17.2°W to 22.8°E, with a resolution of 0.1° in bothdirections. Though July 22 uses 90 pairs less than July 21, we noticeonly negligible differences in the azimuthal fold coverage betweenthese two days. The differences that we observe in Figure 6 are thusnot due to a biased data selection but are genuinely driven by thenonstationary noise sources.

Figure 6. Cumulative noise source kernels for all receiver pairs forJuly 21 (top) and July 22 (bottom), 2014, in the frequency band1.15–1.55 Hz. Being directions of steepest ascent, the kernels pro-vide a first image of the noise source psd. Within the azimuthalrange marked by green tones, the psd should increase relative tothe azimuthally homogeneous initial psd. Within the azimuthalrange marked by pinkish-white tones, the psd should decrease.

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C̄ðxi; xk; tÞ and the artificial data C̄oðxi; xk; tÞ attains its globalmaximum (Luo and Schuster, 1991; Dahlen et al., 2000):

ΔT ikðmÞ ¼ argmax

ZC̄oðxi; xk; tÞC̄ðxi; xk; tþ τÞdt: (6)

The traveltime difference ΔT ikðmÞ is negative when the syn-thetic C̄ðxi; xk; tÞ arrives earlier than the artificial data C̄oðxi; xk; tÞ,and vice versa.Figure 8 displays ΔT ikðmÞ as a function of receiver pair azimuth

for July 21 and July 22, 2014. The traveltime differences oscillatebetween −10 and 10 ms with a quasi-period of approximately 30°.Notable differences on the order of 5 ms exist between the two days.Because artificial data and synthetics have been computed for thesame homogeneous earth model m, the traveltime differences re-present a systematic error induced by the heterogeneous distributionof noise sources.

Static apparent earth structure

As shown in Figure 8, average traveltime differences of approx-imately 5 ms can be observed over an average interstation distanceof approximately 1000 m in the SWIM array. This translates intoapparent velocity variations of approximately 0.5 m/s spread overthe complete interstation distance.To quantify the spatial distribution of apparent velocity hetero-

geneities induced by the heterogeneous noise sources, we invert theapparent traveltimes from Figure 8 for a 2D group velocity model.To be consistent with the finite-frequency traveltime measurementsand to avoid additional systematic errors that may be introducedby the ray approximation (Wielandt, 1987; Montelli et al., 2004;Malcolm and Trampert, 2011), we perform nonlinear wave equationtraveltime tomography, as introduced by Luo and Schuster (1991).The resulting velocity models are displayed in Figure 9.For both days, lateral velocity variations of �1.0 m∕s are re-

quired to optimally explain the traveltime differences. There is noobvious large-scale pattern that may be related intuitively to thenoise source distribution. Heterogeneities outside of the array arecaused by the spatially extended finite-frequency kernels used in thewave-equation traveltime inversion.

Apparent subsurface changes

The apparent velocity heterogeneities on July 21 and July 22 dif-fer markedly, thus giving the incorrect impression of subsurfacechanges. The distribution and magnitude of the apparent velocitychange is displayed in Figure 10, which shows the difference be-tween the velocity maps in Figure 9. Average velocity changes areapproximately �0.5 m∕s, but local extrema exceed �0.75 m∕s(�0.25%). These apparent variations in the subsurface structurefrom one day to the next are solely the result of rapidly movingnoise sources. If we had not known that these changes were due tononstationary noise sources, we would wrongly interpret these assubsurface changes.

REMOVAL OF SYSTEMATIC ERRORSIN NOISE-BASED MONITORING

The sequence of methods introduced in the previous sections,from noise source imaging to finite-frequency traveltime tomogra-phy, can be used constructively to correct for systematic traveltime

Figure 7. Noise source psd for July 21 and July22, 2014. The annulus with nonzero noise sourcestrength is centered around the SWIM array,marked in red.

Figure 8. Traveltime differences ΔT ikðmÞ between synthetic cor-relations C̄ðxi; xk; tÞ computed for an isotropic noise source distri-bution and artificial data C̄oðxi; xk; tÞ computed for the noise sourcedistributions on July 21 and July 22, 2014, as displayed in Figure 7.We observe that the biases show strong anisotropy, a direct result ofthe anisotropic psd.

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and velocity errors. This is achieved by subtracting these quantifiederrors from either the data or model space.

Correcting systematic biases

To minimize the contamination of velocity maps that results fromheterogeneously distributed noise sources, the apparent traveltimedifferences ΔT ikðmÞ computed for an earth model m

ΔT ikðmÞ ¼ T ikðmÞ − T oikðmÞ (7)

introduced in equation 6 and shown in Figure 8, can be subtractedfrom the total observed traveltime differences

ΔTikðmÞ ¼ T ikðmÞ − Toik; (8)

where Toik is the traveltime of the observed noise correlation wave-

form, T ikðmÞ is the traveltime of synthetics with homogeneous psd,and T o

ikðmÞ is the traveltime of synthetics with the imaged hetero-geneous psd inferred from the data. The difference of ΔTikðmÞ andΔT ikðmÞ eliminates the traveltime of the synthetic correlationwaveform with homogeneous source distribution T ikðmÞ and yieldsthe source-corrected traveltime difference,

ΔTcorrik ðmÞ ¼ ΔTikðmÞ − ΔT ikðmÞ ¼ T o

ikðmÞ − Toik: (9)

Equation 9 gives the corrected traveltime difference between syn-thetic noise correlation waveforms computed for an inferred hetero-geneous source distribution and the observed noise correlationwaveforms. It may thus be used to perform traditional ambient noisetomography in which correlations are assumed to equal Green’s func-tions, without the need to modify an existing inversion machinery.For consistency, the traveltime difference ΔTikðmÞ in equation 8

must be computed with the same method used to measureΔT ikðmÞ,i.e., by crosscorrelation in our case. Because the apparent traveltimedifferencesΔT ikðmÞ have a second-order dependence on the under-lying earth model m (Fichtner, 2015; Ermert et al., 2016), theyshould ideally be updated regularly during an iterative inversionof the corrected traveltime differences ΔTcorr

ik ðmÞ.Instead of performing corrections in the data space, corrections in

the model space can be an alternative. For instance, inferences oftrue changes in subsurface structure may be improved by sub-tracting apparent velocity changes. Furthermore, the maps of appar-ent velocity changes can be used to identify regions where we maysafely interpret. However, because the systematic errors exhibit astrong angular dependence that no reasonable model could repli-cate, it is more advisable to remove these errors directly fromthe data space before proceeding to invert for structure. This is fur-ther addressed in a subsequent section “Analysis of noise sourcenonstationarity.”

Synthetic inversion example

To illustrate the basic concept outlined above, we perform a syn-thetic numerical experiment using the SWIM array configuration.Unlike in the real-life case, we now have full control over the struc-

Figure 9. Apparent group velocity variations δv (m∕s) on July 21and July 22, 2014, induced by the heterogeneous distribution ofnoise sources.

Figure 10. Apparent fractional changes of the group velocity var-iations Δδv (%) from July 21 to July 22, 2014, induced by temporalchanges in the noise sources, as rendered in Figure 7. The referencevelocity is 300 m∕s. Without knowing that these are systematic er-rors due to assuming correlation and Green’s function equivalency,we may wrongly attribute these to localized subsurface perturba-tions.

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ture and noise perturbations from one day to the next. We nowpresent three cases.

Case 1: The baseline

On the first day, our structure is homogeneous and allows surfacewaves to propagate at 300 m∕s. The noise source distribution ishomogeneous as well. On the second day, our structure undergoesa regional perturbation of up to 0.3%, as shown in Figure 11a. How-ever, the noise source distribution also undergoes a perturbation onthe second day and becomes inhomogeneous. The true model forthis heterogeneous psd is a smoothed version of the imaged July 21psd in Figure 7. Measuring the traveltime difference in the corre-lation functions between the first and second days, we perform awaveform inversion and image the structural perturbation, assumingperfect knowledge of the heterogeneous noise source distribution.The result is shown in Figure 11b. Because of imperfect coverage,the target perturbation of Figure 11a is, of course, not perfectly re-constructed. However, because the heterogeneous noise sources arefully taken into account, all the differences between Figure 11aand 11b are purely imaging artifacts. In this regard, we have pro-duced the best possible reconstruction of the structural perturba-tions, given our source-receiver setup.

Case 2: Ignoring nonstationary noise sources

Because the time-variable noise source distributions are in prac-tice not known a priori. We therefore repeat the previous inversion,incorrectly assuming that the noise source distribution has remainedhomogeneous from day 1 to day 2. As shown in Figure 11c, the

reconstructed anomaly is distorted substantially relative to the base-line of case 1. Additional artifacts appear in the northwestern part ofthe model in the form of strong positive velocity changes. Theseartifacts are purely the result of the nonstationary noise sources thathave not been taken into account.

Case 3: Applying the method presented in this paper to correctfor the nonstationary noise source traveltime biases

By taking the amplitude ratios of correlation functions from thesecond day, we create a noise source kernel. Using a homogeneousstructure model and a homogeneous noise source model, we theninvert for the noise source psd. Using the imaged noise source psd,we then determine the traveltime bias produced from the inhomo-geneous noise source distribution. We subtract this from thetraveltime differences in case 2 to yield corrected traveltimes.Inverting for these corrected traveltimes, we produce a correctedstructure image, shown in Figure 11d. Because Figure 11d isnearly identical to the baseline in Figure 11b, we conclude thatthe artifacts due to nonstationary noise sources have been removedalmost completely.Although unavoidably simplistic in nature, our synthetic inver-

sion illustrates that the proposed method is self-consistent, andthat artifacts due to nonstationary noise sources may indeed beremoved under favorable circumstances. Nonetheless, results forreal data applications may be different based on the structure wewish to image, the quality of the data, the magnitude of the struc-tural change versus the noise source psd change, etc. This is thesubject of sensitivity studies, an example of which will be pre-sented in the following paragraphs.

Figure 11. Finite-frequency inversion results at-tempting to image the true structure perturbationrendered in panel (a). (b) Case 1 — baseline,in which the nonstationary noise sources are takeninto account and assumed to be known perfectly.This is the best possible reconstruction using thegiven data. (c) Case 2, in which we image time-lapse velocity perturbations assuming that noisesources are homogeneous and not variable in time.(d) Case 3, in which we do correct for noise sourcenonstationarity biases by imaging the noise sourcepsd. Correcting for noise source biases allows usto remove spurious artifacts and better identify thetrue structural change. In case 2, it is difficult toargue for or against whether the imaged positiveanomaly in the northwestern corner is genuine.With the correction applied in case 3, we see thatthis anomaly is merely an artifact.

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Traveltime sensitivity and uncertainties

The extent to which we can remove these traveltime artifacts de-pends upon the prediction error in the traveltime bias ΔT ikðmÞ thatresults from uncertainties in the inferred noise source model. Theseuncertainties are rooted in the measurement error, and would besmaller for measurements that are better in quality and coverage.We estimate the prediction error through forward modeling with

perturbed psds. For this, we perturb the optimal step length σg:m:

that minimizes the misfit to a perturbed step length σpert until it pro-duces synthetic amplitude ratios that differ from the observed am-plitude ratios by more than one standard deviation of themeasurement error. We then compute the traveltime correction fac-tor from this maximally perturbed psd. Comparing the traveltimecorrection factor of the perturbed psd ΔT ikðm; σpertÞ with that ofthe globally minimized psd ΔT ikðm; σg:m:Þ we can deduce a pre-diction error ΔT err

ik ðm; σpert; σg:m:Þ by taking the difference ofΔT ikðm; σpertÞ and ΔT ikðm; σg:m:Þ.Referring back to the SWIM array real data case, we can perturb

the step length by approximately 15% for July 21 and July 22. Thisresults in a traveltime prediction error of approximately �1 ms forJuly 21 and approximately �1.5 ms for July 22, which translates toapproximately 10%–20% uncertainty in the traveltime correctionfactor ΔT ikðm; σg:m:Þ. Thus, noise source nonstationarity may in-duce apparent subsurface changes of up to approximately 30%(and not ∼0.25%), if we account for such uncertainty using thiscriterion. From July 21 to July 22, we may thus argue that wewould not observe any apparent (or trustworthy) subsurfacechanges that are within the range of approximately 10%–20% ofour traveltime correction factor. Moreover, if we cannot readily up-date existing machinery, algorithms, or existing workflows to ac-count for the quantities ΔT ikðm; σg:m:Þ and ΔT err

ik ðm; σpert; σg:m:Þ,we may instead set a minimum threshold (e.g., ∼0.30% apparentvelocity structure change) at which we may observe subsurfacechanges and not attribute them to noise source nonstationarity. Thisminimum threshold will vary and depend upon the standard devia-tion of the amplitude ratio measurements and the heterogeneity ofthe noise source distribution — it is very application dependent.We direct the reader to the supplementary material at the followinglink (s1.pdf) for more information on the computation of the pre-diction errors.

DISCUSSION

Comparison with velocity changes observed inprevious studies

To assess the significance of the apparent velocity variationsshown in Figure 10, we summarize the previous findings of time-variable subsurface structure. Using ambient noise tomography forthe Valhall field, de Ridder et al. (2014) find velocity changes ofapproximately 2% over a six-year period within the same frequencyband used here. Assuming a constant change over time, we translatethis to an average daily velocity change of 10−3%, i.e., two orders ofmagnitude less than the systematic errors observed at Oseberg fromJuly 21–22, 2014. However, the authors make this six-year differ-ence map by choosing only 29 hours from February 2004 and fivedays from December 2010. Though they show that the correlationfunctions are stable within each of these periods, they do not guar-antee that they were produced from the same noise source configu-ration. This may explain discrepancies between their velocity mapsand those produced from an active survey. Moreover, Mordret et al.(2014a) determine yearly velocity variations (also from Valhall) thatvary between �0.08% — less than those presented here. Again,these authors only use 29 hours from 2004 and 6.5 hours from 2005,which also suggests that noise source fluctuations could play anundesired role in these velocity variations. Outside the context ofhydrocarbon reservoir monitoring, time-lapse velocity changes of0.01%–1.0% over several weeks to years have been observed inrelation to volcanic activity (Brenguier et al., 2008b; Obermann et al.,2013), large earthquakes (Brenguier et al., 2008a; Obermannet al., 2014), seasonal variations of the groundwater table (Meieret al., 2010), and the stimulation of geothermal reservoirs (Hillerset al., 2015; Obermann et al., 2015).It follows that the apparent velocity changes induced by nonsta-

tionary noise sources are significant. They limit the quality and timeresolution of ambient noise monitoring based on noise correlationsurface waves and the traditional assumption of perfect Green’sfunction recovery, unless corrections such as those proposed in theprevious sections are applied.

Velocity changes deduced from noise correlation coda

Instead of analyzing the fundamental-mode interface wave,authors in several previous studies use noise correlation coda, as-suming that multiple scattering makes them less susceptible tonoise source variations (Brenguier et al., 2008b; Meier et al.,2010; Hillers et al., 2015). However, as demonstrated by Zhanet al. (2013), coda waveforms are affected by changes in the fre-quency content of noise sources. Furthermore, clearly observablecoda are not generally present, e.g., when attenuation is strong orwhen the medium is smooth. From a theoretical perspective, it iswell-known that only fundamental-mode waves can be recon-structed accurately by noise correlations (Halliday and Curtis,2008; Kimman and Trampert, 2010). Unless noise sources are per-fectly homogeneously distributed or the wavefield is equipartitioned— which is not the case in the earth (Mulgaria, 2012; Sens-Schoen-felder et al., 2015) — the actual physical nature of the noise corre-lation coda is therefore unclear. This complicates the imagingprocedure. It follows from these arguments that the method presentedhere is more widely applicable: It rests on a more solid theoreticalfoundation.

Figure 12. Traveltime differences ΔT ikðmÞ between synthetic cor-relations C̄ðxi; xk; tÞ computed for an isotropic noise source distri-bution and artificial data C̄oðxi; xk; tÞ computed for the noise sourcedistributions on July 21 (blue) and July 22 (red) — now plottedagainst interstation pair distance (in contrast to azimuth in Figure 8).Systematic errors depend on psd azimuths with respect to intersta-tion pair azimuths and are less sensitive to interstation pair distance.

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In the context of our specific data set, we note that clear codawaves are not present in the noise correlations. This may be dueto either insufficiently strong scatterers or attenuation, as discussedabove.

Nature of apparent traveltime differences

We see from Figure 8 that apparent traveltime differences exhibita strong azimuthal dependence with a quasi-period of approxi-mately 30°. On July 21, for instance, traveltimes differ by more than20 ms for azimuths from approximately 120° to approximately150°. This suggests a form of anisotropy with an approximately12-fold symmetry axis that does not exist in minerals composingthe earth. The unphysical nature of the traveltime differences be-comes evident in Figure 12. It reveals that traveltime differencesdo not increase systematically with offset, but decrease slightly.This can be interpreted in terms of sensitivity kernels: As the inter-station distance increases, the noise sourcekernels become narrower, and thus the measure-ments are less sensitive to the source distribution.These traveltime errors are thus systematic, fun-damentally linked to the power and azimuthallayout of the noise sources.It may appear counterintuitive to have arrivals

that occur later than the maximum arrival timeexpected when sources are located within the sta-tionary-phase region (Snieder, 2004; Wapenaaret al., 2010). However, the stationary-phaseregion is not precisely defined, and finite-frequency traveltime sensitivity kernels with re-spect to noise sources for ballistic waves containboth positive and negative contributions. This in-dicates that finite-frequency traveltimes may in-deed be larger and is a facet of apparent finite-frequency paradoxa.This aspect is also considered by Tsai (2009),

in which the author analytically derives a nega-tive-positive oscillatory nature of traveltimebiases for finite-frequency cases even for an iso-tropic distribution of sources. In this case, noisesource frequency, interstation separation, and thevelocity of the medium account for traveltimebiases. Because these parameters would mostlikely remain unchanged from day to day, itwould then be a dynamic noise source distribu-tion that would bring about the pattern that weobserve in Figure 8.The assumption that correlation functions

equal Green’s functions thus leads to unphysicaltraveltime measurements that can hardly be ex-plained without unrealistic anisotropy or an ex-cessively heterogeneous earth structure. In ourtraveltime inversions, we prevent the appearanceof unresolvable subwavelength structure. As aconsequence, we could only reduce the misfit byapproximately 25%, starting from a homoge-neous model on July 21 and 22.When clearly visible, the unphysical nature of

the traveltime differences may be used as an in-dicator of significant systematic errors induced

by nonstationary noise sources. In the specific case of the SWIMarray, the absence of an offset dependence suggests that the errorsmay be reduced by using only the largest offsets.

Analysis of noise source nonstationarity

July 21 and July 22, 2014, are days on which the noise sourceazimuthal mean moves by approximately 90°. Although noisesources vary more slowly throughout most of the year, similarlystrong daily variations are still not uncommon. This is shown mostclearly in Figure 13 in which we plot the noise source angular meanand the correlation functions’ absolute energy ratio as a function ofdaily and likewise of monthly averaging within the frequency bands0.55–0.95 and 1.15–1.55 Hz.For daily averaging, we note that fluctuations in mean noise

source position and psd (inferred via the energy ratio plot) can occurwith little predicability. Indeed, whereas the 1.15–1.55 Hz daily

Figure 13. (a) Weighted angular mean of the noise source azimuth as a function of dailyaveraging (solid line) and of monthly averaging (dotted line) for frequency bands 0.55–0.95 Hz (green) and 1.15–1.55 Hz (magenta). (b) Mean absolute energy ratio of corre-lations at corresponding angular mean. As we invert for noise source psd by using en-ergy ratios, we can use the energy ratios as a psd analog. We observe that noise sourcenonstationarity is spatially, frequency, and temporally dependent and that temporaldependence can be reduced by averaging over longer time periods. Furthermore, wecan infer on which days we are more likely to produce false alarms or fail to correctlyidentify structural changes, should we decide not to set a minimum threshold or removeartificial changes.

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angular mean from April 29, 2014 to May 18, 2015, is approxi-mately 215°, the mean experiences its maximum fluctuation fromAugust 7 to August 8, 2014, shifting by >210°. Furthermore, weobserve that the energy ratio also changes rapidly from day today. This indicates that not only do systematic errors map intodifferent regions within the data and model space, but also withvarying magnitudes. Moreover, these complications are furthercompounded if we decide to use different frequency bands:Whereas the angular mean of the sources within the 1.15–1.55 Hz band is stable in February, this does not hold for the0.55–0.95 Hz band. Thus, to ensure reliable monitoring on a dailybasis, methods that account for the nonstationary nature of noisesources must be implemented.These issues mostly vanish once we choose to stack our corre-

lations on monthly time-frames. Though the noise field remainsinhomogeneous and artifacts leak into static traveltime measure-ments, these cancel out when determining the dynamic traveltimedifferences from month to month. A caveat about this is that theenergy ratio changes per month, notably within the 0.55–0.95 Hzband, where we witness an approximately 68% increase in the en-ergy ratio from August to November. This means that systematicerrors may still appear in terms of magnitude. Because these errorswould not spatially map into a different location, there is a greaterlikelihood to misinterpret these as physically real gradual changesin subsurface conditions, especially if this coincidentally corre-sponds to overburden compaction in a region overlying a producingreservoir unit. By incorporating the correction methodology pro-posed above, we mitigate such dilemmas inherent in passive mon-itoring.It must, however, be noted that we assume a linear relationship

exists between the psd and traveltime errors ΔT ikðmÞ. This behav-ior may break down in regions in which noise and earth structuretrade-off nonnegligibly, e.g., in regions with strong scattering orfocusing.If this happens, we can extend the time averaging to several days

or weeks and then estimate the systematic errors. By computingapparent subsurface changes as a function of the averaging intervallength, we may determine the smallest possible time averaging thatensures that velocity errors do not overwhelm any potential subsur-face changes that we would like to detect. This in turn would help tocounteract any significant mixed or second order effects and reen-able us to linearly remove any errors.It should also be emphasized that our method does not elucidate

the physics or progenitor of these noise sources. The methodologysimply images them, determines the traveltime errors produced bythem, and allows the user to subtract these errors; essentially, a staticerror correction. The procedure thus removes these errors in the de-sired frequency bands, independent of whether these sources stemfrom a single or multiple natural — or even anthropogenic —phenomena.

Extensions of the method: An outlook

The methods and approach proposed in the previous sectionsconstitute a step toward noise-based imaging that fully accountsfor the heterogeneous and time-variable distribution of noisesources, and maximizing monitoring time resolution in a physicallyconsistent way.Future improvements of the method will include (1) the replace-

ment of ray-theory kernels by finite-frequency kernels in the inver-

sion of the ambient noise source distribution, (2) the incorporationof frequency dependence in the noise source and the earth structureinversion, (3) the implementation of this method to phase velocitiesfor arrays with dense receiver coverage (such as SWIM), and (4) thetransition from 2D to 3D wave propagation and inversion. Ulti-mately, it should become possible to jointly invert for ambient noisesources and earth structure in a fully consistent way without theneed to assume equality between Green’s functions and noise cor-relations (Hanasoge, 2014).

CONCLUSIONS

Passive seismic monitoring has many upsides over traditional 4Dtime-lapse surveys: It is significantly cheaper (provided that the per-manent monitoring array system is already installed), does not re-quire active sources, and can be used to provide much faster updatesof the subsurface. In the hydrocarbon context, this cost-conscioustechnology would enable us to move toward the creation of real-time injection monitoring systems that could help to significantlyincrease production, while ensuring the integrity of the overburdenand reservoir. However, we see that the loss of repeatability due tochanges in the location and strength of noise sources jeopardizes thepotential applicability of noise-based monitoring.Therefore, we develop and apply a method to quantify systematic

errors in noise-based monitoring that are due to heterogeneous, non-stationary noise sources and the incorrect assumption that noise cor-relations equal Green’s functions. Our method is based on theinversion of noise correlation asymmetries for the time-dependentdistribution of noise sources, which we then use in the forwardmodeling of noise correlations. This allows us to determine appar-ent traveltime differences and the resulting apparent earth structure.Applying our method to field data acquired at Statoil’s Oseberg

field for daily passive surface wave monitoring, we find that day-to-day changes in the systematic errors would mask any actual subsur-face changes that we would like to observe. Apparent velocitychanges reaching 0.25% within one day are far above the averagedaily changes inferred from longer term monitoring studies ofhydrocarbon reservoirs.With a synthetic inversion and an uncertainty analysis, we dem-

onstrate that the quantified systematic traveltime errors may be usedto correct traveltime measurements of a traditional ambient noisetomography in which correlations are assumed to equal Green’sfunctions. Such a correction would allow practitioners to continueusing existing inversion machineries, while still accounting forheterogeneous, nonstationary noise sources. Should we not wishto correct for these artifacts, we can use this method to set a mini-mum threshold for which we can reliably attribute traveltimechanges to actual subsurface events. This would help to minimizefalse alarms related to noise source nonstationarity. We thus in-crease the repeatability and thereby viability of time-lapse passiveseismic techniques as a valuable means for monitoring subsurfaceconditions at increasingly higher temporal resolutions by ensuringthat any significant changes imaged through our inversion algo-rithm are not due to fluctuations in the power spectral density ofthe noise sources, but rather due to structure.

ACKNOWLEDGMENTS

The authors would like to thank the editors of Geophysics, Jeff-rey Shragge and Alison Malcolm, reviewer Shravan Hanasoge, and

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two additional anonymous reviewers for their constructive com-ments and support. We would also like to thank Statoil for allowingthe lead author to collaborate with its research center in Rotvoll.Additional gratitude goes to Severine Pannetier-Lescoffit andOdd Arve Solheim for handling the funding, security clearances,and hostel arrangements in Trondheim; Marianne Houbiers forher critical feedback and crash course in Seismic Unix; Andy Carterfor his dialogues concerning anisotropy; Florian Wellmann forpropitiously introducing him to Python merely weeks before theproject’s start; Frank Aavnik for his review and input; Ivar Sandøfor his administrative assistance; and particularly to Eirik Aksnesfor getting him up and running so hastily on the clusters.The results here stem from a project that had to be completed

within five months. Given this time-frame, it would have been nighimpossible to meet such deadlines without the fervent support andinput of members from Statoil’s Rotvoll and Oseberg PermanentReservoir Monitoring teams and ETH Zürich’s Computational Seis-mology group — the lead author expresses his appreciation.We would furthermore like to thank the operator and partners of

the Oseberg license (Statoil, Petoro, Total Norge, and ConocoPhil-lips Skandinavia) for permission to publish this paper.Further acknowledgments go to those who provided helpful feed-

back at the 2nd TIDES (Time DEpendent Seismology) TrainingSchool — COST Action ES1401-TIDES, supported by COST(European Cooperation in Science and Technology).This research was supported by the Swiss National Supercomput-

ing Center (CSCS) in the form of the GeoScale and CH1 projects,by the Swiss National Science Foundation (SNF) under grant200021_149143, and by the Netherlands Organization for ScientificResearch (VIDI grant 864.11.008).

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