Refining the pipe depth

Fig. 5. The data of Fig. 5 after spline rubber sheeting to the marks along horizontal traverse and in fill interpola- tion. age. The circle is distorted into an ellipse by vertical exaggeration, as no correction has been performed to make the vertical and horizontal axes the same scale dimension. Also indicated in Fig. 6 is the position of the near-field of the antenna, as the hyperbola shape is fit with a far-field ray tracing model assumption. At this point, the data processing and hyperbola fitting indicate a pipe centered 2.78 m from the begin- ning of the traverse, at a depth of 0.34 m in a soil with relative dielectric permittivity of 4.0, and with a diameter of 0.41 m. These numbers can be further refined. In Fig. 7, an image processing hyperbola mask has been applied to the data to collapse or Ž focus the hyperbola using a process similar to . migration; Yilmaz, 1987 . The image now shows only the scattering cross-section of the visible radius of curvature of the pipe. By looking at Ž other hyperbolas in the image such as those . from the rocks , their over or under migration Ž focusing residual hyperbolic shapes pointed up- . wards or downwards indicates the variability of the velocity and hence dielectric permittivity throughout the section.

4. Refining the pipe depth

In Fig. 8, the data scan under the vertical black line at the location of the peak of the hyperbola in Fig. 6 has been extracted and plotted as the dashed line. This is the original raw data scan before all the processing above. The processing was performed to produce a clear image and better fit to the hyperbola. The solid line in the main portion of the plot is a full waveform model generated through the radar Ž Fig. 6. Hyperbola fitting to estimate permittivity and . hence turn two way travel time into depth and to estimate object size. Ž . Fig. 7. Image processing hyperbola masking migration to focus hyperbolas and to estimate velocity variation throughout the image. Ž equation for a layered earth Duke, 1990; Pow- . ers and Olhoeft, 1995 from the estimated per- mittivity and depth of the hyperbola fit, and using the parameters versus depth shown on the right side of the plot. The key to the parameters is across the top. There are three such sets of parameters, of which one is shown in Fig. 8. This one is for the complex dielectric permittivity in terms of the four Cole–Cole dielectric relaxation parameters Ž . Olhoeft and Capron, 1994; Olhoeft, 1998a : ´ r1 is the low frequency limit of the relative dielec- tric permittivity, ´ is the high frequency limit, r` t´ is the time constant, and a´ is the breadth parameter for the log-normal Cole–Cole distri- bution of time constants. Values of 4.0, 4.0, 0.0, and 1.0 indicate a real permittivity with no imaginary part and no frequency dependence as ´ s ´ . The values of y1.0 in the next layer r1 r` tell the model to use the properties of a metal. There is a similar set for the complex magnetic permeability, which will be assumed to be that of free space in this model, m s m s 1.0. r1 r` The third set describes the low frequency limit- Ž ing electrical conductivity seen in the next . figure . The label at the top of the figure tells where the field scan is located. On the top right are the Ž model frequency using a Ricker wavelet Sheriff, . 1984 , and a display gain in decibels. Across the bottom are the time scale, the Offs s 1.81 indi- cates a 1.81-ns offset from the beginning of data Ž to locate time zero determined from the previ- . ous processing and the C s 1.00 is a coupling r ratio to describe changes in the antenna center Ž frequency as it is loaded by the ground not . used here . In the left half of the main plot, the smaller vertical solid line indicates time zero and the larger vertical solid line indicates the estimated position of the near-field boundary. This model assumes far-field, plane wave prop- agation with vertical incidence at horizontal lay- ering. Thus the air wave between the transmitter Ž and receiver antennas in the data between the . two vertical lines in the near-field is not mod- eled. An example of near-field modeling and Ž . requirements may be found in Kirkendall 1998 . The first excursion in the data at the left-most edge of the plot is an internal radar system sync pulse and is also not modeled. Multiples be- tween the antenna and the pipe are modeled. Further details of this full waveform modeling Ž . are published in Powers and Olhoeft 1995 . In Fig. 8, the first thing noticed about the model attempted from the estimated parameters Ž derived by the previous processing time zero, . permittivity and depth are that the amplitudes are wrong. There is also a lot of high frequency ringing caused by truncation in computing Ž Fourier transforms which are performed only over the frequency range which contains signifi- . cant amplitude in the data . The amplitude com- putation includes the effects of geometric spreading deduced from the processing derived Ž . depth algorithm of May and Hron, 1978 , the Fig. 8. Full waveform modeling of the scan directly over the pipe in the previous figures. This figure shows the modeling only using the estimated permittivity and depth from the previous hyperbola fitting. Fresnel reflection coefficient at the interface, and of the hardware range gain function that was recorded with the data, but it has not yet included the exponential material losses Ž . Olhoeft, 1998a . In Fig. 9, on the right, the electrical conductivity depth profile has been adjusted to more realistically match the decay Ž shown in the data thus including part of the . exponential material losses , and to include the conductivity of the metal pipe. To make the Fig. 9. Improving the model fit by adding the effects of electrical conduction loss compared to the previous figure. zero crossings between the model and the data align better, the original processing depth esti- mate is adjusted from 0.34 to 0.36 m.

5. Estimating the soil density and water con- tent