6 Resolution

75

Chapter 6

Resolution

By definition, resolution is the ability to separate two features that are close together (Sheriff, 2002). Resolution applies to seismic data and to products derived from interpreting seismic data (e.g., maps) in the temporal and spatial domains. We often speak about the resolving power of seismic data and what can be done to increase it because, in doing so, we’ll be able to interpret finer details of subsurface geology. The concept of resolving power of seismic data follows from the sampling theorem, also known as the Nyquist theorem, which formally states that band-limited functions can be reconstructed from equispaced data if there are two or more samples per cycle for the highest frequency present (Sheriff, 2002). On the basis of this theorem, you can describe commonly used measures of temporal and spatial resolving power of seismic data.

The sampling of seismic data is specified by a sample rate, such as 2 or 4 ms, and sampling frequency is defined as the inverse of the sample rate, which is 500 Hz for a sample rate of 2 ms and 250 Hz for a sample rate of 4 ms. The Nyquist frequency is defined as half the sampling fre-quency: For a sample rate of 2 ms, the Nyquist frequency is 250 Hz; and for a sample rate of 4 ms, the Nyquist frequency is 125 Hz. According to the sampling theorem, when there are fewer than two samples per cycle of a given signal, a signal at one frequency yields the same values as those for another frequency, and the one signal can be mistaken for the other. This frequency ambiguity, or aliasing, is illustrated in Figure 1, in which a 200-Hz sine wave is aliased, or misread, as a 50-Hz sine wave when sampled at 4 ms. When acquiring seismic data, you can prevent frequency aliasing by using an antialias filter during recording to attenuate frequencies above the Nyquist frequency.

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76 First Steps in Seismic Interpretation

The temporal resolving power of seismic data is usually described by the tuning thickness of the data, which is based on the fundamental equation that relates velocity V, dominant frequency f, and wavelength l:

V = fl .

The dominant frequency in this equation can be estimated easily from the time separation between adjacent peak and trough reflections on a seismic section (see Figure 5 in Chapter 3); in general, the dominant frequency changes vertically and laterally on a seismic section.

Given estimates of the dominant frequency of the data and the vertical propagation velocity in the vicinity of the features to be resolved, you can calculate the wavelength of the seismic signal from the preceding equa-tion, from which the Rayleigh limit of vertical resolution is derived as l/4. Because propagation velocity and the dominant frequency of the seismic signal change vertically and laterally throughout the subsurface, it follows that temporal resolving power will vary across a given area of investigation.

Tuning thickness usually is visualized with the aid of a diagram known as a wedge or tuning model (Figure 2). The purpose of such a model is to illustrate the seismic response to the wedge and determine the thickness

Figure 1. A 200-Hz sine wave that aliases as a 50-Hz sine wave when sampled at 4 ms. The Nyquist frequency in this case is 125 Hz, so a 125-Hz antialias filter used during recording would attenuate the 200-Hz signal.

50 Hz

200 Hz 4 ms

Nyquist fN = 1 / 2 ∆t = 1 / 2 (4 ms) = 125 Hz Sample rate = ∆t



Chapter 6: Resolution 77

for which the amplitude response is maximum, that is, for which construc-tive interference of the individual responses from the top and base of the wedge is maximum. The point at which this composite amplitude response is maximum is the tuning thickness for the model, with given input wavelet, wedge geometry, and layer impedances. Notice that above the tuning thick-ness, the seismic responses from the top and base of the wedge are separate and distinct (the bed thickness is “resolved” by the time separation between these individual responses). Below the tuning thickness, the waveform of the composite response does not change, but its amplitude decreases as the bed thickness decreases. These observations suggest that with good data quality (based on seismic processing from which wavelet phase and true rel-ative amplitudes can be reliably determined), careful horizon interpretation, and available well data for calibration, you can use seismic data to estimate layer thicknesses, a study commonly referred to as tuning or time-amplitude (time-amp) analysis (see Chapter 3).

Figure 2. Wedge model using a 30-Hz Ricker wavelet and P-wave velocities VP of 6000 and 7000 ft/s (1800 and 2100 m/s) for the wedge (in blue) and encasing medium, respectively. Using the formula V = fλ, the tuning thickness for this model is calculated to be 50 ft (15 m), which corresponds to the point on the model (red line) at which the trough-peak amplitude response is greatest.

Distance (ft) D

epth

(ft)

TW

T (m

s)

No density contrast

VP = 6000 ft/s

VP = 7000 ft/s

30-Hz Ricker wavelet

Tuning thickness = 50 ft

V = f λ




The spatial resolving power of seismic data is usually described in terms of the Fresnel zone, defined as the portion of a reflector from which reflected energy can reach a detector within one-half wavelength of the first reflected energy (Sheriff, 2002). Figure 3 shows the geometry of the first Fresnel zone, which is the smallest and innermost of a succession of higher-order annular Fresnel zones. The equation

FV

fr =

2

12TWT

defines the radius of the first Fresnel zone Fr in terms of the two-way travel-time (TWT) to a reflector, the average propagation velocity V to that reflec-tor, and the dominant frequency f of the seismic signal impinging on the reflector. This formula implies that the size of the first Fresnel zone almost always increases with depth (corresponding to increasing propagation veloc-ity and two-way time) and decreasing dominant frequency of signal (owing to attenuation). Fresnel zones are measured with respect to unmigrated seis-mic data. Seismic migration collapses these zones; however, 2D migration collapses the zones only in the direction of shooting of the 2D line. For 3D data, a full 3D migration collapses the first Fresnel zone to a circle with a diameter of l/2 (radius = l/4), where l is the dominant wavelength of the seismic signal.

Spatial sampling is an important consideration when designing 3D seis-mic surveys. The size of the unit of area into which a 3D survey is subdi-vided, called a 3D bin, ideally should be sufficient in terms of the Nyquist theorem, to properly sample the dip of the steepest reflector and/or the area of the smallest feature of interest within the survey. Figure 4, which represents a 2.5D model of the subsurface (the third dimension in the strike direction

Figure 3. Schematic of the geometry of the first Fresnel zone.

Z Z + /4

Reflector

First Fresnel zone

Z = depth




is perpendicular to the plane of the page), illustrates how the required bin size is related to the average velocity Vavg to the target reflectors, the maxi-mum dip q of those reflectors, and the dominant frequency f of the seismic signal. Notice that the Rayleigh resolution limit (the tuning thickness) also appears in this relationship. It is important to realize that 3D survey design, in addition to addressing technical requirements such as maximum dip and minimum area to be imaged, must also take into account economic consid-erations that can balance or even outweigh technical factors.

Seismic trace displays can exhibit aliasing related to spatial sampling, as shown in Figures 5 and 6. Figure 5 shows four arrays, each consisting of four identical variable-area wiggle traces. Traces in Figure 5a are aligned such that the zero crossing marked in red (the red horizon) is correlated horizontally from trace to trace. On the succeeding arrays (Figure 5b –5d), each trace within the array is shifted downward by a constant amount from the trace on its left, with the amount of shift increasing from array to array. The dip of the red horizon increases in direct proportion to the amount of trace-to-trace shift in each array and in the direction of the shift (from left to right). In Figure 5c and 5d, the dashed blue horizon that dips from right to

Figure 4. Schematic of the maximum 3D bin spacing required to image the maximum dip of target reflectors in terms of average velocity Vavg to the targets, maximum dip q of the targets, and dominant or peak frequency f of the seismic signal.

Maximum bin spacing = Vavg/(4f sinq)

q = True dip4

Bin at surface

q




left is marked as a possible correlation of the same zero crossing; note that as the dip of the red horizon increases, the dip of the dashed blue horizon decreases, and vice versa. This correlation ambiguity is a manifestation of aliasing, which in this example is related to the trace spacing and the mag-nitude of the dip (the trace-to-trace shift) of the red horizon.

Figure 6 illustrates aliasing behavior by changing the interval between traces (effectively, the trace-to-trace sample rate) rather than by trace-to-trace vertical shift, as done in Figure 5. The array of traces in Figure 6a is identical to Figure 5b. In Figure 6b, every other trace has been dropped, effectively doubling the trace interval and halving the trace sample rate (the

Figure 5. Aliasing in an array of four identical traces. The red horizon is the correct trace-to-trace correlation; with increasing vertical shift of adjacent traces, another possible correlation, marked by the dashed blue horizon, appears.

a) b) c) d)

Figure 6. Aliasing caused by deleting every other trace from the original four-trace array in Figure 5. The red horizon is the correct trace-to-trace correlation; the dashed blue horizon is an aliased correlation.

a) b)




left-to-right-dipping red horizon is in the same position on both arrays). In Figure 6b, the under sampled array, the dashed blue horizon dipping from right to left, is shown as a possible correlation of the zero crossing from trace to trace; this aliasing is related to the change in spatial sampling rate.

As stated, the Nyquist theorem applies to reconstruction of band-limited functions from equispaced data, and these can be functions of time or dis-tance. Referring to band-limited functions of distance, the wavenumber k of a waveform is defined as the number of wave cycles per unit distance, which is the inverse of wavelength l. These parameters are the spatial equivalents of the frequency f and period T of a time-domain waveform (as shown in Figure 2 of Chapter 2). The spatial sample rate Dx for a 2D seismic line is the common-depth-point (CDP) interval, and for a 3D survey this rate is the bin size. For a given Dx, the sampling wavenumber is 1/(Dx) and the cor-responding Nyquist wavelength and Nyquist wavenumber are 2Dx and 1/(2Dx), respectively.

When working with a grid of 2D data, you need to think about the size of a feature that can be resolved in terms of the line spacing of your grid. To illustrate this concern, consider a simple 2D anticline that is X units wide and thus has a wavenumber of 1/X. Again for simplicity, assume that you are working with a grid of 2D lines that is Y units by Y units square, and that the axis of the anticline is parallel to one of the directions of the lines in the grid. The Nyquist wavenumber for this grid of lines is 1/(2Y). The Nyquist theorem states that a feature with a wavenumber larger than the Nyquist wavenumber will be aliased, that is, the anticline will be aliased if 1/X > 1/(2Y). Using a numerical example, if the anticline is 2 km wide, it will be aliased by a grid with a line spacing greater than 1 km; said another way, you need a grid spacing of 1 × 1 km or less to map this anticline accurately.

Geologic features come in all shapes and sizes, and you need to under-stand resolution as one of the fundamental elements of seismic data quality to represent geology properly in an integrated interpretation. Each seismic data set has a characteristic resolving power. Good practice dictates that you should investigate each of your data sets carefully to be fully aware of temporal and spatial limits of resolution, especially in the context of your interpretation objectives.





6 Resolution

Documents

sampling of seismic

nyquist frequency

hz signal

sampling frequency

hz figure

hz sample rate

seismic responses

seismic interpretation