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A Wavelet-Enhanced Inversion Method for Water Quality Retrieval From High Spectral Resolution Data for Complex Waters Eva M. Ampe, Dries Raymaekers, Erin L. Hestir, Maarten Jansen, Els Knaeps, and Okke Batelaan
Abstract—Optical remote sensing in complex waters is challenging because the optically active constituents may vary independently and have a combined and interacting influence on the remote sensing signal. Additionally, the remote sensing signal is influenced by noise and spectral contamination by confounding factors, resulting in ill-posedness and ill-conditionedness in the inversion of the model. There is a need for inversion methods that are less sensitive to these changing or shifting spectral features. We propose WaveIN, a wavelet-enhanced inversion method, specifically designed for complex waters. It integrates wavelettransformed high-spectral resolution reflectance spectra in a multiscale analysis tool. Wavelets are less sensitive to a bias in the spectra and can avoid the changing or shifting spectral features by selecting specific wavelet scales. This paper applied WaveIN to simulated reflectance spectra for the Scheldt River. We tested different scenarios, where we added specific noise or confounding factors, specifically uncorrelated noise, contamination due to spectral mixing, a different sun zenith angle, and specific inherent optical property (SIOP) variation. WaveIN improved the constituent estimation in case of the reference scenario, contamination due to spectral mixing, and a different sun zenith angle. WaveIN could reduce, but not overcome, the influence of variation in SIOPs. Furthermore, it is sensitive to wavelet edge effects. In addition, it still requires in situ data for the wavelet scale selection. Future research should therefore improve the wavelet scale selection. Index Terms—Chlorophyll-a, continuous wavelet transforms, dissolved organic matter, hyperspectral remote sensing, multiscale, optically complex waters, suspended matter.
I. I NTRODUCTION
O
PTICALLY active constituents, such as chlorophyll (CHL), total suspended matter (TSM), and colored dissolved organic matter (CDOM), influence the underwater light
Manuscript received January 24, 2014; revised April 22, 2014; accepted June 5, 2014. This work was supported in part by the HOA-HYPERENV Project of the Vrije Universiteit Brussel, by the STEREO II program of the Belgian Science Policy Office in the framework of the MICAS project, and by the CSIRO’s Division of Land and Water, and Water for a Healthy Country Flagship. E. M. Ampe is with the Department of Hydrology and Hydraulic Engineering, VUB, 1050 Brussels, Belgium and also with VITO, Flemish Institute for Technological Research, 2400 Mol, Belgium (e-mail:
[email protected]). D. Raymaekers and E. Knaeps are with VITO, Flemish Institute for Technological Research, 2400 Mol, Belgium. E. L. Hestir is with CSIRO, Land and Water, Canberra ACT 2601, Australia. M. Jansen is with the Departments of Mathematics and Computer Science, Université Libre de Bruxelles, 1050 Brussels, Belgium. O. Batelaan is with the Department of Hydrology and Hydraulic Engineering, VUB, 1050 Brussels and also with the School of the Environment, Flinders University, Adelaide, Australia. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2014.2330251
field through complex optical interactions of light absorption or scattering. Light attenuates through the water column as a result of all absorbing and scattering compounds in the water column [1]. These interactions are expressed by the inherent optical properties (IOPs) of a specific water body at a specific time and are independent of the ambient light field [2]. The amount of upwelling light is linked to IOPs and is determined by the concentration and composition of the optically active constituents. Ground-based, airborne, or spaceborne spectroradiometers measure the ambient light field as water apparent optical properties (AOPs), such as reflectance. AOPs depend both on the IOPs (the medium) and on the directional structure of the ambient light field [1]. These AOPs still show sufficient regular features and stability and contain valuable information regarding the water body [1]. There are a wide range of semiempirical and semianalytical methods to estimate optically active constituents from reflectance spectra [3], [4]. Both approaches link the measured AOPs to the IOPs. Semiempirical techniques are based on a statistical relationship between the water quality variable of interest and the reflectance spectrum. Such approaches are generally easy to implement with existing in situ data and often produce reliable results for the areas and data sets, from which they are derived [3], [4]. Semianalytical approaches can retrieve multiple water quality properties simultaneously from a single reflectance spectrum [5]. These methods often use specific IOPs (SIOPs, i.e., the IOPs per unit concentration). By normalizing the IOPs for their concentration, one can describe the spectral shape and amplitude of the SIOPs and use this information in an inverse method to estimate the constituent concentration. The methods rely on a range of inversion techniques, including lookup tables (LUTs) of complete radiative transfer models (e.g., [6]), neural networks (e.g., [7] and [8]), matrix inversion (e.g., [2] and [9]), and iterative optimization of bio-optical models (e.g., [10]). These inversion methods in remote sensing aim to derive in-water information from the reflectance spectra [11]. A comprehensive list of algorithms is reviewed by Odermatt et al. [4] and references therein. Optically complex waters provide an additional challenge in comparison to ocean waters because the optically active constituents may vary independently [3], [4]. Additionally, a high concentration of one constituent (e.g., TSM) may influence the reflectance spectrum in such a way that it impedes the retrieval of other constituents. Furthermore, depending on the water body or the season, the spectral absorption features may vary both in depth and width and the scattering may vary in magnitude and broad shape [2], [12]–[14]. In addition, an
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erroneous signal can strongly impede constituent retrieval from reflectance spectra [15]. The inversion of bio-optical models introduces three main problems: 1) it is often difficult to obtain a global solution; 2) the problem can be ill-posed where there is no uniqueness or stability of solutions [16]; and 3) the problem can be illconditioned, where small perturbations in the data can lead to large changes in the inverse solution [16]. Problems two and three can occur when there is variation in the reflectance spectra or the SIOPs. Reflectance and SIOP measurements can be affected by instrument noise, and measurement or acquisition errors. SIOPs can express variability due to spatial or temporal variation in composition and quantity of water constituents, i.e., phytoplankton assemblages, organic matter source, and phytoplankton and particle size distribution [2], [13], [17]–[19]. Furthermore, constituent retrieval from high-spectral resolution data is affected by the optical mixture of the optically active constituents [12]. We propose to handle the complexity of the high-spectral resolution remote sensing signal of optically complex waters by using a multiscale approach called wavelet analysis. Wavelet analysis can analyze the spectral signal at various scales, where a scale is the width of a spectral feature, roughly corresponding to the inverse of its frequency [12]. It can evaluate narrow spectral features at fine wavelet scales and broad spectral features at coarse wavelet scales. Therefore, it can potentially decouple the superimposed influence of the confounding factors and the optically active constituents, and be less sensitive to specific types of noise and confounding factors [12], [20], [21]. Wavelets have been used in computational math studies [22], [23] to reduce the influence of noise on the inversion. To our knowledge, wavelets have not been used in the inversion of a bio-optical model for optically active constituent retrieval. Ampe et al. [12] illustrated the potential of wavelets as a powerful multiscale spectral feature detection tool by extracting informative spectral wavelet regions. We will extend this technique toward a bio-optical inversion method for water quality retrieval based on measured SIOPs, and simulated high-spectral resolution reflectance spectra [24], [25]. The objectives of this paper are twofold: 1) To improve bio-optical inversion results in estimating the optically active constituents CHL, TSM, and CDOM. 2) To improve the inversion’s robustness to certain types of noise or confounding factors, specifically contamination due to spectral mixing, sun zenith angle, uncorrelated noise, and SIOP variation.
II. C ONTINUOUS WAVELET T RANSFORM Wavelet analysis is an advanced signal-processing tool that decomposes a signal (i.e., high-spectral resolution data) into localized scales presented as a set of wavelet coefficients [21]. In this paper, we used the continuous wavelet transform (henceforth referred to as “wavelet”) because its output is readily comparable to the original wavebands [20], [26]. All analyses were performed in interactive data language (IDL) 8.0 wavelet toolkit (ITT Visual Information Solutions, Boulder, CO, USA). The wavelet ψa,b (Λ) (Λ = 1, 2, . . . , n, with n the number of wavebands) is constructed by shifting and scaling the mother
wavelet ψ(Λ). This mother wavelet is shifted over different wavebands and scaled by Λ−b 1 (1) ψa,b (Λ) = √ ψ a a where a and b represent the scaling and shifting factor, respectively, both being positive real numbers [21]. Conceptually, it is scaled by stretching ψ(Λ) when it moves from a fine wavelet scale to a coarser wavelet scale. In this paper, we chose the Mexican hat (i.e., second derivative of Gaussian) as the mother wavelet [27]. This decision was based on both the shape of the reflectance spectra and its absorption features [12], [21]. Subsequently, the wavelet ψa,b (Λ) (1) is convolved with the reflectance spectra by +∞ Wf (a, b) = f, ψa,b = f (Λ)ψa,b (Λ)dΛ
(2)
−∞
where f (Λ) is the reflectance signal, and Wf (a, b) are the resulting wavelet coefficients forming a 2-D matrix for the different wavebands and wavelet scales [12], [21], [26]. The resulting wavelet coefficients are a measure of the local correlation between a certain wavelet (for a particular wavelet scale and waveband) and that particular segment of the reflectance signal [12]. This makes wavelet analysis an appropriate tool to capture the change in shape and depth of absorption features, as well as the superimposed overall spectral shape [21]. In literature (e.g., Schmidt and Skidmore [28]), wavelets are often used in wavelet thresholding or related nonlinear smoothing techniques. In contrast to our study, these studies use the discrete or undecimated discrete wavelet transform. Those results are mainly based upon the sparsity property of a wavelet decomposition of a signal. That is, most coefficients in such decomposition are small and thus dominated by the noise. Thresholding removes most of the noise while preserving the few significant coefficients. This holds especially for data that are piecewise smooth, such as images with edges. It is not advised to use these techniques for this current application of uncorrelated noise because this data is not piecewise smooth as it does not show sharp features or edges [29], [30]. Therefore, this paper concentrates on the other important feature of a wavelet decomposition, which is multiscale analysis. III. M ATERIALS AND M ETHODS The materials and methods section describes the IOPs and concentration measurements, and the simulation of reflectance spectra using the measured specific IOPs (SIOPs) as a computer-simulated data set for inversion. It then describes both the standard inversion and our novel inversion method. Finally, it introduces the different simulation scenarios used to investigate the influence of noise and confounding factors on the inversion methods. A. IOP and Concentration Measurements IOP and concentration measurements were made in an optically complex estuary. Our study area is part of River Scheldt, located near the port of Antwerp in Belgium. The Scheldt
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Fig. 1. IOP field stations on the River Scheldt, located between Antwerp and the border between Belgium and the Netherlands.
originates in France and flows into the North Sea (see Fig. 1). It is characterized by strong tidal influence leading to a mixing of river and sea-derived material [31], [32]. The major components of the suspended material are quartz, calcite, clay minerals, and organic matter [31]. From an extensive field campaign in June 2009, we selected three field stations, including concentration and IOP measurements (see Fig. 1). The biomass of phytoplankton was considered correlated to CHL [9]. As TSM consists of algal and nonalgal particles (NAPs), we calculated the NAP concentration as [NAP] = [TSM] − 0.07 · [CHL] (with [NAP] and [TSM] in mg/L and [CHL] in μg/L) [9]. We collected water samples ca. 50 cm below the water surface and stored the samples in dark bottles. These bottles were kept cool on dry ice. Immediately after the field campaign, we filtered the samples and stored them in a freezer for concentration and IOP measurements. We acquired the concentration and IOPs according to the methodology and instrumentology described in [32]. SIOPs are obtained by normalizing the IOPs for their concentration. Due to the limited number of available instruments, we could not obtain a full SIOP set for each station (see Fig. 2). Therefore, we used a subset of the Scheldt SIOP data set that contained only those stations having full SIOP information. Note, we only obtained one usable CDOM sample for these field stations. The resulting data set for subsequent analyses contained four SIOP sets (see Figs. 1 and 2): three SIOP from the field stations (i.e., BA3, BA4, and BA6) and one mean SIOP set. The observed SIOP variability is consistent with but lower than the variability in the North Sea and Western English channel observed by Tilstone et al. [17]. However, they observed a significant geographical difference in both the SIOPs and the SIOP variability [17]. B. Computer-Simulated Input Reflectance Spectra The Ecolight 5 radiative transfer model (Sequoia Scientific, Inc.) was used to simulate a high-spectral resolution data set. Ecolight gives us a controlled environment to simulate data for the desired environmental conditions without the measurement error associated with radiometric measurements [2], [33]. This
Fig. 2. SIOP measurement variability in the River Scheldt for CHL, TSM, NAP, and CDOM. With a∗ph (λ) the specific absorption coefficient of phytoplankton, a∗NAP (λ) the specific absorption coefficient of NAP, a∗CDOM (λ) the specific absorption coefficient of CDOM, and b∗TSM (λ) the specific scattering coefficient of TSM. Black: mean SIOP set, red: BA3 SIOP set, green: BA4 SIOP set and blue: BA6 SIOP set. For CDOM, we obtained only one SIOP set.
provides an appropriate data set for testing the performance of the inversion technique. We used these simulations instead of measured spectra collected via air- or spaceborne remote sensing to be able to test the effects of different confounding factors individually. In Ecolight, we kept the wind speed at 5 m/s and considered no cloud cover. We selected the Petzold phase function and disabled bottom effects and inelastic scattering. From these simulations we extracted both the subsurface irradiance reflectance R(λ, 0−) and remote sensing reflectance Rrs. Henceforth, we will call the simulated R(λ, 0−) the “input reflectance spectra”, except for the scenario that uses Rrs as input reflectance spectra (see Section III-E in the succeeding discussion). We produced three Ecolight runs with many simulations each. Each run was parametrized with specific SIOPs, sun zenith angle, and variation in the concentrations of the three optically active constituents (Table I) to closely represent the biooptical variability we observed in the study site. We selected ten discrete concentration values per constituent, resulting in 1000 concentration combinations. To simulate input reflectance spectra representing the optical properties of the Scheldt River we parametrized Ecolight with the SIOP measurements (i.e., a∗ph , a∗NAP , a∗CDOM , and b∗TSM , see Section III-A). Ecolight run 1 was parametrized with the mean SIOP set and a sun zenith angle of 0◦ (Table I). Ecolight run 2 was like run 1 but with a sun zenith angle of 45◦ . Ecolight run 3 was parametrized with three SIOP sets from the field stations and the mean SIOP set to assess the deviation from the previous (Table I). Consequently, run 3 is a combination of the Ecolight simulations per SIOP set resulting in four times 1000 reflectance spectra. In order to make our study readily applicable to future remote sensing applications, we resampled the input reflectance spectra to the spectral resolution of the airborne prism experiment (APEX) sensor using the ENVI spectral resampling tool (RSI,
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TABLE I D ESCRIPTION OF I NPUT DATA U SED IN E COLIGHT RUN 1 T HROUGH 3
2010). APEX was developed by a Swiss—Belgian consortium on behalf of the European Space Agency. It records highspectral resolution data in around 300 spectral bands, ranging from 380 to 2500 nm [34]. We resampled our input reflectance spectra to 99 bands between 410 and 900 nm. The resulting reflectance spectra can be considered as computer-simulated APEX reflectance spectra and were used as the input reflectance spectra for this study. C. Standard Inversion Method Using a Simplified Bio-Optical Model and Optimization 1) Simplified Bio-Optical Model: To derive the input concentrations of CHL, TSM, and CDOM (aCDOM (440)) for an input reflectance spectrum, we need to invert a forward model. Since it is impractical to invert a full nonlinear radiative transfer model (e.g., Ecolight), common practice is to use simplified bio-optical models. These types of bio-optical models relate R(λ, 0−) or Rrs to IOPs through simplified relations that reduce the model order for easier inversion. For this study, we used the forward-simplified bio-optical model of Albert and Mobley [35]. The model is an approximation of the Ecolight radiative transfer numerical model [36]. It was calibrated for Lake Constance data, but applied to a larger concentration range than observed, which extends the validity of the developed parameterization to a wide range of optically complex waters [10], [36]. Albert and Mobley [35] use the following equation to link the IOPs with the R(λ, 0−): R(λ, 0−) = p1 1 + p2 x + p3 x2 + p4 x3 1 · 1 + p5 (1 + p6 u)x (3) cos θσ with x=
bb (λ) a(λ) + bb (λ)
(4)
where λ the wavelength in nanometers, a(λ) the total absorption coefficient (1/m), bb (λ) the total backscattering coefficient (1/m), θσ the solar zenith angle (rad), u the wind speed (m/s), and p1−6 specific coefficients (see table 3 in Albert and Mobley [35]). Coefficients a(λ) and bb (λ) are calculated as a(λ) = aw (λ) + a∗ph (λ)[CHL] + a∗NAP (λ)[NAP] + a∗CDOM (λ)aCDOM (440) 1 bb (λ) = bw (λ) + b∗b,TSM (λ)[TSM] 2
(5) (6)
where aw (λ) the absorption due to water [37], bw (λ) the scattering due to water [38], a∗ph (λ) the specific (i.e., per unit con-
Fig. 3. Optical closure between the simulated Ecolight spectra (gray) and the forward-modeled spectra (black). The input concentrations are: I. CHL 12 μg/L, TSM 118 mg/L and aCDOM (440) 1.89; II. CHL 18 μg/L, TSM 62 mg/L and aCDOM (440) 2.59; and III. CHL 6 μg/L, TSM 20 mg/L and aCDOM (440) 1.66.
centration) absorption coefficient of phytoplankton, a∗NAP (λ) the specific absorption coefficient of NAPs, a∗CDOM (λ) the specific absorption coefficient of CDOM, b∗b,TSM (λ) the specific backscattering coefficient of TSM, and aCDOM (440) the absorption coefficient of CDOM at 440 nm. We refer the reader to Albert and Mobley [35] for a detailed description on the construction of this forward bio-optical model and the underlying assumptions in the simplification. Henceforth, we will call the reflectance spectra modeled by the simplified bio-optical model of Albert and Mobley the “forward-modeled spectra.” Under perfect conditions, the forward-modeled spectra and the input reflectance spectra should be optically close as they are based on the same SIOPs. In our case, the optical closure between the input reflectance spectra and the forwardsimplified bio-optical model [35] proved to be satisfactory (see Fig. 3). The average root-mean-square error (RMSE) of the optical closure between the input reflectance spectra and the forward-modeled spectra is 0.007 with a standard deviation of 0.001. This closure demonstrates the suitability of the forwardsimplified bio-optical model of Albert and Mobley [35] for this study area. 2) Standard Inversion Method: A common practice for inverting a forward model is by minimizing the error between the forward-modeled spectrum from the simplified bio-optical model and the input reflectance spectrum. In this paper, we matched the spectra using a predictor–corrector optimization scheme [10], [36], which starts with an initial concentration set (CHL, TSM, and CDOM) into the forward-simplified biooptical model. This model is iteratively changed until the total squared error (TSE) between the input reflectance spectrum and the forward-modeled spectrum is minimized (generalized reduced gradient method in IDL [39]). The minimization algorithm was constrained to realistic concentration limits as it is possible that the mathematically optimum solution falls outside
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concentrations of CHL, TSM, and CDOM for this wavelet scale. The scheme is repeated over all wavelet scales from fine to coarse wavelet scales (scales one through twenty). WaveIN then selects the optimal wavelet scale per constituent based on the lowest NRMSE of prediction between the input and modeled constituent concentration (see Section III-C). The NRMSE is also used to evaluate the overall model performance. We call this final step the wavelet scale selection. It is the only step in WaveIN requiring concentration measurements. E. Scenario Development
Fig. 4. WaveIN method flowchart presents the necessary input parameters, the predictor–corrector–optimization scheme, the wavelet filter and the final output.
these limits [10]. Henceforth, we will call this inversion the “standard inversion.” We compared the derived concentrations to the input concentrations [10]. The overall model inversion performance was evaluated using the normalized RMSE (NRMSE). The NRMSE is calculated as RMSE between the input and modeled constituent concentration divided by the concentration range. It is expressed as a percentage, providing a measure of the closure between input and output constituent concentration. D. WaveIN Method The WaveIN method integrates the standard inversion (see Section III-C), and wavelet analysis (see Section II), as illustrated in Fig. 4. Just like the standard inversion, it uses the forward-simplified bio-optical model of Albert and Mobley [35]. The wavelets are introduced as an extra step in the predictor–corrector–optimization scheme of the standard inversion (see Section III-C). The spectra are transformed into the wavelet domain just before the TSE is calculated (see Fig. 4). The key part in WaveIN is that both the input reflectance spectra and the forward-modeled spectra are transformed into the wavelet domain and fed into the predictor corrector–optimization scheme (see Fig. 4). Here, we use the whole wavelet decomposition component at a particular scale, without selecting specific wavelet peaks. Similar to the standard inversion, WaveIN calculates the TSE between the wavelet-transformed input reflectance spectrum and the wavelet-transformed forward-modeled spectrum and iterates this process until the TSE is minimized. It then stores the output
The second objective of the study is to improve the inversion robustness to certain types of noise or confounding factors. We developed five scenarios and applied each scenario to the standard inversion and to WaveIN: reference scenario, contamination due to spectral mixing, sun zenith angle, uncorrelated or white noise, and SIOP variation uncertainty. Table II presents a detailed description of the scenarios. The scenarios were selected according to the expected noise and likely confounding factors that would be present in high-spectral resolution imagery, such as APEX imagery, of optically complex waters (Table II). 1) Reference scenario: this scenario evaluates the overall model performance and thus relates to the first objective of this paper [see Fig. 5(a)]. 2) Contamination due to spectral mixing: Airborne image pixels can be contaminated by spectral mixing of nontarget surfaces such as vegetation [40]. 3) Sun zenith angle: a underestimation of the sun zenith angle will influence the model inversion performance [15]. 4) Uncorrelated noise: this scenario validates WaveIN by checking whether the results are solely due to chance as uncorrelated noise does not possess a particular scale, frequency, or shape. 5) SIOP variation uncertainty [see Fig. 5(b)]: water bodies having the same constituent composition can still display a variety of spectral signatures due to a difference in SIOPs. The SIOP variability can be related to instrument and/or measurement error, and natural variability of optical properties [2], [13], [14], [17], [19]. This scenario allows us to test how well the methods can detect the different SIOPs. These scenarios result in two types of inverse problems: 1) ill-conditioned: water bodies having identical optical properties (SIOP and constituent concentrations) can still express spectral variability due to noise or confounding factors (scenario 2 through 4), which may cause deviations in the constituent estimation; and 2) ill-posed: water bodies that have the same constituent concentrations can still express different spectral signatures due to SIOP variability, and thus have multiple possible solutions (scenario 5). IV. R ESULTS This section starts by introducing the effect of the noise or confounding factors on the input reflectance spectra. Then it describes the overall results of the WaveIN scale selection (see Fig. 7). After this, it presents the inversion outcomes under the five scenarios for the standard inversion and WaveIN.
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TABLE II D ETAILED D ESCRIPTION OF THE D IFFERENT S CENARIOS OF N OISE AND C ONFOUNDING FACTORS
Fig. 5. Schematic representing WaveIN scenario 1 (reference scenario) and scenario 5 (SIOP variation uncertainty).
Figs. 8–12 visually compare the inversion outcomes for each scenario. The gray-scale color indicates samples with a high or a low TSM concentration. WaveIN estimates the constituents for different wavelet scales ranked from one to twenty, for fine wavelet scales (low rank) to coarse wavelet scales (higher rank), respectively. Note that the figures display only two relevant wavelet scales per scenario, this to convey the findings in a consistent and reduced way. A detailed overview of the wavelet scales selected per constituent is listed in Table IV. As an approximation for the deviation between the empirical slope and the 1:1 line, we discuss the slope difference. As an indicator of error variance, we discuss the spread of the data cloud. Models with a slope difference can still provide predictive power, whereas for erroneous models with low predictive power the precision is compromised. A. Effect of Noise or Confounding Factors on the Input Reflectance Spectra This section describes and visualizes the treatment of the different scenarios on the input reflectance spectra. Since the scenarios are the same for all of the 1000 input reflectance
spectra, we will discuss the effect on one example spectrum. The median, maximum and minimum input reflectance spectra are displayed in Fig. 6(a). The spectral range is solely related to the input concentration range used to simulate the input reflectance spectra (see Table I). 1) Scenario 2 Through 4: The contamination due to spectral mixing (scenario 2) caused a global increase in the input reflectance spectra. The most notable increase was mainly in the near infrared (NIR) and relatively limited in the blue wavelengths, which is related to the spectral shape of a vegetation signal. A change in sun zenith angle from 0◦ to 45◦ (scenario 3) caused a global increase in the reflectance signal [see Fig. 6(b)]. The increase is uneven and most pronounced between 550 and 700 nm. The effect of uncorrelated noise (scenario 4) is trivial. Uncorrelated noise introduced a random scattering around the input reflectance spectrum [see Fig. 6(b)]. 2) Scenario 5: The variation uncertainty in SIOPs (scenario 5) caused the largest deviation from the reference scenario (scenario 1). This effect was most pronounced in the green–red wavelengths due to the variation in a∗ph and b∗TSM [see Fig. 6(c)]. BA3 and BA6 differed the most from the reference input spectrum [see Fig. 6(c)], because BA3 had the lowest a∗ph and b∗TSM , whereas BA6 had the highest b∗TSM .
B. Overall Results of the WaveIN Scale Selection Fig. 7 presents the NRMSE of the estimation of CHL, TSM, and CDOM per wavelet scale for WaveIN. The NRMSE varied per wavelet scale for all three constituents. For CHL, except for scenario 4 (uncorrelated noise), the NRMSE plots showed a similar trend for all of the scenarios: the NRMSE decreased after wavelet scale one and increased again after wavelet scale five, reaching a minimum between wavelet scales two and five. For TSM, the NRMSE did not express a clear behavior over the wavelet scales. Only scenario 2 and 3 showed for TSM a specific behavior, where the NRMSE increased with increasing wavelet scale (see Fig. 7). For scenarios 1, 3, and 5, the NRMSE of CDOM showed a similar trend as for CHL. For CDOM, a minimum NRMSE is reached between wavelet scales two and four.
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Fig. 7. NRMSE per wavelet scale for scenarios 1 through 5 using the WaveIN method. The horizontal axis is the wavelet scale, the vertical axis the NRMSE (in percent). Note that some of the vertical axis are different.
Fig. 6. Visualization of (a) the spectral range of the input reflectance spectra (scenario 1), and (b)-(c) the treatment of the different scenarios for an example input reflectance spectrum (CHL 6 μg/L, TSM 20 mg/L, and aCDOM (440) 1.66). The proportional effect on the other input reflectance spectra is the same. b) Displays the effect of scenarios 2 through 4, c) the effect of SIOP variation uncertainty (scenario 5).
C. Scenario 1: Reference Scenario 1) Standard Inversion: Fig. 8(a) and Table III present the inversion results for CHL, TSM and CDOM under scenario 1 (reference scenario) for the standard inversion. The standard inversion proved to be reliable for the River Scheldt; we observed a strong positive relationship between the input and the output constituent concentrations. The NRMSE was highest for CDOM (33.4%), followed by CHL, and lowest for TSM. The CHL and CDOM estimates showed a spread of the data cloud. The error was highest for input reflectance spectra with a high TSM concentration, as indicated by the gray-scale color indication in Fig. 8(a). TSM, on the other hand, was slightly underestimated when TSM concentrations were high. 2) WaveIN: WaveIN decreased the NRMSE for all three constituents [Table IV, Fig. 8(b) and (c)]. Compared with the standard inversion, the NRMSE decreased by 7.3% for CHL, 2.3% for TSM and 21.1% for CDOM. The NRMSE was still highest for CDOM, followed by TSM and lowest for CHL. CHL and CDOM estimation was optimal at fine wavelet scales,
Fig. 8. Inversion results for CHL, TSM, and CDOM in scenario 1 (reference scenario) for the standard inversion (a) and WaveIN (b) and (c). The red line indicates the 1:1 line, the gray and black dots relate to the simulation’s low and high TSM concentration, respectively.
whereas TSM estimation was optimal at coarse wavelet scales (Table IV, Fig. 7). Visually, the improvement was most pronounced for CHL and CDOM [see Fig. 8(b)]. At wavelet scale two, CHL showed no spread, whereas CDOM showed a reduced spread of the data cloud relative to the standard inversion. For CDOM, the
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TABLE III NRMSE FOR S CENARIOS 1 T HROUGH 5 U SING THE S TANDARD I NVERSION
TABLE IV NRMSE FOR S CENARIOS 1 T HROUGH 5 U SING THE WAVE IN M ETHOD
error was highest for input reflectance spectra with a high TSM concentration as indicated by the gray-scale color indication in Fig. 8(b). For high TSM concentrations the error variation increased [Fig. 8(c)]. D. Scenario 2: Contaminated Pixel Due to Spectral Mixing 1) Standard Inversion: The introduced spectral mixing had a considerable influence on the retrieval of all three constituents [see Fig. 9(a)], resulting in a strong increase in the NRMSE compared with scenario 1 (see Table III). Compared with scenario 1, the NRMSE increased by 23.8% for CHL, 12.9% for TSM, and 25.0% for CDOM. The NRMSE is highest for CDOM, followed by CHL and lowest for TSM. CHL estimations were widely spread around the 1:1 line. CHL estimations with a high TSM concentration were mainly underestimated and vice versa, as indicated by the gray-scale color indication in Fig. 9(a). The TSM concentration was overestimated and showed a slope difference. This overestimation increased for increasing TSM concentrations [see Fig. 9(a)]. CDOM retrieval was widely spread around the 1:1 line. CDOM estimations with a high TSM concentration were mainly overestimated and vice versa. 2) WaveIN: WaveIN was influenced by contamination due to spectral mixing. However, it had lower NRMSE for CHL and TSM estimation compared with the standard inversion, although it could not improve the CDOM estimation [see Fig. 9(b) and (c), Table IV]. Compared with the standard inversion the NRMSE decreased by 26.3% for CHL and 13.5% for TSM but increased by 8.1% for CDOM. The NRMSE was highest for CDOM, followed by TSM and the lowest CHL. CHL estimation was optimal at intermediate wavelet scales, CDOM at coarse wavelet scales and TSM at fine wavelet scales. For CHL, WaveIN (wavelet scale 5) reduced the spread around the 1:1 line considerably compared with the standard inversion. For TSM, WaveIN (wavelet scale 1) decreased the slope difference from but increased the spread around the 1:1 line.
Fig. 9. Inversion results for CHL, TSM, and CDOM in scenario 2 (contaminated pixel due to spectral mixing) for the standard inversion (a) and WaveIN (b) and (c). The red line indicates the 1:1 line, the gray and black dots relate to the simulation’s low and high TSM concentration, respectively.
For CDOM, WaveIN increased the NRMSE and failed to improve the retrieval. CDOM was overestimated and showed a large error throughout all the wavelet scales (see Fig. 7). E. Scenario 3: Sun Zenith Angle 1) Standard Inversion: The larger sun zenith angle introduced in the standard inversion had a strong influence on the retrieval of all three constituents [see Fig. 10(a)], resulting in a large increase in the NRMSE compared with scenario 1 (Table III). Compared with scenario 1 the NRMSE increased by 23.7% for CHL, 12.7% for TSM, and 30.8% for CDOM. The NRMSE was highest for CDOM, followed by CHL and lowest for TSM.
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Fig. 10. Inversion results for CHL, TSM, and CDOM in scenario 3 (sun zenith angle) for the standard inversion (a) and WaveIN (b) and (c). The red line indicates the 1:1 line, the gray and black dots relate to the simulation’s low and high TSM concentration, respectively.
Fig. 11. Inversion results for CHL, TSM, and CDOM in scenario 4 (uncorrelated noise) for the standard inversion (a) and WaveIN (b) and (c). The red line indicates the 1:1 line, the gray and black dots relate to the simulation’s low and high TSM concentration, respectively.
CHL showed a wide spread around the 1:1 line, and was underestimated; this underestimation was most pronounced for CHL estimations with a high TSM concentration, as indicated by the gray-scale color indication in Fig. 10(a). The TSM concentration was overestimated and showed a slope difference. This overestimation increased for increasing TSM concentrations [Fig. 10(a)]. CDOM retrieval was widely spread around the 1:1 line [Fig. 10(a)]. 2) WaveIN: WaveIN decreased the NRMSE for CHL, TSM, and CDOM compared with the standard inversion [see Fig. 10(b) and (c), Table IV]. Compared with the standard inversion, the NRMSE decreased by 28.9% for CHL, 15.9% for TSM, and 56.1% for CDOM. The NRMSE was highest for CDOM, followed by TSM and lowest for CHL. CHL, TSM and CDOM estimation was optimal at relatively fine wavelet scales. For CHL, WaveIN does not underestimate the concentration. The estimates lie close to the 1:1 line. WaveIN selected the same wavelet scale as in scenario 1. The TSM estimates lie closer to the 1:1 line than in the standard inversion. Nevertheless, for TSM, the sun zenith angle resulted in an increased error variation for high TSM concentrations. For CDOM, WaveIN strongly decreased the spread around the 1:1 line [Fig. 10(c)].
For CHL the uncorrelated noise caused considerable spread around the 1:1 line. Furthermore, this resulted in a reduced differentiation of samples with respect to high or low TSM concentration, as indicated by the gray-scale color indication [see Fig. 11(a)]. The effect of uncorrelated noise on the TSM estimation was limited. Nevertheless, for TSM, the uncorrelated noise resulted in an increased error variation for high TSM concentrations [see Fig. 11(a)]. For CDOM, the uncorrelated noise resulted in an increased spread around the 1:1 line [see Fig. 11(a)]. 2) WaveIN: WaveIN could not improve the constituent estimation and caused an increase of the NRMSE for all three constituents [see Fig. 11(b) and (c), Table IV]. Compared with the standard inversion, the NRMSE increased by 1.6% for CHL, 3.8% for TSM, and 5.6% for CDOM. The NRMSE was still highest for CDOM, followed by CHL and lowest for TSM. All three constituents were best estimated at intermediate wavelet scales. For CHL and CDOM, we observed similar, but more pronounced spread around the 1:1 line as for the standard inversion. For TSM, WaveIN caused spread around the 1:1 line and a strong increase in the error variation from low to high TSM concentrations.
F. Scenario 4: Uncorrelated Noise
G. Scenario 5: SIOP Variation Uncertainty
1) Standard Inversion: The uncorrelated noise we introduced increased the NRMSE of the standard inversion for all three constituents compared with scenario 1 [see Fig. 11(a)] and Table III). Compared with scenario 1, the NRMSE increased by 1.9% for CHL, 0.1% for TSM, and 9.9% for CDOM. The NRMSE was highest for CDOM, followed by CHL and lowest for TSM.
1) Standard Inversion: The SIOP variation uncertainty introduced in the standard inversion had a considerable influence on the retrieval of all three constituents [see Fig. 12(a), Table III] and resulted in an increase in the NRMSE of all three constituents. Compared with scenario 1, the NRMSE increased by 24.9% for CHL, 39.0% for TSM, and 42.1% for CDOM. The NRMSE was still highest for CDOM, followed
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For CDOM, WaveIN decreased the spread around the 1:1 line [see Fig. 12(c)]. Note, we could not sufficiently evaluate the influence of SIOP variation uncertainty on the CDOM estimation as we only obtained one usable CDOM sample.
V. D ISCUSSION A. WaveIN Scale Selection
Fig. 12. Inversion results for CHL, TSM, and CDOM in scenario 5 (SIOP variation uncertainty) for the standard inversion (a) and WaveIN (b) and (c). The red line indicates the 1:1 line, the colors are related to the input SIOP set (black: average SIOP set, red: BA3 SIOP set, green: BA4 SIOP, and blue: BA6 SIOP set).
by TSM and lowest for CHL. Fig. 12 clearly discriminates the groups of samples derived from a different SIOP set. The best estimation was obtained for the average SIOP set, which is, in fact, scenario 1. The BA6 SIOP set, showed the largest deviation, which was related to the high specific TSM scattering coefficients (see Fig. 2). For CHL, the SIOPs variation uncertainty resulted in considerable spread around the 1:1 line. We could clearly discriminate the different samples according to their input SIOP set. For TSM, the spectral variation resulted in higher error variation for high TSM concentrations. In this case, the error variation is related to the different input SIOPs. Each SIOP input set strongly influenced the slope between the input and output TSM concentration. For CDOM, this spectral variation resulted in considerable spread around the 1:1 line. However, we could not clearly distinguish the different samples according to their input SIOP set. 2) WaveIN: WaveIN decreased the NRMSE for all three constituents and thus improved the constituent retrieval compared with the standard inversion [see Table IV, Fig. 12(b) and (c)]. Compared with the standard inversion, the NRMSE decreased by 24.5% for CHL, 16.0% for TSM, and 38.0% for CDOM. The NRMSE was still highest for CDOM, followed by TSM and lowest for CHL. All three constituents were best estimated at relatively fine wavelet scales. For CHL, WaveIN decreased the spread around the 1:1 line [see Fig. 12(b)]. The optimal wavelet scale was the same as in scenario 1 (reference scenario). Visually, the separation between the SIOP input sets has been reduced. For TSM, WaveIN decreased the error variation from low to high TSM concentrations and reduced the slope difference. However, a considerable difference between the SIOP sets and a high NRMSE compared with scenario 1 still remained.
The NRMSE of the constituent retrieval varied over the wavelet scales and scenarios (see Fig. 7). Here, we will not discuss the results of Scenario 4, since it is developed to verify the assumptions of WaveIN and to check if the solution was solely due to chance (see Section V-E in the succeeding discussion). The consistent behavior of the NRMSE of CHL (see Fig. 7) is straightforward since it is the only variable with distinct and spectrally localized features [12], [41]. CHL is therefore best estimated at fine wavelet scales, which is in agreement with the observations of Ampe et al. [12]. The scale selection of TSM and CDOM is less specific (see Fig. 7). The spectral effect of CDOM is subtle and that of TSM broad, which means the effect will be present at most of the wavelet scales. The reliance on in situ data for wavelet scale selection, and the selection of one optimal wavelet scale per constituent, makes the method a combination of a semianalytical inversion with a semiempirical wavelet scale selection. Fig. 7 shows a gradual change in the NRMSE over the wavelet scales. This indicates a promising potential future development toward an approach that is semianalytical and extensible: a multiscale approach that uses the weighted average of interesting wavelet scales for the different constituents. Such an approach would overcome the problem of picking one single wavelet scale for one specific scenario. Future research should further investigate the consistent behavior of the NRMSE over the different wavelengths and scenarios (see Fig. 7) and verify this for different data sets, both air- and spaceborne. If this behavior persists, one could define a sensor specific wavelet scale selection approach. We envision an improved WaveIN algorithm that takes a weighted average of the wavelet scales, which have previously been defined as interesting for a specific sensor and study area. Due to model errors, noise, or confounding factors, the spectral features of the input reflectance spectrum may change, resulting in imperfect optical closure [6], [9], [12], [42]. Because of poor optical closure, the best fitting solution may not result in accurate estimation of the concentrations [6]. For example, where we had high TSM concentration, the standard inversion could not correctly match reflectance with the correct CHL concentration because the standard inversion optimizes the overall spectral shape. WaveIN was more successful at CHL concentration retrieval under high TSM conditions because WaveIN focuses the optimization per wavelet scale. At fine wavelet scales, the CHL spectral features are enhanced, improving CHL concentration retrieval [12]. In such a case, WaveIN mainly matches the CHL spectral features and, to a lesser extent, the signal magnitude, which may in some cases reduce TSM concentration retrieval. Essentially, in the wavelet domain, certain spectral features are enhanced, and we can find the optimal fit for those features although in the wavelength domain the optical closure might be suboptimal.
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WaveIN’s selective optimization per constituent in the wavelet domain overcomes some of the retrieval errors due to poor optical closure in the wavelength domain. However, this means that WaveIN is not simultaneously retrieving all of the constituents. This is still an ongoing issue; a few studies in inland waters successfully solve for all three constituents [4], [9], and validation is often only a minor part of the study [4], [6], [14]. We hypothesize that in complex waters, it might, in some cases, be difficult to correctly retrieve all the constituents simultaneously. A model that enhances the CHL features may impede the estimation of other constituents. B. Scenario 1: Reference Scenario 1) Standard Inversion: The standard inversion proves to be a reliable method for this study area in the case of good quality reflectance data. The slight increase in error for high TSM concentrations could be related to the calibration of the forwardsimplified bio-optical model of Albert and Mobley [35], which was limited to a TSM concentration of 50 mg/L. Furthermore, at high TSM concentrations, the scattering features of TSM become dominant, impeding CHL and CDOM, resulting in an underestimation of CHL and an overestimation of CDOM. Our results demonstrate the importance of defining the sensitivity, and upper and lower theoretical bounds of both algorithms and sensors [15]. As previously discussed, the selected standard inversion becomes less accurate for high TSM concentrations. In addition, the high TSM concentration influences the estimation of CHL and CDOM. Therefore, we would advise care in using the standard inversion for more turbid waters. It is important to realize this in the development of new algorithms for complex waters. 2) WaveIN: WaveIN improved the estimation of all three constituents because wavelets decouple the combined and interacting influence of the constituents. The results confirm the hypothesis of Ampe et al. [12] that the optically active constituents influence the signal at various wavelet scales. In our case CHL was better estimated by fine wavelet scales because these absorption features are relatively narrow compared with the coarse or broad effect of scattering by TSM (see Fig. 2). WaveIN can decompose the information content and enhance the spectral feature recognition during optimization. This is useful for optically complex waters especially those where high TSM concentration may mask the spectral features of other constituents [3], [4]. C. Scenario 2: Contaminated Pixel Due to Spectral Mixing 1) Standard Inversion: Under scenario 2, the standard inversion had high errors in the estimation of the constituent concentrations (Table III). The global spectral increase plus the deformation of the spectral features caused problems in the spectral matching of the predictor–corrector optimization scheme. Furthermore, the increase in spectral signature due to scenario 2 caused TSM overestimation. 2) WaveIN: WaveIN improved the estimation of CHL and TSM but could not improve the estimation of CDOM. This is related to the way wavelets are calculated. At the edge of the spectrum, the wavelet filter can only partly cover the spectrum as there are not enough spectral bands left. Therefore,
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the wavelet filter uses spectral bands of the other side of the spectrum to fill these gaps. Since the mixing with a vegetation signal caused a higher increase in the NIR than in the visual wavelengths, the wavelet decomposition showed considerable edge effects. For fine wavelet scales, these edge effects only influence the outer wavelengths. This becomes more pronounced at coarser wavelet scales. Hence, WaveIN is sensitive to the location of the spectral features within the overall spectral signature. However, WaveIN can select the wavelet scale, where the feature contamination was minimal. Wavelet edge effects could be identified by defining the cone of influence (COI) of the wavelet filter. The spectral bands influenced by the COI could later on be removed from the analysis. Furthermore, the problem of wavelet edge effects could be mitigated by selecting a wider wavelength range, which could dilute the edge effects or by using an edge avoiding technique, such as second-generation wavelet analysis [43]. CHL spectral features were not located near the edge of the input reflectance spectrum and were best preserved at intermediate wavelet scales. Although the mixture with a vegetation signal caused a deformation in the spectral features, WaveIN could avoid this by selecting a coarser wavelet scale than in scenario 1. Additionally, wavelets are less sensitive to the vertical displacement of the signal and focus on the spectral shape [12]. By definition, wavelet analysis is blind to a vertical displacement of a signal because wavelets must have zero integrals [44]. CDOM absorption is mainly present at short wavelengths (see Fig. 2), where the wavelet edge effects were most pronounced, which explains why WaveIN could not improve the CDOM estimation. In contrast to scenario 1, TSM estimation was optimal here at a fine wavelet scale. This is because TSM influences the overall spectral signature. The wavelet edge effects became more pronounced at these coarse wavelet scales. Therefore, WaveIN avoided these scales by selecting a fine wavelet scale. In the future, we could investigate if WaveIN also improves estimations of spectra influenced by adjacency effects, as this also strongly influences the NIR wavelengths [45]. Therefore, it would be useful to test WaveIN on airborne imagery, for which we need a large set of coupled in-water and remote sensing measurements. D. Scenario 3: Sun Zenith Angle 1) Standard Inversion: Under scenario 3, the standard inversion estimated the constituent concentration with a large error. Analogous to scenario 2, the global spectral increase plus the deformation of the spectral features caused problems in the spectral matching of the predictor–corrector optimization scheme. However, again, the increase in spectral signature is the reason TSM was overestimated. This strong influence of the sun zenith angle demonstrates the importance of algorithms that can cope with noisy data. In many occasions, the influence of the sun angle or atmosphere may overrule the signal-to-noise ratio of the sensor. 2) WaveIN: WaveIN could improve the estimation of all three constituents. As in scenario 2, a change in sun zenith angle caused a variable global increase in the spectral signature. However, in this case, there were less wavelet edge effects as
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there was only a limited difference between the increase in the NIR and the visual wavelengths. Furthermore, if we would concentrate on short wavelength intervals, scenario 3 resembles a vertical bias. As discussed in scenario 2, by definition, wavelet analysis is blind to a vertical displacement of a signal [44]. For CHL, WaveIN selected the same wavelet scale as in scenario 1. The reason for this is twofold: 1) CHL spectral features are localized and not located near the edge of the input reflectance spectrum, they are therefore less affected by wavelet edge effects; and 2) as discussed in the previous paragraph, we expect less influence of the wavelet edge effects compared with scenario 2. For TSM, we can draw the same conclusions as in scenario 2. Due to the wavelet edge effects, the optimal wavelet scale shifted toward fine wavelet scales. As previously discussed, for this scenario, the wavelet edge effect is limited compared with scenario 2. However, for TSM this effect remains strong enough to influence the wavelet scale selection. The estimation of CDOM was less affected by the wavelet edge effects compared with scenario 2. As previously discussed, for this scenario the wavelet edge effect is limited compared with scenario 2. In addition, the shift in optimal wavelet scale was limited. The wavelet scale selection for CDOM was less affected by the edge effect than it was for TSM.
E. Scenario 4: Uncorrelated Noise The random effects in the spectral signature made it difficult for the standard inversion to correctly match the input and forward-modeled spectra. WaveIN could not improve constituent retrieval because wavelets cannot avoid uncorrelated noise as uncorrelated noise does not possess a particular scale, frequency or shape. We added this scenario to verify the assumptions of WaveIN and to check if the solution was solely due to chance. The effect of uncorrelated noise could be tackled by other techniques, such as presmoothing using splines or kernel methods [46].
F. Scenario 5: SIOP Variation Uncertainty 1) Standard Inversion: Under scenario 5, the standard inversion produced a large error in the estimation of the constituents. The variation in SIOPs of the target water body caused spectral variation. This ill-posedness is clearly visible in the result as it impedes correct spectral matching in the predictor–corrector optimization scheme. 2) WaveIN: WaveIN improved the estimation of the three constituents in this ill-posed problem. The reason is similar for all three constituents. WaveIN could avoid the change in spectral features by selecting optimal wavelet scales where this influence is minimal and wavelets are less sensitive to the vertical displacement in the spectral signature. Still, WaveIN could not solve the problem of the influence of SIOP variability. SIOP variability is a key problem in inverse modeling of remote sensing data [2], [18], which cannot be overlooked. For future research, we propose to combine WaveIN with an adaptive approach [2]. In addition, we should test this methodology on an independent data set or different geographical location with different SIOP variability [17].
G. Overall Discussion The generally positive effects of WaveIN on the multiscale estimation of optically active constituents contribute to the present knowledge in semianalytical modeling [4]. Due to its object-oriented nature, WaveIN can easily be integrated with other semianalytical inversion models or LUT approaches [47]. A different (nonwavelet) multiscale approach was introduced by Sadeghi et al. [48] and Bracher et al. [49] and demonstrated the effectiveness of scale space for aquatic optical remote sensing. They improved the identification of phytoplankton functional types by separating the higher frequency absorption structures from broad band features through the subtraction of a polynomial for different wavelength windows. They used the whole spectral wavelength range as opposed to specific wavelengths to calculate the phytoplankton group biomass. Giardino et al. [9] approached the large number of Hyperion bands in the matrix inversion method via a sensitivity analysis [47]. They have found Hyperion too noisy and ill-calibrated in the shortest wavelengths where CDOM variation was the highest and where therefore unable to estimate CDOM. Model inversion using neural networks [7], [8] has been proposed as a method less sensitive to noise [50]. These methods however need ample training data [51] and may have problems in extrapolation far away outside the subspace of the training data [52], [53]. Support vector machine have been proposed to overcome these problems, but further research is needed [50], [51]. For multispectral measurements such as MERIS, the principal component inversion (PCI) [54] has been developed with an internal training scheme based on principal component analysis, which extracts the most informative part of the data set. The method should reduce the dependency on measurement noise [54]. The applicability of PCI on high-spectral resolution data and the influence of other confounding factors should be further be investigated [55]. VI. C ONCLUSION AND O UTLOOK We have proposed WaveIN as a novel inversion technique to estimate the CHL, TSM and CDOM concentration from highspectral resolution data in complex waters. WaveIN is based on a predictor–corrector optimization scheme and uses wavelettransformed spectra, instead of the original high-spectral resolution reflectance signal. WaveIN was able to partly resolve the two objectives of this paper 1) to improve bio-optical inversion results in estimating the optically active constituents CHL, TSM, and CDOM; and 2) to improve the model robustness to certain types of noise or confounding factors. WaveIN had a positive effect on the estimation of CHL, TSM, and CDOM in the case of the reference scenario, and for most of the scenarios. However, although WaveIN reduced the influence of variation in SIOPs, it could not completely resolve it. The multiscale approach is a powerful tool for enhanced information extraction as wavelets can decouple the combined influence of the confounding factors. Additionally, wavelets are less sensitive to a bias in the spectra and can avoid the changing or shifting spectral features by selecting specific wavelet scales. The obvious next step is to test this methodology on APEX imagery. Here, one should first define the optimal wavelet scales based on the in situ concentration measurements. These measurements should be spatially distributed to capture
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different types of image contamination. The waveIN method is therefore not fully analytical as it will need in situ data to select the best wavelet scales. Additional research should focus on various air- or spaceborne optical remote sensing data and additional types of noise and confounding factors, e.g., adjacency effect, high-frequency noise, atmospheric noise, and sky glint. Future studies on real images should also improve the wavelet scale selection. Instead of selecting one optimal wavelet scale, one should investigate a weighted average of a cluster of informative wavelet scales per constituent. ACKNOWLEDGMENT The authors would like to thank the partners in the MICAS project for their significant contributions in the fieldwork and data analysis. The authors also thank L. Bertels, P. Grötsch, K. Hestir, J. Nossent, and the two anonymous reviewers for their useful comments and suggestions. R EFERENCES [1] C. D. Mobley, “Optical Properties ofWater,” in Handbook of Optics, 2nd ed. New York, NY, USA: McGraw-Hill, 1994.. [2] V. E. Brando, A. G. Dekker, Y. J. Park, and T. Schroeder, “Adaptive semi-analytical inversion of ocean color radiometry in optically complex waters,” Appl. Opt., vol. 51, no. 15, pp. 2808–2833, May 2012. [3] M. W. Matthews, “A current review of empirical procedures of remote sensing in inland and near-coastal transitional waters,” Int. J. Remote Sens., vol. 32, no. 21, pp. 6855–6899, 2011. [4] D. Odermatt, A. Gitelson, V. E. Brando, and M. Schaepman, “Review of constituent retrieval in optically deep and complex waters from satellite imagery,” Remote Sens. Environ., vol. 118, pp. 116–126, Mar. 2012. [5] S. Maritorena, D. A. Siegel, and A. R. Peterson, “Optimization of a semianalytical ocean color model for global-scale applications,” Appl. Opt., vol. 41, no. 15, pp. 2705–2714, May 2002. [6] H. J. van der Woerd and R. Pasterkamp, “HYDROPT: A fast and flexible method to retrieve chlorophyll-a from multispectral satellite observations of optically complex coastal waters,” Remote Sens. Environ., vol. 112, no. 4, pp. 1795–1807, Apr. 2008. [7] L. González Vilas, E. Spyrakos, and J. M. Torres Palenzuela, “Neural network estimation of chlorophyll a from MERIS full resolution data for the coastal waters of Galician rias (NW Spain),” Remote Sens. Environ., vol. 115, no. 2, pp. 524–535, Feb. 2011. [8] R. Doerffer and H. Schillera, “The MERIS case 2 water algorithm,” Int. J. Remote Sens., vol. 28, no. 3/4, pp. 517–535, Feb. 2007. [9] C. Giardino, V. E. Brando, A. Dekker, N. Strombeck, and G. Candiani, “Assessment of water quality in Lake Garda (Italy) using hyperion,” Remote Sens. Environ., vol. 109, no. 2, pp. 183–195, Jul. 2007. [10] Z. Lee, K. L. Carder, C. D. Mobley, R. G. Steward, and J. S. Patch, “Hyperspectral remote sensing for shallow waters: 2. Deriving bottom depths and water properties by optimization,” Appl. Opt., vol. 38, no. 18, pp. 3831–3943, Jun. 1999. [11] Z. P. Lee, K. Carder, T. Peacock, C. O. Davis, and J. L. Mueller, “Method to derive ocean absorption coefficients from remote-sensing reflectance,” Appl. Opt., vol. 35, no. 3, pp. 453–462, Jan. 1996. [12] E. M. Ampe et al., “A wavelet approach for estimating chlorophyll-a from inland waters with reflectance spectroscopy,” IEEE Geosci. Remote Sens. Lett., vol. 11, no. 1, pp. 89–93, Jan. 2014. [13] G. Dall’Olmo and A. A. Gitelson, “Effect of bio-optical parameter variability and uncertainties in reflectance measurements on the remote estimation of chlorophyll-a concentration in turbid productive waters: Modeling results,” Appl. Opt., vol. 45, no. 15, pp. 3577–3592, May 2006. [14] T. J. Malthus, E. L. Hestir, A. Dekker, and V. E. Brando, “The case for a global inland water quality product,” in Proc. IEEE IGARSS, Munich, Germany, Jul. 22–27, 2012, pp. 5234–5237. [15] J. D. Hedley, C. M. Roelfsema, S. R. Phinn, and P. J. Mumby, “Environmental and sensor limitations in optical remote sensing of coral reefs: Implications for monitoring and sensor design,” Remote Sens., vol. 4, no. 1, pp. 271–302, 2012. [16] S. I. Kabanikhin, “Defnitions and examples of inverse and ill-posed problems,” J. Inverse Ill-posed Problems, vol. 16, pp. 317–357, 2008.
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Eva M. Ampe received the master’s degree in bioengineering from the Katholieke Universiteit Leuven (KU-Keuven), Leuven, Belgium, in 2009, the master’s degree in water resources engineering from KU-Leuven, Vrije Universiteit Brussel, Brussels, Belgium, in 2010, and the Ph.D. degree in engineering from the Vrije Universiteit Brussel in April 2014.
Dries Raymaekers received the master’s degree in bioengineering from the Katholieke Universiteit Leuven (KU-Leuven), Leuven, Belgium, in 2003. He also received the Advanced Master of Science in earth observation from KU-Leuven and Purdue University, West Lafayette, IN, USA, in 2005. Since 2008, he has been working as a Researcher with the Flemish Institute of Technological Research (VITO), Mol, Belgium on hyperspectral remote sensing.
Erin L. Hestir received the bachelor’s degree in geography from the University of California, Berkeley, CA, USA, in 2004, and the Ph.D. degree in geography from the University of California, Davis, Davis, CA, in 2010. She is currently a Postdoctoral Researcher in Environmental Earth Observation with CSIRO, Australia. She was also a Postdoctoral Researcher with the CSTARS Laboratory. Her research interests include coastal, estuarine, riverine and wetland hydrology, and ecogeomorphology.
Maarten Jansen received the Ph.D. degree in computer science and applied mathematics from the Katholieke Universiteit Leuven (KU-Leuven), Leuven, Belgium, in 2000. He has held postdoctoral positions with Bristol University, Bristol, U.K., and with Rice University, Houston, TX, USA. In 2001–2006, he was an Assistant Professor with the University of Eindhoven, Eindhoven, The Netherlands, and in 2006– 2010, with KU-Leuven . He is currently with the Université Libre de Bruxelles, Brussels, Belgium. He is (co-)author of two books on wavelets. His current research interests include multiscale signal analysis and processing, wavelets, and high-dimensional variable selection.
Els Knaeps received the M.Sc. degree in geography from the Katholieke Universiteit Leuven (KU-Leuven), Leuven, Belgium, in 2003. She also received the M.Sc. degree in remote sensing and Geographic Information Systems from the KULeuven and Purdue University, West Lafayette, IN, USA, in 2005. She is currently working with the Flemish Institute for Technological Research (VITO), Mol, Belgium, where she is coordinating the activities of the hyperspectral remote sensing research group.
Okke Batelaan is a Strategic Professor of hydrogeology with Flinders University, Adelaide, SA, Australia. He is also connected to his former university, Vrije Universiteit Brussel, Brussels, Belgium. His expertise and teaching is in hydrogeology, hydrological modeling, GIS and remote sensing for water resources, and ecohydrology. He is the Promotor of the Ph.D. of the first author.
