Correlation based imaging

Correlation based imaging George Papanicolaou Stanford University International Conference on Applied Mathematics Heraklion, Crete September 17, 2013 G. Papanicolaou, ACMAC-Crete Correlation based imaging 1/33

Imaging with ambient noise Key words in imaging: Passive sensors, coherent imaging, noise source illumination, deterministic or random medium, auxiliary arrays. 1. Recovering information (travel times, the Green s function) from cross correlations of recorded noise signals. 2. Using stationary phase to understand the role of directivity in noise sources. Background velocity estimation. 3. Virtual source imaging with auxiliary arrays. Minimizing the effect of medium inhomogeneities with correlation-based imaging using auxiliary arrays. G. Papanicolaou, ACMAC-Crete Correlation based imaging 2/33

Passive sensor imaging with noise cross correlations 1. When there is enough ambient noise, cross correlations can determine wave energy inter-sensor travel times (singular support of time domain Green s function) 2. If sensor locations are known, a background speed of wave propagation can be estimated tomographically, as has been done very successfully with regional seismic data (San Diego group and Grenoble 2005) 3. If background speed is known then sensor locations can be estimated (passive topological self-awareness of sensor networks) 4. How can we detect and possibly image changes in the environment, such as geophysical changes or moving reflectors, with noise cross correlations? G. Papanicolaou, ACMAC-Crete Correlation based imaging 3/33

Southern California Seismic Network (a) Map of the 150 stations of the SCSN; (b) Shot record generated by cross correlations over one month of noise measurements; (c) Surface-wave group velocity map Figure from: A. Curtis, P. Gerstoft, H. Sato, R. Snieder, and K. Wapenaar, Seismic Interferometry -Turning Noise to Signal, The Leading Edge, September 2006. G. Papanicolaou, ACMAC-Crete Correlation based imaging 4/33

European Seismic Network (Alps) Left the seismic sensor network with 3000 selected paths. Right the Rayleigh surface wave velocity map obtained tomographically from the correlation-estimated travel times. Figures from: P. Gouiédard, L. Stehly, F. Brenguier, M. Campillo, Y. Colin de Verdière, E. Larose, L. Margerin, P. Roux, F. J. Sánchez-Sesma, N. M. Shapiro and R. L. Weaver, Cross-correlation of random fields: mathematical approach and applications, Geophysical Prospecting, 2008, 56, 375393 G. Papanicolaou, ACMAC-Crete Correlation based imaging 5/33

Hellenic Unified Seismic Network From: Ambient Noise Levels in Greece as Recorded at the Hellenic Unified Seismic Network by C. P. Evangelidis and N. S. Melis, Bulletin of the Seismological Society of America, Vol. 102, No. 6, pp. 25072517, December 2012. G. Papanicolaou, ACMAC-Crete Correlation based imaging 6/33

The empirical cross correlation and geophysics The empirical cross correlation of the signals recorded at x 1 and x 2 for an integration time T is C T (τ, x 1, x 2 ) = 1 T T 0 u(t, x 1 )u(t + τ, x 2 )dt. (1) Here u(t, x j ) is the signal recordded at x j and T is chosen appropriately (a non-trivial statistical signal processing issue in applications). What is generating the wave energy recorded in the noise signals? Frequency band of microseisms from wave breaking: 0.5 2.0Hz. Dispersiveness of the underlying wave propagation. Field, regional and continental seismic data. What is the relevant geophysics? G. Papanicolaou, ACMAC-Crete Correlation based imaging 7/33

Simulated cross correlation of noise signals piece of signal recorded at x 1 t 0 100 200 300 400 500 600 700 800 900 1000 x 1 x 2 piece of signal recorded at x 2 t 0 100 200 300 400 500 600 700 800 900 1000 cross correlation x 1!x 2 t!500!400!300!200!100 0 100 200 300 400 500 When the spatial support of the noise sources (circles) extends over all space then the cross correlation function is symmetric. The positive and negative parts correspond to the Green s function between x 1 and x 2 and its anti-causal counterpart, respectively. For the scalar wave equation used here all the information about the Green s function is contained in the travel time. G. Papanicolaou, ACMAC-Crete Correlation based imaging 8/33

A model correlation-gf calculation The solution u(t, x) of the wave equation in an inhomogeneous medium: 1 2 u c 2 (x) t 2 xu = n ɛ (t, x). (2) The term n ɛ (t, x) models a random distribution of noise sources. It is a zero-mean stationary (in time) Gaussian process with autocorrelation function n ɛ (t 1, y 1 )n ɛ (t 2, y 2 ) = F ɛ (t 2 t 1 )θ(y 1 )δ(y 1 y 2 ). (3) We assume that the decoherence time of the noise sources is much smaller than typical travel times between sensors. If we denote with ɛ the (small) ratio of these two time scales, we can then write the time correlation function F ɛ in the form ( F ɛ t2 t ) 1 (t 2 t 1 ) = F, (4) ɛ where t 1 and t 2 are scaled relative to typical inter-sensor travel times. G. Papanicolaou, ACMAC-Crete Correlation based imaging 9/33

The model GF calculation continued The stationary solution of the wave equation has the integral representation u(t, x) = = t n ɛ (s, y)g(t s, x, y)dsdy n ɛ (t s, y)g(s, x, y)dsdy, (5) where G(t, x, y) is the time-dependent Green s function. It is the fundamental solution of the wave equation 1 2 G c 2 (x) t 2 xg = δ(t)δ(x y), (6) starting from G(0, x, y) = t G(0, x, y) = 0 (and continued on the negative time axis by G(t, x, y) = 0 t 0). G. Papanicolaou, ACMAC-Crete Correlation based imaging 10/33

Wave equation in a homogeneous medium and cross correlations When the medium is homogeneous with background velocity c 0 and the source distribution extends over all space, i.e. θ 1, the signal amplitude diverges because the contributions from noise sources far away from the sensors are not damped. So we consider the solution u of the damped wave equation with T a an absorption time: 1 ( 1 + ) 2u x u = n ɛ (t, x). (7) T a t c 2 0 In a three-dimensional open medium with dissipation and with noise sources over all space, θ 1, then x 1 x 2 [ ] c 0 Ta F ɛ G(τ, x 1, x 2 ) F ɛ G( τ, x 1, x 2 ), τ C(1) (τ, x 1, x 2 ) = c2 0 T a 4 e (8) where stands for the convolution in τ and G is the Green s function of the wave equation in a homogeneous medium without dissipation: G(t, x 1, x 2 ) = 1 ( 4π x 1 x 2 δ t x 1 x 2 ). c 0 G. Papanicolaou, ACMAC-Crete Correlation based imaging 11/33

The basic GF calculation continued If the decoherence time of the sources is much shorter than the travel time (i.e., ɛ 1), then F ɛ behaves like a Dirac delta function and we have τ C(1) (τ, x 1, x 2 ) e x 1 x 2 [ ] c 0 Ta G(τ, x 1, x 2 ) G( τ, x 1, x 2 ), up to a multiplicative constant. We see from this simple model calculation how it is theoretically possible to estimate the travel time τ(x 1, x 2 ) = x 1 x 2 /c 0 between x 1 and x 2 from the cross correlation, with an accuracy of the order of the decoherence time of the noise sources. With real geophysical data the connection with (elastic) wave propagation models is exceedingly complex and has not been carried out. G. Papanicolaou, ACMAC-Crete Correlation based imaging 12/33

What do we want from models? What can we expect from data? From models: How does the directivity of the wave energy affect cross correlation calculations? From models: How can we interpret the observed dispersiveness of the waves? From models: Do we really want to fit surface wave models to the data? From DATA: Can we identify other interesting travel times or peaks in the cross correlations? From DATA: What about absorption and multiple scattering? G. Papanicolaou, ACMAC-Crete Correlation based imaging 13/33

Using stationary phase to understand energy directivity y x 1 x 1 x 1 x 2 x 2 x 2 If the ray going through x 1 and x 2 (solid line) enters into the source region (left figure), then the travel time can be estimated from the cross correlation. If this is not the case, then the cross correlation does not have a peak at the travel time (middle figure). Right figure: The main contribution to the singular components of the cross correlation is from pairs of ray segments issuing from a source y going to x 1 and to x 2, respectively (solid and dashed lines, respectively), and starting in the same direction. G. Papanicolaou, ACMAC-Crete Correlation based imaging 14/33

Noise signal cross correlations piece of signal recorded at x 1 t 0 100 200 300 400 500 600 700 800 900 1000 x 1 x 2 piece of signal recorded at x 2 t 0 100 200 300 400 500 600 700 800 900 1000 cross correlation x 1!x 2 t!500!400!300!200!100 0 100 200 300 400 500 piece of signal recorded at x 1 t 0 100 200 300 400 500 600 700 800 900 1000 x 1 x 2 piece of signal recorded at x 2 t 0 100 200 300 400 500 600 700 800 900 1000 cross correlation x 1!x 2 t!500!400!300!200!100 0 100 200 300 400 500 When the distribution of noise sources is spatially localized then the cross correlation function is not symmetric G. Papanicolaou, ACMAC-Crete Correlation based imaging 15/33

Noise signal cross correlations, continued piece of signal recorded at x 1 x 1 t 0 100 200 300 400 500 600 700 800 900 1000 piece of signal recorded at x 2 x 2 t 0 100 200 300 400 500 600 700 800 900 1000 cross correlation x 1!x 2 t!500!400!300!200!100 0 100 200 300 400 500 When the distribution of noise sources is spatially localized then the coherent part of the cross correlation function can be difficult or even impossible to distinguish if the axis formed by the two sensors is perpendicular to the main direction of energy flux from the noise sources. G. Papanicolaou, ACMAC-Crete Correlation based imaging 16/33

Cross correlations of Long Beach array seismic data A 4 8 kilometer array with an average 100m spacing (From Chang and Biondi, SEP, Stanford 2013). Note the highway (405) crossing the array. G. Papanicolaou, ACMAC-Crete Correlation based imaging 17/33

Cross correlations of Long Beach array seismic data II Snapshots of a virtual source in the center of the array at 5s lag on left and +5s lag on right for frequencies 0.5 2.0Hz. Speed is 250m/s. (From Chang and Biondi 2013). Note the directivity of the wave energy. G. Papanicolaou, ACMAC-Crete Correlation based imaging 18/33

Cross correlations of Long Beach array seismic data III Snapshots of a virtual source in the center of the array at 5s lag on left and +5s lag on right for frequencies 2.0 4.0Hz. Speed is 500m/s. (From Chang and Biondi 2013). Note the relative isotropy of the wave energy. G. Papanicolaou, ACMAC-Crete Correlation based imaging 19/33

Imaging with passive auxiliary arrays schematic x s x r y Active sensor array for imaging a reflector at y through a scattering medium. Here x s, x r, and y denote source, receiver and reflector locations, respectively. x s x q x q y An auxiliary passive array for imaging through a scattering medium. x s is the source location (primary array), x q and x q are receiver locations (auxiliary array) below the scattering medium, and y is the reflector. G. Papanicolaou, ACMAC-Crete Correlation based imaging 20/33

How is such an imaging setup realized? Original setup in seismic exploration imaging 1 : Use passive arrays placed in drill holes to image below them. Typical wave propagation parameters: Speed of propagation: 3km/s, wavelength: 100m (30Hz peak frequency), bandwidth: broadband (more than 50%), synthetic array size: up to 10 10km in marine surveys, range (depth) 5 6km. Placement of auxiliary arrays at depths of 5 6km is being contemplated for enhanced imaging to well below 6km. Proposed radar application: Ground based radar arrays to image and track (small) satellites using passive auxiliary (synthetic) arrays on high-flying platforms. Potential elimination of atmospheric turbulence effects below the flying platform (at 15 20km hight). 1 A. Bakulin and R. Calvert, The virtual source method: Theory and case study, Geophysics, 71 (2006), pp. SI139-SI150 G. Papanicolaou, ACMAC-Crete Correlation based imaging 21/33

Travel-time migration imaging with the primary, active array Travel-time migration imaging using the array data p(t, x s, x r ), with x s the source location and x r the receiver location, recorded at the active array on top, is done with N r N s I( y S ) = p ( ) T( x s, y S ) + T( y S, x r ), x r ; x s, r=1 s=1 where the travel time, assuming a homogeneous medium, is given by T( x, y) = x y c 0, G. Papanicolaou, ACMAC-Crete Correlation based imaging 22/33

The correlation based imaging algorithm with the passive auxiliary array Step 1: Compute cross-correlations of the recorded signals p(t, x q ; x s ) on the auxiliary array receiver located at x q when the illumination is from the primary array at x s, through the inhomogeneous medium: C T (τ, x q, x q ) = 1 T T 0 N s p(t, x q ; x s )p(t+τ, x q ; x s )dt, q, q = 1,..., N q s=1 Step 2: Migrate the cross-correlations as if they were primary data (that is, actively generated) on the auxiliary array: I( y S ) = N q q,q =1 C T ( T( xq, y S ) + T( y S, x q ), x q, x q ). where, assuming a homogeneous medium below the auxiliary array, T( x, y) = x y c 0, G. Papanicolaou, ACMAC-Crete Correlation based imaging 23/33

Simulations (with geophysical parameters) 2 Imaging through random media with an auxiliary array: Left: The active, primary imaging array is at the top, the random medium below it, the auxiliary array is just below the random medium, and the object to be imaged is the black dot at the bottom. Middle: The image obtained with standard migration with active array at the top. Right: The image obtained when the auxiliary array is used. It is remarkable that the effects of the random medium have been taken out almost entirely. 2 Signal to Noise Ratio estimation in passive correlation-based imaging, J. Garnier, G. Papanicolaou, A. Semin and C. Tsogka, SIAM J. Imaging Sci. 2013 G. Papanicolaou, ACMAC-Crete Correlation based imaging 24/33

Remarks on the simulations Use of a 2D finite element code (high resolution) that can handle about numerical domains 200 200 wavelengths. This explains why the scatterer is about 180 wavelengths (1800m) deep. One dimensional layer codes can go much deeper but they are too special and mostly of theoretical interest today. 2D (3D) random effects are very important because they increase the lateral diversity of the illumination from the primary array. If this is properly quantified it can be exploited to reduce data acquisition costs by reducing the size of the primary array. Secondary array need not be horizontal and can be synthetic. G. Papanicolaou, ACMAC-Crete Correlation based imaging 25/33

Why does the auxiliary array imaging with correlations work so well? There are two important effects that we need to understand and exploit When imaging with ambient noise sources (spatially uncorrelated), the cross-correlations on the auxiliary passive array behave like the Green s function between the two points on the array. This is true only in a homogeneous or slowly varying medium. It is not true in a strongly scattering medium. Clearly the random medium helps make surface illumination more diverse when it gets to the auxiliary array. This is good for correlation based imaging. But it does not help if the medium is layered! So why does correlation based imaging work? It is time-reversal that makes the random medium effects go away. G. Papanicolaou, ACMAC-Crete Correlation based imaging 26/33

How does time reversal work? Basic fact: The cross correlation of two signals is the convolution of one with the time-reversed version of the other. Recall that C T (τ, x q, x q ) = 1 T T 0 N s p(t, x q ; x s )p(t+τ, x q ; x s )dt, q, q = 1,..., N q s=1 Interpretation: The cross correlation of signals on the auxiliary array is the same as having an impulsive point source source at one location x q, a time reversal mirror at the primary array on top, and an observation point x q of the time-reversed field at the other location on the auxiliary array. Main TR facts 3 : The TR field is peaked very near the source and is statistically stable (self-averaging) relative to the random medium 3 Theory and applications of time reversal and interferometric imaging, L. Borcea, G. Papanicolaou and C. Tsogka. Inverse Problems, vol 19, (2003), pp. 5139-5164 G. Papanicolaou, ACMAC-Crete Correlation based imaging 27/33

Why does it work, continued Conclusion: The cross-correlations computed on the auxiliary array are very different from the ones in a homogeneous medium. They do not peak at the travel time between the two points. They peak near zero lag, and only when the points are near each other. Effect: This is perfect for imaging below the auxiliary array, where the medium is (assumed) homogeneous, because there, which means at later lag times, the cross correlation will have peaks as we expect from the theory in homogeneous media. The effect of the random medium has thus been removed! This has been proven in detail for (i) randomly layered media and, (ii) for media for which the paraxial approximation can be used 4. The paraxial approximation is particularly useful for microwave propagation in a turbulent atmosphere. 4 Correlation based virtual source imaging in strongly scattering media, Josselin Garnier and George Papanicolaou. Inverse Problems, vol 28, (2012). G. Papanicolaou, ACMAC-Crete Correlation based imaging 28/33

Concluding remarks Passive sensor imaging with noise sources and in scattering media has both theoretical and applied potential. It is mathematically challenging and draws on many related research areas (wave propagation, random media, statistical signal processing, optimization,...). Passive auxiliary arrays can have a huge impact in imaging through inhomogeneities. It is the first time in imaging that the real benefits of time-reversal are being used. Theoretical issues, resolution theory for example, can often be reduced to already established results in active sensor imaging, both for distributed sensors and for arrays. G. Papanicolaou, ACMAC-Crete Correlation based imaging 29/33

Predicting volcanic eruptions The schematic of the seismic sensor deployment around the Piton de la Fournese volcano (by M. Campillo in Grenoble and collaborators, 2012). G. Papanicolaou, ACMAC-Crete Correlation based imaging 30/33

Echolocation in nature: Dolphins Ultrasonic echolocation by the dolphin is about ten times more accurate than what one expects from the basic physics of array imaging. (From: W. Au and J. Simmons, Echolocation in dolphins and bats, Physics Today, 2007) G. Papanicolaou, ACMAC-Crete Correlation based imaging 31/33

Echolocation in nature: Bats Ultrasonic echolocation by bats is much more accurate than what one expects from the basic physics of array imaging. (From: W. Au and J. Simmons, Echolocation in dolphins and bats, Physics Today, 2007) G. Papanicolaou, ACMAC-Crete Correlation based imaging 32/33

Basic resolution theory in array imaging Imaging formation key words: Coherent, array-data structure, active-passive, broadband-narrowband illumination-signals, homogeneous-inhomogeneous (random) medium. In small aperture regime, λ a L: Cross-range resolution: λl λl Range resolution: 2 a narrowband, and c 2 B broadband. For full (surrounding) arrays resolution is λ 2, narrowband and full recovery in broadband, in 3D. 2 λ (L/a) a. a y a λ L/a L L G. Papanicolaou, ACMAC-Crete Correlation based imaging 33/33