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. In general, one can visualize the problem as depicted in Figure 4-23. An electromagnetic field is incident on a localized object which gives rise to a secondary field being scattered outward from the object. The source of the secondary field is a movement of the electrical charges in the structure in response to the incidence fields impinging on the object.

Figure: 4-23 A localized object interacts with an incident electromagnetic field. The field causes movement of elec-trical charge which gives rise a re-emitted (scattered) electromagnetic field.

Full analysis in even the simplest case in quantitative form is a very involved mathematical problem. The essential

Incident

Field Scattered

Field

- +

physical behavior is discussed here and readers should refer to other texts on electromagnetics to get more details on the subject.

For most analysis one can view the incidence field as a field which carries a certain amount of power per unit area PI. The object acts like a mask which absorbs energy and casts a shadow behind it as depicted in Figure 4-24. In essence the object can be treated optically to first order with the incidence signal power being extracted. This extracted power may be absorbed internally into the object or it may give rise to re-radiated energy which makes it detectable at some distance (the GPR goal).

Figure: 4-24 An incident field illuminates a target with a finite amount of power per unit area. The cross section of the target interrupts the power flow. If the target absorbs the power intercepted it will cast a shadow.

In general, one defines the power extracted from the incident field from the object as follows

(4-55)

where A is the cross sectional area for the target.

To first order, A is the physical area cross section presented to the field. In practice, however, the area is only a way of representing the effect and is not always equal to the geometrical cross section of the object. Only in the optical limit of infinitely short wavelengths will the geometrical and target cross section be equal. When the object is finite compared to the wavelength then A is called the effective cross section, Ae.

The target may emit some or all of the power which it extracts from the incident field. Mathematically the re-emitted signal, Ps, is expressed as

(4-56)

where indicates the fraction of the power which is reemitted . The combination of is called the scattering cross section for the target.

Incident field with P power per unit

area^I cross sectional area

shadow zone

P

= AP

P

= ξA

P

ξ ( 0 ≤ ≤ ξ 1 ) ξA

Figure: 4-25 Variation of the scattering cross section of a spherical target as a function of dimension normalized against wavelength (Skolnik (1970)). While specific to a sphere, similar behaviour is displayed by any object of finite dimension. The response increases until the object is on the same order of size as the wavelength. For GPR to pene-trate through a heterogeneous material, it is highly advantageous that the GPR wavelength be large compared to the

scale of heterogeneity.

Perhaps the most informative study of a scattering object is the work by Mie described in Skolnik (1970) who studied the behavior of a perfectly conducting sphere of arbitrary dimension for monochromatic (sinusoidal) excitation. The basic result is depicted in Figure 4-25. Here the cross section is normalized to the geometric cross section

(4-57)

and is plotted against the normalized dimension

(4-58)

where the factor a is radius of the sphere and is the wavelength of the excitation signal.

The characteristics of the plotted cross section versus dimension are shown in Figure 4-25. When the dimensions of the object are small compared to the excitation wavelength, the response is called the Rayleigh response. This represents the response of a small object and the response varies as the fourth power of the excitation frequency.

When the object dimension approaches the same size as the excitation wavelength, the response is called the Mie or the resonance response. Here effective area can exceed unity and will oscillate in amplitude. Physically the charges (currents) on the sphere have transit times which are comparable to the transit time or the period of the excitation signal. As a result, a response enhancement through resonance or a response suppression when anti-resonance can occur. As the dimension of the object gets larger, the normalized cross section approaches unity. At this point the response is referred to as the optical response and the geometrical cross section essentially matches the scattering cross section. The object is much larger than the wavelength and the sphere has a cross section equal to that of a disc with the same diameter.

A

A

πa

---=

D

2πa

--- λ

=

λ

It is interesting to compare the Mie scattering response with the behavior of the thin layer reflection coefficient discussed in section 4.8. The behavior is quite similar which is not surprising given that the same basic physics and phenomena are involved.

Of much interest in GPR and many other physical problems is the case where the object is small compared with the wavelength which is referred to as the Raleigh scattering regime. The strength of the scattered signal and hence the scattering cross section very strongly with frequency. Typically the scattering cross section can be expressed as

(4-59)

where C is a constant, a is the radius of the object and f is the excitation frequency. One can see the scattering cross section increases rapidly with frequency and slight changes in GPR frequency will give rise to much more intense response from a localized object.