INTRODUCTION

The term "resolution" refers to how much detail we can see on a seismic reflection record. What is the thinnest bed for which we can clearly discern echoes from the top and from the bottom? How narrow a feature clearly displays signs of the sides?

It is important to have some idea of resolution before planning a survey. There’s no point in doing a survey that is too coarse to reveal the features we are interested in. By the same token we need to know about the resolution on a profile so we won’t delude ourselves about the significance of features too small to be meaningful.

VERTICAL RESOLUTION

What is the thinnest bed that we can resolve? To answer this question let’s consider an infinite plane wave at normal incidence on a layer of slightly lower acoustic impedance imbedded in an infinite medium. Recall that velocity equals wavelength divided by period or wavelength multiplied by frequency.

For "thick" beds we’ll get distinct echoes from the top and bottom of the bed. The reflection from the top is 180 degrees out of phase with the incident wave (remember the Rayleigh reflection coefficients). The echo from the bottom trails behind and is in phase with the incident wave. Thick beds are thicker than half a wavelength.

The thickness of the thinnest thick bed equals the wavelength divided by two. In this case the bottom echo comes directly after the top echo with no time delay at all. For all thick beds, the top and bottom echoes do not overlap in time and their amplitudes are independent of thickness.

For "thin" beds (thickness less than half a wavelength) the top and bottom echoes overlap in time and can’t be separated or timed individually. We can not determine bed thickness from arrival times. For a thickness of a quarter wavelength the later half of the top echo overlaps the first half of the bottom echo. This "tuning thickness" leads to a very strong echo; it’s actually doubled in amplitude.

As beds get even thinner than the tuning thickness, the waveform stays about the same but the amplitude decreases. For a thickness of zero we get complete destructive interference and there’s no echo at all!

So, in summary, we use arrival times of echoes to determine the thickness of thick beds. For thin beds we use amplitudes. Conventionally we say that the limit of vertical resolution is one-quarter wavelength. A normal seismic signal is not monochromatic (single frequency) so we conventionally use the dominant frequency and wavelength.

HORIZONTAL RESOLUTION

Where on an infinite planar reflector do echoes come from? We say that the normal incidence travel-time is just twice the depth divided by the velocity. This formula suggests that the echo comes only from a mathematical point. How can a non-physical feature have a physical result? To answer this question and to learn about horizontal resolution let’s try a little lab experiment.

For equipment we’ll need a loudspeaker to transmit and receive signals, a circular plywood disk as a reflector and an oscilloscope to display and time the signals. We can keep the transmitted pulse constant and use targets of different sizes to see what happens.

For small targets the echo amplitude is directly proportional to target radius. For targets above some given size, however, the amplitude is constant. The radius at which the amplitude versus radius plots starts to flatten out is called the (first) Fresnel zone radius.

The basic physical explanation is that echoes from successively larger rings or circular zones of the target are progressively delayed in time with respect to the normal incidence ray. For small targets, only the inner ring contributes and the echo amplitude grows as the target does. Beyond the Fresnel zone radius we have roughly equal destructive and constructive interference from successive rings and the echo amplitude stays constant.

A simple calculation based on the Pythagorean Theorem shows that the radius of the first Fresnel zone is approximately equal to the square root of half the depth times the dominant wavelength.

As the formula implies, the resolution gets worse (larger Fresnel Zone) as we look deeper into the earth. In addition, resolution gets worse because the dominant wavelength increases with depth for two reasons. One is that usually the velocity increases with depth and thus the wave is stretched. The second reason is that the short wavelengths are preferentially attenuated in the earth. For these three reasons resolution decreases with depth.

EFFECTS OF REFLECTOR WIDTH

For echoes from "narrow" targets less than one Fresnel zone width (diameter) we don’t clearly see a flat surface with distinct edges. Instead the echo is just a hyperbolic diffraction. The amplitude is directly proportional to target width and, of course, is zero for zero width.

For echoes from "wide" targets greater than one Fresnel zone width, there is a clear, distinct flat echo. In addition there are hyperbolic diffractions with apices at each edge of the target. The outer limbs of the diffractions have the same phase as the reflection whereas the inner limbs are reversed in phase. The amplitudes of the reflections and diffractions are independent of target width.

EFFECTS OF TARGET SPACING

For target spacings less than one Fresnel zone the gap is largely filled by overlapping in-phase diffractions, one from each side. Thus the gap does not show clearly and we’ll think we have a continuous reflector.

For spacings greater than one Fresnel zone we see distinct edges on either side as well as a distinct hyperbola on each side.

READINGS

Meckel and Nath, 1977, in "Seismic Stratigraphy-applications to hydrocarbon exploration", (C. E. Payton, editor), American Association of Petroleum Geologists, Tulsa, 516 pp.

Widess, 1973, Geophysics, 38, 1176-1180.