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Surveys in Geophysics

, Volume 40, Issue 4, pp 803–837 | Cite as

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

  • Matteo PardiniEmail author
  • John Armston
  • Wenlu Qi
  • Seung Kuk Lee
  • Marivi Tello
  • Victor Cazcarra Bes
  • Changhyun Choi
  • Konstantinos P. Papathanassiou
  • Ralph O. Dubayah
  • Lola E. Fatoyinbo
Article
  • 89 Downloads

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.

Keywords

Forest Structure Height Lidar Synthetic-aperture radar (SAR) Full waveforms Polarimetric SAR interferometry (Pol-InSAR) SAR tomography (TomoSAR) 

1 Introduction

The development and functionality of forests are widely reflected in their biophysical structure. According to a commonly used definition, forest (canopy) structure is “the organisation in space and time, including the position, extent, quantity, type and connectivity, of the aboveground components of vegetation” (Parker 1995; Lefsky et al. 1999; Spies 1998; Harding et al. 2001; Snyder 2010). Thus, it expresses forest state, functionality, biodiversity and evolution (Hall et al. 2011; Brokaw and Lent 1999; McElhinny et al. 2005). Accordingly, forest structure is an indicator of forest successional stage and development as well as sustainability and habitability, and is therefore an important parameter for assessing forest productivity (Bohn and Huth 2017), biomass and biodiversity (Bergen et al. 2009; Goetz et al. 2007; Turner et al. 2003). Forest structure changes are associated with dynamic processes such as growth, regeneration, decay, and natural or anthropogenic disturbance. Knowledge about such processes is important for modeling the function and development of forest ecosystems, and for developing accurate and robust forest biomass estimators (Frolking et al. 2009). Mapping forest structure is therefore critical for understanding the history, function, and future of forest ecosystems.

Traditionally, forest structure characterisation relies on sampling at local scales by means of either field inventory plots or more recently terrestrial laser scanning techniques able to catch the 3D arrangement of vegetation compartments. However, any extrapolation to larger scales is limited by the ability of these measurements to represent larger areas. Moreover, the fact that the temporal continuity of such plot measurements is in many cases difficult to be established limits even more their ability to characterise forest structure change. In this context, remote sensing techniques offer the potential to overcome, at least partially, these limitations (Hall et al. 2011; Bergen et al. 2009; Goetz et al. 2007; Turner et al. 2003; Frolking et al. 2009). Today, only two remote sensing techniques can provide 3D information and contribute to the characterisation of forest structure at large scales (Hall et al. 2011): lidar and synthetic-aperture radar (SAR).

Lidar systems are active configurations usually operating in the infrared or visible region of the electromagnetic spectrum at wavelengths in the nm range. They transmit laser pulses in nadir-looking geometry which “illuminate” a given footprint on the ground whose diameter is typically in the order of decimeters for small-footprint lidar up to tens of meters for large-footprint configurations. The transmitted pulses are reflected by the vegetation elements within the footprint back to the sensor, where they are usually “incoherently” recorded, i.e., only the energy (amplitude) of the reflected light is detected as a function of the signal runtime. A single transmitted laser pulse can be reflected by different vegetation elements located at different heights within the footprint, leading to multiple reflections and an extended distribution of light energy that returns to the sensor. The distribution of light energy that returns to the sensor is known as waveform and is directly related to the 3D distribution of the intercepted vegetation surfaces within the footprint (Dubayah and Drake 2000).

SAR configurations also transmit electromagnetic pulses, but differently from the lidar sensors in the microwave region of the electromagnetic spectrum (wavelengths in the order of cm to dm) in a side-looking geometry. The transmitted pulses interact with Earth’s surface, and only a portion of them is scattered back to the sensor. The backscattered pulses are “coherently” recorded by the receiver, i.e., both their amplitude and phase are measured. The amplitude and phase of the backscattered signal depend on the geometric and dielectric properties of the scatterers and their distribution within the antenna footprint, i.e., the illuminated area/volume. With increasing wavelength (decreasing frequency) radar pulses penetrate more and more into and through vegetation layers, and interact with vegetation elements located at different heights within the forest volume and with the underlying ground. However, a single SAR image, even if it results from the interaction of the transmitted pulse(s) with the whole 3D forest structure, does not allow a reconstruction of the 3D distribution of scatterers. For this, a set of SAR images acquired under (slightly) different angular directions (i.e., incidence angles) is required in the context of interferometric (InSAR) and tomographic SAR measurements (Moreira et al. 2013). The side-looking geometry complicates additionally the interpretation of a single SAR image, but at the same time, it allows the separation of the scatterers in the ground range direction and the realisation of wide swath widths up to hundreds of kilometers. Spaceborne SAR configurations of the latest generation can therefore realize revisit times between 1 and 2 weeks with resolution on ground between 1 m and 10 m.

Accordingly, the combined use of lidar and SAR data has the potential to enhance the quality of forest structure characterization by improving the accuracy of physical forest structure parameter estimates, (e.g., forest height and biomass). The combination of SAR and lidar data becomes even more relevant in view of a multitude of SAR and lidar spaceborne missions that are curently operational or planned to be launched and operated in the coming years to meet a number of forest-related objectives. Lidar missions like NASA’s GEDI (Dubayah 2015) and IceSAT-2 (Neuenschwander and Magruder 2018) or JAXA’s MOLI (Mitsuhashi et al. 2018) and SAR missions like ESA’s BIOMASS (Le Toan et al. 2011) NASA’s NISAR (Rosen et al. 2017) and/or DLR’s TanDEM-X (Krieger et al. 2013) and Tandem-L (Moreira et al. 2015) are expected to operate and acquire data simultaneously or contiguously in time. For this, the understanding of the complementarities and possible synergies between lidar and SAR (i.e., polarimetric SAR interferometry, Pol-InSAR, and/or SAR tomography, TomoSAR) measurements is required.

The similarity of information content of lidar and SAR measurements arises from their common sensitivity to the size and location of vegetation elements in the forest volume. The difference, thus complementarity, arises from the different wavelengths and the different acquisition geometries (i.e., nadir-looking vs. side-looking) defining  the interaction of the transmitted pulses with the forest elements. For lidar, the penetration into and through the canopy layer is supported by the nadir-looking geometry that facilitates the penetration through vegetation gaps, while longer wavelengths allow SAR pulses to penetrate into and through canopy layers even in side-looking geometries. The structure information revealed by lidar measurements primarily implies the geometry of trunk and branch structure forming and supporting the canopy. On the other hand, the relative importance of the different tree components (e.g., leaves, branches, trunks) in SAR measurements depends on their vertical and horizontal distribution and the wavelength. Further, in contrast to lidar, SAR measurements are also sensitive to the dielectric properties of the forest which depend strongly on the amount and distribution of water across the vegetation elements (Hall et al. 2011; Moreira et al. 2013).

The difference in information content is one dimension that can be exploited for the development of lidar-SAR fusion algorithms. In this case, lidar and SAR data are combined in a commonly valid scattering or allometric model. One example is the direct use of lidar measurements or lidar-derived parameters to support and/or improve model-based inversions of forest structure parameters from SAR data. The use of lidar data allows to employ more complex physical models and to decouple the geometric from the dielectric information of the SAR data. This could be particularly important for instance to support smaller SAR observations spaces repeated in time to detect structural or dielectric changes. This concept has been demonstrated to estimate top height from X-band SAR interferometric coherence measurements by means of the Random-Volume-over-Ground (RVoG) model using the lidar ground topography (Kugler et al. 2014). Similarly, the use of lidar ground and top forest height in the RVoG model allows the estimation of the attenuation of SAR pulses through the vegetation again from interferometric coherences (Praks et al. 2012; Pardini et al. 2016; Qi and Dubayah 2016). Furthermore, higher-dimensional frameworks for the extraction of structure information from radar reflectivity profiles can benefit from the use of lidar vegetation profiles for parameterisation (Brolly et al. 2016). However, these approaches rely not only on the validity of the model(s) used to combine the two data sets, but also require that lidar and SAR configurations share a set of physically equivalent parameters. However, this is not always the case as, especially with increasing difference in the lidar and radar wavelengths, the differences in the scattering and propagation make lidar and SAR pulses interact differently with the physical vegetation elements. The loss of physical equivalence between SAR and lidar parameters makes the lidar-based parameterisation of SAR models lose significance (Treuhaft et al. 2009; Brolly et al. 2016; Pardini et al. 2018a, b).

The second dimension that can be used to combine lidar and radar data is their complementarity in terms of acquisition and measurement characteristics driven by the individual technologies and acquisition strategies. With particular reference to spaceborne implementations, lidar measurements are typically acquired by means of footprint samples along rather narrow stripes so that large scale coverage and short revisit times become challenging to achieve. In contrast, SAR measurements are continuous with high spatial resolution, and can be implemented by means of wide swaths that allow large scale coverage and short revisit times (Bamler and Hartl 1998; Moreira et al. 2013). At the same time, lidar configurations allow direct measurement of vegetation reflectance profiles with a fairly high vertical resolution, while SAR systems measure backscattering amplitudes and/or interferometric coherences at different polarizations (Pol-InSAR) or radar reflectivity profiles (TomoSAR) with a lower vertical resolution, which are not always straightforward to be interpreted in terms of physical forest structure.

In this case, the combination of lidar and SAR measurements strongly depends on the ability to establish a (physical or statistical) link between them at the different scales/resolutions performed. This link can be established by means of statistical regression(s), and several examples for such an approach have been reported in the literature, for example aiming at biomass estimation by combining lidar heights and SAR backscatter and/or interferometric parameters [see, e.g., (Kellndorfer et al. 2010; Sun et al. 2011; Tsui et al. 2013; Fatoyinbo and Simard 2013; Kaasalainen et al. 2015)]. On the other hand, lidar and SAR measurements can be linked more systematically by means of a scattering model or by means of common structure-dependent indices. The choice between the two frameworks strongly depends on the individual information content. The applicability of the model-based framework relies on the common sensitivity to geometric parameters. For instance, the lidar ground height has been used to bridge the spatial (Qi and Dubayah 2016, here together with forest heights at lidar footprint locations) or the temporal (Persson et al. 2017) dissimilarities of lidar and SAR measurements to obtain continuous estimates of forest height and of its changes over large scales. On the other hand, the structure-based framework aims at establishing a correspondence between lidar and Pol-InSAR/TomoSAR measurements by means of their capability in reflecting physical 3D forest structure through appropriate indices without necessarily exchanging parameters (Cazcarra-Bes et al. 2017; Tello et al. 2018; Pardini et al. 2018a). In this way, not only the commonalities, but also the complementarities of the different configurations and wavelengths can be exploited. Although promising, the latter framework is in a very early stage of development, considering also that the interpretation of 3D SAR reflectivity in terms of physical forest structure attributes is not as advanced as for lidar profiles.

The objective of this paper is to review (i) the differences, thus indicating the complementarities, in terms of information content of lidar and SAR measurements at different wavelengths, and (ii) the model-based and the structure-based frameworks in which their combination appears promising. Section 2 presents from a theoretical point of view how lidar and SAR measurements are linked to forest structure, and the factors affecting this relationship. The complementarity between the information content of lidar and SAR measurements is then addressed in Sect. 3 by using reflectance and reflectivity profiles in different forest types. The dependency of the SAR profiles on wavelength, dielectric properties, 3D structure, and temporal changes is shown. For this, real airborne SAR (DLR E-SAR and F-SAR platforms) and lidar data acquired over the temperate Traunstein forest (South of Germany) are used. As an example of a tropical forest site lidar and SAR results over the ForestGEO (Smithsonian Institution Forest Global Earth Observatories) plot in Rabi (Gabon) are also shown. The model-based framework is reviewed in Sect. 4 with a case study concerning forest height estimation. With reference to a spaceborne implementation, one InSAR coherence is used to fill the spatial gaps of a lidar acquisition that, in turn, parameterises the RVoG model. Different parameterisation strategies are compared. Concerning the structure-based framework, Sect. 5 reviews a case study on the Rabi plot in which indices expressing horizontal and vertical heterogeneity of structure are calculated from lidar waveforms and TomoSAR profiles, in correspondence to structure indices with the same meaning already established in forestry and ecology. Finally, Sect. 6 draws the conclusions.

2 Basics of Lidar and SAR Measurements of Forest Structure

2.1 Lidar Measurements

Lidar measurements rely on the transmission of laser pulses of finite length towards a scene. The pulses are intercepted, attenuated and reflected (i.e., backscattered) by the branches and leaves of the vegetation canopy, and by the ground, and return back to the sensor. The receiver records the reflected distribution of (light) energy (i.e., the waveform) as a function of time. The time a pulse takes after being transmitted to return to the sensor depends on the range and the scattering properties of intercepted objects within the illuminated part of the scene (i.e., the footprint) as well as the distance between the laser source and the receiver, and the actual atmospheric conditions. The range between the sensor and an intercepted object is calculated as \(r_{i} = c(t_{i} - t_{0} ) /2\), where \(c\) is the speed of light, \(t_{0}\) is the timing of the transmitted pulse and \(t_{i}\) is the timing of a peak or discretised range bin of the received waveform. Accordingly, the received waveform depends directly on the 3D distribution of the intercepted vegetation elements within the footprint (Dubayah and Drake 2000).

The reflected energy from a vegetation layer of thickness (i.e., height) \(H_{V}\) located over a reference elevation \(z_{G}\)—ignoring for simplicity the propagation through the atmosphere and assuming a random distribution of scattering elements and penetration until the ground level \(z_{G}\)—can generally be modelled as
$$P(x,y,z) = P_{0} \int\limits_{{z_{G} }}^{{z_{G} + H_{V} }} {{\text{d}}z} \iint {\beta (x,y,z) \cdot {\text{e}}^{{ - \tau \left( {z - z_{G} } \right)}} {\text{d}}y{\text{d}}x,}$$
(1)
where P0 is the total returned power, where \(\beta (x,y,z)\) is the volumetric reflection of the vegetation layer, \(\tau\) the volumetric extinction coefficient that accounts for the two-way attenuation within the vegetation layer.
Vegetation lidar instruments use in general wavelengths of 1064 nm or 1550 nm and are designed to maximise the measurement signal-to-noise ratio considering atmospheric attenuation, eye-safey, and the reflectance of ground and vegetation elements at those wavelengths. The way the distribution of light energy that returns to the sensor is received and recorded categorises lidar instruments as discrete return or waveform:
  1. 1.

    “Discrete return” lidar systems record only individual (discrete) peaks (i.e., time-stamped ranges triggered real-time) of the waveform. They identify peaks and record a point at each peak location in the waveform curve. These individual or discrete points are called returns. A discrete system may record 1–4 (and sometimes more) returns from each laser pulse.

     
  2. 2.

    “Full-waveform” lidar systems record the distribution of transmitted and returned light energy. Accordingly, (full) waveform lidar data contain more information compared to discrete return lidar systems, but are more complex to process. Waveform lidar systems sample the returned waveform at a higher frequency and record the energy returned over equal time intervals.

     

Discrete return lidars are typically designed for fine-scale topographic airborne mapping, therefore their footprint is typically kept to 0.1 to 2 m by employing a small beam divergence angle. However, large-footprint (> 5 m) waveform lidar have several advantages over small footprint ones for mapping vegetation (Dubayah and Drake 2000). First, with a footprint comparable to the average crown diameter (10–25 m), there is a higher probability to contain both ground and tree top at the same time in a waveform. Second, larger footprints can cover wider areas at a lower cost. Large-footprint full-waveform lidar systems from airborne platforms such as the NASA Laser Vegetation, and Ice Sensor (LVIS; Blair et al. 1999) are being used not only for many vegetation studies, but also for calibration and validation of structure parameter extraction algorithms for instruments on spaceborne platforms, like the GEDI (Global Ecosystem Dynamics Investigation).

Central to the use of lidar measurements for estimating vegetation structure parameters is the separation of  reflections (returns) from canopy and ground surfaces. Simple waveforms are common from bare ground and complex waveforms from heterogeneous forest canopies. In addition to the characteristics of canopy structure, waveform shapes are affected by ground slope, pulse width and the spatial distribution of wavefront energy (Dubayah et al. 2000). The first and last modes along the profile of a recorded waveform are typically associated with the highest and lowest reflecting surfaces within the footprint, respectively. Identification of these modes within the waveform is required to precisely geolocate the corresponding reflecting surfaces and calculate statistical quantile metrics from the integrated waveform between these ranging points. These statistical quantiles are typically referred to as relative height (RH) metrics (sketched in Fig. 1), which are often used in models to predict aboveground biomass density (Dubayah et al. 2010). The derivation of vertical structure of vegetation canopies from lidar waveform data, though, depends on the knowledge of the relationship between lidar waveforms and the spatial structure and optical properties of vegetation canopies.
Fig. 1

Example of a lidar waveform simulated from airborne discrete return lidar data acquired over a complex rainforest canopy at La Selva Biological Station, Costa Rica, showing the energy returned from the ground surface (brown) and the energy returned from the canopy (green) as a function of elevation. The reference ground elevation is shown along with the 25th, 50th, 75th, and 100th percentile relative height (RH) metrics extracted from the cumulated energy profile (dashed line)

In addition to “lidar perceived” statistical metrics, large-footprint waveform lidars enable the estimation of biophysical metrics, as they are a direct measurement of the vertical distribution of the intercepted surfaces between the canopy top and the ground. In the absence of multiple scattering, a received waveform is the product of the projected area of canopy/background materials along the path of the laser pulse, their single scattering albedo, and phase function. The projected area of canopy/background materials is often expressed as the canopy gap probability (\(P_{\text{gap}}\)), which is a parameter that is fundamental to linking lidar measurements, vertical and horizontal canopy structure, and the radiation regime of a plant canopy (Armston et al. 2013; Ni-Meister et al. 2001). The directional gap probability, \(P_{\text{gap}}\) (\(\theta\)), is defined as the probability of a beam of infinitesimal width at zenith angle \(\theta\) to the local normal, being directly transmitted through a canopy. The vertical distribution of lidar-derived \(P_{\text{gap}}\) is indirectly related to the vertical distribution of plant area, enabling the estimation of Plant Area Index (PAI), vegetation cover and their vertical profiles (Ni-Meister et al. 2001; Tang et al. 2012).

Surface elevation, height, and vertical profiles of PAI and vegetation cover are all structure parameters that are being generated from GEDI waveform lidar measurements. However, in contrast to airborne scanning systems such as LVIS, spaceborne lidar instruments such as NASA’s ICESat/GLAS and GEDI are profiling instruments and may only measure a fraction of the Earth’s surface. For example, during GEDI’s 2-year mission life, waveform measurements are made over the Earth’s surface between 51.6°N and 51.6°S, resulting in approximately 10 billion observations of the Earth’s land surface in absence of clouds. The sampling pattern is optimised to maximise the geographic coverage of these observations. In the case of GEDI, shown in Fig. 2, three lasers (two full power and a coverage one) generate four beams that are dithered to allow data collection along 8 tracks separated by 600 m in across-track direction. This way, a total across-track width of 4.2 km is covered by a single GEDI overflight. Along each track, footprints with a nominal diameter of 25 m are illuminated on ground and separated by 60 m in the along-track direction.
Fig. 2

The sampling pattern of the NASA Global Ecosystems Dynamics Investigation (GEDI) mission, illustrating the arrangement of ground tracks generated from the full power and coverage laser beams. Vertical and horizontal spacing of footprints are not to scale

Lidar acquisitions are limited by the presence of clouds and dense atmospheric haze: in these cases, the signal is often attenuated before it reaches the ground. In addition, the estimation of the sub-canopy topography, vegetation height and the interpretation of the waveform in terms of physical structure is affected directly by multiple scattering, sloped terrains and off-nadir pointing (Yang et al. 2011). Multiple scattering occurring in optically thick media distort the waveform making their interpretation more difficult. Additionally, off-nadir pointing and/or a sloped terrain may also deform the vertical waveform shape with respect to a nadir pointing and/or flat terrain. Sloped terrains typically extend the ground return and make it closer to, or even overlapping with, the vegetation returns. This effect becomes more critical at larger footprint sizes. For an off-nadir pointing angle, a larger amount of light energy tends to remain trapped in the upper canopy layers. If penetration to the ground occurs, the ground peak can disappear even on a flat terrain, similarly to the sloped terrain and nadir pointing case. However, in an off-nadir pointing lidar the path length is tilted with respect to the tree growth direction, and this has to be accounted for in the height estimation process. Finally, it is worth remarking that spaceborne lidar sensor are further limited due to the sampling on ground, that leads to a gapped coverage and hampers high spatial resolution (e.g., 1 ha) mapping of forest structure characteristics. In this case, high-resolution mapping can be achieved by combining the lidar data with those from different sensors (e.g., radar or optical).

2.2 SAR Measurements

SAR systems are installed on moving platforms, typically airborne or spaceborne, and transmit electromagnetic pulses in side-looking geometry (see Fig. 3). The radiation backscattered from the illuminated footprint is recorded continuously. The illuminated width of the footprint, i.e. the swath width, extends for 1–20 km in the airborne case and for 30–500 km in the spaceborne case, with typical resolution between 1 and 10 m (Moreira et al. 2013).
Fig. 3

SAR and (Pol-)InSAR acquisition geometry in the basic single baseline configuration. The azimuth axis is orthogonal to the plane of the page. Angles and distances are not to scale

The received signals are coherently processed in order to focus 2D SAR (amplitude and phase) single-look complex (SLC) images in the slant range (\(r\), parallel to the line of sight)—azimuth (\(x\), parallel to the flight direction) plane. According to the Born approximation, the backscattered radiation is the linear superposition of the contributions arising from the individual scatterers (i.e., the physical vegetation elements). Multiple scattering phenomena from their mutual interactions are neglected. As a consequence, the SLC complex amplitude for a fixed polarization channel at the generic range–azimuth \(\left( {r^{{\prime }} ,x^{{\prime }} } \right)\) coordinates can be expressed as a volume integral (Bamler and Hartl 1998; Fornaro et al. 2003):
$$y\left( {r^{{\prime }} ,x^{{\prime }} } \right) = \iiint {h\left( {r^{{\prime }} - r,x^{{\prime }} - x} \right) \cdot \xi (r,x,s) \cdot {\text{e}}^{{ - j\frac{4\pi }{\lambda }R(r,s)}} {\text{d}}r{\text{d}}x{\text{d}}s,}$$
(2)
where \(s\) is the elevation coordinate, \(R\left( {r,s} \right)\) is the distance between the antenna and the elementary volume defined by \({\text{d}}r{\text{d}}x{\text{d}}s\), \(\lambda\) is the wavelength, and \(h(r,x)\) is the (end-to-end) system point-spread function after focusing which depends on the range and azimuth resolutions (Bamler and Hartl 1998). \(\xi (r,x,s)\) is the (unknown) 3D complex radar reflectivity, resulting from the interaction between the transmitted pulse and the scatterer. From (2), it is apparent that a SAR image alone contains backscatter amplitude information that depends on the full 3D forest structure, but it does not provide structure reconstruction capability as a consequence of the integration in \(s\).
The sensitivity to the vertical dimension can be recovered in an InSAR configuration constituted by two SAR images \(y_{1} \left( {r^{{\prime }} ,x^{{\prime }} } \right)\) and \(y_{2} \left( {r^{{\prime }} ,x^{{\prime }} } \right)\) acquired under a (slight) angular diversity (incidence angle) from tracks or orbits separated in space by a baseline of length \(B\). With reference to Fig. 3, InSAR can localize a scatterer by measuring the phase difference in the SLCs induced by the difference of the distances from the two SAR platform positions to the scatterer. The sensitivity (i.e., the derivative) of the InSAR phase difference with respect to the vertical height corresponds to the so-called vertical wavenumber, which is given by (Reigber and Moreira 2000; Papathanassiou and Cloude 2001):
$$k_{Z} : = \frac{4\pi }{\lambda r}\frac{{B_{ \bot } }}{{\sin \theta_{0} }},$$
(3)
where \(B_{ \bot }\) is the orthogonal baseline, i.e., the length of the projection of the InSAR baseline in the direction orthogonal to the line of sight, and \(\theta_{0}\) is the incidence angle. As any phase value, the interferometric phase is ambiguous by integer multiples of \(2\pi\). As a consequence, the related height measurements are ambiguous over a height interval \({\text{HoA}} = 2\pi /k_{Z}\).
The scattering from natural media, including forests, is typically regarded as a stochastic process. Although Gaussianity may not hold depending on wavelength and/or resolution, the use of second-order statistics has been proven to be valuable for the extraction of scatterer information. Thus, the complex coherence is considered and it is defined as:
$$\gamma_{1,2} \left( {k_{Z} } \right): = \frac{{E\left\{ {y_{1} \left( {r^{{\prime }} ,x^{{\prime }} } \right)y_{2}^{*} \left( {r^{{\prime }} ,x^{{\prime }} } \right)} \right\}}}{{\sqrt {E\left\{ {\left| {y_{1} \left( {r^{{\prime }} ,x^{{\prime }} } \right)} \right|^{2} } \right\}E\left\{ {\left| {y_{2} \left( {r^{{\prime }} ,x^{{\prime }} } \right)} \right|^{2} } \right\}} }},$$
(4)
where \(E\{ \cdot \}\) indicates the statistical expectation operator, and the asterisk indicates the complex conjugate. The dependence of the coherence on \(k_{Z}\) has been explicitly indicated at the left-hand side of (4), while range and azimuth have been dropped for simplicity. The estimation of \(\gamma_{1,2} \left( {k_{Z} } \right)\) in (4) would require several realizations of the underlying stochastic process, which are generally not available in reality. Therefore, assuming spatial ergodicity, reliable measurements of the statistical expectations in (4) are typically obtained by averaging neighboring pixels within a range–azimuth (multilook) cell.
By following all the algebraic development (see Appendix 1), and by considering the presence of noise, the complex coherence (3) becomes (Zebker and Villasenor 1992; Bamler and Hartl 1998):
$$\gamma_{1,2} \left( {k_{Z} } \right) = \gamma_{\text{Sys}} \gamma_{\text{SNR}} \gamma_{S} \left( {k_{Z} } \right) \gamma_{\text{vol}} \left( {k_{Z} } \right).$$
(5)
In (5), \(\gamma_{\text{Sys}}\) includes a wide range of decorrelation effects induced by the SAR system and processing inaccuracies (e.g., quantization, co-registration, range and azimuth ambiguities, …), while \(\gamma_{\text{SNR}}\) is the decorrelation induced by the presence of noise, and therefore by a finite signal-to-noise ratio. \(\gamma_{S} (k_{Z} )\) is a (real-valued) decorrelation term that depends on \(h(r,x)\) (Zebker and Villasenor 1992; Bamler and Hartl 1998). The former two terms can be compensated by exploiting the available SAR system characterisation, while \(\gamma_{S} (k_{Z} )\) is compensated by applying the so-called spectral shift filtering (Bamler and Hartl 1998). The remaining term \(\gamma_{\text{vol}} (k_{Z} )\) is the volume decorrelation, and expresses the spectral decorrelation properties of the 3D reflectivity:
$$\gamma_{\text{vol}} \left( {k_{Z} } \right): = {\text{e}}^{{jk_{Z} z_{G} }} \frac{{\int_{0}^{{H_{V} }} {F_{\text{vol}} \left( z \right) {\text{e}}^{{jk_{Z} z}} {\text{d}}z} }}{{\int_{0}^{{H_{V} }} {F_{\text{vol}} \left( z \right) {\text{d}}z} }}.$$
(6)
In (6), \(F_{\text{vol}} \left( z \right)\) is the spatial density of backscattered power along height (i.e., a projection of \(\xi (r,x,s)\), see Appendix 1) for the fixed range cell, called also vertical reflectivity profile. \(z_{G}\) is a reference volume bottom height [see also (1)] and \(H_{V}\) is the volume extension above \(z_{G}\). If the wavelength allows penetration down to the underlying ground, then \(z_{G}\) is the ground height with respect to the global or local height considered as a reference.
The vertical reflectivity \(F_{\text{vol}} (z)\) reflects the physical (geometric) 3D distribution of vegetation elements through scattering and attenuation processes that depend on the SAR wavelength, incidence angle, polarisation, and the dielectric properties of the canopy layers:
  • The wavelength defines directly the scattering of and the propagation through the vegetation. Longer wavelengths (e.g., P- and L-bands) are more sensitive to larger vegetation elements. At the same time, a (high) penetration through them provides propagation and sensitivity down to the ground. The sensitivity to smaller elements, which can still be relevant from an ecological point of view, increases with decreasing wavelength (from S- up to X-band). However, the attenuation of the vegetation increases. Hence, at shorter wavelengths the propagation through (even small) vegetation gaps contributes to the penetration to a larger extent than at longer wavelengths.

  • For a fixed slant range, wavefronts at steeper (i.e., closer to nadir) incidence angles have a wider projection on the horizontal plane, thus \(F_{\text{vol}} (z)\) results more sensitive to the projection of the horizontal distribution of the effective scatterers. In contrast, shallower incidence angles (i.e., far from nadir), increase the sensitivity to the vertical distribution. At the same time, the incidence angle defines the length of the propagation path through the vegetation volume, hence the (two-way) attenuation of the transmitted wave as a function of height. Steeper incidence angles are less attenuated than shallower ones for the same stand characteristics. Not only this, but steeper incidence angles facilitate penetration through gaps at shorter wavelengths.

  • The orientation of the scattering vegetation elements as a function of height affects the related distribution of the backscattered power in the different SAR polarisation channels. Therefore, the relative power levels of scattering components at different heights in \(F_{\text{vol}} (z)\) can be changed by changing polarisation channel.

  • The dielectric properties of the vegetation elements affect their scattering and attenuation, which depend on the water content of the tree tissuesand on its distribution. Dielectric changes can occur due to both environmental (e.g., rains, droughts, snowfalls…) and seasonal changes, and \(F_{\text{vol}} (z)\) can change accordingly although the extent and type of change can be wavelength-dependent (Pardini and Papathanassiou 2018; Bai et al 2018; Kugler et al 2014).

Equation (6) shows that the volume coherence \(\gamma_{\text{vol}} (k_{Z} )\) and the vertical reflectivity profile \(F_{\text{vol}} (z)\) form a Fourier pair, i.e., an InSAR acquisition corresponds to one Fourier component of the reflectivity profile at a spatial frequency equal to the vertical wavenumber. One single-baseline InSAR measurement makes it possible to extract the height of the backscattering phase center within the volume, i.e., the Digital Elevation Model (DEM), corresponding (at first order) to the “center of mass” of \(F_{\text{vol}} (z)\). It is calculated as \(z_{\text{DEM}} (k_{Z} ) = \varphi_{{{\text{vol}},u}} (k_{Z} ) /k_{Z}\), with \(\varphi_{{{\text{vol,}}u}} (k_{Z} )\) being the phase of the volume coherence after unwrapping. In order to be able to estimate a larger number of parameters, one can increase the observation space by acquiring interferometric pairs in different polarisation channels, and/or by acquiring multiple images from multiple track/orbit displacements (multi-baseline acquisition). Two different approaches can then be followed:
  1. (A1)

    The first one is to parameterise \(F_{\text{vol}} (z)\) in terms of geometrical and scattering properties and to use then the set of measured \(\gamma_{\text{vol}} (k_{Z} )\) to estimate the individual model parameters, as it is done with Pol-InSAR inversions (Papathanassiou and Cloude 2001). The scattering model, whose validity is in general frequency-dependent, is essential for the significance and accuracy of the estimated parameters. The model should contain enough physical structure to interpret the interferometric measurements, and at the same time, it must be simple in terms of number of parameters in order to be determinable with the available (in general limited) number of measured coherences.

     
  2. (A2)

    The second approach is the most direct one, and attempts to invert the Fourier relationship (6) provided that a set of measurements of \(\gamma_{\text{vol}} (k_{Z} )\) is available at suitable \(k_{Z}\), according to the TomoSAR imaging principle (Reigber and Moreira 2000). This approach is more expensive in terms of acquisitions needed, but it has the advantage that no assumption on the shape of \(F_{\text{vol}} (z)\) is required, allowing the reconstruction of arbitrary vertical reflectivity profiles. The smallest available \({\text{HoA}}\) (i.e., the largest \(k_{Z}\)) is the TomoSAR vertical resolution (Rayleigh limit).

     
Pol-InSAR model-based inversion approaches have been demonstrated, e.g., for forest height estimation in boreal, temperate and tropical forests at multiple frequencies from P- up to X-band, considering both airborne and spaceborne platforms (Papathanassiou and Cloude 2001; Garestier et al. 2008; Neumann et al 2010; Lee et al. 2013; Kugler et al. 2014; Lavalle and Hensley 2015; Simard and Denbina 2018). The TomoSAR reconstruction of \(F_{\text{vol}} (z)\) has been demonstrated in airborne experiments across different forest types (Reigber and Moreira 2000; Frey and Meyer 2011; Tebaldini and Rocca 2012; Mariotti et al. 2012; Pardini et al. 2018a, b). Several algorithmic solutions have been found in the spectral estimation theory (Gini et al. 2002; Frey and Meyer 2011; Aguilera et al. 2013) to improve the imaging performance against the typical low number and the irregular distribution of the flight tracks/orbits. Among the several alternatives, the model-free Capon spectral estimator (Lombardini and Reigber 2003) is widely employed, and can provide vertical resolution beyond the Rayleigh limit (Cazcarra et al. 2019).

As a final remark, a fundamental prerequisite for a successful estimation of 3D structure parameters from SAR acquisitions is the stationarity of \(F_{\text{vol}} (z)\) within the total acquisition time. Changes can be induced by, e.g., the movement of the vegetation under the action of wind, or changes in the dielectric properties, and introduce an additional (wavelength-dependent) temporal decorrelation factor in (4). If not properly accounted for, the inversion problem may become ambiguous if not unfeasible (Lee et al. 2013; Lavalle and Hensley 2015). On the other hand, (bistatic) single-pass (Pol-)InSAR acquisitions provide temporal decorrelation-free measurements of \(\gamma_{\text{vol}} (k_{Z} )\), although dielectric changes may still occur between acquisitions. In this frame, a full separation between geometric and dielectric contributions within the same vertical reflectivity profile or its parameters would only be possible with single-pass multistatic TomoSAR constellations.

3 A Comparison Between Lidar and SAR Profiles at Different Wavelengths

In this Section, the dependency of the TomoSAR vertical reflectivity profiles on wavelength, dielectric properties, and structure types is discussed by comparing profiles extracted in the temperate Traunstein (Germany) and the tropical Rabi (Gabon) forest sites. Details on the used lidar and SAR data sets are reported in Appendix 2. TomoSAR vertical profiles have been estimated in the HV channel in square range–azimuth multilook cells measuring 20 m × 20 m, corresponding to the profile resolution in the horizontal range–azimuth plane. The profiles have been obtained by means of the Capon spectral estimator (Lombardini and Reigber 2003). Since lidar waveforms were available only in Rabi, lidar profiles have been generated in Traunstein as the histogram of all the recorded ALS returns as a function of height.

The profiles in Fig. 4 have been calculated in the Traunstein forest, and refer to different growth stages (“transition”, “close-to-nature”, “mature” and “young”) which are typical of this forest site. The profiles in Fig. 5 have been estimated in the Rabi forest in correspondence of a ForestGEO inventory plot (see Appendix 2).
Fig. 4

Traunstein forest (south of Germany). SAR vertical reflectivity profiles and lidar profiles (histograms of the lidar returns as a function of height) over a representative azimuth transect at constant range, HV polarization channel. The height axis is relative to the ground height (0 m), indicated with the horizontal white dashed line. The vertical white dashed lines delimit stands of interest in different growth stages. SAR profile intensities are normalised by the total backscattered power. a P-band (2009); b L-band (2009); c L-band (2013); d X-band (2013); e lidar profiles

Fig. 5

Rabi ForestGEO plot (Gabon). Vertical profiles along a North–South transect. The height axis is relative to the ground height (0 m), indicated with the white dashed line. SAR profile intensities are normalised by the total backscattered power. a P-band and b L-band TomoSAR vertical reflectivity profiles, c LVIS waveforms

The ability of the different wavelengths of penetrating into and through the canopy layers can be assessed directly by detecting the presence and measuring the power of the ground scattering in the profiles. In both Traunstein and Rabi, the ground scattering is well visible in almost all the P-band profiles (see Figs. 4a, 5a). A larger variation of ground power across forest stands occurs at L-band due to the higher sensitivity to the spatial variations of the attenuation levels induced by the changes of both canopy density and water content (see Figs. 4b, 5b). In Traunstein, the L-band profiles in Fig. 4c have been obtained with data acquired around 3.5 years after the ones used to generate the profiles in Fig. 4b, and show a change of ground power. This can be due to a change of canopy attenuation caused by a change of both canopy density, as a result of forest management, and dielectric properties, as a result of seasonality-induced redistributions of water content. The effect of forest management is particularly visible in the “transition” stands, in which old taller trees are cut to allow the growth of the younger shorter ones below. This is recognisable also in the sparsity of the lidar returns in Fig. 4e. The effect of seasonal dielectric changes is likely visible in the forest stand beyond a horizontal distance of 2 km (Autumn 2009 vs. Spring 2013). Indeed, an increase of ground power occurs in the presence of the thick canopy cover shown by the lidar profiles. Finally, the X-band profiles (Fig. 4d) are the ones characterized by the lowest ground contribution due to the largest attenuation of the shortest X-band wavelength. However, the ground scattering becomes dominant in the “transition” stands. Here, the sparser canopy facilitates penetration through its largest gaps even in the slanted SAR geometry.

In both Traunstein and Rabi, P- and L-band waves exhibit similar penetration, but different sensitivity to the physical canopy elements. In Traunstein, the P-band canopy contributions are weaker than the ground ones in many stands in contrast to L-band, similarly to other temperate and boreal forests (Frey and Meyer 2011; Tebaldini and Rocca 2012). However, there are cases in which the P- and L-band profiles are more similar (e.g., the forest stand beyond a horizontal distance of 2 km). The spatial variation of the canopy backscattering distribution (which in general occurs at very small scales) is reflected in different ways by the wavelengths. At P-band, the visibility of the canopy layers changes in space faster than at L-band. At L-band, canopy scattering contributions are either concentrated closer to the canopy top, or spread along the whole volume extension. Seasonal dielectric changes affect not only the distribution of the power at the ground, but also within the canopy layers (e.g., the forest stand beyond a horizontal distance of 2 km). Due to the larger attenuation, at X-band the canopy scattering contributions tend to be closer to the canopy top than at the other wavelengths. In contrast to Traunstein, P- and L-band profiles tend to be more similar in Rabi. However, large differences can occur, as shown for the stand corresponding to the horizontal distances between 600 and 800 m (delimited by the vertical dashed lines in Fig. 5). Here, stronger canopy contributions are close to the canopy top at P-band, while they are distributed along height at L-band. The distribution of the physical size of the involved scatterers might again explain this difference: the bigger ones might be close to the top, while smaller ones (which are semi-transparent at P-band) might be distributed down to the ground. On the contrary, the most powerful contributions in the lidar profiles are close to the canopy top due to a reduced penetration.

The profiles in Fig. 6 are extracted from Fig. 4 (Traunstein forest), and can be used to gain some additional insights about the capability of each wavelength to reflect 3D structure properties. Each plot shows 10 profiles extracted every 10 m in one of the four growth stages highlighted in Fig. 4:
  • In the “transition” stands, all wavelengths show similar vertical profiles, with relevant vegetation backscattering contributions up to 10 m and between 20 and 40 m. However, the relative profile amplitudes and their spatial variation across profile coordinates are wavelength- and time-dependent. The contributions close to the canopy top appear in 2009 more heterogeneous at P- than at L-band. This might be due to electromagnetic semitransparency at P-band, affecting any interpretation in terms of horizontal heterogeneity. As a result of the sparsity induced at the canopy top by management activities, in 2013 the upper canopy scattering contributions become weak at L-band, but stronger at X-band. In addition, the sparsity of the vegetation elements makes X-band penetrate well down to the ground, and the related reflectivity profiles reflect closely the geometric distribution of the lidar returns.

  • The “close-to-nature” stands are characterized by the largest heterogeneity in terms of tree species and heights. The resulting structural complexity is reflected by the lidar profiles, which are composed by returns at all heights. In parallel, all the SAR frequencies show this heterogeneity, but at different extents. P-band shows still large penetration capabilities, and the profiles contain larger or even dominant scattering contributions close to ground. These contributions are present also in the L-band profiles together with additional stronger ones in correspondence of the upper canopy layers. The difference between the L-band profiles in 2009 and 2013 is probably caused by the difference in TomoSAR vertical resolution (better in 2013) and dielectric differences at a lower extent, as no management action is documented in these stands. At X-band, the reduced penetration makes the profiles sensitive to the upper canopy variations between 15 m and 30 m.

  • The “mature” stands are constituted by (tall) trees uniformly distributed in space. This uniformity characterizes the lidar and SAR profiles at all wavelengths. The P-band profiles show the least sensitivity to the canopy top, having their main peak located closer to the ground. Similarly to the “close-to-nature” stands, the change of the L-band profiles from 2009 to 2013 is mostly due to an improvement in the vertical resolution and to dielectric changes. In 2013, beyond the difference in attenuation, the L- and X-band profiles convey similar structural (i.e., geometric) information.

  • Finally, the “young” stands are composed of mainly a layer of short trees with low density and some sparser taller trees. The shorter trees are visible at all the frequencies, while the taller trees are less visible at P-band. In 2013, the L-band profiles show a significant spatial variability of the canopy layer close to the ground corresponding to the shorter trees. Such variability is not shown either by the lidar or the X-band profiles. The spatial distribution of the X-band profiles is very similar to the lidar ones, and once more this could indicate their increased sensitivity to the geometric properties of the canopy.

Fig. 6

Traunstein forest (south of Germany). Examples of lidar and TomoSAR profiles (HV channel) in the forest stands delimited in Fig. 4. The height axis is relative to the ground height (0 m), indicated with the horizontal black dashed line. For the sake of visualization the profiles have been normalized by their maximum

To summarize, these examples have shown that, together with lidar profiles, SAR profiles at all wavelengths contain information about 3D structure. P- and L-band can penetrate through the canopy layers down to the ground even in dense tropical forest environments, and can complement lidar profiles in mapping vertical heterogeneity. At P-band, vegetation elements can be semitransparent in some stands, and the fast and strong spatial changes of attenuation might make the canopy appear more heterogeneous horizontally. At L-band, the scattering from smaller vegetation elements becomes more significant, and different degrees of vertical heterogeneity could be more distinguishable than, e.g., at P-band. The complementarity between these two frequencies seems to depend on the imaged forest structures. The L-band examples clearly show also the dependency of the reflectivity profiles on the dielectric properties, which might affect the interpretation of the profiles and their spatial changes in terms of physical structure. Finally, the X-band profiles resemble more closely the geometric distribution of the canopy layers in the lidar profiles, at least in those sparser stands where penetration until the ground occurs. This might be an indication that penetration is facilitated by vegetation gaps along the SAR line of sight. In the absence of these, the resulting lower X-band penetration increases the sensitivity to the top canopy spatial variations, therefore complementing and improving the capability of longer wavelengths in the characterization of horizontal heterogeneity.

4 The Lidar-SAR Model-Based Framework for Forest Height Estimation

In this section, a case study is considered in which the common sensitivity of lidar and SAR measurements to geometric structure parameters is exploited in the parameterisation of the Random-Volume-over-Ground Pol-InSAR backscattering model using sparsely sampled lidar full waveforms for obtaining accurate, continuous and high-resolution forest (top) height estimates.

4.1 The Random-Volume-Over-Ground Model

The Random-Volume-over-Ground (RVoG) model describes the variation of the reflectivity profiles with the polarization channel \({\varvec{\upomega}}\) (Papathanassiou and Cloude 2001). First of all, it is assumed that \(F_{\text{vol}} \left( {z} \right)\) is the sum of the ground-only and the volume-only vertical reflectivity profiles (two-layer model). Next, it is assumed that the backscattered power at the ground level is the only component of \(F_{\text{vol}} \left( {z} \right)\) that changes with polarisation. In other words, the volume-only profile is the same across all polarimetric channels except for a scalar factor. Additionally, the ground-only profile is modelled by a Dirac-\(\delta\) function (Papathanassiou and Cloude 2001). As a consequence:
$$\gamma_{\text{vol}} \left( {k_{Z} ,\varvec{\omega}} \right) = {\text{e}}^{{jk_{Z} z_{G} }} \frac{{\gamma_{V} \left( {k_{Z} } \right) + \mu \left(\varvec{\omega}\right)}}{{1 + \mu \left(\varvec{\omega}\right)}}$$
(7)
where the dependency of the individual parameters on the polarisation channel \(\varvec{\omega}\) has been explicitly indicated. \(\gamma_{V} (k_{Z} )\) is the (polarisation-independent) volume-only coherence and \(\mu (\varvec{\omega})\) is the ground-to-volume amplitude ratio. \(\gamma_{V} (k_{Z} )\) is obtained from the volume-only vertical reflectivity profile \(F_{V} (z)\) with a relationship formally identical to (6). The dependency of \(\gamma_{V} (k_{Z} )\) on the forest top height \(H_{V}\) is made explicit by modeling \(F_{V} (z)\). A widely and very successful model is an exponential distribution of scatterers, i.e.
$$F_{V} \left( z \right) = {\text{e}}^{{ - 2\left( {H_{V} - z} \right)\tau /\cos \theta_{0} }} \quad {\text{for}}\;\;0 \le z \le H_{V}$$
(8)
where defines the attenuation rate of the profile, which is a function of both density of scatterers and their dielectric constant. The exponential profile tends to fit better at shorter wavelengths. At longer wavelengths, or more in general in the cases in which more effective scatterers are located closer to the ground, the exponential decay in (6) may not be valid. Different parameterisations might be used instead [see, e.g., (Garestier et al. 2008)].

4.2 Case Study: Fusion of GEDI Waveforms and TanDEM-X Coherences

The GEDI lidar acquires footprints according to the spatial sampling pattern illustrated in Fig. 2. The use of single-pass bistatic X-band InSAR coherences acquired during the DLR’s TanDEM-X mission has been proposed to bridge the spatial gaps between the footprints in order to obtain continuous and high-resolution forest mapping, and the inversion of the RVoG model (7, 8) is a possible solution. Only one TanDEM-X coherence in one polarimetric channel is most typically available for inversion. On the other hand, fixed a polarimetric channel, the RVoG model (7) presents 4 unknown parameters, i.e., \(H_{V}\), \(\tau\), \(\mu (\varvec{\omega})\) and \(z_{G}\). As one complex coherence can account for only two real parameters, the inversion of a full RVoG model results underdetermined. Additionally, the X-band penetration is not guaranteed in all forest types, and the use of an external \(z_{G}\) is needed to avoid any penetration-dependent height estimation bias (Kugler et al. 2014). The spatially sparse set of GEDI footprints can then be used to initialize some of the RVoG parameters, thus making the inversion determined (Qi and Dubayah 2016, 2017) and correcting penetration-induced biases. Depending on the model assumptions, the following strategies can be adopted:
  1. (S1)

    The sparse GEDI ground topography values are interpolated on the TanDEM-X spatial grid of the complex InSAR coherences. Given the interpolated topography, the RVoG model is fitted to the complex coherences to estimate \(H_{V}\) and \(\tau\) assuming \(\mu = 0\) (no ground contribution).

     
  2. (S2)

    Assuming \(\mu = 0\), the GEDI ground topography and top height are used in the RVoG model to estimate extinction values from TanDEM-X complex coherences in correspondence of the footprints. Then, both topography and extinction are interpolated over the TanDEM-X grid. Finally, a continuous and high-resolution estimate of \(H_{V}\) is obtained by fitting the RVoG to the coherences using the interpolated topography and extinction.

     
  3. (S3)

    As in (S2), but allowing \(\mu \ne 0\), i.e., values of \(\mu\) and \(\tau\) are jointly estimated in correspondence of the footprints, interpolated and used in the inversion of the remaining parameters.

     
The interpolation of the three parameters (i.e., ground topography, extinction and ground-to-volume ratio) estimated in correspondence of the footprints can be carried out, e.g., by means of a simple kriging interpolation (Qi et al. 2017; Oliver and Webster 1990), in which it is assumed that samples are correlated according to a negative exponential spatial correlation function. Alternatively, the TanDEM-X high-resolution DEM can be used to obtain a ground topography with higher resolution than the one that a simple interpolation would provide (Lee et al. 2018). A wavelet transformation is used to decompose the TanDEM-X DEM into three high spatial frequency components (describing local changes of topography) and a low-frequency one, according to the methodology outlined in (De Grandi et al. 2016). The low-frequency part of the TanDEM-X DEM is substituted by the GEDI interpolated ground topography, and a GEDI-TanDEM-X combined ground topography is obtained by inverting the wavelet transformation.
Real data results have been generated over the two tropical forest sites of La Selva (Costa Rica) and Lopé (Gabon) by using TanDEM-X HH acquisitions and simulated GEDI acquisition patterns (Qi and Dubayah 2016, 2017). Details about the radar and lidar data sets are reported in Appendix 2. Over La Selva, the lidar data were processed to simulate GEDI full waveforms following the method in Blair and Hofton (1999) with measurement noise added following Hancock et al. (2011). GEDI acquisition tracks were simulated also over the Lopé site. In both cases, the nominal lidar sampling after 2 years of mission has been considered. In the experiments, TanDEM-X interferometric coherences have been calculated over 25 m × 25 m (ground range) multilook cell approximately corresponding to the GEDI footprint diameter (25 m). Therefore the TanDEM-X estimates of \(H_{V}\) have been obtained with 25 m resolution. To reduce statistical variations, the estimates have been averaged on a threefold bigger cell. As a consequence, the final resolution of the height map is 75 m. Figure 7 shows the estimated TanDEM-X height maps and their validation against the lidar top height (2D histograms) for different configurations of the model parameters over La Selva. The adoption of the strategy (S1) (Fig. 7b) improves both the estimation RMSE and the bias with respect to the case in which the TanDEM-X absolute coherence values alone are used to estimate \(H_{V}\) with \(\tau = \tau_{0}\) fixed a priori (Fig. 7a). However, the overall estimation performance is not satisfactory, due to the still large bias (around 4 m). A noticeable improvement is obtained by applying (S2), as the bias reduces to around 0.5 m (Fig. 7c). The use of (S3) keeps almost unaltered the already very small bias and further reduces the RMSE to 4.3 m (13%) (Fig. 7d). Such estimation performance is definitely better than just interpolating the GEDI top heights over the TanDEM-X grid. Moreover, the residual RMSE can be reduced by averaging further the final estimates to a coarser resolution.
Fig. 7

La Selva Biological Station. TanDEM-X forest top height maps before and after combination with GEDI simulated data and 2D histograms against lidar canopy height a Inversion from TanDEM-X data only; b TanDEM-X inversion using the interpolated GEDI terrain height; c TanDEM-X inversion using the GEDI interpolated terrain height and \(\sigma\), for \(\mu = 0\); d TanDEM-X inversion using the GEDI interpolated terrain height, \(\sigma\) and \(\mu\); e reference lidar top height

The combination strategy (S1) with the refined wavelet-based interpolation of the ground topography has been applied to the Lopé acquisition. The mean value of the interpolation error between the original topography and the interpolated one (Fig. 8a) is 0.8 m and the standard deviation is around 8 m. The forest height (Fig. 8b) could be estimated with a correlation coefficient of 0.81 and an RMSE of 5.47 m (i.e., around 10%) with respect to the lidar RH95 heights (Fig. 8c). This performance is still comparable to the one obtained for the forest height inverted using the original lidar fully sampled topography (correlation coefficient of 0.92 and RMSE of 3.58 m).
Fig. 8

Lopé test site. a LVIS terrain height (upper) and terrain height resulting from the simulated GEDI-TanDEM-X combination; b TanDEM-X top height inversion results from the LVIS terrain height (upper) and the one resulting from the GEDI-TanDEM-X combination (bottom); c validation plots against LVIS RH95; e reference LVIS RH95 canopy height

As a final remark, the effectiveness of the parameterisation of SAR interferometric models by means of lidar waveform parameters is influenced by the difference of wavelengths, the difference in the looking geometry, and the resolution of the parameterisation itself. Concerning the resolution of the parameterisation, in the example above \(\tau\) and \(\mu\) are radar reflectivity parameters, which depend on the (relatively) fast spatial variations of dielectric properties and physical arrangement of the vegetation elements. In general, any mismatch between the resolution of an external parameterisation and the resolution of the TanDEM-X height inversion causes a loss of estimation performance. A constant value of \(\tau\) makes the inversion of forest height feasible even from single-baseline coherences without external parameterisations, but can lead to an unsatisfactory estimation performance. The interpolation of the values of \(\tau\) and \(\mu\) retrieved at the footprint locations improves the estimation performance, but it can track only a low-resolution component of these variations depending on the spatial sampling rate provided by the footprints. A way to reduce the consequential mismatch between the interpolated parameters and the SAR data could be to reduce the resolution of both the lidar parameterisation (for instance by aggregating multiple footprints) and at the same time of the inversion (multilook cell) from the interferometric coherences, for instance as noted by Brolly et al. (2016).

5 The Lidar-SAR Structure-Based Framework

The applicability of the lidar-SAR model-based framework presented in Sect. 4 becomes limited with increasing difference of the wavelength involved. This is a consequence of the fact that different wavelengths “see” different effective scatterers, as shown by the examples in Sect. 3. A direct relationship between lidar profiles and SAR vertical reflectivity profiles becomes difficult to be established at the increase of the wavelength difference, resolution and look angle (Brolly et al. 2016; Pardini et al. 2018a, b). In the structure-based framework, lidar and SAR measurements are used to calculate one or more structure indices independently, i.e., without exchanging a parametrization of the underlying backscattering models. The structure indices considered here are defined with the aim of describing (components of) physical structure, and they use features extracted from the vertical reflectance/reflectivity profiles that reflect physical heterogeneity within a certain scale. In this Section, we review meaningful structure indices both from field data and lidar and SAR profiles, and we show in an example with real data how they can be used to relate the structure information content of the profiles.

5.1 Structure from Ground Measurements

Forest structural information is often quantified by means of scalar indices whose definition depends on the specific application. Basing on field inventory data, structure indices are defined at plot level by means of single-tree attributes such as diameter, height, basal area, canopy dimension, species composition, and stand density. A number of indices have been proposed in the literature, like, e.g., the aggregation index (Clark and Evans 1954), the diametric differentiation index, the mingling index, the contagion index, or the complex stand index (Pastorella and Paletto 2013). However, there is not yet an overall and unequivocal measure able to express forest structure in terms of forestry and ecology appropriate for a wide variety of application, spatial scales, and forest types (Zenner and Hibbs 2000; Pommerening 2002; Pretzsch 2009; del Rio et al. 2016). Despite this difficulty, there is a common understanding that the structural heterogeneity in both the horizontal and the vertical dimensions are two complementary aspects that should be considered.

One way to quantify horizontal structure or heterogeneity is to use the concept of density. High density means low heterogeneity, and vice versa. The stand density index \({\text{SDI}}\) (Reineke 1933) (measured in trees per hectare) links space utilization to tree size. It is closely related to basal area and by definition relates the stand density with the equivalent density of a stand with a quadratic mean diameter of 25 cm:
$${\text{SDI}} = N_{S} \left( {\frac{{D_{g} }}{25}} \right)^{1.605}$$
(9)
where \(N_{S}\) is the number of stems per hectare and \(D_{g}\) is the quadratic mean diameter at breast height (dbh) in cm within the scale at which \({\text{SDI}}\) is calculated. A horizontal (dimensionless) structure index can be defined as
$${\text{HS}}_{\text{field}} : = 1 - {\text{SDI}}_{\text{norm}} ,$$
(10)
where \({\text{SDI}}_{\text{norm}}\) is the normalized \({\text{SDI}}\) within an area. \({\text{HS}}_{\text{field}}\) varies between 0 and 1, indicating the minimum and maximum horizontal heterogeneity, respectively. Additional refinements can be applied, e.g., considering the taller trees to better characterize canopy closure.
Quantifying vertical structure is often treated as quantifying the degree of heterogeneity in the vertical direction. In this sense, vertical structure has been expressed by means of indices such as the Gini coefficient, the Shannon index or the standard deviation of single tree heights (Liang et al. 2007; Barbeito et al. 2009; Bohn and Huth 2017). However, tree height measurements are often not performed so that these indices cannot be calculated directly from inventory data. An alternative way to express tree size variability is to use the diameter at breast height (dbh), which is a standard inventory measurement and can be related to tree height through models. Therefore, a vertical structure index can be defined as (McElhinny et al. 2005):
$${\text{VS}}_{\text{field}} : = {\text{std}}\left( {\left\{ {\text{dbh}} \right\}} \right) ,$$
(11)
in cm, where \(\left\{ {\text{dbh}} \right\}\) is the ensemble of diameter at breast height of all the trees included in a stand, given in cm. Without loss of generality, the vertical structure index can be normalized to its maximum within the area under consideration, becoming a dimensionless index as the horizontal one.

5.2 Structure from Lidar and TomoSAR Profiles

The definition of indices from lidar and TomoSAR vertical profiles with the same structural meaning as the ones from field inventory data in Sect. 5.1 is challenged by the fact that the resolution of conventional remote sensing configurations (especially SAR) does not allow single trees to be distinguished directly. An additional complication in the interpretation of the profiles in term of geometric structure is caused by the specific dependencies of their information content (see Sect. 2). Consequently, the direct translation of the indices in Sect. 5.2 is not possible, and the use of physically significant profile features is desired to express the same heterogeneity. Recently, TomoSAR profiles have been interpreted in terms of forest structure by considering the distribution of their peaks. Even if the correspondence between the 3D distribution of peaks and the variability in the distribution of trees may not be given at all frequencies and/or spatial scales, it is supported in a number of models and experiments (Lin and Sarabandi 1999; Frey and Meyer 2011; Thirion et al. 2006; Brolly and Woodhouse 2013; Whitehurst et al. 2013; Cazcarra-Bes et al. 2017; Tello et al. 2018).

Canopy height variation above the ground height within a certain scale can be considered as a proxy for horizontal structure (Couteron et al. 2005; Carabajal and Harding 2006; Neumann et al. 2012; De Grandi et al. 2016). Here, this idea is applied to the top peaks (i.e., the peaks at the highest height) of the waveforms and the vertical reflectivity profiles. Let \({\text{Z}}_{\text{TOP}} = \left\{ {z_{T1} ,z_{T2} , \ldots ,z_{TM} } \right\}\) be the ensemble of the \(M\) unique top peak heights within the structure window. The horizontal structure index \({\text{HS}}\) can be calculated as:
$${\text{HS}}\text{ := }\text{var} \left\{ {Z_{\text{TOP}} } \right\},$$
(12)
where \(\text{var} \{ \cdot \}\) denotes the variance of a set of values. Along this line, a way to quantify vertical structure is to consider the ensemble \(Z = \left\{ {z_{1} ,z_{2} , \ldots ,z_{N} } \right\}\) of all the unique peak heights (excluding the ground peak, if present) between the ground and the canopy top. Then, a vertical structure index can be calculated as:
$${\text{VS}}\text{ := }N\text{var} \left\{ Z \right\} .$$
(13)
Similarly to \({\text{HS}}_{\text{field}}\) and \({\text{VS}}_{\text{field}}\), \({\text{HS}}\) and \({\text{VS}}\) can be normalized to their maximum value over an area. The increase in their values toward 1 corresponds to increasing heterogeneity.

5.3 Real Data Examples

The structure indices defined in the previous Sections have been applied to the field inventory measurements LVIS lidar and L- and P-band TomoSAR (Capon spectral estimator) profiles extracted over the ForestGEO plot in the Rabi test site (see Appendix 2). The profiles have horizontal resolution 20 m × 20 m and have been used to calculate the structure indices within a window measuring 100 m × 100 m. This means aggregating information from 25 statistically independent profile cells. A smaller window would reduce the statistical validity of the number of independent cells, while a larger window could provide biased values of the structure indicators as different structure types would be mixed together, and therefore lose ecological significance.

Figure 9 shows the maps of the top peak heights of LVIS profiles, and the L- and P-band TomoSAR profiles (Capon spectral estimator). Due to penetration differences, the L- and P-band top peak heights are in general lower than the LVIS ones, as shown by the profiles in Fig. 5. Nevertheless, there is a limited number of cases in which the L-band top peak height can be even higher than the LVIS one, and they are more likely focusing artifacts (e.g., sidelobes) coming from very high decorrelations. The generated \({\text{HS}}_{\text{field}}\) and \({\text{HS}}\) maps are shown in Fig. 10. All of them can well locate heterogeneous areas in the North and the South parts of the plot, and some discrepancies are visible in the remaining areas. The temporal difference (around 4 years) between the field measurements and the lidar and SAR acquisitions may also affect the comparison between field and remote sensing indices. The LVIS and the L-

band \({\text{HS}}\) maps are the most similar ones, although in some stands the L-band profiles lead to higher values of \({\text{HS}}\) as a consequence of the mentioned artifacts. At P-band, many stands result in more homogeneity as a consequence of the lower sensitivity to the top canopy variations.
Fig. 9

Rabi ForestGEO plot. Top peak heights maps extracted from a LVIS waveforms, b L-band and c P-band vertical reflectivity profiles

Fig. 10

Rabi ForestGEO plot. Maps of a\({\text{HS}}_{\text{field}}\), and \({\text{HS}}\) calculated from b LVIS waveforms, c L-band and d P-band vertical reflectivity profiles

Figure 11 shows the \({\text{VS}}_{\text{field}}\) and the \({\text{VS}}\) maps. All maps highlight an area of high vertical structure at the southeastern part of the site, which seems to be the mean feature in the plot. More to the North, another high structure area is indicated in the map derived from the field data. This area is also visible in the LVIS map and in the L-band map, while it becomes hardly visible in the P-band one. The correlation plot of Fig. 12 clearly shows that L-band reconstructs the wider range and distribution of structure indices obtained from the field data. The profiles of Fig. 5 offer some insights to understand these differences. In the LVIS profiles, the majority of peaks appear within 20 to 45 m, while peaks below 20 m are rather seldom. On the other hand, the TomoSAR peaks at L- as well as at P-band are distributed across the whole canopy extent down to the ground. The peaks from the P-band TomoSAR profiles are particularly numerous closer to the ground level. Occasionally, gaps in the canopy layer become visible at P-band which are not noticeable at L-band or in the LVIS profiles. Therefore, LVIS profiles are more sensitive than TomoSAR reflectivity profiles to variations mostly concentrated in the highest part of the canopy and can better predict \({\text{VS}}_{\text{field}}\) in those areas. However, TomoSAR long-wavelength profiles are more sensitive to variations of vertical structure when these are driven by the presence of sub-canopy elements.
Fig. 11

Rabi ForestGEO plot. Maps of a\({\text{VS}}_{\text{field}}\), and \({\text{VS}}\) calculated from b LVIS waveforms, c L-band and d P-band vertical reflectivity profiles

Fig. 12

Rabi, ForestGEO plot. Comparison between \({\text{VS}}\) and \({\text{VS}}_{\text{field}}\). The square symbols indicate (conditioned) mean values, while the vertical bars delimit the \({\text{VS}}\) interval where 75% of values are found

6 Conclusions

In this paper, first, a comparison between lidar waveforms and TomoSAR reflectivity profiles at multiple wavelengths has been attempted by means of real data in a temperate and a tropical test site. It has been shown that the wavelength affects directly penetration through the vegetation volume down to the ground. While lidar penetration occurs through vegetation gaps and is facilitated by the nadir-looking geometry, SAR penetration typically reduces from P- up to X-band, being however further affected by forest type, density, and dielectric properties (attenuation induced by water content). At the same time, it has been shown, even when the penetration is comparable, the measurements provided by lidar and by SAR systems at different wavelengths are sensitive to different components of forest structure, and therefore all of them can complement each other for structure characterization. This comparison has then be used as a basis to gain some early insights on the capabilities of two most promising combination frameworks, i.e., the model-based and the structure-based frameworks, based on two recent case studies.

The common sensitivity to geometric parameters drives the parameterisation of SAR backscattering models by means of lidar measurements in the model-based framework. A fusion algorithm could then be based on the complementarity of type and/or spatial distribution and resolution of the measurements. With reference to forest height estimation, some of the experimented model-based combination methodologies have been presented by means of a case study developed in the context of the NASA GEDI lidar mission. The TanDEM-X coherences can be used to continuously extend the coverage of the top heights measured by GEDI on the sparsely sampled waveform footprints. The RVoG model has been shown to be simple and representative enough to allow accurate parameterisation and inversion. Depending on the SAR wavelength and configurations, additional parameters can be accommodated (Simard and Denbina 2018). In general, (i) the choice of an appropriate backscattering model and its dimensionality, (ii) the choice a meaningful lidar parameterisation, and (iii) its interpolation from the lidar footprint to the continuous SAR coverage are three critical steps that contribute to the effectiveness of the combination.

The structure-based framework can directly accommodate both commonalities and complementarities in the information content of lidar and SAR measurements, bridging the limitations of the model-based framework in the cases in which the physical equivalence between lidar and SAR parameters is not given. An example has been shown in which indices expressing horizontal and vertical structure (heterogeneity) have been applied to lidar waveforms and TomoSAR vertical reflectivity profiles, in correspondence to structure indices established in ecology and forestry with ground inventory measurements. Lidar and P- and L-band wavelengths fully complement each other in both the horizontal and vertical structure if significant penetration differences exist and/or structural differences are driven by element sizes. For sure, the sensitivity of lidar waveforms to vertical structure variations mostly close to the canopy top is complemented by the higher sensitivity of long-wavelength TomoSAR profiles to variations closer to the ground. The definition of ecologically significant structure indices is of crucial importance. Nevertheless, the structure space defined by the two indices (whose independence is not guaranteed) seems to be able to relate lidar and SAR measurements and field measurements, therefore providing a biophysical interpretation. Limitations and ambiguities of this framework are still under investigation, remembering that indices might only describe an incomplete “subspace” of the whole structure information.

It is worth noting that a combination framework that unifies both the model-based and the structure-based ones might also be developed. For instance, the structure-based framework could be used to identify the spatial regions of validity of a certain parameterisation, thus defining models and supporting inversion strategies in the model-based framework.

A critical factor affecting both frameworks is the scale at which structure parameters and indices are retrieved. Indeed, it affects both the meaningfulness of the parameterisation with respect to the final inversion performance and the structure indices with respect to the underlying stand or forest. Highresolution sensors allow to explore multiple scales, representing an additional degree of freedom for structure characterisation to be exploited in the combination between lidar and SAR, and SAR at multiple wavelengths. Existing models, algorithms and indices should be assessed also in this perspective, and new ones developed, if necessary. At the present stage, addressing these issues and understanding the underlying relationships is a unique challenge that still needs to be mastered toward the development of effective fusion techniques for forest structure observation, and to optimize system and acquisitions configurations.

Footnotes

  1. 1.

    The dependence on the azimuth direction can be readily included. Here it has not been considered for simplicity.

  2. 2.

    If the dependence of the problem on the azimuth coordinate is taken into account, a similar term would arise also in azimuth. However, the related decorrelation is normally lower than the range one, thus it is considered negligible.

Notes

Acknowledgements

This review originates from the workshop “Space-based Measurement of Forest Properties for Carbon Cycle Research” held at the International Space Science Institute in Bern in November 2017. We thank P. Biber and M. Heym (Technische Universität München, TUM, Munich, Germany), and A. Huth and R. Fischer (Helmholtz Zentrum für Umweltforschung, UFZ, Leipzig, Germany) for inputs and discussions on forest structure and structure metrics. Part of this work was supported by the Helmholtz Alliance Remote Sensing and Earth System Dynamics funded by the Initiative and networking Fund of the Helmholtz Association. Laser Vegetation and Ice Sensor (LVIS) data sets were provided by the team in the Laser Remote Sensing Branch at NASA Goddard Space Flight Center with support from the University of Maryland, College Park. Finally, we thank the two anonymous reviewers that helped to improve the manuscript with their constructive comments.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Matteo Pardini
    • 1
    Email author
  • John Armston
    • 2
  • Wenlu Qi
    • 2
  • Seung Kuk Lee
    • 2
    • 3
  • Marivi Tello
    • 1
  • Victor Cazcarra Bes
    • 1
  • Changhyun Choi
    • 1
  • Konstantinos P. Papathanassiou
    • 1
  • Ralph O. Dubayah
    • 2
  • Lola E. Fatoyinbo
    • 3
  1. 1.Microwave and Radar InstituteGerman Aerospace CenterWesslingGermany
  2. 2.Department of Geographical SciencesUniversity of MarylandCollege ParkUSA
  3. 3.Biospheric Sciences LaboratoryNASA Goddard Space Flight CenterGreenbeltUSA

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