Early Lessons on Combining Lidar and Multi-baseline SAR Measurements for Forest Structure Characterization

Abstract

The estimation and monitoring of 3D forest structure at large scales strongly rely on the use of remote sensing techniques. Today, two of them are able to provide 3D forest structure estimates: lidar and synthetic aperture radar (SAR) configurations. The differences in wavelength, imaging geometry, and technical implementation make the measurements provided by the two configurations different and, when it comes to the sensitivity to individual 3D forest structure components, complementary. Accordingly, the potential of combining lidar and SAR measurements toward an improved 3D forest structure estimation has been recognised from the very beginning. However, until today there is no established framework for this combination. This paper attempts to review differences, commonalities, and complementarities of lidar and SAR measurements. First, vertical lidar reflectance and SAR reflectivity profiles at different wavelengths are compared in different forest types. Then, current perspectives on their combination for the generation of enhanced structure products are discussed. Two promising frameworks for combining lidar and SAR measurements are reviewed. The first one is a model-based framework where lidar-derived parameters are used to initialize SAR scattering models, and relies on both the validity of the models and on the physical equivalence of the used lidar and SAR parameters. The second one is a structure-based framework based on the ability of lidar and SAR measurements to express physical forest structure by means of appropriate indices. These indices can then be used to establish a link between the two kind of measurements. The review is supported by experimental results achieved using space- and airborne data acquired in recent relevant mission and campaigns.

Appendices

Appendix 1: Derivation of the Relationship Between InSAR Volume Coherences and Vertical Reflectivity Profile

The purpose of this Appendix is to summarize the algebraic steps that lead to the Fourier relationship between the InSAR coherence and the vertical reflectivity profile starting from the volume integral (2).

First of all, it is commonly assumed that the (end-to-end) system point-spread function after focusing is separable in the range and azimuth dimensions, i.e.

$$h\left( {r,x} \right) = h_{r} \left( r \right) \cdot h_{x} \left( x \right)$$

(14)

being $h_{r} \left( r \right)$ and $h_{x} \left( x \right)$ the range and azimuth system point-spread functions, usually approximated as ideal rectangular functions with width equal to the respective resolution. The formulation in (14) implies that the viewing geometry is independent of the azimuth coordinate,¹ therefore the integral on ${\text{d}}x$ can be solved, and (1) becomes:

$$y\left( {r^{{\prime }} } \right) = \iint {h_{r} \left( {r^{{\prime }} - r} \right) \cdot \bar{\xi }(r,s) \cdot {\text{e}}^{{ - j\frac{4\pi }{\lambda }R(r,s)}} {\text{d}}r{\text{d}}s},$$

(15)

$\bar{\xi }\left( {r,s} \right)$ being the integral along azimuth of the complex reflectivity weighted by the azimuth point-spread function. The dependency on the azimuth has been dropped for notational simplicity. From (15), a single SAR image provides a single tomographic projection of the 3D reflectivity filtered by the range response (Bamler and Hartl 1998). The interpretation of the convolution integral in (15) is facilitated in the domain of the 2D range-elevation spatial frequencies after a Fourier transform. Indeed, the SAR image spectrum is a 2D slice out of the complete 3D reflectivity spectrum (Bamler and Hartl 1998).

In general, an additional 2D spectral slice with minimum overlap increases the amount of information available on the reflectivity spectrum toward its reconstruction. Conventional InSAR configurations realize this by varying the complex exponential in (15), i.e., by collecting one additional SAR image in spatial (incidence angle) diversity. First of all, the two images are co-registered, i.e., both images are resampled onto a common grid so that the co-registered pixels refer to a common position in the 3D space. After co-registration, and in absence of system (including noise) and processing non-idealities, the SLC complex amplitudes for the same pixel become:

$$\begin{aligned} y_{1} \left( {r^{{\prime }} } \right) = \iint {h_{r} \left( {r^{{\prime }} - r} \right) \cdot \bar{\xi }\left( {r,s} \right) \cdot {\text{e}}^{{ - j\frac{4\pi }{\lambda }R_{1} \left( {r,s} \right)}} {\text{d}}r{\text{d}}s}, \hfill \\ y_{2} \left( {r^{{\prime }} } \right) = \iint {h_{r} \left( {r^{{\prime }} - r} \right) \cdot \bar{\xi }\left( {r,s} \right) \cdot {\text{e}}^{{ - j\frac{4\pi }{\lambda }R_{2} \left( {r,s} \right)}} {\text{d}}r{\text{d}}s}, \hfill \\ \end{aligned}$$

(16)

under the additional assumption that $\bar{\xi }\left( {r,s} \right)$ does not change with the difference of incidence angle induced by the track/orbit displacement and with the (possible) time difference between the two acquisition. The interferogram is defined as

$$\varphi \left( {r^{{\prime }} } \right) = \angle y_{1} \left( {r^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} } \right),$$

(17)

which for an ideal deterministic point scatterer located at $(r^{{\prime }} ,s^{{\prime }} )$ becomes:

$$\varphi \left( {r^{{\prime }} } \right) = \frac{4\pi }{\lambda }\left[ {R_{2} \left( {r^{{\prime }} ,s^{{\prime }} } \right) - R_{1} \left( {r^{{\prime }} ,s^{{\prime }} } \right)} \right]$$

(18)

Let $B_{\parallel }$ and $B_{ \bot }$ be the spatial track/orbit displacements in the direction parallel and orthogonal to the line of sight, respectively. Therefore, one can write:

$$R_{2} \left( {r,s} \right) = \sqrt {\left( {r - B_{\parallel } } \right)^{2} + \left( {s - B_{ \bot } } \right)^{2} } .$$

(19)

Some processing steps and approximations can be applied in order to reach a convenient formulation, which are detailed in Bamler and Hartl (1998) and Fornaro et al (2003). In these operations, $y_{1} \left( {r^{{\prime }} } \right)$ is taken as a phase reference, and for this reason it is called master; $y_{2} \left( {r^{{\prime }} } \right)$ is then called slave. The small variation of look angle between the two acquisitions allows to retain the paraxial approximation. By developing (19), a residual $s^{2}$-dependent phase contribution is originated by the curvature of the wavefront which can be included in the complex reflectivity, as its (second-order) amplitude distribution is normally of interest. Alternatively, it can simply be neglected if the distance between the sensor and the scatterers is large enough (as it typically is) to make the plane-wave approximation hold. Finally, the phase of the slave image is “flattened” with respect to a reference height by using the interferometric phase (18). In this case, the reference height can be chosen constant for the entire scene, or varying locally according to a reference external DEM. After these manipulations, by further assuming $B_{\parallel } \ll r$, it results:

$$\begin{aligned} y_{1} \left( {r^{{\prime }} } \right) = \iint {h_{r} \left( {r^{{\prime }} - r} \right) \cdot \xi^{{\prime }} \left( {r,z} \right) {\text{d}}r{\text{d}}z,} \hfill \\ y_{2} \left( {r^{{\prime }} } \right) \cong \iint {h_{r} \left( {r^{{\prime }} - r} \right) \cdot \xi^{{\prime }} \left( {r,z} \right) \cdot e^{{jk_{Z} z}} {\text{d}}r{\text{d}}z.} \hfill \\ \end{aligned}$$

(20)

In (20), a change of variable from elevation to vertical height $z$ has been performed, with $s = z /\sin \theta_{0}$. $\xi^{{\prime }} \left( {r,z} \right)$ is the final complex reflectivity after the change of variables and the inclusion of phase residuals $\psi$, i.e., $\xi^{{\prime }} \left( {r,z} \right) = \bar{\xi }\left( {r,z/\sin \theta_{0} } \right){\text{e}}^{j\psi } /\sin \theta_{0}$, being $\theta_{0}$ the local incidence angle. $k_{Z}$ is the so-called vertical wavenumber, defined in (3).

Being composed of distributed scatterers, the scattering from forest scenarios is better described by its statistics, particularly by the complex coherence defined in (4). This results in:

$$\begin{aligned} E\left\{ {y_{1} \left( {r^{{\prime }} } \right)} \right\} = E\left\{ {y_{2} \left( {r^{{\prime }} } \right)} \right\} = 0, \hfill \\ E\left\{ {\left| {y_{1} \left( {r^{{\prime }} } \right)} \right|^{2} } \right\} = E\left\{ {\left| {y_{2} \left( {r^{{\prime }} } \right)} \right|^{2} } \right\} = P_{y} , \hfill \\ E\left\{ {y_{1} \left( {r^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} } \right)} \right\} = P_{y} \gamma_{1,2} \left( {k_{Z} } \right), \hfill \\ \end{aligned}$$

(21)

and using (20):

$$E\left\{ {y_{1} \left( {r^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} } \right)} \right\} = \iint {\left| {h_{r} \left( {r^{{\prime }} - r} \right)} \right|^{2} \cdot F_{\xi } \left( {r,z} \right) \cdot {\text{e}}^{{jk_{Z} z}} {\text{d}}r{\text{d}}z}.$$

(22)

$F_{\xi } \left( {r,z} \right)$ represents the spatial density of backscattered power in the height-range plane. It is interesting to consider the case of surface scattering at a constant height $z = z_{0}$. Its complex reflectivity can be written as a Dirac-$\delta$ with power $P_{S}$. After appropriate algebraic manipulations, it provides the following interferometric coherence:

$$\gamma_{1,2} \left( {k_{Z} } \right) = \frac{{E\left\{ {y_{1} \left( {r^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} } \right)} \right\}}}{{P_{S} }} = \gamma_{S} \left( {k_{Z} } \right) {\text{e}}^{{jk_{Z} z_{0} }} .$$

(23)

Therefore, the coherence phase is proportional to the surface of the height. $\gamma_{S} (k_{Z} )$ is a positive (real-valued) geometric decorrelation term that depends on the range point-spread function.² However, forest scatterers are distributed within a volume, that can be seen as the coherent superposition of the number of surfaces with different total powers. As a consequence, it simply follows:

$$E\left\{ {y_{1} \left( {r^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} } \right)} \right\} = \gamma_{S} \left( {k_{Z} } \right)\sum\limits_{n} {P_{S} \left( {z_{n} } \right){\text{e}}^{{jk_{Z} z_{n} }} } .$$

(24)

In the most general case, by letting the number of surfaces tend to infinity, the sum in (24) becomes an integral, and it is readily obtained

$$\gamma_{1,2} \left( {k_{Z} } \right) = \gamma_{S} \left( {k_{Z} } \right) \gamma_{\text{vol}} \left( {k_{Z} } \right),$$

(25)

where:

$$\gamma_{\text{vol}} \left( {k_{Z} } \right): = {\text{e}}^{{jk_{Z} z_{G} }} \frac{{\int_{0}^{{H_{V} }} {F_{vol} \left( z \right) {\text{e}}^{{jk_{Z} z}} {\text{d}}z} }}{{\int_{0}^{{H_{V} }} {F_{\text{vol}} \left( z \right) {\text{d}}z} }},$$

(26)

which provides the Fourier relationship between vertical reflectivity profile and volume coherence anticipated in Sect. 2.1.

Appendix 2: Descriptions of Test Sites and Data Sets

2.1 Traunstein Forest (South of Germany)

This temperate forest site is located close to the city of Traunstein in the South of Germany. Most forest stands are structurally complex due to species richness and management practices, with a majority of conifer stands (around 70%). The forest top height can reach 40 m, and mean biomass level is about 200 t/h, significantly higher than other managed forests in the same ecological zone. In this test site, TomoSAR data sets at different frequencies have been collected by means of the German Aerospace Center (DLR) E-SAR and F-SAR airborne platforms since 2008. The data sets used in this paper were acquired at P- and L-band in 2009, and at X-band in 2013 with approximately the same flight tracks and viewing (master) geometry. Due to the 4-year time difference between the P- and X-band data sets, an additional L-band TomoSAR data set of 2013 has been considered. In this way, two direct comparisons can be performed, i.e., P- versus L-band in 2009, and L- versus X-band in 2013. All acquisitions share the same (master) geometry. While the acquisitions at P- and L-band are in repeat-pass mode, the one at X-band is in repeated single-pass mode (with changing $k_{Z}$ for different passes) to avoid temporal decorrelation (Pardini et al. 2018b). The most relevant acquisition parameters are reported in Table 1. On November 18, 2012, discrete return lidar data have been acquired as well. During the total 4-year time, structural changes occurred in some stands due to forest management activities. Additionally, all acquisitions are in leaf-off conditions, but differences of water content and of its distribution in the tree trunk (Gates 1991) could have affected the SAR backscattering.

Table 1 Summary of parameters characterizing the airborne SAR acquisitions over the Traunstein and Rabi forests

2.2 Rabi and Lopé (Gabon)

The Rabi forest site is located at the southwestern Gamba Complex of Gabon, and consists of a diverse mix of upland and wet-forest with a mean tree height of about 40 m on a fairly flat topography. Within this forest site, a ForestGEO (Smithsonian Institution Forest Global Earth Observatories) plot was established between 2010 and 2012 in which the position and the diameter at breast height (dbh) of more than 175,000 trees belonging to 340 species with dbh ≥ 1 cm were inventoried across 25 ha. P- and L-band TomoSAR acquisitions were carried out with the F-SAR airborne platform during the AfriSAR campaign in 2016 (see Table 1 for the relevant acquisition parameters) (Pardini et al. 2018a). During the same campaign, the Rabi site has been covered also by NASA LVIS lidar waveforms. For AfriSAR, LVIS operated at 7315 m altitude, with a wavelength equal to 1064 nm, ~20 m footprint diameter and 10 m footprint spacing along latitude and longitude. Together with the waveforms, ground topography (Digital Terrain Model, DTM) and relative height metrics have been processed.

Table 2 Summary of parameters characterizing the TanDEM-X SAR acquisitions over Lopé and La Selva

The Lopé site lies within the Lopé National Park, and it consists of a mosaic of savannah and (dense) forest, with varying species richness and tree density across stands. The maximum tree height exceeds 50 m in many stands. The forest (above ground) biomass ranges between 10 t/ha in savanna areas up to ~ 400 t/ha in mature forest stands. The terrain is hilly, with many local slopes steeper than 20°. This test site was covered during the AfriSAR campaign by both airborne SAR and LVIS lidar acquisitions similarly to the Rabi site. For the experiments in Sect. 4, a TanDEM-X bistatic single-pass single polarisation (HH) interferometric acquisition has been considered (see Table 2) that was performed a few weeks apart from the LVIS acquisitions.

2.3 La Selva (Costa Rica)

La Selva Biological Station (LSBS) forest site is located in northeastern Costa Rica. The site is a protected low-land region of tropical rain forest, covering an area of 1600 ha. The area contains a mixture of old-growth, secondary and selectively logged forests with height up to 60 m. The mean biomass of old-growth forest, which is the major components of total LSBS biomass, is around 169 t/ha (Clark et al. 2011). Airborne discrete return lidar data were collected over LSBS in September and October 2009. In 2011, the LSBS was covered by a dual polarisation (HH/VV) TanDEM-X bistatic acquisition on December 5, whose characteristic parameters are reported in Table 2.