Robustness of Vegetation Optical Depth Retrievals Based on L-Band Global Radiometry

Microwave vegetation optical depth (VOD) and soil moisture (SM) can be simultaneously retrieved based on L-band radiometry with polarization information. VOD is indicative of the vegetation water content (VWC) because it captures the extinction of land surface emission. If the connectivity of VOD to VWC is robust, the pair of VWC-SM observations can be viable bases for understanding soil–plant–atmosphere water relations, providing new perspectives on ecosystem science. Simultaneous SM–VOD retrievals are feasible by inverting the <inline-formula> <tex-math notation="LaTeX">$\tau -\omega $ </tex-math></inline-formula> model with two independent datasets in dual-channel algorithms. However, given correlated satellite vertical and horizontal brightness temperatures (TBs; TB<italic>v</italic> and <inline-formula> <tex-math notation="LaTeX">${{\mathrm {TB}}}_{h}$ </tex-math></inline-formula>), an ill-posed inverse problem arises where TB errors result in high uncertainties of retrievals. In this study, we apply the degrees-of-information (DoI) metric and propose a signal-to-noise ratio (SNR) metric to assess the “retrievability” of VOD given the Soil Moisture Active Passive (SMAP) TB<italic>v</italic>–TB<italic>h</italic> linear dependence. The application of these metrics allows determining where the VOD retrievals are robust and reliable. This is a necessary step in supporting the applications of VOD in ecology and hydrology. Results show that regions with mainly nonwoody vegetation have the best potential for VOD retrievals, though regularization is necessary. We then assess VOD time variations from two regularization products that reduce the impact of underdetermined inversions: the L3 dual-channel algorithm (L3-DCA) and the multitemporal dual-channel algorithm (MTDCA), which constrain VOD time dynamics with and without using <italic>a priori</italic> VOD climatology, respectively. Though they both reduce noise, especially in the VOD retrievals, they result in differences in VOD seasonal amplitude and coupling to SM at high frequencies as we outline here.

) 38 satellites measure the Earth's microwave surface emission 39 at a low frequency (L-band, 1.4 GHz). Over land, such 40 measurements are sensitive to the rough surface reflectivity 41 and to the attenuation and scattering that the entire vege-42 tation canopy exerts over the surface emission. The rough 43 surface reflectivity is related to the soil dielectric constant and 44 electromagnetic roughness. The inversion of estimated surface 45 reflectivity results in estimates of surface soil moisture (SM). 46 A by-product of the retrieval is the amount of vegetation 47 attenuation and scattering that together are captured by the 48 vegetation optical depth (VOD). VOD is known to be related to 49 the vegetation water content (VWC), the vegetation biomass, 50 and the plants' structure [3], [4], [5]. 51 SM and VOD are valuable hydrologic and ecological 52 indicators important for a breadth of applications and studies. 53 These include biomass estimation (e.g., [6], [7], [8]), crop 54 yield assessment [9], [10], development of drought indicators 55 (e.g., [11]), study of drought-derived tree mortality [12], 56 estimation of vegetation moisture [13], [14], and 57 analyses of water exchange in the soil-plant-atmosphere 58 continuum [15], [16], [17]. 59 The estimates of global SM fields based on SMAP and 60 SMOS L-band measurements are routinely assessed against 61 widely available in situ SM probe measurements. In contrast, 62 there are only few studies reporting how well VOD represents 63 in situ plant physiology and phenology (e.g., [18]). Studies 64 of how well VOD represents VWC at the satellite scale are 65 becoming more prevalent [4], [5], [15], [19], [20], [21]. These 66 assessments are based on sparse tower measurements and 67 crop models, which are highly informative. However, VOD 68 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ in situ measurements are sparse, leaving only limited or affected. 117 In order to quantify how well-posed the inversion is, pre-118 vious work estimates the degrees-of-information (DoI) metric 119 (DoI is defined in [22] and measures the fractional amount 120 of information, which is between 1 and 2 in a pair of 121 measurements). Because TB v and TB h as observation pairs 122 are correlated, DoI is below 2 and differences between TB v 123 and TB h can become influenced by random instrument error. 124 Thus, retrievals of SM and VOD by using two polarizations 125 are not fully independent and errors can potentially affect one 126 or both retrievals. The spurious noise reduces the robustness 127 of the method [22], [32]. For multiple angles, this effect is 128 expected to be less pronounced due to the higher amount of 129 information available [33], [34], although the depolarization 130 with more dense vegetation will still reduce the amount of 131 information across the angles. Also, note that other error 132 sources are present, for example, from errors in the assump-133 tions about the roughness parameter (h, e.g., [35], [36], [37]) 134 and single-scattering albedo (ω). They are explored only 135 complementarily in this work because our main focus here 136 is on how TB errors impact SM and VOD. 137 To overcome these issues, SMAP-based VOD-SM retrievals 138 have introduced various regularization approaches that aim to 139 reduce retrieval noise by incorporating a priori information 140 mainly about variations in VOD. The multitemporal DCA 141 (MTDCA) is based on the premise that changes in the vege-142 tation biomass occur on time scales that are longer than SM 143 fluctuations due to storms and interstorms [32], [38]. Based on 144 this assumption, the MTDCA uses two consecutive overpasses 145 to retrieve two SM values and a single VOD output for each 146 time-adjacent overpass pair. It also uses model selection over 147 the entire record to estimate the effective single-scattering 148 albedo as a static feature of the dominant vegetation type. 149 Hence, four TB values (two for each overpass) are available 150 to retrieve three unknowns. This increases DoI above three 151 [32], [39]. As DoI is the upper limit on the number of 152 possibly retrieved parameters, the problem is not necessarily 153 overdetermined. This procedure results in two VOD values 154 retrieved for each overpass (one using information from the 155 overpass before and one using information from the over-156 pass after). In averaging these two VOD values together, 157 information from multiple overpasses ultimately constrains the 158 VOD retrieved at a given overpass. Recently, other algorithms 159 have also included time aggregation with a priori decision 160 of the degree of regularization: the SMOS L3 algorithm [40]  Despite these advances on new information metrics and 177 regularization techniques, SMAP DCAs and regularization 178 approaches still need to be evaluated. We recognize that the 179 DoI metric is useful to quantify the information available in 180 satellite measurements, but it does not uniquely indicate the 181 robustness of the retrievals. For example, DoI may increase 182 with more random noise (i.e., independent TB v and TB h 183 values), which paradoxically suggests more robustness to 184 noise. Therefore, here, we introduce an additional metric of 185 retrieval susceptibility to noise (and hence robustness): the 186  [44]. Off-sets between the two VOD series are partially due to different algorithmic treatments of surface electromagnetic roughness and effective vegetation scattering albedo. Different behaviors in the two VOD products (i.e., smoother versus sharper changes along the time series) are likely due to the fact of using different regularization approaches which we investigate in this study. signal-to-noise ratio (SNR). It complements DoI in order 187 to provide a holistic understanding of the VOD retrieval 188 algorithm robustness.     Fig. 1(b) shows that seasonal variations in a woody savannah 204 are captured by both approaches, but with a smaller L3-DCA 205 seasonal amplitude. We aim to understand if either of the 206 regularization approaches may be under or overregularizing 207 the VOD variability both in seasonal and high-frequency vari-208 ations. Fig. 1(c) and (d) show how high-frequency MTDCA 209 VOD variation increases with biomass (i.e., the largest rapid 210 changes are found in the tropical forest pixel), while the 211 L3-DCA VOD time series is smooth in both cases (especially 212 in the tropical forest). For dense vegetation, this motivates 213 investigating whether the MTDCA approach may be carrying 214 excess noise in its retrievals in high plant biomass regions. 215 It also motivates determining whether the L3-DCA is overreg-216 ularizing VOD variability, creating an unintended smoothing 217 effect. 218 Therefore, in this study, we first assess the retrievability 219 of VOD by examining observed TB v -TB h differences and 220 using DoI and the proposed SNR. Second, we compare the  2) The SMAP L3-DCA that contains SM, VOD, and con-  observations can be potentially used to reliably retrieve 277 SM and VOD with reduced noise. More details on this 278 regularization method are provided in [26] and [51]. 279 3) The SMAP MTDCA SM, VOD, and ω datasets 280 [38], [52]. Note that ω is constant for the study period. 281 The product is also derived from the SMAP L1C 9-km 282 TBs and applies the MTDCA retrieval algorithm for two 283 consecutive overpasses. The algorithm is based on a time 284 series method, which uses all TB values within a pre-285 defined time window. The default window length is two 286 overpasses (i.e., 2-3 days depending on latitude). VOD 287 is held constant between the two overpasses, but this is 288 repeated for each time-adjacent overpass pair such that 289 information from both time-adjacent overpasses is used 290 (averaged) in the VOD retrieval. Therefore, the VOD 291 variations, especially those due to noise, are reduced. 292 Ultimately, this approach penalizes large changes in 293 VOD between overpasses, eliminating noise more than 294 the physical VOD signal [30].

295
In order to analyze the results, datasets on vegetation density 296 and type are used. This includes the VWC product [42] that 297 is used in the SCA. This VWC product is derived from NDVI 298 seasonal climatologies from the NASA MODIS satellite for 299 use within the SMAP algorithms. Complementarily, we also 300 include the original MODIS NDVI [48]. In addition, land 301 cover (LC) data from the MODIS International Geosphere-302 Biosphere Program (IGBP, MCD12C1 product v.6; 3-km 303 resolution) is used to define homogeneous vegetation classes 304 in two steps: 1) only the fully homogeneous 9-km pixels 305 (i.e., those containing all 3-km pixels of the same LC 306 class) are considered and 2) latitude and homogeneous LC 307 pixels are applied to define seven different vegetation classes: 308 tropical forests, temperate forests, boreal forests, savannahs, 309 shrublands, grasslands, and croplands (Table S1,  Retrievals of SM and VOD from passive microwave mea-314 surements rely on the inversion of a "zeroth-order" radiative 315 transfer model, commonly known as the τ − ω model [23]. 316 In this model, the L-band TBs are represented as the sum 317 of three terms: 1) the upwelling vegetation emission; 2) the 318 downwelling emission from vegetation, which is reflected by 319 the soil and then attenuated by the canopy; and 3) the direct 320 soil emission and its attenuation through the vegetation where γ is the vegetation transmissivity, which depends on the 324 VOD (algebraically represented by τ ) according to Beer's law 325 where θ is the incidence angle. Then, τ (used synony-327 mously with VOD here) is one of the two unknowns to be 328 retrieved.
where the two values of surface reflectivity (r p : one for TB v 359 and one for TB h ) will lead to two equations. In (3)   Second, we use the DoI as proposed in [22]. DoI is a mea-391 sure of how much independent information exists in several 392 measurements (e.g., TB v and TB h ) when the measurements 393 are correlated. DoI is computed as where N is the number of parameters (here, N = 2 if 396 considering a single snapshot with TB v and TB h ) and C n 397 represents the total correlation among the different parame-398 ters X 1 -X n (here, TB v and TB h ). The total correlation is a 399 generalization of the mutual information, which consists of 400 the Kullback-Leibler divergence between the joint and the 401 marginal entropies of the datasets. C n captures the amount of 402 information shared between any of the measurements in a set 403 [22], [55]. Higher total correlation suggests less independent 404 information between two parameters.

405
Third, we introduce an SNR metric to quantitatively assess 406 retrievability. It measures how much TB v and TB h (two cor-407 related measurements) are different relative to the instrument 408 noise. The dispersion (standard deviation) of the polarization 409 difference relative to the TB v -TB h linear dependence is com-410 puted. In Fig. 2, this polarization difference is distance L and 411 is represented by a red solid line.

412
Once σ (L) is obtained, then the SNR metric is 413 SNR = σ 2 (L)   To evaluate the impacts of multitemporal and Tikhonov 441 regularizations on the resulting VOD and noise, both the 442 MTDCA and the L3-DCA products will be assessed and 443 compared at different time scales. Results will be interpreted 444 in the context of DoI and SNR. First, the mean annual 445 VOD and the seasonal amplitude of the raw VOD signal are 446 estimated and compared between the products. Second, the 447 near-Nyquist frequency of VOD (NyVOD) and SM (NySM) 448 will be computed by subtracting the seven-day moving aver-449 ages from both raw variables. The high-frequency changes in 450   6. Summary map of uncertainty of joint SM and VOD retrievals with regularization based on SNR and DOI. Dark green (higher confidence) spans 44.2% of global vegetated land, light green (medium confidence) spans 24.2%, and light yellow (lower confidence) spans 31.6%. DOI * 2 represents the degrees of information of two overpass pairs (or four TB measurements), which should be greater than 3 to be able to retrieve VOD without being an underdetermined problem. We use DOI * 2 here (instead of DOI) for a clearer interpretation of the retrievability, especially in the multitemporal case.   Note that VOD differences between algorithms will not 460 solely be a function of the differences in regularization 461 approaches in (7) and (8)  we also conduct complementary analyses on the impact of 467 different albedos in the VOD differences between algorithms. 468 In addition, we discuss the role that roughness may have on 469 these differences.

472
The capacity of the τ -ω framework to provide accurate 473 VOD retrievals depends on the availability of at least two 474 independent pieces of information. However, Fig. 3 shows 475 that TB v and TB h are highly correlated. Indeed, the TB v -TB h 476 difference narrows as VWC increases [see (2) and (3)]. In the 477 densest canopies where VWC > 9 kg/m 2 (i.e., tropical forests, 478 Figs. 3 and S1) and NDVI > 0.8 (Fig. S2), the differences and 479 their variability become small. This illustrates why TB v and 480 TB h do not represent two independent data sources (Fig. 3) and 481 their differences can be dominated by instrument error. In con-482 trast, regions with less vegetation density (VWC < 1, i.e., 483  The DoI metric can be used to quantify the amount of inde-488 pendent information in the two measurements. Fig. 4 shows 489 the DoI map where, generally, DoI ranges between 1.5 and  . This indicates 523 that the signal is greatly influenced by noise and suggests 524 caution in interpreting DoI alone. A low SNR shows that 525 DoI is likely only higher in tropical forests because the total 526 correlation is low due to noise. This quantifies the problem 527 and shows, geographically precise, where retrievals of VOD 528 may be problematic (Figs. 4 and 5). In other forest types, 529 as well as in savannas, the value of DoI decreases and the SNR 530 increases compared to tropical forests (median DoI and SNR 531 in temperate forests: 1.72 and 1.44, respectively; median DoI 532 and SNR in savannas: 1.60 and 1.71, respectively; Fig. S3). 533 In the case of boreal forests, a higher SNR is found (median 534 SNR = 2.24). Overall, this shows that VOD and SM retrievals 535 based on DCAs should be robust in terms of available infor-536 mation in most land regions, including nontropical forests.

537
In contrast, lightly vegetated, nonforested regions have the 538 largest variations in the difference in horizontal and vertical 539 polarization TBs (Fig. 3). As shown in Figs. 5 and S3, 540 this results in SNR values over 3 in semiarid regions 541 (e.g., the Sahel and central Australia), grasslands (e.g., Central 542 Asia, the U.S. Great Plains, and the Pampas), and croplands 543 (e.g., the U.S. Corn Belt, Ukraine, Argentina, and the SW 544 and SE areas of Australia). In these areas, the low vegetation 545 density permits a good retrievability with high SNR, but in 546 need of regularization as shown by DoI values closer to 547 1.5 (Figs. 4 and S3). Hence, VOD and SM retrievals will 548 be achievable where DoI and SNR are both high. However, 549 based on these SMAP measurements, DOI is well below 2 550 where SNR is high, meaning that some degree of regular-551 ization is needed to stabilize retrievals. DoI can be adjusted 552 with regularization. However, SNR is intrinsic to the satellite 553 measurements and thus cannot be directly altered. Therefore, 554 we anticipate retrieval difficulty of VOD in wooded regions 555 with low SNR, but an improvement on VOD retrievals after 556 regularization in grasslands, croplands, and shrublands as 557 well as in few cases of boreal and temperate forest areas 558 (Figs. 6 and S3).  (Fig. 6). If DoI is doubled in a regularization approach 564 (see Section IV), in regions where SNR is also high, both 565 SM and VOD can be retrieved with lower risk of estimation 566 instability within the optimization (Fig. 6). Fig. 6 should be 567 used as a guide in determining where dual retrievals of SM 568 and VOD are most certain using regularization approaches.  Fig. 7 for MTDCA (multitemporal regularization) and 574 L3-DCA (Tikhonov regularization). The spatial patterns are 575 similar, with Pearson's correlation coefficient (r ) equal to 576 0.89, although mean VOD values for MTDCA are generally 577 higher than those for L3-DCA (Fig. 7). These differences are 578 partially attributable to choices of the roughness parameter (h) 579 in both algorithms, where higher h inputs generally reduce 580 mean VOD. In particular, differences in h explain 67% of 581 variance of the difference between the average VODs of both 582 products [ Fig. 8(a)], where lower L3-DCA's mean VOD can 583 be partially explained by its higher h inputs. Again, note 584 that the L3-DCA includes the SMAP product with Tikhonov 585 regularization and is not a traditional DCA snapshot retrieval.

586
In addition, note that we have found no relationship between 587 differences in average VOD and those in ω [ Fig. 8(b)].

588
The components of VOD variability at low frequency 589 (i.e., seasonal amplitude) are shown in Fig. 9. Note that since 590 the L3-DCA is constrained by NDVI climatology, the seasonal 591 amplitude may partially be driven by the NDVI climatol-   in comparison to that of MTDCA. 635 We now evaluate the higher frequency variability near the 636 Nyquist frequency (periods of 4-7 days for SMAP), which is 637 Fig. 11. DCA estimation cost function using synthetic data. The blue symbol is the true solution, while the green symbol is the noisy solution when the TB error on the order of 1 K is added to the measurements. Adapted from [27, Fig. 1]. more sensitive to noise, but the robustness of which can be 638 improved through regularization [30]. The Nyquist variability 639 removes some of the influence of the NDVI climatology prior, 640 and thus, the L3-DCA NyVOD results are more of a function 641 of the degree of regularization choice (λ). Fig. 10(a) and (b) 642 show the standard deviation of the MTDCA and L3-DCA 643 high-frequency values. A pattern emerges, which provides 644 insight into how retrievable VOD is under different regular-645 ization approaches. The variability of MTDCA NyVOD is 646 higher than that of L3-DCA NyVOD distinctly in tropical 647 and boreal forests. Outside of these forests and in croplands, 648 the NyVOD standard deviation tends to be lower for both 649 algorithms. This implies that the forest biomes need additional 650 constraints on VOD than the TBs alone can provide (such as an 651 input VOD climatology), even with multitemporal persistence 652 assumptions.

653
The estimation of the covariance between SM and VOD 654 high-frequency variabilities provides insights into how much 655 compensation may be taking place in inverting for SM and 656 VOD simultaneously. Fig. 11 shows a representation of the 657 joint VOD-SM cost function for an example of a dual-658 channel retrieval problem without regularization for a given 659 overpass. In this example, the cost function has an elongated 660 valley. Small amounts of noise will result in variations in 661 retrievals following the contours of the valley. In the presence 662 of noise, this compensation will result in positive covariance 663 at the Nyquist frequency as shown by the positive VOD-SM 664 relationship at the minimum cost function values (Fig. 11).

665
Given that SM and VOD errors are typically positively 666 correlated ( Fig. 11 and [30]), we evaluate systematic compen-667 sation between VOD and SM by computing the covariance 668 between the SM and VOD at the Nyquist frequency. If the 669 problem is significantly underdetermined, this may manifest 670 itself as random variability (and thus as positive covariance) 671 in the signal at high frequencies.
672 Fig. 12 shows the resulting covariance between VOD 673 and SM at the Nyquist frequencies for each product. The 674 covariance rather than the correlation is used to normalize out 675 differences between the L3-DCA's and MTDCA's SM-VOD 676 coupling that are due to the standard deviations of SM 677  in croplands (Fig. S5b).

721
Despite differences in the algorithms and coupling with SM, 722 the high-frequency variability of VOD from both products 723 tends to be positively correlated (except in tropical forests, 724 see Fig. 13). This indicates that the aforementioned differences 725 may be playing a larger role on the amplitude of variations 726 across frequencies of the variations. Ultimately, the detection 727 of increases and decreases in VOD generally tends to be 728 similar and in phase.

730
This study evaluates the robustness of VOD retrievals 731 based on SMAP horizontal and vertical polarization TB 732 measurements in the context of instrument noise. The study 733 also assesses how two different SMAP VOD regularization 734 techniques impact this robustness. Toward these goals, first, 735 an SNR metric is proposed to capture variability above instru-736 ment noise; it is used as a complementary metric to the DoI 737 that measures the statistical independence of measurements. 738 Second, VOD retrievals and noise from two different VOD 739 regularization approaches using SMAP observations are qual-740 itatively compared across different time scales (annual mean, 741 seasonal amplitude, and high-frequency variability). Namely, 742 the SMAP MTDCA and L3-DCA products are compared 743 based on multitemporal and Tikhonov regularization tech-744 niques, respectively. 745 We show that VOD can be robustly retrieved with regular-746 ization in regions with lower vegetation density and with more 747 uncertainty in regions with greater vegetation density. Regions 748 with the highest DoI values correspond to high vegetation 749 densities (i.e., tropical forests, VWC > 9 kg/m 2 ), but these 750 values are inflated due to random noise. Moreover, SNR is 751 low based on high TB v -TB h correlations, which indicates that 752 the VOD signal in tropical forests is highly impacted by noise 753 due to TB depolarization. Therefore, interpreting DoI alone is 754 misleading in dense vegetation areas: values of DoI ∼ 2 are 755 due to random noise, which gives the false impression of hav-756 ing two independent TB sources. We conclude that DoI must 757 be interpreted along with SNR for a holistic understanding of If so, it can be reduced to capture VOD dynamics where 815 adequate polarization information exists based on TBs, DoI, 816 and SNR. This reduction could be addressed by using a 817 spatially varying λ (instead of a global constant value) accord-818 ing to a map of VOD noise [see [26, Fig. 2(a)]. Third, 819 we recommend potentially increasing the regularization in 820 the multitemporal retrievals or imposing a priori climatology 821 for noise-dominated regions (mainly tropical forests). Also, 822 future work should be addressed to evaluate the robustness of 823 VOD retrievals for other algorithms (e.g., CMCA [28]) and 824 sensors (the L3 algorithm of SMOS [40]), which are based 825 on time-aggregation approaches using a priori regularization. 826 We expect that the implementation of these changes can lead 827 to more accurately retrieving SM-VOD dynamics.