Gradient-Based Automatic Lookup Table Generator for Radiative Transfer Models

Abstract

Physically based radiative transfer models (RTMs) are widely used in Earth observation to understand the radiation processes occurring on the Earth’s surface and their interactions with water, vegetation, and atmosphere. Through continuous improvements, RTMs have increased in accuracy and representativity of complex scenes at expenses of an increase in complexity and computation time, making them impractical in various remote sensing applications. To overcome this limitation, the common practice is to precompute large lookup tables (LUTs) for their later interpolation. To further reduce the RTM computation burden and the error in LUT interpolation, we have developed a method to automatically select the minimum and optimal set of input–output points (nodes) to be included in an LUT. We present the gradient-based automatic LUT generator algorithm (GALGA), which relies on the notion of an acquisition function that incorporates: 1) the Jacobian evaluation of an RTM and 2) the information about the multivariate distribution of the current nodes. We illustrate the capabilities of GALGA in the automatic construction and optimization of MODTRAN-based LUTs of different dimensions of the input variables space. Our results indicate that when compared with a pseudorandom homogeneous distribution of the LUT nodes, GALGA reduces:1) the LUT size by >24%; 2) the computation time by 27%; and 3) the maximum interpolation relative errors by at least 10%. It is concluded that an automatic LUT design might benefit from the methodology proposed in GALGA to reduce interpolation errors and computation time in computationally expensive RTMs.

I. Introduction

SINCE the advent of optical remote sensing, physically based radiative transfer models (RTMs) have deeply helped in understanding the radiation processes occurring on the Earth’s surface and their interactions with water, vegetation, and atmosphere [1]–[3]. RTMs are physically based computer models that describe scattering, absorption, and emission processes in the ultraviolet to microwave region. Continuous improvement in the accuracy of these models has diversified them from simple turbid medium toward advanced RTMs that allow for explicit 3-D representations of complex media. A few examples of these advanced RTMs are: HydroLight [1], DART [4], Raytran [5], SCOPE [6], and MODTRAN [7]. This evolution has resulted in an increase of complexity, intepretability, and computational requirements to run the model on a pixel-per-pixel basis, which bears implications toward practical applications, such as: 1) numerical inversion of atmospheric and vegetation properties from remotely sensed data [8], [9]; 2) sensitivity analysis of RTM outputs by each input variable [10], [11]; and 3) generation of artificial scenes as would be observed by a sensor [12], [13]. To overcome this limitation, large lookup tables (LUTs) are precomputed for their later interpolation [14], [15]. However, little information is available in the remote sensing scientific literature about the criteria that should be adopted to design these RTM-based LUTs and about the errors derived from their interpolation. In addition, the computation of these LUTs is still time-consuming, requiring techniques of parallelization and execution in computer grids [16], [17].

In order to further reduce the RTM computation time, a possible strategy is used to select the minimum and optimal set of points (nodes and anchors) to be included in an LUT that reduce the error in its interpolation. This problem is known as an experimental optimal design [18], [19] of interpolators of arbitrary functions f, and it aims at reducing the number of direct evaluations of f (RTM runs in the context of the LUT design). A possible approach is to construct an approximation of f starting with a set of initial points. This approximation is then sequentially improved, incorporating new points given a suitable selection rule until a certain stop condition is satisfied. Another interesting alternative approach is based on adaptive gridding, which aims to construct a partitioning of the input variable space, X, into cells of equal size, where the cell edges have different lengths depending on their spatial direction [20]. In order to find such lengths, the adaptive gridding method uses a Gaussian process (GP) model with an automatic relevant determination kernel so that the performance is similar to pseudorandom uniform approaches [21]. A clear problem of such an approach is that the number of hyperparameters to be estimated increases as the input dimension grows. The topic of the experimental optimal design has received attention from (apparently unrelated) research areas, such as optimal nonuniform sampling, quantization and interpolation of continuous signals [22], Bayesian optimization (BO) [23], [24], and active learning [25]. However, some of these techniques are not sequential procedures since they try to optimize the locations of a fixed number of nodes.

The main objective of this paper is to present a simpler method for the automatic generation of RTM-based LUTs through the evaluation of the underlying function gradient. As a proof of concept, the proposed method is applied here to the construction of an MODTRAN-based atmospheric LUT with the ultimate goal of reducing errors in its interpolation. Our method is sequential and automatically builds the LUT based on the notion of the acquisition function, similar to the BO approach. Such an acquisition function acts as a sort of oracle that tells us about the most informative regions of the input space to sample. Essentially, starting from a set of initial points, the LUT is automatically built with the addition of new nodes, maximizing the acquisition function at each iteration. Unlike in BO, our goal is not the optimization of the minimum/maximum of the unknown underlying function f but its accurate approximation $\widehat{f}$ through minimization of its interpolation error δ. Thus, the experimental optimal design problem is converted into a sequential optimization problem of the acquisition function, regardless of the dimensionality of the input space.

The remainder of this paper is structured as follows. Section II details the implemented gradient-based automatic LUT generator algorithm (GALGA). Section III describes the experimental simulation setup including the methodology to evaluate the performance of the proposed algorithm. Section IV shows the functioning of the algorithm and its performance for LUTs of different dimensionalities. Finally, in Section V, we conclude this paper with a discussion of the proposed method in the context of Earth observation applications and an outlook of future research lines.

II. Gradient-Based Automatic Lut Generator

This section describes the developed GALGA. We start in Section II-A by giving a schematic overview of the proposed algorithm and the employed notation. We then detail in Sections II-B–II-D the specificities of the algorithm through the implemented interpolation and the concepts of the acquisition function and the stop condition.

A. Method Overview

The basic component of GALGA is the acquisition function based on the geometric and density terms and was originally introduced in [21], [26], and [27]. See Fig. 1 for an illustrative processing scheme of the method. Notationally, let us consider a D-dimensional input space X, i.e., x ∈ X ⊂ ℝ^D in which a costly K-dimensional object function f (x; λ) = [f(x; λ₁), …, f(x; λ_K): X ↦ ℝK is evaluated. In the context of this paper, X comprises the input space of biogeophysical and geometric variables [e.g., leaf area index, aerosol optical thickness (AOT), and viewing zenith angle (VZA)] that control the behavior of the function f (x; λ), i.e., an RTM. Here, λ represents the wavelengths in the K -dimensional output space. For the sake of simplicity, this wavelength dependence is omitted in the formulation in this paper, f (x; λ) ≡ f(x). Given a set of input variables in the matrix X_i = [x₁, …, x_mi] of dimension D × m_i, we have a matrix of K -dimensional outputs Y_i = [y₁, …, y_mi], being y _j = f(x _j) for j ∈[1, m_i]. At each iteration i ∈ ℕ⁺, GALGA first performs an interpolation, ${\widehat{y}}_i\equiv {\widehat{f}}_i(x|X_i,Y_i)$, of the function f (x). Second, the algorithm follows with an acquisition step that creates/updates the acquisition function, A_i (x), and increases the number of LUT nodes from [X_i;Y_i] to X_i+1 = [x₁, …, x_mi+1 and Y_i+1 = [y₁, …, y_mi+1. This two-step procedure is repeated until a suitable stopping condition is met based on the difference between f(x) and ${\widehat{f}}_i(x)$.

Fig. 1. Schematic representation of GALGA’s processing chain.

The algorithm starts (i=0) by choosing N₀ 5.2^D pseudo-random nodes based on a Latin hypercube sampling [28] of the input variable space. The choice of N₀ is a compromise between avoiding unnecessary RTM computations (small N₀) and having a reasonable good initial sampling for the calculation of the underlying gradient (large N₀). However, even if N₀ is too small, GALGA will be able to construct a suitable acquisition function encoding the gradient information after a certain number of iterations. This initial set of LUT nodes is complemented with the addition of all the 2^D vertex of the input variable space (where the input variables get the minimum/maximum values). With this set of m₀ = N₀ + 2^D nodes, we ensure to have an initial homogeneous and bounded distribution of the input variable space so that no extrapolations are performed.

B. Interpolation Method

GALGA relies on the use of an interpolation method $\widehat{f}(x)$ in order to provide an approximation of the underlying function f(x) within X. In our previous work [21], [26], [27], we considered a GP interpolator [29], widely used in various remote sensing applications [30]. Interpolation in GPs is trivially implemented by considering that there is no noise in the observed signal, and hence, only the kernel hyperparameters need to be learned. However, the use of GP for multioutput functions (i.e., K > 1) is not straighforward, which most of the times requires conducting first a dimensionality reduction [31] followed by individual GPs for each projection. Not only the model complexity increases but also the risk of falling in local minima increases because of the problems of learning hyperparameters in multiple GP models. In GALGA, we use, instead, a multidimensional piecewise linear interpolation, commonly applied in remote sensing applications [14], [32]. The implementation of the linear interpolation is based on the Quickhull algorithm [33] for triangulations in multidimensional input spaces. For the scattered LUT input data in X_i, the linear interpolation method is reduced to find the corresponding Delaunay simplex [34] (e.g., a triangle when D = 2) that encloses a query D-dimensional point x_q[see (1) and Fig. 2]

$${\widehat{f}}_i\left(x_q\right)={\displaystyle \sum _{j=1}^{D+1}{\omega }_{j}f(x_j)}$$ (1)

where ω_j are the (scalar) barycentric coordinates of x_q with respect to the D-dimensional simplex (with D + 1 vertices) [35].

Fig. 2. Schematic representation of a 2-D interpolation of a query point x_q (white *) after Delaunay triangulation (solid lines) of the scattered LUT nodes X_i (*).

Since f (x) is a K-dimensional function, the result of the interpolation is also K -dimensional. The Delaunay triangulation, in turn, provides the partitions of the input space in simplices. The use of these simplices will help us to define the acquisition function (see Section II-D).

C. Stop Condition

The purpose of the stop condition is to end the iterative process of the algorithm when a suitable condition in the LUT data is met. In the proposed algorithm, the stop condition is based on the evaluation of the interpolation error through the error metric δ_i (x)

$${\delta }_i({\overline{X}}_i)=\underset{\lambda }{\max }\left(100\cdot \left|\frac{{\widehat{f}}_i({\overline{X}}_i)-f({\overline{X}}_i)}{f({\overline{X}}_i)}\right|\right)$$ (2)

where ${\overline{X}}_i$ is a subset of X_i that comprises all the LUT nodes at the i th iteration with the exception of the 2^D vertex of the input variable space. The error metric, therefore, evaluates the interpolation relative error over each node in the subset ${\overline{X}}_i$ by using the leave-one-out cross-validation technique (see the green * in Fig. 3) [36]. Among all the spectral channels ( λ), this error metric takes the most critical spectral channel (max_λ).

Fig. 3.

Schematic process for the calculation of δ_i in (2) within the ${\overline{X}}_i$ subset (colored *). Notice that how the “leave-one-out” cross-validation technique modifies the Delaunay simplices with respect to the complete ${\overline{X}}_i$ subset in Fig. 2.

The iterative process finishes when the 95% percentile of ${\delta }_i({\overline{X}}_i)$ is below an error threshold, ε_t.

By taking the spectral channel under which the interpolation relative error is maximum, the stop condition ensures that all the spectral channels will have an interpolation error lower than ε_t. In this way, GALGA will be valid for (and independent of) all remote sensing applications. With respect to the error threshold, this can be user-defined according to some precalculated conditions as, e.g., a factor 10 over an instrument absolute radiometric accuracy.

It should be noted that the leave-one-out cross-validation technique does not provide the “true” error of the interpolation over all the input spaces X but an approximation. Since the cross-validation technique leaves some LUT nodes out, it is expected that the calculated interpolation relative error in (2) will be higher than the “true” error. However, as the LUT nodes are also used to determine the interpolation error, using this cross-validation technique allows us to avoid generating an external (i.e., not included in the final LUT) validation data set.

D. Acquisition Function

The acquisition function, A_i (x), is the core of the proposed algorithm since it allows determining the new LUT nodes to be added at each iteration. This function incorporates: 1) the geometric information of the unknown function f through the evaluation of its gradient and (b) the density information about the distribution of the current nodes. Indeed, areas of high variability of f(x) require the addition of more LUT nodes, and the areas with a small concentration of nodes require the introduction of new inputs. Accordingly, we define the acquisition function conceptually in (3) as the product of two functions: a geometric term G_i (x) and a density term D_i (x)

$$A_i(x)=G_i(x{)}^{{\beta }_i}{D}_i(x{)}^{1-{\beta }_i}$$ (3)

where β_i is a discrete function that alternates the acquisition function between the geometry and density terms for every T=3 iterations

$${\beta }_i={\beta }_{i+T}=\{\begin{array}{ll}1\hfill & \text{ if }i\le T-1\hfill \\ 0\hfill & \text{ if }i=T.\hfill \end{array}$$ (4)

The geometric term G_i (x) is based on the calculation of the gradient of the underlying function f. However, since f is unknown in all the input variable space X, the gradient can only be approximated and calculated at the current LUT nodes X_i. Therefore, G_i (x) is calculated according to the following steps, as shown in Fig. 4.

Fig. 4.

Schematic representation for the calculation of G_i (x). Gradients are calculated between the nodes x_k (*) and the selected LUT node x_j (green *). A new LUT node (white o) is added at the barycenter of the Delaunay simplex with the highest average gradient (shaded in dark gray).

• 1)Among the LUT nodes in X_i = [x₁, … , x_mi], we select only those m_g,_i nodes whose interpolation error δ_i(xj) [see (2)] is higher than the error threshold ε_t. By choosing this subset, the new LUT nodes will only be added in areas with high interpolation error.
• 2)
  The gradient, Δ_kf_i (xj), is calculated according to (5) between the current node xj (* in Fig. 4) and the N_k remaining nodes (x_k with k ∈[1, N_k]) of the Delaunay simplices for which xj is a vertex (* in Fig. 4)

  $${\nabla }_k{f}_i{\left(x_j\right)}_{{\lambda }_{\max }}=|y_j-y_k{|}_{{\lambda }_{\max }}.$$ (5)

  The subindex λ_max indicates that, out of the K-dimensional output values in y, only the most critical spectral channel (see Section II-C) is used to calculate the gradient.

• 3)
  For each Delaunay simplex (l) containing x_j as vertex, we calculate the root-mean-square of the corresponding D gradients in the previous step according to the following equation:

  $$g_l=\sqrt{\frac{1}{D}\underset{n=n_1}{\overset{n_D}{{{\displaystyle \sum }}^{\text{}}}}{({\nabla }_n{f}_i{(x_j)}_{{\lambda }_{\max }})}^2}$$ (6)

  where the index n (from n1 to nD) identifies the D nodes, among x_l, that conform a Delaunay simplex together with x_j (see n1 and n2 tagged nodes in Fig. 4).
• 4)The gradient term finally adds a new LUT node at the barycenter of the Delaunay simplex with a higher value of gl.

Following the previous steps, G_i (x) will place a new node in the vicinity of each current LUT node in X_i with an interpolation error higher than ε_t in the direction of the highest gradient. Therefore, the LUT size will increase from m_i nodes to m_i+1 = m_i + m_g,i nodes.

Since the gradient term is based on the existing LUT nodes (X_i), the computed interpolation errors and gradients might not be representative in empty areas of the input variable space, particularly in those with low density of nodes. Thus, the acquisition function includes a density term, D_i (x), which aims at proofing these lower sampled areas at every T iterations [see (3) and (4)]. The density term identifies these poorly sampled areas by calculating the volume of each Delaunay simplex according to [37]

$$V=\frac{1}{D!}\det ([x_{n_2}-x_1],\dots ,[x_{n_{D+1}}-x_{n_1}])$$ (7)

where the indices n₁–n_D+1 identify the D + 1 nodes that conform each D-dimensional Delaunay simplex. The density term will then place a new LUT node in the barycenter of the m_d,i = 5 · 2^D simplices with higher volume. Therefore, the LUT size will increase from m_i nodes to m_i+1 = m_i + m_d,i nodes.

III. Experimental Setup And Analysis

In order to analyze the functioning and performance of the proposed algorithm, we run three simulation test cases for the optimization of an MODTRAN5-based atmospheric LUT. MODTRAN5 is one of the most widely used atmospheric RTMs due to its accurate simulation of the coupled absorption and scattering effects [38], [39]. Atmospheric LUTs are also commonly used in physically based atmospheric correction algorithms [40]–[42]. Following the notation in Section II, the underlying function f consists of top-of-atmosphere (TOA) radiance spectra, calculated at a solar zenith angle (SZA), θ_il, and for a Lambertian surface reflectance,¹ ρ, according to

$$L=L_0+\frac{\left(T_{\text{dir}}+T_{\text{dif}}\right)\left(E_{\text{dir}}\cos {\theta }_{il}+E_{\text{dif}}\right)\rho }{\pi (1-S\rho )}$$ (8)

where L0 is the path radiance, T_dir/dif are the target-to-sensor direct/diffuse transmittances, E_dir/dif are the direct/diffuse at-surface solar irradiances, and S is the spherical albedo. These terms are often called atmospheric transfer functions and are obtained using the MODTRAN5 interrogation technique developed in [15]. Unless otherwise specified, all simulations are carried out for a near nadir-viewing satellite sensor (VZA = 0.5°), target at 0-km altitude, rural aerosols, and midlatitude summer atmosphere.

The three simulation test cases consist of LUTs of increasing dimensionality of the input space, i.e., D = [2; 4; 6], in the wavelength range 400–550 nm at 15-cm⁻¹ spectral sampling (≈0.4 nm). The input variables (see Table I) range typical variability in the AOT, the Ångström exponent (α), the Henyey–Greenstein asymmetry parameter (gHG), and the single scattering albedo [44]–[46]. Despite their low number of input dimensions and short spectral range, the selection of input variables and a spectral range are realistic for applications in aerosol characterization and atmospheric correction. In addition, the selection of the interpolation error thresholds is a factor 1–10 higher than the typical absolute radiometric calibration for satellite optical instruments (e.g., OLCI/ Sentinel-3 [47]). Their choice as a function of the number of dimensions limits MODTRAN simulations to a reasonable computation time.

Table I. Input Variables and Spectral Configuration for the Visualization Test Scenario.

Case	Input variables (range)	Error threshold, εt (%)
#1	AOT (0.05-0.4) SZA (20-70°)	0.2
#2	As in Case #1 plus…     α (1-2)    gHG (0.60-0.99)	1
#3	As in Case #2 plus…     SSA (0.85-0.99)    VZA (0.5-20°)	2

We start the analysis of the data by visualizing the functioning of the algorithm in terms of: 1) the evaluation of the stop condition through cross-validation error and 2) the distribution of new nodes according to G_i and D_i. To do so, we exploit the 2-D data in Case #1, showing the cross-validation and true error maps. These two maps are shown at two consecutive iterations, which correspond to the actuation of each term (geometry and density) of the acquisition function. On the one hand, the cross-validation error maps are based on δ_i [see (2)], calculated through the leave-one-out cross validation of each subset ${\overline{X}}_i$ as introduced in Section II-C. To create a bidimensional map, the scattered values of ${\delta }_i({\overline{X}}_i)$ are linearly interpolated over a grid of 100 × 100 linearly spaced values of the input variables. Since this cross-validation method reduces locally the LUT nodes density (thus the name leave-one-out), the resulting error maps should not be understood as an estimation of the underlying LUT interpolation errors. Instead, the purpose of the cross-validation error maps is to illustrate the distribution and magnitude of the cross-validation errors, which are the ones used to determine the distribution of new LUT nodes. overlapped with these error maps, the current LUT nodes X_i and their Delaunay triangulation are shown together with the nodes added at the iteration i + 1. On the other hand, the true error maps correspond to δ_i calculated over a grid of 100 × 100 linearly spaced values of the input variables where TOA radiance spectra are precalculated. Namely, this thin grid represents the true value of f (x).

We continue the analysis of the data by assessing the performance of the proposed algorithm in the test cases #1, #2, and #3. For each test case, we calculate the 95% percentiles (P₉₅) of δ_i obtained from the cross-validation subset ${\overline{X}}_i$. Since the initial node distribution in our algorithm is pseudorandom, we calculate the mean and standard deviation of P₉₅ in the cross-validation subset after 10 independent runs. The performance of the proposed algorithm is shown by plotting these statistics against the number of LUT nodes m_i, fitted by a double exponential function. For comparison, we also show the performance obtained after a homogeneous pseudorandom node distribution following Sobol’s sequence [48]. The performance assessment through the evaluation of the true interpolation error is not considered here, given the large number of ground truth nodes (~100^D) needed to have a representative discrete sampling of the underlying TOA radiance in the input variable space.

The LUT computation time and gain in LUT size to achieve the selected interpolation error threshold will also be reported. These calculations are performed in an i7-4710MQ CPU at 2.5 GHz with 16 GB of RAM and 64-bit operating system.

IV. Results

First, we visualize the functioning of GALGA through the 2-D error maps from the test case #1 (see Figs. 5 and 6). For the actuation of the geometry term (iteration i = 5), the new nodes are added in areas where the interpolation error is estimated to be higher than the ε_t = 0.2% error threshold (see cross-validation error map in Fig. 5). Most of these nodes are located in the areas of low TOA radiance (i.e., at SZA > 60°), thus where higher relative interpolation errors are expected. The addition of these new nodes reduce the areas with errors above the threshold as observed in the change of the true error map between iterations i = 5 and i = 6 (see Fig. 6). This indicates that the method is functioning correctly under the geometry term.

Fig. 5.

(Top) Cross-validation and (Bottom) true error maps for the Case #1 test at iteration i = 5, illustrating the functioning of the geometry term of the acquisition function. The light blue lines indicate the underlying Delaunay triangulation.

Fig. 6.

(Top) Cross-validation and (Bottom) true error maps for the Case #1 test at iteration i = 6, illustrating the functioning of the density term of the acquisition function. The light blue lines indicate the underlying Delaunay triangulation.

Since GALGA approximates the interpolation error based on the leave-one-out cross-validation technique, we can also observe that the cross-validation error map has systematically higher error values than the true error map. Consequently, GALGA leads to an oversampling or undersampling of some areas of the input variable space. On the one hand, some areas have a true interpolation error at i = 5 that is already below the error threshold (e.g., SZA ≈ 45° and AOT = 0.2–0.25). However, GALGA adds new nodes, leading to a local oversampling of the input space (see Fig. 5). On the other hand, undersampled areas (e.g., at SZA ≈ 52° and AOT ≈ 0.4) still remain with a high interpolation error [see Fig. 6 (bottom)]. The density term of the acquisition function intends to reduce the amount of undersampled areas. Indeed, at iteration i = 6, the new added nodes are located in the barycenter of the simplices with the largest (undersampled) areas. Additionally, for this particular case, we can observe a pattern of low interpolation errors connecting the nodes with similar SZA but different AOTs (see dark red vertical pattern in the true error map). This indicates that linear interpolation derives larger errors when interpolating between SZA values than between AOT values.

We continue by assessing the performance of the proposed method against a Sobol pseudorandom homogeneous distribution of LUT nodes. The analysis is done for LUTs of increasing input dimensions (2-D, 4-D, and 6-D), and the results are shown in Fig. 7. When evaluating the algorithm performance, we observe that GALGA outperforms the Sobol pseudorandom distribution in terms of LUT size, computation time, and interpolation error, and this for any dimensionality of the input variable space (see Table II). We also observe that GALGA needs a minimum set of LUT nodes to start outperforming homogeneous pseudorandom distributions. For a given dimensionality of the input space, higher values of the error threshold would give nearly the same accuracy as the Sobol distribution.

Fig. 7.

Performance of GALGA (in blue lines) and Sobol distribution (in red lines) for (Top) 2-D, (Middle) 4-D, and (Bottom) 6-D test cases estimated through cross-validation error. Mean (solid lines) and standard deviation (shaded areas) are obtained after averaging N = 10 independent runs. The error thresholds, ε_t, are indicated with the black horizontal dashed lines in each subplot.

Table II. Comparison Between GALGA and Sobol Methods in Terms of LUT Size Reduction, Computation Time, and Error Reduction.

	Size reduction	Time (min)		Error reduction
		GALGA	Sobol
2 D	33%	40	60	40%
4 D	26%	160	220	15%
6 D	24%	520	700	10%

V. Conclusion

in this paper, we have proposed GALGA, a new method to optimize the node distribution of RTM-based LUTs. The construction of LUTs is particularly useful in remote sensing problems with complex dependences of the input–output biogeophysical and radiometric variables, such as SCOPE [6]. LUTs are also useful to replace pixel-by-pixel execution of computationally expensive RTMs [1], [7]. The proposed method is based on the exploitation of the gradient/Jacobian information of the underlying function and the concept of an acquisition function, divided into its geometry and density terms. The proposed algorithm was applied here to the construction of MODTRAN LUTs of atmospheric transfer functions in order to reduce: 1) the errors in atmospheric correction and 2) the computation time to build these LUTs. Through the experimental setup, we have verified that GALGA functions as expected, observing that the use of the acquisition function identifies the areas in the input variable space with high interpolation errors. In this experiment, GALGA reduced the number of LUT nodes from 33% (2-D) to 24% (6-D) when compared against a pseudorandom homogeneous distribution. The performance of GALGA was also evaluated by calculating the cross-validation interpolation error in atmospheric LUTs of 2-D, 4-D, and 6-D. The LUTs constructed with the proposed method achieve an interpolation relative error that is at least ~10% lower than the obtained with an LUT of homogeneously distributed nodes of the same size. It was observed that the gain in LUT-size reduction and interpolation accuracy using GALGA is lower at higher dimensions of the input space due to the so-called “curse of dimensionality” and the insufficient sample representativeness in medium-/high-dimensional problems [49]. We also observed that GALGA needs a minimum set of LUT nodes in order to have a good approximation of the underlying Jacobian and thus to perform better than the homogeneous pseudorandom distribution.

GALGA has been implemented in the ALG v1.2 software [50] (downloaded at http://ipl.uv.es/artmo/index.php/tools/80-tools/24-alg). ALG allows generating LUTs based on a suite of atmospheric RTMs for any input configuration. In combination with ALG, GALGA facilitates users generating optimized LUTs, reducing computation time in the execution of atmospheric RTMs and improving the accuracy of LUT interpolation. The proposed algorithm can eventually be implemented for the generation of LUTs in a wider range of remote sensing applications, including vegetation and water RTMs [1], [3]. Compact and informative LUTs give rise to interesting possibilities, such as optimization of biophysical parameters retrieval algorithms [51], atmospheric correction [15], RTM emulation [11], [31], and satellite mission design [52].

Future research will focus on the use of statistical methods to improve the reconstruction of the underlying interpolation error, which have been demonstrated to be suitable for both canopy and atmospheric RTMs [53]. Altogether, we are aiming at further optimizing the distribution of LUT nodes and reducing the errors in LUT interpolation.

Biographies

Jorge Vicent Servera received the B.Sc. degree in physics from the University of Valencia, Valencia, Spain, in 2008, the M.Sc. degree in physics from the École Polytechnique Fédérale de Lausanne, Lau-sanne, Switzerland, in 2010, and the Ph.D. degree in remote sensing from the University of Valencia in 2016.

Since 2017, he has been with the Earth Observation Department, Magellium, Toulouse, France, as a R&D Engineer. He is currently involved in developing the level-2 processing chain for ESA’s FLEX Mission. His research interests include the modeling of Earth observation satellites, system engineering, radiative transfer modeling, atmospheric correction, and hyperspectral data analysis.

Luis Alonso received the B.Sc. degree in physics and the M.S. degree in environmental physics from the University of Valencia, Valencia, Spain, in 1999 and 2002, respectively, with a focus on the geometric correction of airborne and spaceborne remote sensing imagery.

He is currently a member of the Laboratory for Earth Observation, Image Processing Laboratory, University of Valencia, where he studies the remote sensing of chlorophyll fluorescence at canopy level. He has participated in several scientific studies for the European Space Agency’s Fluorescence Explorer Mission as well as for Copernicus Sentinel-2 and Sentinel-3.

Luca Martino was born in Palermo, Italy, in 1980. He received the M.Sc. degree in electronic engineering from the Politecnico di Milano, Milan, Italy, and the Ph.D. degree in statistical signal processing from the Universidad Carlos III de Madrid, Madrid, Spain, in 2011.

He spent two years with the Department of Statistics, University of Helsinki, Helsinki, Finland. He carried out a Post-Doctoral Research at the São Paulo Research Foundation (FAPESP), São Paulo, Brazil, and the University of Valencia, Valencia, Spain.

Neus Sabater received the B.Sc. degree in physics, the M.Sc. degree in remote sensing, and the Ph.D. degree in remote sensing from the University of Valencia, Valencia, Spain, in 2010, 2013, and 2018, respectively.

Since 2012, she has been with the Laboratory for Earth Observation, Image Processing Laboratory, University of Valencia, as a Research Technician. Her main activities during this period were related to the development of the preparatory activities of the FLEX Mission. Her research interests include atmospheric correction, atmospheric radiative transfer, meteorology, and hyperspectral remote sensing.

Dr. Sabater received the Award of the University of Valencia to the best student records for her M.Sc. degree from 2012 to 2013. In 2013, she received a Ph.D. Scholarship from the Spanish Ministry of Economy and Competitiveness associated with the Ingenio/Seosat Spanish Space Mission.

Jochem Verrelst received the M.Sc. degrees in tropical land use and in geoinformation science and the Ph.D. degree in remote sensing from Wageningen University, Wageningen, The Netherlands, in 2005 and 2010, respectively. His dissertation focused on the spaceborne spectrodirectional estimation of forest properties.

Since 2010, he has been involved in the preparatory activities of FLEX Misson. He is the Founder of the ARTMO software package. His research interests include the retrieval of vegetation properties using airborne and satellite data, canopy radiative transfer modeling and emulation, and hyperspectral data analysis.

Dr. Verrelst received the H2020 ERC Starting Grant (755617) in 2017 to work on the development of vegetation products based on synergy of FLEX and Sentinel-3 data.

Gustau Camps-Valls (M’04–SM’07–F’18) received the Ph.D. degree in physics from the University of Valencia, Valencia, Spain, in 2002.

He is currently a Full Professor in electrical engineering and a Coordinator of the Image and Signal Processing Group, University of Valencia. He has published more than 150 journal papers, 200 conference papers, and 4 books on machine learning, remote sensing, and signal processing. He research interests include the development of machine learning algorithms for geoscience and remote sensing data analysis.

Dr. Camps-Valls received the prestigious ERC Consolidator Grant in 2015 to advance in statistical inference for Earth observation data analysis. He is an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, the IEEE SIGNAL PROCESSING LETTERS, and the IEEE GEOSCIENCE AND REMOTE SENSING LETTERS. He entered the list of highly cited researchers by Thomson Reuters in 2011 and holds an h-index of 55. Visit http://isp.uv.es for more information.

José Moreno (M’18) is currently with the Department of Earth Physics and Thermodynamics, Faculty of Physics, University of Valencia, Valencia, Spain, as a Professor of earth physics, teaching and working on different projects related to remote sensing and space research as responsible for the Laboratory for Earth Observation (LEO). He is also the Director of LEO, Image Processing Laboratory/Scientific Park. He has been involved in many international projects and research networks, including the preparatory activities and exploitation programs of several satel-lite missions, such as ENVISAT, CHRIS/PROBA, GMES/Sentinels, and SEOSAT and the Fluorescence Explorer, ESA’s 8th Earth Explorer Mission. His research interests include the modeling and monitoring of land surface processes by using remote sensing techniques.

Dr. Moreno was a member of the ESA Earth Sciences Advisory Committee from 1998 to 2002, the Space Station Users Panel, and other international advisory committees. He has served as an Associate Editor for the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING from 1994 to 2000.