R. Snieder

How do we understand and visualize uncertainty

Geophysicists are often concerned with reconstructing subsurface properties using observations collected at or near the surface. For example, in seismic migration, we attempt to reconstruct subsurface geometry from surface seismic recordings, and in potential field inversion, observations are used to map electrical conductivity or density variations in geologic layers. The procedure of inferring information from indirect observations is called an inverse problem by mathematicians, and such problems are common in many areas of the physical sciences. The inverse problem of inferring the subsurface using surface observations has a corresponding forward problem, which consists of determining the data that would be recorded for a given subsurface configuration. In the seismic case, forward modeling involves a method for calculating a synthetic seismogram, for gravity data it consists of a computer code to compute gravity fields from an assumed subsurface density model. Note that forward modeling often involves assumptions about the appropriate physical relationship between unknowns (at depth) and observations on the surface, and all attempts to solve the problem at hand are limited by the accuracy of those assumptions. In the broadest sense then, exploration geophysicists have been engaged in inversion since the dawn of the profession and indeed algorithms often applied in processing centers can all be viewed as procedures to invert geophysical data.

A probabilistic approach for estimating the separation between a pair of earthquakes directly from their coda waves

[1] Coda wave interferometry (CWI) can be used to estimate the separation between a pair of earthquakes directly from the coda recorded at a single station. Existing CWI methodology leads to a single estimate of separation and provides no information on uncertainty. Here, the theory of coda wave interferometry is revisited and modifications introduced that extend the range of applicability by 50% (i.e., 300–450 m separation for 1–5 Hz filtered coda waves). Synthetic experiments suggest that coda wave separation estimates fluctuate around the actual separation and that they have an increased tendency to underestimate the actual separation as the distance between events increases. A Bayesian framework is used to build a probabilistic understanding of the coda wave constraints which accounts for both the fluctuations and bias. The resulting a posteriori function provides a conditional probability distribution of the actual separation given the coda wave constraints. It can be used in isolation, or in combination with other constraints such as travel times or geodetic data, and provides a method for combining data from multiple stations and events. Earthquakes on the Calaveras Fault, California, are used to demonstrate that CWI is relatively insensitive to the number of recording stations and leads to enhanced estimates of separation in situations where station geometry is unfavorable for traditional relative location techniques.

Solving Spatial Sampling Problems in 2D-CSEM Interferometry using Elongated Sources

With interferometry by multidimensional deconvolution (MDD), all effects from the air-water interface in marine Controlled Source Electromagnetics (CSEM) data can be removed. Unfortunately, to apply interferometry byMDD, a very dense receiver sampling is necessary. We show, that the critical sampling distance is equal to the larger of the two parameters: source height and source length. Consequently, by using an elongated source, the sampling criteria can be relaxed also for small vertical source-receiver distances.

Autofocusing for Retrieving the Green's Function in the Presence of a Free Surface

Recent work on autofocusing with the Marchenko equation has shown how the Greens function for a virtual source in the subsurface can be obtained from reflection data. The response to the virtual source is the Greens function from the location of the virtual source to the surface. The Greens function is retrieved using only the reflection response of the medium and an estimate of the first arrival at the surface from the virtual source. Current techniques, however, only include primaries and internal multiples. Therefore, all surface-related multiples must be removed from the reflection response prior to the Greens function retrieval. Here, we present a new scheme that includes primaries, internal multiples, and free-surface multiples. In other words, we retrieve the Greens function in the presence of the free surface. The information needed for the retrieval are the reflection response at the acquisition surface and an estimate of the first arrival at the surface from the virtual source. The reflection response, in this case, includes the free-surface multiples; this makes it possible to include these multiples in the imaging operator and it obviates the need for surface-related multiple elimination.

Creating Virtual Sources Inside an Unknown Medium from Reflection Data - A New Approach to Internal Multiple Elimination

It has recently been shown that the response to a virtual source in the subsurface can be derived from reflection data at the surface and an estimate of the direct arrivals between the virtual source and the surface. Hence, unlike for seismic interferometry, no receivers are needed inside the medium. This new method recovers the complete wavefield of a virtual source, including all internal multiple scattering. Because no actual receivers are needed in the medium, the virtual source can be placed anywhere in the subsurface. With some additional processing steps (decomposition and multidimensional deconvolution) it is possible to obtain a redatumed reflection response at any depth level in the subsurface, from which all the overburden effects are eliminated. By applying standard migration between these depth levels, a true amplitude image of the subsurface can be obtained, free from ghosts due to internal multiples. The method is non-recursive and therefore does not suffer from error propagation. Moreover, the internal multiples are eliminated by deconvolution, hence no adaptive prediction and subtraction is required.

CSEM Interferometry Using a Synthetic Aperture Source

With interferometry by multidimensional deconvolution for Controlled Source Electromagnetics, the medium above the receivers is replaced with a homogeneous halfspace and the source is redatumed to the receiver level. The resulting retrieved reflection response is completely free of any airwave effect. Since the source is redatumed, the original source position becomes irrelevant, which is a useful property for time-lapse surveys. So far, interferometry by multidimensional deconvolution required a densely spaced receiver array. We show that this restrictive sampling requirement can be overcome by combining a lot of source positions to one very long source. By creating such a synthetic aperture source, we were able to apply interferometry for a receiver spacing as large as 1280 m.

Data-Driven Green's Function Retrieval from Reflection Data - Theory and Example

Recently we introduced a new approach for retrieving the Greens response to a virtual source in the subsurface from reflection data at the surface. Unlike in seismic interferometry, no receiver is needed at the position of the virtual source. Here we present the theory behind this new method. First we introduce the Greens function G and a so-called fundamental solution F of an inhomogeneous medium. Next we derive a relation between G and F, using reciprocity theorems. This relation is used as the basis for deriving a 3D single-sided Marchenko equation. We show that this equation is solved by a 3D autofocusing scheme and that the Greens function is obtained by combining the focusing wave field and its response in a specific way. We illustrate the method with a numerical example.

Creating the Green's Response to a Virtual Source Inside a Medium Using Reflection Data with Internal Multiples

Seismic interferometry is a technique that allows one to reconstruct the full wavefield originating from a virtual source inside a medium, assuming a receiver is present at the virtual source location. We discuss a method that creates a virtual source inside a medium from reflection data measured at the surface, without needing a receiver inside the medium and, hence, presenting an advantage over seismic interferometry. An estimate of the direct arriving wavefront is required in addition to the reflection data. However, no information about the medium is needed. We illustrate the method with numerical examples in a lossless acoustic medium with laterally-varying velocity and density. We examine the reconstructed wavefield when a macro model is used to estimate the direct arrivals and we take into consideration finite acquisition aperture. Additionally, a variant of the iterative scheme allows us to decompose the reconstructed wave field into downgoing and upgoing fields. These wave fields are then used to create an image of the medium with either crosscorrelation or multidimensional deconvolution.

Synthesized 2D CSEM-interferometry Using Automatic Source Line Determination

Interferometry by multidimensional deconvolution applied to Controlled-Source Electromagnetic data replaces the medium above the receivers by a homogeneous halfspace, suppresses the direct field and redatums the source positions to the receiver locations. In that sense, the airwave and any other interactions of the signal with the air-water interface and the water layer are suppressed and the source uncertainty is reduced. Interferometry requires grid data and cannot be applied to line data unless the source is infinitely long in the crossline direction. To create such a source, a set of source lines is required. We use an iterative algorithm to determine the optimal locations of these source lines and show that more source lines are required if the source is towed closer to the sea bottom and closer to the receivers.

Increasing the sensitivity of controlled source electromagnetics by using synthetic aperture

Controlled-source electromagnetics (CSEM) has been used as a de-risking tool in the hydrocarbon exploration industry. Although there have been successful applications of CSEM, this technique is still not widely used in the industry because the limited types of hydrocarbon reservoirs CSEM can detect. In this paper, we apply the concept of synthetic aperture to CSEM data. Synthetic aperture allows us to design sources with specific radiation patterns for different purposes. The ability to detect reservoirs is dramatically increased after forming an appropriate synthetic aperture antenna. Consequently, the types of hydrocarbon reservoirs that CSEM can detect are significantly extended. In this paper, we mainly show one type of synthetic aperture antenna whose field can be steered into a designed angle. Consequently, the field concentrates on the target reservoir and the airwave is reduced. We show a synthetic example and a data example to illustrate the increased sensitivity obtained by applying synthetic aperture CSEM source. Because synthetic apertures are constructed as a data processing step, there is no additional cost for the CSEM acquisition. Aside from the applications to marine CSEM, synthetic aperture can be widely applied to other electromagnetic methods such as on land electromagnetics and bore hole electromagnetics.