Vegetation modelling for height inversionusing InSAR/Pol-InSAR data.

Franck Garestier & Thuy Le ToanCESBIO

18, av. Edouard BelinF-31 400 Toulouse

Email: Franck.Garestier@cesbio.cnes.frTelephone: +33 5 61 55 85 35

Abstract— The Random Volume over Ground model has beenextensively used for forest height inversion using Pol-InSAR data.Two different forest models are proposed to better representthe forest structure for height inversion using Pol-InSAR. Thefirst one takes into account a vertically varying mean extinctioncoefficient and the second one considers contributions localizedin a finite height interval.

I. INTRODUCTION

Interferometry and Polarimetric SAR Interferometry haveshown a good potential for vegetation height estimation. Acoherent model assuming the vegetation layer as a homo-geneous volume constituted by randomly oriented particles[1], [2] has been developed for forest height inversion usinginterferometric data. The Pol-InSAR inversion procedures usethe interferometric signal diversity in the different polarimetricchannels, reflecting different ground to volume power ratios,to estimate the vegetation layer height using the RandomVolume over Ground model [3], [4], which has been validatedat different frequencies [5]-[8].

This work consists in the establishment of an interferometriccoherence expression associated with a more realistic forestrepresentation. First, a varying mean extinction coefficientin the vertical direction is considered, in order to take intoaccount the size variation of the forest structural element.Then, the impact of scatterers located in a finite height intervalon the interferometric coherence is investigated, correspondingto a more accurate forest structural representation [9].

II. VARYING EXTINCTION VOLUME INTERFEROMETRIC

COHERENCE

The mean extinction coefficient σ is assumed to varylinearly with height z with a slope α:

σ(z) = αz with α > 0 (1)

Volume interferometric coherence can then be expressed as:

γv =

∫ hv

0e−

2αz2

cos θ+jkz(hv−z)dz

∫ hv

0e−

2αz2

cos θ dz(2)

where hv is the volume height, θ is the incidence angle andkz is the vertical wavenumber.

Interferometric coherence associated with a linearly varyingmean extinction coefficient can be then expressed as:

γv = e−cos θk

2z

8α+jkzhv .

erf(

j cos θkz+4αhv

2√

2α cos θ

)

− erf(

12

√

cos θ2α

jkz

)

erf(√

2αcos θ

hv

)

(3)

Fig. 1. Interferometric coherence representation in the complex plane for arandom volume in yellow (height ranging from 0 to 40 m and mean extinctioncoefficient from 0 to 2 dB/m), and for a varying extinction volume in red,green and blue, corresponding respectively to 0,0057, 0,0028 and 0,0014 αvariation rate values.

Fig. 1 represents in the complex plane the interferometriccoherence associated with a random volume with a heightranging from 0 to 40 m and a mean extinction coefficientranging from 0 to 2 dB/m in yellow, and with a varyingextinction volume for three variation rates as a function ofthe canopy height in red, green and blue colors. For thissimulation, the ambiguity height is equal to 40 m (kz =

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0.157080) and the incidence angle θ is 45 degrees. Each yellowsegment corresponds to a given random volume height. Alongthe segment, mean extinction coefficient increasing wheninterferometric coherence comes closer to the trigonometriccircle. Concerning the varying extinction volume, the top ofthe canopy is constrained by a null extinction. Red, greenand blue colors correspond respectively to 2, 1 and 0,5 dB/mvariation in a 40 m height interval. The α variation rate takesrespectively the following values: 0,0057, 0,0028 and 0,0014.Since the variation rate is assumed constant, when the volumeheight increases, the mean extinction coefficient associatedwith the lowest volume layer increases. Indeed, it can beseen in Fig. 1 that interferometric coherence of the varyingextinction volume is close to null extinction random volumecoherence for low heights and differentiates more significantlybetween the different extinction variation rates when the forestheight increases.

Fig. 2. Linear varying extinction volume coherence and interferometric phasefunction of the volume height for three extinction gradients (red, green andblue). Coherence and interferometric phase associated with a random volumeis represented in black dotted lines for mean extinction coefficients equals to0 and 2 dB/m.

Fig. 2 represents coherence and phase of a varying ex-tinction volume versus its height and indicates that it isalways comprised between null and high extinction randomvolume coherences, as expected. Sensitivity of coherence andinterferometric phase to the extinction gradient is observed forvegetation layers higher than 10 m.

Fig. 3. Three different scattering center height standard deviations represen-tation in context of a 40 m volume height hv example. Standard deviationsare set to hv/5, hv/7, and hv/12, coded respectively in red, green and bluecolors. Scattering center mean height is fixed to hv/2.

III. GAUSSIAN BACKSCATTER VOLUME INTERFEROMETRIC

COHERENCE

Contributions generated in a finite height interval are nowconsidered. Interferometric coherence solutions have beenfound using assumption of Gaussian height distribution. Thebackscattered power height distribution is expressed as:

P (δ, σh) = e−

(z−δ)2

2σ2h (4)

where δ represents the scattering center mean elevation andσh, its corresponding standard deviation.

Interferometric coherence associated with a volume wherethe backscattered power varies vertically following a Gaussianfunction over a hv height is written as the integral ratio:

γv(hv, δ, σh) =

∫ hv

0 e−

(z−δ)2

2σ2h

+jkzzdz

∫ hv

0 e−

(z−δ)2

2σ2h dz

(5)

The final interferometric coherence expression becomes:

γv(hv, δ, σh) = e−σ2h

k2z

2 +jδkz .

erf(√

2σ2

h

jσ2h

kz+δ

2

)

− erf(√

2σ2

h

jσ2h

kz−hv+δ

2

)

erf(√

2σ2

h

hv−δ2

)

+ erf(√

2σ2

h

δ2

)

(6)

Scattering center height and standard deviation effect oncoherence and interferometric phase are now investigated.

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A. Scattering center height standard deviation effect

Three scattering center standard deviations are first consid-ered assuming a mean elevation half of the volume height.Simulated standard deviations are hv/5, hv/7 and hv/12corresponding respectively to red, green and blue colors inthe representation of the scattering center height distributionof Fig. 3, under the example of a 40 m volume height.

Fig. 4 represents the associated interferometric coherencesin the complex plane, overplotted on the random volume in-terferometric coherence computed for volume heights rangingfrom 0 to 40 m and mean extinction coefficients comprisedbetween 0 and 2 dB/m.

Fig. 4. Interferometric coherence representation in the complex plane fora random volume in yellow (height ranging from 0 to 40 m and meanextinction coefficient from 0 to 2 dB/m), and for a Gaussian distributedbackscattered power volume in red, green and blue. These three colorscorrespond respectively to hv/5, hv/7, and hv/12 scattering center heightstandard deviations and to a hv/2 mean elevation.

Since the scattering center mean elevation is fixed to a com-mon value, the angular position of the three interferometriccoherences is the same for the different distributions.

Fig 5 represents coherence and interferometric phases as-sociated with these three distribution standard deviations asa function of the volume height. The black dotted linescorrespond to 0 and 2 dB/m random volume boundaries. Itfirst appears that the coherence behavior when the volumeheight increase varies with the scattering center height standarddeviation. The volume decorrelation is generated by the heightdiversity of the scatterers and is then proportional to the scat-terer height interval. Since the mean scattering center heightelevation is the same, set to hv/2, the three interferometricheights are similar, superimposed to the 0 dB/m randomvolume phase center height.

Fig. 5. Coherence (top) and interferometric phase (bottom) representationsas a function of the volume height for the three different scattering centerheight standard deviations.

B. Scattering center elevation effect

Three relative scattering center elevations in the vegetationlayer are now considered assuming a given standard deviation.Elevations are set to three different volume height fractions, asrepresented in Fig. 6: hv/4, hv/2, and 3hv/4. The backscat-tered power height distribution standard deviation is fixed tohv/12.

Fig. 7 represents the corresponding interferometric coher-ences in the complex plane, overplotted on the random volumeinterferometric coherence computed for volume heights rang-ing from 0 to 40 m and mean extinction coefficients comprisedbetween 0 and 2 dB/m. It appears that the relative scatteringcenter mean elevation in the volume influences the angularinterferometric coherence position in the complex plane, asexpected.

Fig. 8 represents coherence and interferometric phase asso-ciated with the three different relative scattering center meanelevations. Since scattering center height standard deviationsremain the same, the volume decorrelation is equal for thethree simulated elevations. Concerning the increase of theinterferometric phase, its slope is directly related to the relativemean elevation of the scatterers.

IV. CONCLUSION

The Random Volume over Ground model has been usedsuccessfully to estimate the forest height at different frequen-cies using Pol-InSAR data. In this work, two different forestmodels have been developed in order to increase the forest

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Fig. 6. Three different scattering center height elevations representationin context of a 40 m volume height hv example. Elevations are set tohv/4, hv/2, and 3hv/4, coded respectively in red, green and blue colors.Backscattering power height distribution standard deviation is fixed to hv/12.

Fig. 7. Interferometric coherence representation in the complex plane fora random volume in yellow (height ranging from 0 to 40 m and meanextinction coefficient from 0 to 2 dB/m), and for a Gaussian distributedbackscattered power volume in red, green and blue. These three colorscorrespond respectively to hv/4, hv/2, and 3hv/4 mean scattering centerheights and to a hv/12 height standard deviation.

representation complexity and to suit better with the naturalstructure of the vegetation media. The first model assumes avertically varying mean extinction coefficient, which reflectsthe varying structural element size in the vertical direction(trunks and branches). The second model considers vegetation

Fig. 8. Coherence (top) and interferometric phase (bottom) representationsas a function of the volume height for the three different scattering centermean elevations.

contributions located at a finite height interval.These two models have to be now evaluated over different

forest types and over different datasets acquired at differentfrequencies. These two forest representations can also be com-bined together, and with the random volume, in the differentpolarimetric channels to estimate forest height.

REFERENCES

[1] R. N. Treuhaft, S. Madsen, M. Moghaddam, J. J. van Zyl, ”VegetationCharacteristics and Underlying Topography from Interferometric Data,”Radio Science, Vol. 31, pp. 1449-1495, December 1996.

[2] J. I. H. Askne, P. B. G. Dammert, L. M. H. Ulander, G. Smith, ”C-Band Repeat-Pass Interferometric SAR Observations of the Forest,” IEEETrans. on Geosci. Remote Sensing, vol. 35, No. 1 pp. 25-35, January 1997.

[3] S. R. Cloude and K. P. Papathanassiou, ”Single-Baseline PolarimetricSAR Interferometry,” IEEE Trans. on Geosci. Remote Sensing, Vol. 39,No. 11, pp. 2352-2363, November 2001.

[4] K. P. Papathanassiou and S. R. Cloude, ”A 3-Stages Inversion Processfor Polarimetric SAR Interferometry,” IEE Proceedings, Radar, Sonar andNavigation, Vol. 150, No. 03, pp. 125-134, June 2003.

[5] F. Garestier, K. Papathanassiou, I. Hajnsek, P. Dubois-Fernandez and X.Dupuis, ”Analysis of forest parameters and agricultural field structurefrom high resolution PolInSAR X band data,” Proc. of POLInSAR 05,Frascati, Italy, 2005.

[6] F. Garestier, T. Le Toan and P. Dubois-Fernandez, ”Forest height esti-mation using P-band Pol-InSAR data,” Proc. of POLInSAR 07, Frascati,Italy, 2007.

[7] P. Dubois-Fernandez, F. Garestier, X. Dupuis ,I. Champion, ”Ramses PBand Campaign over the Nezer Forest : Calibration and PolarimetricAnalysis,” Proceedings of EUSAR 06, Dresden, Germany, 2006.

[8] F. Kugler et al., ”Forest Parameter Estimation in Tropical Forests byMeans of Pol-InSAR: Evaluation of the INDREX II Campaign,” Proc. ofPOLInSAR 07, Frascati, Italy, 2007.

[9] I. Woodhouse., ”On the vertical distribution of backscatter from aforest canopy for improving polinsar retrievals,” Proc. of POLInSAR 07,Frascati, Italy, 2007.

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