It is known that waves generated by ambient noise sources and recorded by passive receivers can be used to image the reflectivities of an unknown medium. However, reconstructing the reflectivity of the medium from partial boundary measurements remains a challenging problem, particularly when the background wave speed is unknown. In this paper, we investigate passive correlation-based imaging in the daylight configuration, where uncontrolled noise sources illuminate the medium and only ambient fields are recorded by a sensor array. We first analyze daylight migration for a point reflector embedded in a homogeneous background. By introducing a searching wave speed into the migration functional, we derive an explicit characterization of the deterministic shift and defocusing effects induced by wave-speed mismatch. We show that the maximum of the envelope of the resulting functional provides a reliable estimator of the true wave speed. We then extend the analysis to a random medium with correlation length smaller than the wavelength. Leveraging the shift formula obtained in the homogeneous case, we introduce a virtual guide star that remains fixed under migration with different searching speeds. This property enables an effective wave-speed estimation strategy based on spatial averaging around the virtual guide star. For both homogeneous and random media, we establish resolution analyses for the proposed wave-speed estimators. Numerical experiments are conducted to validate the theoretical result.
Passive correlation-based imaging reconstructs information about a medium or subsurface using only wavefields transmitted by opportunistic or ambient noise sources. This technique was first demonstrated experimentally in [1] and has since gained widespread use in various fields, including helioseismology [2], volcano monitoring [3], and reservoir monitoring [4]. For a broad interdisciplinary review of correlation-based methods, we refer the reader to [5]; an extensive overview of the mathematical foundations can be found in the monograph [6].
It is now well understood that key information-such as travel times and Green functions-is encoded in the cross-correlation of recorded signals [7]- [10]. This observation underlies the success of imaging reflectors using migration of the empirical cross-correlation. In [11], the authors developed a unified mathematical framework for analyzing passive imaging across diverse configurations. The uniqueness of passive imaging inverse problems has also been established in both the time and frequency domains; see, for instance, [12], [13].
In this paper, we focus on passive imaging in the daylight configuration, where a sensor array is located between randomly distributed noise sources and the medium of interest. This setting is common in the literature on virtual-source imaging, where the receiver acts as a virtual source. The scattered waves recorded at the array are numerically back-propagated as if emitted from the receiver, allowing wavefield reconstruction and improved focusing in complex media. This method is widely used in seismic exploration [14]- [16], and
We consider the scalar wavefield u governed by the inhomogeneous acoustic wave equation
where c(x) denotes the spatially varying wave speed and n(t, x) is a random noise field. The noise is assumed to be zero-mean and stationary in time, with frequency band B centered at ω c and bandwidth B.
Its second-order statistics are specified by
where F ε encodes the temporal correlation and satisfies F ε (0) = 1. Its Fourier transform,
is real-valued, even, nonnegative, and proportional to the source power spectral density. We assume that the decoherence time of the sources is short relative to the typical travel times. Introducing a small scale parameter ε ≪ 1, we write
so that the central frequency scales as ω c = ω 0 /ε, and the bandwidth scales as B = B H /ε. Spatially, the sources are taken to be delta-correlated:
where K describes their spatial distribution.
Medium model. The wave speed is described as a small perturbation of a homogeneous background:
(1 + ρ(x)) .
We consider two types of perturbations:
• Point-like reflector. A compact reflector located near z r is modeled by
where Ω r has characteristic size ℓ r . The point-reflector regime corresponds to ℓ r ≪ λ, and the reflector is assumed weak, σ r ≪ 1. Neither the reflector position z r nor the background velocity c 0 is assumed to be known.
• Random medium. In this regime, the perturbations are supported in a region D and modeled by a rapidly oscillatory random field:
where ℓ c ≪ λ is the correlation length. The random field µ is assumed to be stationary, mean-zero, and has covariance function Σ(x) = Cov(µ(x), µ(0)), decaying at the rate |Σ(x)| ≲ (1 + |x|) -4-η for some η > 0. The fluctuations are assumed weak, |Σ(0)| ≪ 1. While the mean-zero assumption is restrictive theoretically, it is not an obstacle in practice because the boundary reflections due to a nonzero mean can be mitigated or are negligible in the imaging geometry presented below.
Data acquisition geometry. We work in a passive daylight configuration in which an array A records the signals u(t, x j ), j = 1, . . . , N , and is positioned between the random noise sources (i.e. the support of K) and the medium of interest (i.e. z r or D). The short decoherence time implies a high central frequency. The array radius a is assumed larger than the wavelength λ, but smaller than the distance to the reflector |z r |.
Under the scaling (2.3), we write
and parameterize the array as A = a A 0 for some reference geometry A 0 . Here, we assume that the typical travel distance is much larger than the wavelength with dist(z r , A) ∼ 1 or dist(D, A) ∼ 1. The primary goal of this paper is to propose estimators for the background wave speed c 0 for both configurations and to localize the point-like reflector in the first configuration. Figure 1
We start the introduction to the cross-correlation with the representation of the wave function (2.1). By the formulation provided previously, the solution to the wave equation has the following integral representation:
dsdy n(t -s, y)G(s, x, y).
(2.4)
The function G is the time-dependent causal Green’s function, i.e., the fundamental solution to the equation
(2.5)
We also introduce the time-harmonic Green’s function G 0 for the background homogeneous medium, i.e., the solution to the Helmholtz equation
We consider the empirical cross-correlatio
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