Resolution and Zone of Influence

Since GPR detects objects at a distance, the question that always arises is how accurately can the object be located and what detailed information can be extracted about the geometry of the object. Resolution indicates how precisely the position can be determined. Closely related to the question of resolution is the question of what geometrical attributes of the target can be extracted. Geometrical attributes include such factors as the size, shape, thickness, etc.

Resolution divides into two topics; there is longitudinal (range or depth) resolution and there is the lateral (or angular) resolution. The basic concepts are depicted in Figure 4-19.

Figure: 4-19 Resolution for GPR divides into two parts; namely range resolution and lateral (or angular) resolu-tion.

Understanding resolution gets into the fundamental issues of GPR detection concepts. These concepts are also common to seismic measurements and in fact they are applicable to any technique where wave phenomenon are used to detect objects at a distance.

In the current discussion we will work in the time domain. We will be considering systems which generate a pulse and detect the echoes from distant targets. Echoes may arrive simultaneously, overlap or be separated in time as depicted in Figure 4-20.

Figure: 4-20 Temporal pulses with 1/2 width of W. Pulses are clearly separable when T>>W (a). Two pulses are said to be distinguishable until (b). When T<<W then two events are not distinguishable (c).

T R

When two responses are present, how closely spaced in time can they be and still be discerned as distinct separate events. If two pulses are coincident in time, the amplitude will be enhanced. The result will be one event with a larger amplitude. As time separation between the events increase they can be recognized distinctly as two events when they no longer overlap. This subject in raised again in chapter 5 when we talk about system bandwidth and resolution.

Generally speaking a pulse is characterized by its width at half amplitude, W. While there are many definitions of resolution, the widely accepted definition of resolvable pulses requires that the two pulses be separated by half their

“half width” in order to be distinguishable as two events. If they are separated in time by less than this amount then they will most likely be interpreted as a single event.

These temporal pulse concepts must be translated into the spatial domain to define spatial resolution. The approach is best understood by examining the response of two point targets as illustrated in Figure 4-21.

(a) (b)

Figure: 4-21 Range and lateral resolution can be determined by considering the response of two localized targets either in-line (a) or side-by-side (b).

For two targets denoted 1 and 2, as shown in Figure 4-21, the longitudinal difference in travel time (which is observed on a GPR record) is directly related to the difference in radial or longitudinal distance between the targets.

The travel time for the first target will be

(4-43)

and the travel time for the second target will be

(4-44)

where v is the propagation velocity of the signal.

∆ r

The differential time is expressed as

(4-45)

and this time difference must be greater than half the pulse half width in order that the responses be detected as two events. The spatial separation of targets in the radial direction from the system must be greater than or equal to

(4-46)

One can see from this analysis that the pulse width and the velocity in the material combine to dictate the radial resolution. The radial resolution is essentially independent of distance from the source in an ideal world.

The lateral resolution is analyzed in a similar fashion. Following along from the geometry in Figure 4-21, the travel time for target 1 is

(4-47)

and the travel time for target 2

(4-48)

The time difference between the two events is expressed as

(4-49)

In most situations the target is a substantial distance away from the measurement system permitting the use of an approximation. When this approximation is employed, the time difference is the square root in Equation (4-49).

(4-50)

The lateral resolution (minimum separation of two side-by-side targets to be distinguishable) must be

∆t t

t

2∆r

(4-51)

From this result one can see that the lateral resolution depends on the velocity and the pulse width as well as the distance from the system. The larger the distance to the targets, the lower the lateral resolution.

The lateral resolution is closely related to the Fresnel zone concept which expresses the resolution concepts for interference of monochromatic (sinusoidal) signals.

With GPR, the pulse width, W, in time is directly related to the bandwidth, B, which is normally related to the center frequency, fc. Using the relationship

(4-52)

and

(4-53)

where is the wavelength of the GPR center frequency, one finds that the lateral resolution can be expressed as

(4-54)

This result is identical to the expression for the Fresnel zone for monochromatic signals.

Lateral resolution actually defines an area or zone of resolution since all targets encompassed by a radius of perpendicular to r can not be resolved.

∆l vrW

--- 2

≥

W 1

B ---- 1 f

=

----=

λ

= v f ⁄

λ

∆l dλ

--- 2

=

12

∆

2

∆l

Figure: 4-22 The zone of influence is defined as the area which can contain a second target that can not be uniquely resolved.

The zone of influence extends into a “tube” of influence when the concepts are extended to transillumination GPR techniques. These concepts are not unique to GPR and an excellent discussion of these same concepts from a seismic perspective is presented by Knoll (1991).