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(a)

(b)

Figure: 4-15 wavefronts spreading out from a localized source a) located above the ground and b) located on the air-ground interface.

If one examines the wavefront in the ground, it is no longer spherical as bending occurs with differing degreesdepending on the varying incidence angle. To understand what is happening in the ground and near the interface, thelimiting case of the source right at the interface is informative (see Figure: 4-15b). The incident and reflected wavesin the air coalesce into an upgoing spherical wave. In the ground, the transmitted signal divides into two parts, aspherical wave and a planar wavefront travelling at the critical angle which links the direct spherical air wave and thespherical ground wave. Near the interface, the spherical ground wave extends into the air as an evanescent field.

The derivation of the mathematical form is based on the plane wave field expansion discussed in the previous section.For those interested, references such as Sommerfield (1949), Wait (1962), Brekhovskik (1960) and Annan (1973)have discussions. The various wave fields are clearly visible at large distances from the source and/or very shortwavelengths. For short distances from the source or long wavelengths, the separation of the events is blurred but theessential concepts are valid.

For a localized target in the ground, there can be several possible paths that energy can travel from a transmitter to thereceiver. The concepts are depicted in Figure 4-16 and the ray paths are shown in Figure 4-17.

Figure: 4-16 Concept drawing of a localized target beneath the ground with a transmitter and receiver located at the air ground interface.

Air

Ground

Critical Angle

Air

G round

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a) Direct air signal b) Direct ground signal

c) Direct reflected (scattered) signal d) Transmitted signal critically refracted and target scatte rs towards receiver.

e) Transmitted signal scattered and f) Critically refracted transmitted critically refracted to receiver. signal is scattered and critically refracted to the receiver.

Figure: 4-17 For the simple model in Figure 4-15, there are many paths that signals can travel between a transmit-ter and a receiver.

The relative importance of each path depends on the target depth, the separation between the transmitter and receiverand elevation of the transmitter and receiver. One should also note that one can not distinguish between d) and e).

In most GPR cases, the transmitter receiver separation is small and the predominant paths are a), b) and c). Paths d)and e) can still be important if both the transmitter and receiver are a substantial distance from the target even if thetransmitter and receiver are close together.

As indicated earlier, the intent here is to provide a basic overview. For the curious, dig into the references. Many ofthe ideas will be revisited in later discussions where some aspects are explored in more detail.

4.11 RESOLUTION AND ZONE OF INFLUENCE

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

Resolution essentially breaks into two components when dealing with GPR. There is the longitudinal (range or depth)resolution and there is the lateral resolution which relates to angular or lateral displacement resolution. The basicconcepts are depicted in Figure 4-18.

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T R

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Figure: 4-18 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 alsocommon to seismic measurements and in fact they are applicable to any technique where wave type phenomenon areused to detect objects at a distance.

In the current discussions we will work in the time domain. We will be considering systems which generate a pulseand detect the echoes that are observed and are close replicas of the pulse. These echoes may arrive simultaneously,overlap or be separated in time. The basic concepts are depicted in Figure 4-19.

Figure: 4-19 Temporal pulses with 1/2 width of W. Pulses are clearly separable when T>>W (a). To pulses are said to be distinguishable until (b) when T<<W then two events are not distinguishable (c).

TR

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P ulsesC oinc iden t

T

P ulsesoverlap

W

P ulsesC learly

S eparate

T W≈

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When two responses are present, one has to address the question of how closely spaced in time can they be in order tobe distinguished from one another. If one has two pulses that are coincident in time, the amplitude will be enhanced.The result will be one event with a larger amplitude. The question then arises as to how much separation between theevents must occur before they can be recognized distinctly as two events. This subject in raised again in chapter 5when we talk about system bandwidth and resolution.

Figure 4-19 depicts the extremes for two pulses. Generally speaking a pulse is characterized by its width at halfamplitude, W. While there are several definitions of resolution or separation of two temporal pulses, the widelyaccepted definition is that if the two pulses are separated by half their “half width” then they are distinguishable astwo events. If they are separated in time by less than this amount then they will most likely be interpreted as a singleevent.

The preceding is a discussion of the detection and presentation of information for a system. These concepts must betranslated into the spatial domain in order that the temporal pulse separation issue can be translated into spatialresolution. The approach is best understood by examining the response of two point targets for the cases illustrated inFigure 4-20.

(a) (b)

Figure: 4-20 Range and lateral resolution can be determined by considering the response of two localized targets either in-line (a) or side-by-side (b).

If one has two targets denoted to T1 and T2, and these targets line in the same direction away from the measurementsystem, then the difference in travel time (which is observed on the radar record) is directly related to the differentialdistance 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)

∆∆∆∆ r

r

T R T R

r

∆∆∆∆ L

t12dv

------=

t22d 2∆r+

v----------------------=

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The differential time is expressed as

(4-45)

and we require that this time difference be greater than half the pulse half width in order that the responses bedetectable. From this one can directly translate the spatial separation in the radial detection from the system must begreater than or equal to

(4-46)

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

One can examine lateral resolution in a similar fashion. Following along from the geometry in Figure 4-20, one hasthe travel time for target one (T1)

(4-47)

and the travel time for target two (T2)

(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 and one can use theapproximation that the lateral offset is small compared to the distance from the system. When this approximation isemployed, the time difference can be expressed approximately as

(4-50)

∆t t2 t12∆r

v---------=–=

∆r Wv4

---------≥

t12dv

------=

t22 d

2 ∆l2

+( )1 2⁄

v-----------------------------------=

∆t2 d

2 ∆l2

+( )1 2⁄

d–[ ]v

-------------------------------------------------=

∆t ∆l2

vd--------≈

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This leads to the result that the lateral (resolution or separation of the two side-by-side targets to be distinguishable)must be

(4-51)

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

The lateral resolution is closely related to the Fresnel zone concept which expresses the same idea for interference ofmonochromatic (sinusoidal) signals.

With GPR, the pulse width, W, in time is directly related to the bandwidth, B, which is also directly related to thecenter frequency, fc. If one uses this relationship

(4-52)

and

(4-53)

one finds that the lateral resolution can be expressed as

(4-54)

where is the wavelength of the GPR center frequency. This is the expression for the Fresnel zone formonochromatic signals.

The issue of lateral resolution is one which is commonly discussed in seismic measurements. Quite often the circulararea encompassed with radius equal to lateral resolution defines the zone of influence at a given distance. Anexcellent discussion of these concepts from the seismic point of view is given by Knoll (1991)

4.12 SCATTERING FROM A LOCALIZED OBJECT

Time and again with GPR, one has to address the issue of energy returned from a localized feature in the subsurface.Trying to quantify this behavior is complex and full analysis is beyond the scope of this introductory set of notes. Ingeneral, one can visualize the problem as depicted in Figure 4-21. An electromagnetic field is incident on a localizedobject which gives rise to a secondary field being scattered outward from the object. The source of the secondaryfield is a movement of the electrical charges in the structure in response to the incidence fields impinging on theobject.

∆l vdW2

------------≥

W 1B----

1fc----==

λ c fc v⁄=

∆ldλc

2--------=

λc