# A Phase-Offset Estimation Method for InSAR DEM Generation Based on Phase-Offset Functions

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Method

_{m}in Figure 1. The grid of the wrapped phase values is transformed into a grid of unwrapped phase values by using a phase-unwrapping algorithm that adds to each measured phase value a constant integer multiple of 2π, represented by ϕ

_{unw}in Figure 1.

_{off}in Figure 1, is a constant phase component for the whole scene that must be estimated and added to the unwrapped phase in order to obtain the absolute interferometric phase ϕ

_{abs}, from which a digital elevation model (DEM) can be generated. The phase-offset value can be positive or negative, and is sometimes greater than 2π, depending on the interferometric SAR processing strategy used.

_{1}as the reference (Figure 1), the following equations can be written

_{1}.

_{p}of a point P in the overlapped area can be written, for both acquisitions, based on the Equation (5), by:

_{abs1}and ϕ

_{abs2}are the absolute phase of the first and second acquisition, respectively.

_{p}, are only related by their incident angles. From this equation, the phase-offset ϕ

_{off 1}can be related as a function of ϕ

_{off 2}, as follows:

_{θ}and R

_{ph}are constant values at the position P with the height h

_{p}, the phase-offset value ϕ

_{off 1}can be seen as a linear combination of ϕ

_{off 2}, depending on the relation of the incident angle and the unwrapped phase difference between the two acquisitions.

_{i}selected, based on its geographic coordinate, with a Cartesian coordinate of (x

_{p}, y

_{p}, z

_{p}) and height equal to h

_{p}relative to an ellipsoid, as shown in Figure 3, whose value depends on the knowledge of the slant range distances, r

_{1}and r

_{2}, where for a monostatic SAR system, is given by

_{1}and r

_{2}, can be found through the Cartesian distance from P

_{i}(x

_{p}, y

_{p}, z

_{p}) to A

_{1}(x

_{1}, y

_{1}, z

_{1}) and from P

_{i}(x

_{p}, y

_{p}, z

_{p}) to A

_{2}(x

_{2}, y

_{2}, z

_{2}), respectively, where A

_{1}(x

_{1}, y

_{1}, z

_{1}) and A

_{2}(x

_{2}, y

_{2}, z

_{2}) represent the coordinates of the two antenna phase centers, derived from the state vector of the platform, (V⃗, S⃗), shown in Figure 3. The corresponding unwrapped phase value of P

_{i}(x

_{p}, y

_{p}, z

_{p}), ϕ

_{unwPi}, in the grid of unwrapped phase ϕ

_{unw i}, can be determined by using the backward geocoding technique [14].

_{i}(x

_{p}, y

_{p}, z

_{p}), with height h

_{p}, can be calculated by taking the difference from the absolute phase value calculated by using the slant range difference and the unwrapped phase value at this point, as follows

_{i}true height were h

_{p}, which is the case when it uses the position and height of a corner reflector to calculate the phase-offset value. As the true height of the point P

_{i}is unknown, a procedure based on function that relates phase-offset to height, applied in a height interval of h

_{min}to h

_{max}(Figure 2), was adopted to estimate the true phase-offset values for both acquisitions, assuming that the true height h

_{i}of the point P

_{i}is within [h

_{min}, h

_{max}].

_{i}from h

_{min}to h

_{max}in N steps, as shown in Figure 2, and applying the same procedure discussed previously to calculate the phase-offset value for each step, one can create a set of N phase-offset values for the first acquisition. The same procedure can be applied for the second acquisition, creating another set of phase-offset values related to the same height interval. The two set of phase-offset values calculated for the point P

_{i}can be represented by functions, the phase-offset functions (POFs), as follows

_{min}to h

_{max}in N height steps.

_{t}of the point P

_{i}is unknown, the phase-offset values cannot be determined from the POFs${f}_{1}^{Pi}$ and ${f}_{2}^{Pi}$. In order to overcome this problem, the following procedure was used to estimate the phase-offset values of both acquisitions:

- - Firstly, as the POFs represented by the Equations (17) and (18) are generated within the same height interval, [h
_{min}, h_{max}], one can combine them creating a new function g_{i}, the combined phase-offset function (CPOF), which relates the phase-offset values in the space Φ_{off 1}× Φ_{off 2}for the point P_{i}. A CPOF can be represented by a function that relates the phase-offset values of both acquisitions, for each height step, through the relation based on the Equation (13), as follows$${\mathrm{\Phi}}_{\mathit{off}1}(h)={\mathrm{\Phi}}_{\mathit{off}2}(h)\hspace{0.17em}.\hspace{0.17em}{K}_{{\theta}_{(h)}}+\hspace{0.17em}{R}_{p{h}_{(h)}},$$$${K}_{{\theta}_{(h)}}=\hspace{0.17em}\text{cos}{\theta}_{2(h)}/\text{cos}{\theta}_{1\hspace{0.17em}(h)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\text{and}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{R}_{p{h}_{(h)}}=\hspace{0.17em}{\varphi}_{\mathit{unw}2}\hspace{0.17em}{K}_{{\theta}_{(h)}}-\hspace{0.17em}{\varphi}_{\mathit{unw}1}$$_{θ}and R_{ph}of the Equation (19) can be considered quite constant when computed in a short height interval for the same point P_{i}. Based on that, one can suppose without loss of generality, that Equation (19) represents a linear function of Φ^{i}_{off 1}with respect to^{i}_{off 2}, where the term K_{θ}represents the angular coefficient and the term R_{ph}represents the constant value of this linear function. The coefficients K_{θ}and R_{ph}change according to the range position selected for a point in the overlapped area. - - Secondly, considering another point P
_{k}in the overlapped area, with a different range position from P_{i}, another two POFs, one for each acquisition, can be created for the point P_{k}. These two new functions can be combined, creating another CPOF, the g_{k}, that relates the phase-offset values of both acquisition in the space Φ_{off 1}× Φ_{off 2}for the point P_{k}. - - Finally, as the CPOFs g
_{i}and g_{k}are generated using the same height interval in different range positions, represented in the space Φ_{off 1}× Φ_{off 2}, they have different angular coefficients, ensuring an intersection point between them, from where the phase-offset values for both acquisitions can be estimated, as illustrated in Figure 4. The coordinates of the intersection point in Φ_{off 1}× Φ_{off 2}, shown in Figure 4(c), represent the estimate values of the phase-offset for the first acquisition, ϕ̂_{off 1}, and for the second acquisition, ϕ̂_{off 2}.

_{off 1}× Φ

_{off}, with a common intersection point. Due to noise presence in the interferometric unwrapped phase, or to abrupt variation of the phase, the common point of the intersection is not unique but has a cluster of points, very close together, from where the phase-offset values can be estimated.

## 3. Processing Sequence

_{unw1}and ϕ

_{unw2}are filtered in three steps. Firstly, they are filtered, based on a coherence threshold from Coh

_{1}and Coh

_{2}images, respectively, to discard points with low coherence—the threshold value is an input parameter which should be chosen according to the characteristics of the terrain and the acquisition; this filter is crucial to improve the accuracy of the estimation. Secondly, a morphological erosion filter [15] is applied on the interferometric unwrapped phases to discard very small regions in order to decrease the disrupting effects during the POF generation. Finally, an average filter with window size fixed according to the image resolution is used to reduce phase noise on the interferometric unwrapped phase images. To gain computation time, only boxes around the points of interest are filtered. After the filtering steps, the positions of the set of points in the overlapped area are selected to carry out the generation of the POFs.

_{1}and f

_{2}for a selected point in the overlapped area, shown in Figure 5, is based on the approach previously described and represented by Equations (17) and (18); in the next step of the processing chain, the POFs are combined through a linear combination, creating a CPOF g

_{comb}in the space Φ

_{off 1}× Φ

_{off 2}.

_{comb}can be created from where the intersection point can be estimated. Figure 7 shows the set of CPOF for 100 points scanned in the overlapped area in a height interval [h

_{max}− h

_{min}] equal to 200 m, based on the knowledge of the terrain topography from the SRTM DEM data, and a height step δ

_{h}equal to 2 m.

_{comb}(Figure 5) was introduced. This filter performs a linear approximation of the curves and uses the chi-square goodness of fit statistics [17] to decide which ones will be used for the estimation and which ones will be discarded.

_{comb}, into a set of values of uncorrelated variables called Principal Components.

_{comb}functions, allowing an easy determination of the crossing point by finding the minimum dispersion value of these transformed functions on the horizontal axis, from where the transformed phase-offset values can be determined. Finally, the transformed phase-offset values are transformed back using the Principal Components Coefficients, providing an estimation of the phase-offset values for both acquisitions, ϕ̑

_{off}

_{1}and ϕ̑

_{off}

_{2}. From these estimated phase-offset values, the absolute interferometric phase for both acquisitions can be determined by using the Equation (1).

_{max}− h

_{min}] and a height step δ

_{h}, according to the knowledge of the terrain mean height from the SRTM DEM data. Secondly, the coarse phase-offsets estimated values, ϕ̑

_{off}

_{1}and ϕ̑

_{off}

_{2}(Figure 5) are used to decrease the height interval and the height step for the second iteration to allow a fine estimation. In some cases, a third iteration can be tried to improve the estimation. For each iteration the accuracy of the phase-offset values can be evaluated through the root mean square error between the DEMs in the overlapped area, ɛ

_{DEM}

_{12}, of both acquisitions (Figure 5), which should be very small when the phase-offset values are well estimated.

## 4. Results and Discussion

#### 4.1. Error Analysis

_{ϕu}is the phase uncertainty, B

_{n}is the normal baseline, θ the incident angle, r the slant range distance and λ the wavelength.

_{ϕ}is the standard deviation of the interferometric phase, N is the number of measurement and μ

_{ϕ}is the mean phase error.

#### 4.2. Test Result without Using Corner Reflectors

## 5. Conclusions

## Acknowledgments

## References

- Graham, L.C. Synthetic interferometer radar for topographic mapping. Proc. IEEE
**1974**, 62, 763–768. [Google Scholar] - Zebker, H.A.; Goldstein, R.M. Topographic mapping from interferometric synthetic aperture radar observations. J. Geophys. Res
**1986**, 91, 4993–4999. [Google Scholar] - Hagberg, J.O.; Ulander, L.M.H. On the optimization of interferometric SAR for topographic mapping. IEEE Trans. Geosci. Remote Sens
**1993**, 31, 303–306. [Google Scholar] - Rosen, P.A.; Hensley, S.; Joughin, I.R.; Li, F.K.; Madsen, S.N.; Rodriguez, E.; Goldstein, R.M. Synthetic Aperture Radar Interferometry. Proc. IEEE
**2000**, 88, 333–382. [Google Scholar] - Goldstein, R.M.; Zebker, H.; Werner, C.L. Satellite radar interferometry: Two-dimentional phase unwrapping. Radio Sci
**1988**, 23, 713–720. [Google Scholar] - Ghiglia, D.C.; Romero, L.A. Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J. Opt. Soc. Am
**1994**, 11, 107–117. [Google Scholar] - Fornaro, G.; Franceschetti, G.; Lanari, R. Interferometric SAR phase unwrapping using Green’s formulation. IEEE Trans. Geosci. Remote Sens
**1996**, 34, 720–727. [Google Scholar] - Pritt, M.D. Phase unwrapping by means of multigrid techniques for interferometric SAR. IEEE Trans. Geosci. Remote Sens
**1996**, 34, 728–738. [Google Scholar] - Madsen, S.N. Absolute phase determination techniques in SAR interferometry. Proc. SPIE
**1995**, 2487, 393–401. [Google Scholar] - Scheiber, R.; Fischer, J. Absolute Phase Offset in SAR Interferometry: Estimation by Spectral Diversity and Integration into Processing. Proceedings of EUSAR, Ulm, Germany, 25–27 May 2004.
- Brcic, R.; Eineder, M.; Bamler, R. Absolute Phase Estimation from TerraSAR-X Acquisitions using Wideband Interferometry. Proceedings of IEEE Radar Conference, Pasadena, CA, USA, 4–8 May 2009.
- Eineder, M.; Adam, N. A maximum likelihood estimator to simultaneously unwrap, geocode and fuse SAR interferograms from different viewing geometries into one digital elevation model. IEEE Trans. Geosci. Remote Sens
**2005**, 43, 24–36. [Google Scholar] - Fornaro, G.; Sansosti, E.; Lanari, R.; Tesauro, M. Role of processing geometry in SAR raw data focusing. IEEE Trans. Aerosp. Electron. Syst
**2002**, 38, 441–454. [Google Scholar] - Schreier, G. Standard Geocoded Ellipsoid Corrected Images. In SAR Geocoding: Data and Systems; Chapter 6; Wichmann Verlag: Kalsruhe, Germany, 1993; pp. 159–163. [Google Scholar]
- Maragos, P. Morphological Filtering for Image Enhancement and Feature Detection. In Handbook of Image and Video Processing, 2nd ed.; Bovik, A.C., Ed.; Academic Press: New York NY, USA, 2005; pp. 135–156. [Google Scholar]
- Pinheiro, M.; Rosa, R.; Moreira, J.R. Multi-Path Correction Model for Multi-Channel Airborne SAR. Proceedings of 2009 IEEE International Geoscience & Remote Sensing Symposium (IGARSS’09), Cape Town, South Africa, 12–19 June 2009; 3, pp. 729–732.
- Taylor, J.R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd ed.; University Science Books: Sausalito, CA, USA, 1997. [Google Scholar]
- Richards, J.A.; Jia, X.P. Multispectral Transformation of Image Data. In Remote Sensing Digital Image Analysis, 2nd ed.; Springer-Verlag: Berlin, Germany, 1993; pp. 137–163. [Google Scholar]
- Rodriguez, E.; Martin, J.M. Theory and design of interferometric synthetic aperture radar. IEE Proc.-F
**1992**, 139, 147–159. [Google Scholar] - Greenwalt, C.R.; Schultz, M.E. Principles and Error Theory and Cartographic Applications; ACIC Thechnical Report No 96; Aeronautical Chart and Information Center: St. Louis, MO, USA, 1968. [Google Scholar]

**Figure 3.**Positioning of the point P

_{i}and the antennas phase centers A

_{1}and A

_{2}in Cartesian coordinates.

**Figure 4.**(

**a**) The illustration of the POF (relating phase-offset to height) for the first acquisition in two points, P

_{i}and P

_{k}, where h

_{i}and h

_{k}represent the true height of these points respectively. (

**b**) The illustration of the POF for the second acquisition for the same points. (

**c**) The CPOFs g

_{i}and g

_{k}in the space Φ

_{off 1}× Φ

_{off 2}are used to estimate the phase-offset values ϕ

_{off 1}and ϕ

_{off 2}through the intersection point of g

_{i}and g

_{k}.

**Figure 6.**OrbiSAR-1 X-band InSAR data; (

**a**) unwrapped phase grid and (

**c**) coherence image of the first acquisition; (

**b**) unwrapped phase grid and (

**d**) coherence image of the second acquisition.

**Figure 10.**OrbiSAR-1 data: (

**a**) Geocoded X-band image, (

**b**) X-band DEM, (

**c**) Geocoded P-band image, (

**d**) P-band DEM.

**Figure 11.**DEMs of OrbiSAR-1 X-band data for all tracks generated through the use of the proposed method for phase-offset estimation in the Amazon test site.

Phase-offset | Percentage of Overlapped the Area in Range Direction | |||||||
---|---|---|---|---|---|---|---|---|

20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | |

Mean value [rd] | 1.05 | 0.97 | 0.98 | 0.99 | 0.99 | 0.97 | 0.99 | 1.00 |

Std. dev. [rd] | 0.06 | 0.04 | 0.03 | 0.03 | 0.03 | 0.03 | 0.02 | 0.05 |

Parameter | Band | |
---|---|---|

X | P | |

Wavelength (λ) [m] | 0.031228 | 0.713791 |

Flight altitude (H) [Km] | 5.6 | 5.6 |

Incident angle (θ)—mid swath [deg] | 50 | 50 |

Normal Baseline (Bn)—mid swath [m] | 2.16 | 35.3 |

Swath width [Km] | 7.0 | 7.0 |

Chirp bandwidth [MHz] | 200 | 50 |

DEM spatial resolution [m] | 2.0 | 2.0 |

Statistics | Phase-Offset Value Difference (Corner Reflector–Proposed Method) | |
---|---|---|

X-band | P-band | |

Mean difference [rd] | 0.047 | 0.051 |

Std. dev. [rd] | 0.0152 | 0.0191 |

Statistics Using 16 GCPs | Phase-Offset Estimation | |||
---|---|---|---|---|

Corner Reflectors | Proposed Method | |||

X-Band DEM Difference | P-Band DEM Difference | X-Band DEM Difference | P-Band DEM Difference | |

Mean: μ_{h} [m] | 1.1127 | 0.7641 | 1.1378 | 0.8265 |

Std. dev.: σ_{h} [m] | 1.4720 | 0.6999 | 1.4709 | 0.7094 |

**Table 5.**Statistics regarding the phase-offset estimation error of the proposed method for X- and P-band.

Parameters and Statistics | Band | |
---|---|---|

X | P | |

Wavelength: (λ) [m] | 0.0313918 | 0.713791 |

Measurement (GCPs) : (N) | 16 | 16 |

Mean phase error : μ_{ϕ} [rd] | 0.0470 | 0.0510 |

Phase Std. dev. σ_{ϕ} [rd] | 0.0152 | 0.0191 |

Phase uncertainty (95%): Max(|σ_{ϕu}|) [rd] | 0.0508 | 0.0558 |

Mean height error: μ_{h}[m] | 0.327 | 0.702 |

Height Std. dev. σ_{h} [m] | 0.059 | 0.203 |

Height uncertainty (95%): Max(|σ_{hu}|) [m] | 0.445 | 1.108 |

Computed Statistics between the DEMs | DEM | |
---|---|---|

X-Band | P-Band | |

Mean value of the difference: μ_{DEM}[m] | 0.520 | 0.640 |

Std. dev. of the difference: σ_{DEM} [m] | 3.170 | 2.520 |

Std. dev. of the range slope: σ_{rg} [m] | 0.003 | 0.360 |

Std. dev. of the azimuth slope: σ_{az} [m] | 0.008 | 0.310 |

Height standard error: σ_{he} [m] | 1.124 | 1.127 |

Height uncertainty (95%): Max(|σ_{hu}|) [m] | 2.768 | 2.904 |

## Share and Cite

**MDPI and ACS Style**

Mura, J.C.; Pinheiro, M.; Rosa, R.; Moreira, J.R.
A Phase-Offset Estimation Method for InSAR DEM Generation Based on Phase-Offset Functions. *Remote Sens.* **2012**, *4*, 745-761.
https://doi.org/10.3390/rs4030745

**AMA Style**

Mura JC, Pinheiro M, Rosa R, Moreira JR.
A Phase-Offset Estimation Method for InSAR DEM Generation Based on Phase-Offset Functions. *Remote Sensing*. 2012; 4(3):745-761.
https://doi.org/10.3390/rs4030745

**Chicago/Turabian Style**

Mura, José Claudio, Muriel Pinheiro, Rafael Rosa, and João Roberto Moreira.
2012. "A Phase-Offset Estimation Method for InSAR DEM Generation Based on Phase-Offset Functions" *Remote Sensing* 4, no. 3: 745-761.
https://doi.org/10.3390/rs4030745