# Kirchhoff and F-K migration to focus ground penetrating radar images

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## Abstract

Ground penetrating radar (GPR) based land mine detection has a main challenge of having an accurate image analysis method that is capable of reducing false alarms. However this image analysis depends on having sufficient spatial resolution in the backscattered signal. This paper aims at getting better resolution by applying two migration algorithms. One is by Kirchhoff’s migration using geometrical approach and other one is F-K migration algorithms with Fourier transform. The algorithms are developed using MATLAB simulations over different scenarios for stepped frequency continuous wave (SFCW) GPR.

### Keywords

GPR Migration Hyperbola B-scan## Introduction

Ground penetrating radar (GPR) is widely used in detecting subsurface objects such as buried landmines. It is sensitive to changes in all three electromagnetic characteristics of the medium: namely the electric permittivity, electric conductivity and magnetic permeability. Depending on the application, different scanning schemes, namely, A-scan, B-scan, and C-scan, are being employed [1]. In the B-scan measurement situation, a downward looking GPR antenna is moved along a straight path on the top of the surface while the GPR sensor is collecting and recording the scattered field at different spatial positions. This static measured data collected at single point is called an A-scan [2].

In a typical B-scan GPR image, the scatterers within the image region show up as a hyperbolic curves because of the different trip times of the EM wave while the antenna is moving along the scanning direction. Such an image construction may be sufficient when the aim is only to detect the targets. In most situations, however, the indicative information about the location, shape and size of the targets is required [1, 3]. Therefore, one of the most applied problem for the B-scan GPR image is to transform (or to migrate) the unfocused space–time GPR image to a focused one showing the object’s true location and size with corresponding EM reflectivity. The common name for this process is called *migration* or *focusing* [3, 4]. The migration methods were primarily developed for processing seismic images [5] and have been widely used for the focusing of GPR data because of the similarities between the acoustic and the electromagnetic wave equations [4, 5, 6, 7, 8, 9, 10, 11, 12]. Various focusing or migration techniques have been developed to increase the spatial resolution of GPR images. These are; diffraction summation [12], Kirchhoff wave-equation migration [11], phase-shift migration [7, 9], and frequency-wave number (F-K) migration [6, 8].

Use of non-uniform fast fourier transform (NUFFT) algorithm to migrate GPR data as discussed in [9] has resulted in high signal to noise ratio in the migrated data. The non-uniform nature of the wavenumber space requires linear interpolation before the regular fast Fourier transform (FFT) could be applied. However, linear interpolation usually degrades the quality of reconstructed images. The NUFFT method mitigates such errors by using high-order spatial-varying kernels. The phase-shift migration (PSM) method is first introduced and applied by Gazdag [7] to handle lateral velocity variation. This algorithm iteratively puts a phase shift to migrate the wave field to the exploding time of *t* = 0, such that all the scattered waves are drawn back to object site to have a focused image. The performance of focusing algorithms over different parameters are compared over various migration algorithm shows F-K migration is the fastest one and the Back-Projection based Migration (BPA) is the second best while the Kirchhoff migration is the slowest [4]. From the signal to clutter ratio results, the clutter suppression ability of BPA is the best. Synthetic aperture radar (SAR) based focusing techniques have been applied to GPR data in [13] based on F-K migration. It also suggests that F-K domain SAR algorithm better focuses than hyperbolic summation method and also more efficient in computation time.

## Kirchhoff migration algorithm

The transmitter and receiver must me placed such that for all positions of transmitter and receiver for a particular shot gather, the midpoint of the offset must remain the same. Here the offset refers to the distance between the transmitter and receiver. For each shot, we obtain the reflection which is referred to as the A-scan and the collection of such several A-scans gives the 2-D B-scan image which is used for processing. The B-Scan image turns out to be the hyperbola.

Non-zero offset data is characterized by a travel time increase with increase in offset distance from the source to the reflector. The non-zero offset to zero offset conversion is achieved through a correction called NMO (normal move out) correction. For the single constant horizontal velocity layer the trace time curve of a function of offset is a hyperbola.

*x*

_{2}is the distance (offset) between the source2 and receiver2 position in Fig. 2. \(v_{2 }\) is the velocity of the medium above the reflecting interface. \(t_{2} \left( 0 \right)\) is twice the travel time along the vertical path. Generally, the NMO correction is given by the difference between t(x) and t(0)

Each CMP gather after NMO correction is summed together to yield a stacked trace. Stacking enhances the in-phase components and reduces the random noise. It yields zero offset section (in the absence of dipping layers in the subsurface) [5].

In this way, each point on the migrated section is treated independently from the other points. Each point on the output migrated section is produced by adding all data values along a diffraction that is centered at that point. The diffraction summation method sums the seismic event amplitudes as recorded; *Kirchhoff migration* corrects the amplitudes and phase for three factors before summing.

The migration principles are

*x*,

*z*,

*t*). Solving the wave equation, the far-field approximation considered in Kirchhoff migration can be expressed as:

_{0}, t

_{0}) is usually dropped, because it is proportional to 1/r

^{2}[4].

*O*. At point A transmitter—receiver pair gets a reflection from the dipped reflector or the object. By continuing scanning in x direction, assume that the reflection ends at point B. Now, we have the apparent position of the object as C’D’. It is necessary to apply the migration algorithm on this geometry of Fig. 5a to get true position of the object as CD [5].

The length BD’ and AC’ are known from reflected time. Also, from geometry in Fig. 5a, Angle BDO = 90° and Angle ACD = 90°. Consider the origin O(x_{o}, z_{o}), location of transmitter–receiver pair at A(x_{a}, z_{0}) and apparent position of object C’(x_{a}, z_{a}). Since we are considering the point object, only C’ location in the Fig. 5a is the apparent position of target.

- 1.
Calculate OA = x

_{a}−x_{o}and AC’ = z_{a}−z_{0} - 2.
Assign length AC = length AC’

- 3.
Calculate apparent angle, θ

_{a}= tan^{−1}(AC’/OA) [from Fig. 5a] - 4.
Calculate true angle, θ

_{t}= sin^{−1}(AC/OA) [from Fig. 5b] - 5.
Calculate OC = OA* cos(θ

_{t}) - 6.
Obtain x

_{p}= OC*cos(θ_{t}) - 7.
Obtain z

_{p}= OC*sin(θ_{t})

Thus true location is P(x_{p}, z_{0}) and apparent location is A(x_{a}, z_{0}) over the ground surface as in Fig. 5b. C’(x_{a}, z_{a}) over depth of z_{a} from ground is the apparent position and C(x_{p}, z_{p}) is the true location over a depth of z_{p} from ground surface.

## F-K migration algorithm

*ω*for pre-migrated event and it is the vertical wavenumber k

_{z}, associated with the depth axis for post-migrated event. We assume velocity equal to 1. F-K migration maps lines of constant frequency AB in the (ω, k

_{x}) plane to circles AB’ in the (k

_{z}, k

_{x}) plane. Therefore, migration maps point B vertically onto point B’. Note that in this process, the horizontal wave number k

_{x}does not change as a result of mapping. When this mapping is completed, the dipping event OB is mapped along OB’ after migration. Thus, the dip angle after migration (θ’) is greater than before migration (θ). For comparison, these two radial lines are shown on the same plane (k

_{z}, k

_{x}), Fig. 6.

The phase shift method can only handle vertically varying velocities. To handle lateral velocity variations, first the input wave field is extrapolated by the phase-shift method using a multiple number of laterally constant velocity functions and a series of reference wave fields are created. The imaged wave field then is computed by interpolation from the reference wave fields. This migration method is known as phase-shift-plus-interpolation. If the medium velocity is constant, then migration can be expressed as a direct mapping from temporal frequency ωω to vertical wavenumber k_{z.}

### Mathematical model

_{z}term by substituting the results of the following equation,

_{x}, k

_{y}, w) domain should be transformed to (k

_{x}, k

_{y}, k

_{z}) domain to be able to use the FFT. Hence the mapping is done in order to do this transformation. The value of k

_{z}is found out by using the following relation,

_{z}value is found out for every corresponding k

_{x}and w and substituted in the following equation;

After the above procedures have been carried out, the inverse Fourier Transform is taken to obtain the migrated or focused image.

_{y}terms should be dropped respectively from the above equations. The resulting equation in that case will be as under:

Although the mapping procedure slows down the algorithm, yet it is speedier than any other migration algorithm used for humanitarian demining. This is due to the use of FFT for fast computation of the data.

### Step by step description of F-K migration algorithm

- 1.
Obtain the time traces of wave-field amplitudes E(x, z = 0, t)

- 2.
Discretize E(x, z = 0, t) and obtain MxN matrix

- 3.
Take 2-D FFT on E(x, z = 0, t)

- 4.
Interpolate E(k

_{x}, f) to obtain desired depth E(k_{x}, k_{z}) - 5.
Apply 2-D IFFT on E(k

_{x}, k_{z}) and select absolute values to get focused image.

## Simulation results

As a case 2, B-scan data was generated considering three targets in different locations numbered as 4, 5 and 6 and having true coordinates (17, 156) (53, 121) (65, 139) respectively resulting in Fig. 7b. After applying Kirchhoff migration algorithm, output data with migrated targets 4, 5 and 6 are in locations (21, 157) (53, 119) (65, 139) respectively as shown in Fig. 8b.

Results for Kirchhoff Migration with multiple targets

Target number | X co-ordinate (true) | X true Location true (cm) | Z co-ordinate (true) | Z true location (cm) | X co-ordinate (migrated) | X obtained location (cm) | Z co-ordinate (migrated) | Z obtained location (cm) | Error in lateral position (cm) | Error in depth (cm) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 50 | 117.18 | 105 | 82.03 | 50 | 117.18 | 105 | 82.03 | 0 | 0 |

2 | 100 | 234.37 | 115 | 89.84 | 10.0 | 234.37 | 115 | 89.84 | 0 | 0 |

3 | 80 | 187.5 | 110 | 85.95 | 80 | 187.5 | 110 | 85.95 | 0 | 0 |

4 | 17 | 39.84 | 156 | 121.87 | 21 | 49.22 | 157 | 122.66 | −9.38 | −0.79 |

5 | 53 | 124.21 | 121 | 94.53 | 53 | 124.21 | 121 | 94.53 | 0 | 0 |

6 | 65 | 152.23 | 139 | 108.59 | 65 | 152.34 | 139 | 108.59 | 0 | 0 |

As a case 4, the Fig. 7b is considered as input to F-K migration algorithm. Figure 9b shows the focused image of migrated data after applying F-K migration algorithm, resulting in output data with migrated targets of 4, 5 and 6 in locations (18, 155) (53, 119) (66, 136) respectively.

Results for F-K Migration with multiple targets

Target number | X co-ordinate (true) | X true location (cm) | Z co-ordinate (true) | Z true location (cm) | X co-ordinate (migrated) | X obtained location (cm) | Z co-ordinate (migrated) | Z obtained location (cm) | Error in lateral position (cm) | Error in depth (cm) |
---|---|---|---|---|---|---|---|---|---|---|

1 | 50 | 117.18 | 105 | 82.03 | 50 | 117.18 | 103 | 80.46 | 0 | −1.57 |

2 | 100 | 234.37 | 115 | 89.84 | 100 | 234.37 | 112 | 87.5 | 0 | −2.34 |

3 | 80 | 187.5 | 110 | 85.95 | 80 | 187.5 | 108 | 84.375 | 0 | −1.57 |

4 | 17 | 39.84 | 156 | 121.875 | 18 | 42.18 | 155 | 121.09 | +2.3 | −0.78 |

5 | 53 | 124.21 | 121 | 94.53 | 53 | 124.21 | 119 | 92.96 | 0 | −1.56 |

6 | 65 | 152.23 | 139 | 108.59 | 66 | 154.68 | 136 | 106.25 | +2.3 | −2.34 |

## Conclusions

We have considered different cases for SFCW GPR with multiple targets, homogeneous ground conditions and no clutter. Both Kirchhoff and F-K migration methods showed promising results in improving resolution of GPR in focusing the images. The Kirchhoff migration is done after doing hyperbolic summation as NMO and stacking. Maximum error in lateral position is of one coordinate corresponding to 2.3 cm and maximum error in depth position is of 3 coordinates corresponding to 2.34 cm in F-K migration. This indicates that the actual and migrated depths are very close to each other. Under various scenarios, both migration algorithm provide very less error, hence these methods can be implemented on data from GPR in all cases. In F-K migration algorithm, before applying the FFT routine, a mapping procedure from frequency-wavenumber domain to wavenumber–wavenumber domain is necessary. Although this mapping procedure may slow down the execution time of the algorithm, it is still fast thanks to the FFT step. Here, F-K migration algorithm takes an average time of 2.05 s and Kirchhoff migration takes 3.45 s. It clearly says that F-K migration algorithm results in less computation time than Kirchhoff migration algorithm and also results in slightly less error in lateral and depth position.

## Notes

### Authors’ contributions

NS helped in performing data processing, analysis and interpretation and also drafted the manuscript. DRUB and SA conducted experiments and collected various results. SNS developed investigation plan, helped in algorithm development and reviewed the manuscript. VS gave research idea, helped in algorithm development and reviewed the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

The authors are thankful to late Prof. RB Kulkarni for providing invaluable suggestions and to project students for their contributions.

### Competing interests

The authors declare that they have no competing interests.

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