Olivier Merlin
email: olivier.merlin@cesbio.cnes.fr

Ahmad Al Bitar

Jeffrey P Walker

Yann Kerr

A sequential model for disaggregating near-surface soil moisture observations using multi-resolution thermal sensors

Keywords:

A sequential model is developed to disaggregate microwave-derived soil moisture from 40 km to 4 km 27 resolution using MODIS (Moderate Imaging Spectroradiometer) data and subsequently from 4 km to 500 m 28 resolution using ASTER (Advanced Scanning Thermal Emission and Reflection Radiometer) data. The 1 km 29 resolution airborne data collected during the three-week National Airborne Field Experiment 2006 30 (NAFE'06) are used to simulate the 40 km pixels, and a thermal-based disaggregation algorithm is applied 31 using 1 km resolution MODIS and 100 m resolution ASTER data. The downscaled soil moisture data are 32 subsequently evaluated using a combination of airborne and in situ soil moisture measurements. A key step 33 in the procedure is to identify an optimal downscaling resolution in terms of disaggregation accuracy and 34 sub-pixel soil moisture variability. Very consistent optimal downscaling resolutions are obtained for MODIS 35 aboard Terra, MODIS aboard Aqua and ASTER, which are 4 to 5 times the thermal sensor resolution. The root 36 mean square error between the 500 m resolution sequentially disaggregated and ground-measured soil 37 moisture is 0.

Introduction

Predicting the spatio-temporal variability of hydrological processes requires models that operate at different scales: evapotranspiration and infiltration at paddock-scale, run-off and drainage at catchment-scale, and atmospheric circulation at meso-scale. Due to the complexity of interacting processes [START_REF] Chehbouni | An integrated modelling and remote sensing approach for hydrological study in arid and semi-arid regions: The SUDMED Programme[END_REF], the reliability of model predictions is intimately related to the ability to represent dominant processes in space and time using observations. Remote sensing has shown promise for this application due to its multi-resolution and multi-spectral capabilities [START_REF] Choudhury | Synergism of multispectral satellite observations for estimating regional land surface evaporation[END_REF].

Among the variables observable from space, soil moisture is one of the most crucial parameters that control hydrometeorological processes from paddock-to meso-scale. However, current and near-future spaceborne soil moisture products have a spatial resolution of several tens of kilometers [START_REF] Crow | Upscaling of field-scale soil moisture mea ^surements using distributed land surface modeling[END_REF] -about ~40 km resolution for the forthcoming Soil Moisture and Ocean Salinity (SMOS, [START_REF] Kerr | Soil moisture retrieval from space: The soil moisture and ocean salinity (SMOS) mission[END_REF] mission-, which make their application to hydrological and agricultural models challenging.

U N C O R R E C T E D P R O O F

86 brightness temperature fields prior to soil moisture retrieval.

87

Similarly, Merlin et al. (2008a) developed a deterministic downscaling 88 algorithm that combines 1 km resolution MODIS (MODerate resolu-89 tion Imaging Spectroradiometer) data and a semi-empirical soil 90 evaporative efficiency model. The main advantage of those 91 approaches [START_REF] Bindlish | Subpixel variability of remotely sensed soil moisture: An inter-comparison study of SAR and ESTAR[END_REF]Merlin et al., 2008a) over the 92 purely empirical ones based on log-log plots (e.g. Kim & Barros,93 2002a) is that some physical considerations are used to build a 94 relationship between soil moisture and an ancillary observable; radar 95 backscatter in [START_REF] Bindlish | Subpixel variability of remotely sensed soil moisture: An inter-comparison study of SAR and ESTAR[END_REF] and soil evaporative 96 efficiency in Merlin et al. (2008a).

97

In Merlin et al. (2008a), the disaggregation scale was fixed to 10 98 times the spatial resolution of MODIS thermal data to reduce the 99 random uncertainties in disaggregated soil moisture. The authors 100 observed that the sub-pixel variability of disaggregated soil moisture 101 was significantly correlated with the observed fine-scale soil moisture 102 variability, suggesting that the downscaling algorithm could be 103 applied to spatial resolutions finer than 10 km. Nevertheless, that 104 study did not apply the downscaling approach at multiple resolutions.

105

As a follow-up of Merlin et al. (2008a), this paper seeks to identify 106 optimal downscaling resolutions in terms of disaggregation accuracy In situ measurements of 0-5 cm soil moisture were made using 145 HDAS (Hydraprobe Data Acquisition System) on 16 November over three 9 km 2 sampling areas (denoted as Y2, Y9 and Y12) included in the 40 km by 60 km Yanco area (Merlin et al., 2008b). Within each 9km 2 sampling area, an average of three successive measurements was made ~1 m apart at each node of a 250 m resolution grid.

PLMR-derived soil moisture

The near-surface soil moisture was retrieved from the 1 km resolution brightness temperature collected by the Polarimetric L-band

Multibeam Radiometer (PLMR) on eleven days over the 40 km by 60 km a r e a :3 1O c t o b e r ,2 ,3 ,4 ,5 ,7 ,9 ,1 3 ,1 4 ,1 6 ,1 8N o v e m b e r [START_REF] Merlin | Assessing 726 the SMOS soil moisture retrieval parameters with high-resolution NAFE'06 data[END_REF]. The surface temperature data used for the PLMR soil moisture inversion came from MODIS data on clear sky days, and from in situ measurements on overcast days. The root mean square difference between PLMR-derived and ground-measured soil moisture at 1 km resolution was estimated to 0. ^03 vol./vol. in non-irrigated areas. A bias of about-0.09 ^vol./vol. was obtained over pixels including some irrigation.

This bias was explained by a difference in sensing depth between the L-band radiometer (~0-3cm)andinsitumeasurements(0-5.7 cm), associated with a strong vertical gradient in the top 0-6cm of the soil. Moreover on 3 November, which followed a rainfall event, the PLMR-derived soil moisture seemed to be affected by the presence of water intercepted by vegetation (Merlin et al., 2008b,a). In this study, data from this date were discarded.

MODIS data

The MODIS data used in this paper are the Version 5 MODIS/Terra (10:30 am) and MODIS/Aqua (1:30 pm) 1 km resolution daily surface temperature, and MODIS/Terra 250 m resolution 16-day Normalized Difference Vegetation Index (NDVI). The 16-day NDVI product was cloud free. In between the first (31 October) and last day (18 November) of 1 km resolution PLMR flights over the Yanco area, sixteen MODIS Version 5 surface temperature images with 0% cloud cover were acquired including nine aboard Terra (3,5,7,8,9,10,11,17,18 November) and seven aboard Aqua (31 October,3,4,6,8,9,17 November). Note that more cloud free images were obtained than from Version 4 surface temperature (Merlin et al., 2008a). The overestimation of cloud cover in Version 4 products and the subsequent increase of coverage in Version 5 land surface temperature products are discussed in [START_REF] Wan | New refinements and validation of the MODIS land-surface temperature/ 737 emissivity products[END_REF].MODIS data were re-sampled on the same 1 km resolution grid as PLMRderived soil moisture, and MODIS surface temperature was shifted of (+1 km E; -0.5 km N) and (+2 km E; 0 N) for Terra and Aqua respectively to maximize the spatial correlation with 1 km resolution MODIS NDVI, which was used as a reference for the co-registration.

ASTER data

The ASTER/Terra overpass of the NAFE'06 site was on 16 November 2006 at 10:30 am. Radiometric surface temperature was estimated from 90 m resolution L1B thermal radiances using the emissivity normalization method develop ^ed by [START_REF] Gillespie | Lithologic mapping of silicate rocks using TIMS[END_REF] and [START_REF] Realmuto | Separating the effects of temperature and emissivity: Emissivity 729 spectrum normalization[END_REF] and implemented in ENVI (ENvironment for Visualizing Images, http://www.ittvis.com/envi/) image processing software.

Temperature was computed for each of the five thermal channels using a ^uniform emissivity set to 1, and the actual radiometric temperature was assumed to be equal to the highest computed temperature. Pre-processing of ASTER-derived radiometric temperature consisted of (i) registering the image with an accuracy better than 90 m from reference points (ii) extracting data over three 12 km by 12 km areas centered over the three 9 km 2 sampling areas, (iii) removing data that were visually identified as cloud or as cloud shade on the ground (note that the scene was cloud free over the three 9km 2 sampling areas Y2, Y9 and Y12)and (iv) re-sampling data at 100 m resolution. An important point is that ASTER-derived radiometric surface temperature was not corrected for atmospheric effects.

U N C O R R E C T E D P R O O F 206
The rationale is that only the spatial variability of surface temperature The trade-off between downscaling resolution and accuracy within a 217 disaggregation framework was already mentioned in a previous study 218 (Merlin et al., 2008a). However, Merlin et al. (2008a) did not apply the The approach adopted is to (i) aggregate reference (either PLMR- resolutions, the pixels including no ground measurement were discarded from the analysis and only pixels immediately over the ground measurement sites included. For simplicity, the different spatial resolutions will be denoted using the subscript n, varying from 1 (native resolution) to 12 (for instance, SM PLMR,4 refers to PLMRderived soil moisture aggregated at 4 km resolution and SM mHDAS,5 refers to HDAS-measured soil moisture aggregated at 500 m resolution).

Disaggregation method

The thermal-based disaggregation approach used in this paper is that developed in Merlin et al. (2008a). The equations below represent the case of disaggregation using MODIS data for SMOSresolution pixels simulated by aggregating PLMR-derived soil moisture. Note that all equations also apply for disaggregation using ASTER data.

The soil moisture SM

with T MODIS,n being the soil temperature estimated using MODISderived NDVI and surface temperature, T MODIS,40 its average within the SMOS pixel, and T min,1 the minimum MODIS-derived soil temperature at 1 km resolution. Note that the minimum soil temperature was approximated to the minimum MODIS surface temperature. In [START_REF] Komatsu | Towards a robust phenomenological expression of evaporation 705 efficiency for unsaturated soil surfaces[END_REF], the param ^eter SM C was calibrated for three different soils as function of wind speed

SM C =S M C0 1+ γ r ah ð4Þ
with SM C0 (vol./vol.) being a soil-dependent parameter (ranging from about 0.01 ^vol./vol. for sand to 0.04 ^vol./vol. for clay), and r ah (s m -1 ) the aerodynamic resistance over bare soil. Aerodynamic resistance can be estimated from wind speed measurements u (m s -1 ) at reference height Z (m) given the soil roughness z 0m (m)

r ah = 1 k 2 u ln Z z 0m 2 ð5Þ
with k being the von Karman constant. The soil temperature in Eq. ( 3) is estimated as

T MODIS;n = T surf ;MODIS;n À f v;MODIS;n T v;n 1 À f v;MODIS;n ð6Þ
with T surf,MODIS,n being the MODIS-derived surface temperature, T v,n the vegetation temperature, and f v,MODIS,n the fractional vegetation cover. In Merlin et al. (2008a), the vegetation temperature was approximated to derived soil moisture, and -SD n;1 the mean standard deviation of 1 km 320 resolution PLMR-derived soil moisture computed within each n 2 km 2 321 pixel. The n km resolution error is computed as

RMSE n;n = ½ 1 N =n 2 ∑ðSM MODIS;n À SM PLMR;n Þ 2 0:5 ð9Þ
322 323 with N being the number of 1 km resolution pixels within the 40 km 324 by 60 km study area. The mean sub-pixel variability is computed as

- SD n;1 = 1 N = n 2 ∑SD n;1 ð10Þ
325 326 = 1 N = n 2 ∑ 1 n 2 À 1 ∑ðSM PLMR;n À SM PLMR;1 Þ 2 0:5 ð11Þ
327 328

329

The second criterion denoted C2 is the condition that the error 330 evaluated at the native resolution (n = 1) is minimum. In other words,

331

C2 is satisfied when the downscaling resolution makes the disag-332 gregation output the most accurate with respect to the reference soil 333 moisture data obtained at the thermal sensor native resolution. C2 can 334 be formulated as

RMSE n;1 = 1 N ∑ðSM MODIS;n À SM PLMR;1 Þ 2 0:5 is minimum ð12Þ
335 336 with RMSE n,1 being the root mean square error evaluated at 1 km 337 resolution between the n km resolution disaggregated and 1 km 338 resolution PLMR-derived soil moisture.

339

The criteria C1 and C2 can be applied to the three farms Y2, Y9 and 340 Y12 by replacing in Eqs. ( 8) ^to (12) ^PLMR and MODIS by HDAS and 341 ASTER respectively.

Application to MODIS

The disaggregation algorithm of Eq. ( 2) is applied to each of the eight MODIS/Terra images (5,7,8,9,10,11,17 and 18 November) and to each of the six MODIS/Aqua images (31 October, 4, 6, 8, 9 and 17 November), with a downscaling resolution ranging from 1 to 12 km. However, the range of soil moisture values is also reduced and consequently the larger the resolution, the more limited the spatial representation of the actual soil moisture heterogeneity is.

As MODIS data were used for the PLMR soil moisture inversion, PLMR-derived and MODIS-disaggregated soil moisture are theoretically not fully independent on clear sky days. However, it is argued that the cross-correlation of errors in the PLMR soil moisture measurements and the disaggregated soil moisture fields is not responsible for the good results in Fig. 1. One simple reason is that MODIS temperature has a positive impact on PLMR soil moisture retrievals (increasing with MODIS temperature) and a negative impact on disaggregated soil moisture (decreasing with MODIS temperature). Consequently, the cross-correlation of errors in PLMRderived and MODIS-disaggregated soil moisture would actually make the results poorer. Aqua and Terra. The standard deviation is equal to 0 at 1 km resolution because only one PLMR measurement is available per downscaled pixel at 1 km resolution. Following criterion C1 in Eq. ( 8), an optimal downscaling resolution exists where the RMSE and spatial variability lines cross. Inspection of Fig. 2 shows that the mean optimal resolution is about 3.7 km for MODIS aboard Aqua and 4.2 km for MODIS aboard Terra. Although relatively similar for both sensors, the RMSE of disaggregated soil moisture are remarkably more spread about the mean for Terra than for Aqua. The more consistent disaggregation results using MODIS/Aqua compared to MODIS/Terra was already mentioned in (Merlin et al., 2008a) when applied to 10 km resolution. This is due to the stronger coupling between SEE and soil moisture at 1:30 pm than at 10:30 am. (evaluated at the thermal sensor native resolution) as a function n for Aqua and Terra data. The mean error is higher for Terra than for Aqua, which is consistent with previous results. For both Terra and Aqua, the mean error slightly decreases as spatial resolution increases from 1 to 5 km, and slightly increases for spatial resolutions greater than 5 km.

Following criterion C2, an optimal downscaling resolution is identified at about 5 km for both MODIS/Terra and MODIS/Aqua. Nevertheless, the minimum of RMSE n,1 is not very well defined since the dynamics of the mean value are smaller than the variability observed within the In summary, the application of criteria C1 and C2 to MODIS/PLMR data demonstrates that the optimal downscaling resolution in terms of disaggregation accuracy (using the NAFE'06 data set) is about 4 to 5 km. Also, criterion C1 is better defined than C2 since it smooths out the uncertainties associated with random errors in PLMR-derived soil moisture. Since rice crops were flooded during NAFE'06, no HDAS measurement was made. Consequently, the nearby ground measurements did not represent well the overall "wetness" (including both soil moisture and standing water) of the surface that the disaggregation algorithm actually represents.

When comparing Figs. 1 and4, one observes that the disaggregation approach is much more accurate when applied to MODIS data than when applied to ASTER data. In particular for n =8, the correlation coefficient is about 0.80 for MODIS and0.60 for ASTER.

The relatively poor results obtained using ASTER data can be interpreted as a consequence of the spatial variability of soil moisture at fine scale. As the typical crop size in the study area was about 100-300 m, soil moisture fields were much more heterogeneous at 100 m resolution than at 1 km and above. It is suggested that point-scale measurements aggregated at 100-1000 m resolution were generally more uncertain than 1 km resolution remotely-sensed PLMR-derived soil moisture.

U N C O R R E C T E D P R O O F

Fig. 5 also plots the error RMSE n,1 evaluated at the ASTER native resolution (100 m) as a function of n. Although one observes a minimum of the error for Y12 at n = 7, no minimum is observed for the other farms (Y2 and Y9). Several hypotheses can be postulated to explain these constrasting results. First, when using ground measurements instead of airborne L-band data, reference soil moisture data are representative of the point-scale and may not be representative of the scales integrated to several hundreds of meters, especially over highly heterogeneous irrigated areas like in Y9. Second, the farm-scale variability in Y2 was about the same as the local-scale variability (uncertainty in a single HDAS measurement). Consequently, the disaggregation over that farm was not expected to improve the accuracy of soil moisture at fine scale.

Third, it was seen in the case of MODIS/PLMR that criterion C2 was not very stable from date to date, so no clear result can be expected from only one date with ASTER/HDAS. In summary, the application of criteria C1 and C2 to ASTER/HDAS data suggests that the optimal downscaling resolution in terms of disaggregation accuracy (using the NAFE'06 data set) is about 4 to 5 times the thermal sensor resolution. Criterion C1 is again found to be better defined than C2.

Sequential disaggregation

The general approach of the sequential disaggregation using multiresolution thermal sensors is presented in Fig. 6

SM S i +1 =SM S i + ∂SM ∂SEE ΔSEE S i+1 ð13Þ
484 485

with S i being the sensor of index i. In our case, S 0 , S 1 and S 2 486 corresponds to SMOS, MODIS and ASTER respectively. By using this 487 notation, Eqs. ( 2) and (3) become

SM S i +1 =SM S i +SM C × SMP S i+1 ð14Þ
488 489

with

SMP S i+1 = T S i À T S i +1 T S i À T min ð15Þ
490 491

From the above equations, one is able to identify the parameters that 492 do not vary with scale. In particular, the minimum soil temperature T min and the soil property SM C are assumed to be scale-invariant. An important point is that these assumptions might not be valid in the case of heterogeneous soil within the SMOS-scale pixel. In particular, Merlin et al. (2008a) demonstrated that estimating SM C at high resolution improved significantly the disaggregation accuracy. However, the scale-invariance of SM C was not tested in this paper since only one ASTER image was available whereas a time series would be required (Merlin et al., 2008a).

Application

Based on the results of the previous section, the intermediate resolution is set to four times the MODIS native resolution (4 km) and the target resolution to five times the ASTER native resolution (500 m). In practice, three data sets were derived by defining a 4 km resolution pixel centered on each of the three sampling areas (see black outlines in Fig. 6). This pixel was used to create over the SMOS-scale pixel a 4 km resolution grid, on which the 1 km resolution MODIS and PLMR data were aggregated. The sequential model of Eq. ( 14) was finally applied to each data set. were used to simulate coarse-scale pixels, and a thermal-based disaggregation algorithm was applied using 1 km resolution MODIS and 100 m resolution ASTER data. A key step in the procedure was to identify an optimal downscaling resolution in terms of disaggregation accuracy and sub-pixel soil moisture variability by using two criteria.

The first criterion C1 was to look for the spatial resolution such that the RMSE evaluated at the downscaling resolution be equal to the subpixel soil moisture variability, while the second criterion C2 was to look for the spatial resolution that minimized the RMSE evaluated at the thermal sensor native resolution (1 km for MODIS or 100 m for ASTER). Very consistent optimal downscaling resolutions were obtained for MODIS aboard Terra, MODIS aboard Aqua and ASTER, which were 4 to 5 times the thermal sensor resolution. Note that the operational application of thermal-based methods would require high-spatial-resolution thermal data acquired at hightemporal-resolution, typically 2-3 days. However, high-spatial-resolution (ASTER-like) thermal data are currently available on a monthly basis, which raises the issue of disaggregating low-spatial-resolution (MODIS-like) thermal data at high-temporal-resolution [START_REF] Agam | A vegetation index based technique for spatial sharpening of thermal imagery[END_REF].

Refinements of the sequential disaggregation method would include a physical calibration of the soil evaporative efficiency model, which is at present semi-empirical. Moreover, the disaggregation accuracy is affected by the non-linearity of that exponential function. Recent developments have accounted for the non-linearity of the models used in the disaggregation of remote sensing data with the projection technique [START_REF] Merlin | A downscaling method for 716 distributing surface soil moisture within a microwave pixel: Application to the 717 Monsoon'90 data[END_REF] or the Taylor series including derivative terms at orders superior to 1 citepmerlin08c. The applicability of those approaches and their stability still need to be confirmed at a range of spatial resolutions.

  107 and sub-pixel spatial variability, and demonstrate the utility of this 108 approach for sequential disaggregation of spaceborne surface soil 109 moisture observations using multi-resolution thermal sensors. The 110 development of a sequential approach is motivated by (i) the fact that 111 high-resolution thermal data such as ASTER (Advanced Scanning 112 Thermal Emission and Reflection Radiometer) data generally have a 113 swath width smaller than the SMOS pixel and (ii) the hypothesis that 114 the use of an intermediate resolution provides a better linearized 115 approximation to a non linear function (e.g. soil evaporative efficiency 116 model). One objective of the paper is to assess this hypothesis using 117 data collected during the three-week National Airborne Field Experi-118 ment 2006 (NAFE'06). Airborne L-band data are used to simulate the 119 40 km resolution pixels expected from SMOS, and a thermal-based 120 disaggregation algorithm is applied using MODIS and ASTER data. 121 While the first part of the paper focuses on estimating optimal 122 downscaling resolutions with MODIS and ASTER data, the second part 123 takes advantage of these results to develop a sequential model for 124 disaggregating ~40 km resolution microwave-derived soil moisture 06 was conducted from 31 October to 20 November 2006 128 over a 40km by 60km area near Y anco (-35°N; 146°E) in 129 southeastern Australia. While a full description of the data set is 130 given in Merlin et al. (2008b), a brief overview of the most pertinent 131 details are provided here. The data used in this study are comprised of 132 wind speed measurements, L-band derived soil moisture and MODIS 133 data collected over the Yanco area on twelve days, and ground 134 measurements of 0-5 cm soil moisture and ASTER data collected over 135 three 9 km 2 areas included in the Yanco area on one day (16 136 November) of the experiment.

  monitored at 2 m by a meteorological station 139 (located in the southwestern corner of the Yanco area, see Fig. 1 of 140 Merlin et al. (2008b)) continuously during NAFE'06 with a time step 141 of ^20 min ^. The time series is illustrated in Fig. 1 of Merlin et al. 142 (2008a).

  143 2.2. Ground soil moisture 144

219

  downscaling approach at multiple resolutions. One objective of this 220 paper is to identify the optimal downscaling resolution(s) in terms of 221 disaggregation accuracy when using data from three sensors: MODIS 222 aboard Terra, MODIS aboard Aqua and ASTER.223 3.1. Approach 224

  225 derived or HDAS-measured) soil moisture to the maximum spatial 226 extent (40 km by 60 km for PLMR and 3 km by 3 km for HDAS), (ii) apply 227 the disaggregation method at a range of resolutions, and (iii) compare 228 the disaggregated soil moisture to the reference data for each down-229 scaling resolution. The disaggregation of soil moisture thus requires 230 simultaneous observations of surface temperature and NDVI. Moreover, 231 v a l i d a t i o nr e q u i r e ss o i lm o i s t u r eo b s e r v a t i o n sa tac o m m o ns p a t i a l 232 resolution. Among the twelve dates with at least one (either Terra or 233 Aqua) MODIS image with 0% cloud cover, seven are concurrent with 234 PLMR 1 km resolution flights. For the other five dates (6, 8, 10, 11 and 17 235 November), the PLMR-derived soil moisture data of the day before are 236 used. This extrapolation is valid because no rainfall occurred between the 237 PLMR flight and MODIS overpass on each date. Data are listed in Table 1. 238 Data derived from MODIS, PLMR, ASTER and HDAS are then 239 aggregated to a range of spatial resolutions. MODIS surface tempera-240 ture, MODIS NDVI and PLMR soil moisture are aggregated successively 241 from 1 to 12 km in 1 km increments over the 40 km by 60 km area. 242 Similarly, ASTER surface temperature, ASTER NDVI and HDAS soil 243 moisture are aggregated successively from 100 to 1200 m in 100 m 244 increments over the three 9 km 2 sampling areas. One should note that 245 the spacing between ground measurements (250 m) was smaller than 246 the two first aggregation resolutions (100 and 200 m). For these two 247

Fig. 1

 1 Fig. 1 plots the n km resolution disaggregated soil moisture versus the n km resolution PLMR-derived soil moisture for n =1 ,2,4,8and12.It is apparent that the noise on disaggregated soil moisture is successively reduced by increasing the downscaling resolution.

Fig. 2

 2 Fig. 2 plots the RMSE n,n evaluated at the downscaling resolution as a function of n for each MODIS overpass date, separated according to Aqua and Terra data. The average for all dates is also plotted for each platform. The mean error decreases from about 0.045 ^vol./vol. at 1 km resolution to about 0.015 ^vol./vol. at 12 km resolution for both Aqua and Terra. On the same graph is plotted the mean sub-pixel variability -SD n;1 for all dates. The mean sub-pixel variability increases from 0 to about 0.04 ^vol./vol. at 1 and 12 km resolution respectively for both

Fig. 3

 3 Fig. 3 plots the average and standard deviation of the error RMSE n,1

U

  N C O R R E C T E D P R O O F 397 data set (shown on Fig. 3 by the standard deviation σ). One limitation 398 of the criterion C2 is that it includes both the uncertainty in the 399 disaggregation output and the uncertainty in PLMR-derived soil 400 moisture at the observation scale, so that the RMSE n,1 can never be 401 lower than the measurement error at the native resolution.

Fig. 1 .

 1 Fig. 1. Scatterplots of the MODIS-disaggregated versus PLMR-derived soil moisture using all twelve days of data for different downscaling resolutions: 1 km, 2 km, 4 km, 8 km and 12 km. The correlation coefficient R2 is indicated on each plot.

Fig. 2 .

 2 Fig. 2. Estimating an optimal downscaling resolution by comparing the root mean square error (RMSE) and the sub-pixel soil moisture variability at the disaggregation scale. The mean (thick line) RMSE is equal to the mean sub-pixel variability at about 4 km for both MODIS/Aqua and MODIS/Terra. The other lines represent the different dates.

Fig. 3 .

 3 Fig.3. Root mean square error (RMSE) evaluated at 1 km resolution for downscaling resolutions increasing from 1 to 12 km. Although the standard deviation (σ) between dates is high, the RMSE is minimum at 5 km for both MODIS aboard Aqua and MODIS aboard Terra.

Fig. 5

 5 Fig. 5 plots the RMSE n,n evaluated at the downscaling resolution as a function of n. It is apparent that the error is approximately constant at 100 m and 200 m resolution, which is consistent with the fact that the spacing (250 m)of HDAS measurements was larger than the thermal sensor native resolution so that the spatial variability of HDAS measurements is not represented below 300 m. For all farms, the error is maximum at 200 m, and is minimum at 1200 m resolution with a value of about 0.02 ^vol./vol. On the same graph is plotted the mean sub-pixel variability -SD n;1 for each farm. The mean variability is about 0.02 vol./vol. at n = 1 and is generally maximum at n = 12. Note that

Fig. 4 .

 4 Fig. 4. Scatterplots of the ASTER-disaggregated versus ground-measured soil moisture on 16 November for different downscaling resolutions: 100 m, 200 m, 400 m, 500 m, 800 m and 1200 m. Highlighted pluses correspond to pixels containing standing water (flooded rice fields). The correlation coefficient R2 is indicated on each plot.

Fig. 5 .

 5 Fig. 5. Root mean square error (RMSE) evaluated at the downscaling resolution (top) and at 100 m resolution (bottom) for downscaling resolutions increasing from 100 to 1200 m.

  .The~40kmresolution SMOS-scale soil moisture generated from PLMR data on 16 November is 477 disaggregated at an intermediate resolution (4 km in Fig. 6)u s i n g 478 MODIS data and the MODIS-disaggregated soil moisture is disaggre-479 gated again at a finer resolution using ASTER data. Note that the MODIS 480 data on 16 November were not cloud free over the 40 km SMOS-scale 481 pixel so that the MODIS data on 17 November were used instead.

Fig. 6 .

 6 Fig.6. Schematic diagram presenting the sequential disaggregation of SMOS-scale soil moisture using MODIS and ASTER data.

U

  N C O R R E C T E D P R O O F

Fig. 7

 7 Fig. 7 plots the 4 km resolution MODIS-disaggregated soil moisture versus the 4 km resolution PLMR-derived soil moisture for each of the three data sets. The root mean square error is 0.026 ^vol./vol. ^Fig. 7 also plots the 500 m resolution ASTER-MODIS-disaggregated soil moisture versus the 500 m resolution HDAS-measured soil moisture in each farm. The sequentially disaggregated soil moisture has a RMSE of 0.062 ^vol./vol. and a bias of -0.045 ^vol./vol ^. Results are degraded compared to the case when the ASTER-disaggregated soil moisture was based on HDAS-aggregated measurements and not on MODISdisaggregated soil moisture. The increase of uncertainty could be due

Fig. 8 .

 8 Fig. 8. Scatterplots of the 1 km resolution ASTER-MODIS-disaggregated soil moisture versus HDAS measurements for three different intermediate resolutions: 4 km, 8 km and 12 km, and for the case of "no intermediate resolution".

Fig. 9 .

 9 Fig. 9. Root mean square error on the 1 km resolution ASTER-MODIS-disaggregated soil moisture for an intermediate resolution increasing from 3 to 12 km. The error obtained in the case of "no intermediate resolution" is also indicated.

  SMOS-scale soil moisture generated from airborne L-band data on 16 November was disaggregated at an intermediate resolution (4 km) using MODIS data and the MODISdisaggregated soil moisture was disaggregated again at 500 m resolution using ASTER data. The RMSE between the 500 m resolution sequentially-disaggregated and ground-measured soil moisture was 0.062 ^vol./vol. with a bias of -0.045 ^vol./vol. and soil moisture values ranging from 0.08 to 0.40 ^vol./vol. To assess the impact of the intermediate resolution on disaggregation accuracy, a different approach was proposed by setting the target resolution to 1 km and by increasing the intermediate resolution from 3 to 12 km. The optimal intermediate resolution was found to be 3-5 km, meaning that the use of MODIS data reduced the non-linearity effects across scales between SMOS and ASTER resolutions, despite the increase of uncertainties associated with the combination of MODIS and ASTER data.Beyond the application of multi-resolution soil moisture data to a range of environmental sciences, such an approach could greatly facilitate the validation of coarse-scale microwave-derived soil moisture data using point-scale ground measurements. The sequential model is being implemented over the Valencia Anchor Station area[START_REF] Lopez-Baeza | Representativity of the Valencia 711 and the Alacant anchor stations in the context of validation of remote sensing 712 algorithms and low-resolution products[END_REF] in the SMOS calibration/validation framework.

  MODIS,n disaggregated at n km resolution at first

	order around the SMOS-resolution soil moisture SM PLMR,40 is written as
	SM MODIS;n =S M PLMR;40 +	∂SM ∂SEE	ΔSEE MODIS;n	ð1Þ

with ∂SM/∂SEE being the partial derivative (evaluated at SM SMOS,40 ) of soil moisture to soil evaporative efficiency (SEE), and ΔSEE n the difference between the MODIS-derived SEE estimated at n km resolution and its average within the SMOS pixel. Eq. (

1

) can be further simplified by using the simple expression of SEE from

[START_REF] Komatsu | Towards a robust phenomenological expression of evaporation 705 efficiency for unsaturated soil surfaces[END_REF]

. The downscaling relationship becomes

SM MODIS;n =S M PLMR;40 +S M C × SMP MODIS;n

ð2Þ

with SM C being a semi-empirical parameter that depends on soil type and boundary layer conditions and SMP a normalized soil moisture proxy. In

Merlin et al. (2008a)

, the SMP was defined as SMP MODIS;n = T MODIS;40 À T MODIS;n T MODIS;40 À T min;1

  T min,1 by assuming that vegetation was not undergoing water stress, and 297 fractional vegetation cover was estimated asf v;MODIS;n = NDVI MODIS;n À NDVI min NDVI maxÀNDVI minð7Þ298 299 with NDVI min and NDVI max being the NDVI value that corresponds to 300 bare soil and fully vegetated pixels respectively.

	301	In this study, parameters SM C0 ,N D V I min and NDVI max , as well as
	302	wind speed (r ah ) are assumed to be uniform within the SMOS pixel
	303	(model parameters are listed in Table 2). This invariance assumption
	304	will be further assessed in view of the disaggregation results obtained
	305	at a range of spatial resolutions.
	306	3.3. Downscaling resolution versus disaggregation accuracy
	t1:1 t1:2 t1:3 t1:4 t1:5 t1:6 t1:7 t1:8 t1:9 307 308 309 310 319	Table 1 List of acquisition dates, mean PLMR-derived soil moisture, wind speed measured at Terra (T) or Aqua (A) overpass time (10:30 am/1:30 pm), and minimum MODIS/Terra, MODIS/Aqua and ASTER surface temperature. Date SM PLMR,40 u (m s -1 ) T min,1 (°C) ASTER vol./vol. T A MODIS/T MODIS/A 31 October 0.046 6.0 36.2 4 November 0.11 7.6 36.5 5 November 0.065 5.0 35.0 6 November 0.065⁎ 7.5 37.6 7 November 0.043 7.4 33.3 Two different criteria are developed to estimate an optimal downscaling resolution for each of the three sensors. The first criterion denoted C1 is the condition that the disaggregation error evaluated at the downscaling resolution is equal to the observed sub-U N C O R R E C T E D P R O O F
	t1:10	8 November	0.043⁎	9.4	6.3	31.7	35.4
	t1:11	9 November	0.040	10.5	4.1	31.4	37.7
	t1:12	10 November	0.040⁎	11.9		36.1
	t1:13	11 November	0.040⁎	5.3		36.8
	t1:14	16 November	0.11	13.0			19.0
	t1:15	17 November	0.11⁎	4.5	3.6	32.2	36.3
	t1:16	18 November	0.055	5.1		34.7
	t1:17	⁎ PLMR data from the day before.

296 311 pixel variability. Intuitively, if the error on disaggregated soil moisture 312 is smaller than the sub-pixel variability, then the downscaling 313 resolution is too coarse to represent the actual variability; and 314 conversely if the error is larger, then the downscaling resolution is 315 too fine. C1 can be formulated as RMSE n;n = -SD n;1 ð8Þ 316 317 with RMSE n,n being the root mean square error evaluated at the 318 (n km) disaggregation resolution between disaggregated and PLMR-

Please cite this article as: Merlin, O., et al., A sequential model for disaggregating near-surface soil moisture observations using multiresolution thermal sensors, Remote Sensing of Environment (2009), doi:10.1016/j.rse.2009.06.012

Acknowledgements

The NAFE'06 participants are gratefully acknowledged for their participation in collecting this extensive data set. The National Airborne Field Experiments have been made possible through infrastructure (LE0453434 and LE0560930) and research (DP0557543) funding from the Australian Research Council, and the collaboration of a large number of scientists from throughout Australia, United States and Europe. Initial setup and maintenance of the study catchments was funded by a research grant (DP0343778) from the Australian Research Council and by the CRC for Catchment Hydrology. This work was funded by the French program Terre-Océan-Surface-Atmosphère and the Centre National de la Recherche Scientifique.

U N C O R R E C T E D P R O O F

541 particular, output errors are expected to be reduced by setting the 542 downscaling resolution to a value larger than the resolution that was 543 found to be optimal when using one sensor (MODIS or ASTER) 544 independently from the combination of both.


Gerardo Salinas
email: gerardo.salinassanchez@enscbp.fr

Cara Lozon

Alexander Kuhn
email: kuhn@enscbp.fr

Unconventional Applications of the Magnetohydrodynamic Effect in Electrochemical Systems

Keywords: Magnetic field, magnetohydrodynamic effect, microfluidics, electrocatalysis, selfpropelled swimmers

The theory of electromagnetism has made countless contributions to the understanding of physicochemical phenomena and continues to be relevant in many scientific fields. The use of electromagnetism to induce physical flux in a fluid containing ions is known as magnetohydrodynamics (MHD). In electrochemistry, MHD has been explored, among others, to produce for example chiral structures on the surface of electrodes. However, the Lorentz force responsible for inducing hydrodynamic convection in an electrochemical system also allows bypassing mass transport limitations in novel and unconventional processes. This review aims to highlight the principles of MHD convection and to discuss recent and unconventional approaches to the integration of magnetic fields in electrochemical systems.

Introduction

The synergy between electric and magnetic fields goes well beyond Maxwell's electromagnetism theory, since such an interplay can be extended to many different areas of science, and among others also to electrochemistry [START_REF] Monzon | Magnetic fields in electrochemistry: The Kelvin force. A mini-review[END_REF][START_REF] Monzon | Magnetic fields in electrochemistry: The Lorentz force. A mini-review[END_REF][START_REF] Kumar | Chiral Magneto-Electrochemistry[END_REF][START_REF] Luo | Electrochemistry in Magnetic Fields[END_REF][START_REF] Bau | Applications of Magneto Electrochemistry and Magnetohydrodynamics in Microfluidics[END_REF]. Due to the electric and ionic currents involved in an electrochemical process, it is possible to assume that the presence of a magnetic field can affect the overall kinetics of an electrochemical conversion. From magnetic hyperthermia [START_REF] Gatard | Use of magnetic fields in electrochemistry: A selected review[END_REF] to magnetic attraction [START_REF] Tang | An internal magnetic field strategy to reuse pulverized active materials for high performance: a magnetic three-dimensional ordered macroporous TiO2/CoPt/α-Fe2O3 nanocomposite anode[END_REF], magnetoelectrochemistry has passed from an experimental curiosity to a wellstablished field of research. For example, in recent years the study of spin polarization effects has become a hot topic of research [START_REF] Mogi | Unusual Spin Polarization in the Chirality-Induced Spin Selectivity[END_REF][9][10]. Among the different magnetoelectrochemical approaches, the most popular one is probably the induction of hydrodynamic convection by the Lorentz force, as it is a powerful tool to overcome mass transport limitations [START_REF] Monzon | Magnetic fields in electrochemistry: The Lorentz force. A mini-review[END_REF][START_REF] Kumar | Chiral Magneto-Electrochemistry[END_REF]. Briefly, when an homogeneous magnetic field (B ) is imposed perpendicular to an ion flux (J ), associated to an electrochemical process, a Lorentz force ((F_L ) ) perpendicular to both B and J is produced. This is a well-known phenomenon, the so-called magnetohydrodynamic effect (MHD). One specific type of research related to the MHD effect is focused on the electrodeposition of metals, alloys or conducting polymers, in order to induce surface chirality [START_REF] Naaman | Chiral Induced Spin Selectivity and Its Implications for Biological Functions[END_REF][START_REF] Mogi | Breaking of Odd Chirality in Magnetoelectrodeposition[END_REF]. However, since the MHD convection is a surface localized phenomenon, it has gained considerable attention also in different other fields of research, such as redox microfluidics or electrocatalysis. The aim of this review is to describe the physico-chemical principles governing the MHD effect and to discuss some unconventional trends in this field.

Fundamental concepts behind the magnetohydrodynamic effect

In electrochemistry, mass transport is governed by the diffusion of electroactive species towards the electrode surface. Under classic conditions, when a potential difference is applied between two electrodes a concentration gradient is formed, due to the continuous consumption of the analyte (Figure 1a). The thickness of this so-called Nernst diffusion layer (δD) is a function of time, the electrode potential, the concentration (C*) and the diffusion coefficient of the analyte (D). Thus, as the size of the diffusion layer increases, the current obtained by the electrolysis of the electroactive species is governed by the δD value. However, in the presence of a homogeneous B , a local Lorentz force is induced on the ion flux associated to the electrochemical process (Equation 1).

F _L=J ×B

(1)

Under these conditions a hydrodynamic boundary layer (δ0), with an ion flux velocity gradient across the layer (U(y)) is formed (Figure 1b). This leads to a decrease of the thickness of δD with a concomitant acceleration of mass transport, which causes a continuous supply of electroactive species towards the electrode surface, improving the mass-transport limited current. Willner et al.

proposed a theoretical model that describes the influence of the (F_L ) on the limiting current in the presence of a homogeneous magnetic field oriented parallel to the electrode surface (equation 2) [START_REF] Lioubashevski | Magnetic Field Effects on Electrochemical Processes: A Theoretical Hydrodynamic Model[END_REF][14][START_REF] Katz | Magnetic Field Effects on Bioelectrocatalytic Reactions of Surface-Confined Enzyme Systems: Enhanced Performance of Biofuel Cells[END_REF].

i_L=0.63〖(ρR)〗^(-1⁄3) D^(8⁄9) v^(-2⁄9) 〖(nFC^*)〗^(4⁄3) B ^(1⁄3) ( 2
)
where n is the number of electrons, F is the Faraday constant, R is the size of the electrode, ρ is the specific density of the fluid, v is the kinematic viscosity, D and C* are the diffusion coefficient and bulk concentration of the electroactive species, respectively. Since the obtained Lorentz force acts only on moving charged species and strongly depends on their velocity and the strength of B , its maximum value is obtained when J and the magnetic field are perfectly orthogonal to each other. For example, the magnitude of a Lorentz force produced by a magnetic field of 1 T, orthogonal to a current density of 100 mA/cm2, is comparable to the buoyancy force (≈ 103 N/m3), but inferior to mechanical stirring (≈ 105 N/m3). However, the limiting current obtained in the presence of a homogeneous magnetic field is comparable to the one produced by electrochemical measurements performed with a rotating disk electrode [START_REF] Hinds | Magnetic Field Effects on Copper Electrolysis[END_REF][START_REF] Hinds | Influence of magnetic field forces on electrochemical mass transport[END_REF]. For example, Hinds et al. observed that a limiting current of 9.4 A/m2 produced in the presence of a 0.5 T magnetic field, is roughly equivalent to the current obtained with a rotation rate of 10 rpm [START_REF] Hinds | Influence of magnetic field forces on electrochemical mass transport[END_REF]. The hydrodynamic convection induced by the Lorentz force is intimately related to the spatial arrangement of the magnetic field with respect to the position of the electrode. This interaction can cause two different types of MHD flow in the presence of a homogeneous magnetic field. When the magnetic field is placed parallel to the surface of the electrode, the entire ion flux is perpendicular to B (B ⊥J ), producing a macro-MHD flow (Figure 1c). Under these conditions, it is possible to predict the limiting current according to the above mentioned theoretical model. However, when B is placed perpendicular to the electrode surface (B ∥J ), only the ionic current at the electrode edges is able to generate a Lorentz force, causing the formation of macro-MHD flow along the electrode edges (Figure 1d). In this case, the effect of the magnetic field on the overall electrochemical kinetics is strongly related to the intensity of B and the distance between the electrode and the magnet [START_REF] Mogi | Effects of vertical magnetohydrodynamic flows on chiral surface formation in magnetoelectrolysis[END_REF]. In addition, in such a configuration, due to the presence of local inhomogeneities, micro-MHD vortices are formed at the electrode surface (Figure 2a). This micro-MHD flow pattern induces localized high current density spots at a considerably smaller scale, impossible to produce via mechanical stirring. Finally, it is important to highlight that a magnetic field can induce other forces in addition to (F_L ) , such as a paramagnetic force ((F_p ) ), a field gradient force ((F_B ) ) and an electrokinetic force ((F_E ) ) (Table 1) [START_REF] Hinds | Magnetic Field Effects on Copper Electrolysis[END_REF][START_REF] Hinds | Influence of magnetic field forces on electrochemical mass transport[END_REF]. From a physical point of view, their origin is intimately related to the properties of the electrolyte ((F_p ) ) and the applied magnetic field ((F_B ) ) or the dynamic behavior of the charged species at the electrode/electrolyte interface ((F_E ) , (F_L ) ). However, (F_B ) is absent in the presence of a homogeneous magnetic field, and (F_p ) is negligible compared to the driving force of diffusion ((F_D ) ). Therefore, it is possible to assume that the hydrodynamic convection is driven by the Lorentz force in the vicinity of the electrode surface, and on a smaller scale, in the diffuse layer, by the electrokinetic force.

Table 1. Typical forces acting in aqueous electrolytes.

Force Typical value (N/m3) Physical origin Definition Driving force for diffusion ((F_D ) ) 1010 Force due to the presence of a concentration gradient.

Paramagnetic force ((F_P ) ) 104

Magnetic properties Paramagnetic susceptibility of species in the diffusion layer Field gradient force ((F_B ) ) 101

Magnetic properties Force due to the presence of a magnetic field gradient.

Lorentz force ((F_L ) ) 103

Motion of charges Motion of charges orthogonal to a magnetic field.

Electrokinetic force ((F_E ) ) 103

Motion of charges Force on charge carriers under the influence of a magnetic field in the diffuse layer.

Unconventional applications based on the magnetohydrodynamic effect

After these theoretical considerations, we would like to present in this paragraph a series of rather unconventional applications of the MHD concept. The enhancement of the faradic current due to MHD induced convection has been quite well-studied over the last 40 years, but some studies have also focused on more exotic phenomena, such as the electro-induction of surface chirality. In the present contribution, all the different examples of the MHD effect involving electro-deposition or dissolution of metals and polymers are considered as conventional applications. [START_REF] Monzon | Magnetic fields in electrochemistry: The Kelvin force. A mini-review[END_REF][START_REF] Monzon | Magnetic fields in electrochemistry: The Lorentz force. A mini-review[END_REF][START_REF] Kumar | Chiral Magneto-Electrochemistry[END_REF][START_REF] Luo | Electrochemistry in Magnetic Fields[END_REF][START_REF] Gatard | Use of magnetic fields in electrochemistry: A selected review[END_REF][START_REF] Mogi | Unusual Spin Polarization in the Chirality-Induced Spin Selectivity[END_REF][START_REF] Mogi | Breaking of Odd Chirality in Magnetoelectrodeposition[END_REF] However, recent attempts have been made to couple the MHD effect with alternative electrochemical systems, where this magnetoelectrochemical phenomena is the driving force to produce dynamic behavior. For example the hydrodynamic convection induced by the Lorentz force has been successfully used to boost the directional pumping and mixing of fluids (redox magnetohydrodynamic microfluidics), to promote mass transfer and facilitate bubble ejection (electrocatalysis) and to generate motion (Lorentz force-driven swimmers). In the following, we summarize the fundamental concepts concerning the above mentioned electrochemical processes.

Electrocatalysis

A first and rather straightforward idea is to benefit from the MHD effect in the context of the design of highly efficient electrocatalysts, which are of importance to develop novel energy conversion and storage devices, i.e., water electrolysers, fuel cells or metal-air batteries. Commonly the efficiency of an electrocatalytic process can be fine-tuned by increasing the exposed amount of active sites or by enhancing the intrinsic activity of the catalysts. However, an interesting alternative is to enhance the overall performance of an electrocatalytic system also by improving the efficiency of local mass transport. In this context, the MHD effect is presented as a powerful tool in electrocatalysis, due to the surface-localized mixing of the solution and the efficient gas bubble removal (Figure 2a) [START_REF] Zhang | Recent Advances in Magnetic Field-Enhanced Electrocatalysis[END_REF]. It is important to highlight that also alternative magnetic field-based methods have been developed to enhance the electrocatalytic performance such as hyperthermia, Maxwell stress effect or spin selectivity [START_REF] Zhang | Recent Advances in Magnetic Field-Enhanced Electrocatalysis[END_REF]. Nonetheless, in this section we focus exclusively on the basic concepts of MHDenhanced electrocatalysis.

The macro-and micro-MHD convection in electrocatalysis impacts primarily the removal of gas bubbles from the electrode surface [START_REF] Matsushima | Gas bubble evolution on transparent electrode during water electrolysis in a magnetic field[END_REF]. In a first order approximation, at the macroscopic level, the theoretical residence time and the size of the bubbles released from the electrode surface, is intimately related to the net magnetic force ((F_net ) ). For example, when (F_L ) and the buoyancy force ((F_B ) ) are in the same direction, the upward pumping effect of the hydrodynamic convection theoretically reduces the average release size and residence time (Figure 2b), whereas the opposite effect is expected when (F_L ) and (F_B ) are opposite to each other (Figure 2c) [START_REF] Lin | The effect of magnetic force on hydrogen production efficiency in water electrolysis[END_REF][START_REF] Kaya | Investigation of alkaline water electrolysis performance for different cost effective electrodes under magnetic field[END_REF][START_REF] Liu | Experimental and numerical investigation of gas-liquid flow in water electrolysis under magnetic field[END_REF]. However, at the microscopic scale, the micro-MHD vortex triggers a more complex scenario, since the release of bubbles depends on the spatial distribution of low pressure regions within the bubble [START_REF] Liu | Hydrogen bubble growth at micro-electrode under magnetic field[END_REF]. According to Liu et al., at a microelectrode, the hydrodynamic convection induced by the Lorentz force generates low pressure regions at the lower half part of the bubble [START_REF] Liu | Hydrogen bubble growth at micro-electrode under magnetic field[END_REF]. This is caused by the interplay between the macro-and the micro-MHD vortex. When using a microelectrode (d ≈ 0.2 mm), the macro-MHD vortex produces a more violent convection at the electrode interface, in comparison to the micro-MHD flow, which decreases the pressure at the bottom part of the bubble. Such a different fluid flow induces a pressure difference around the bubble, which nails it to the electrode surface. However, at a macro-electrode the edge induced MHD flow enhances the upper part of the flow, producing a low pressure region above the top of the bubble, thereby enabling its release. In addition, the low pressure regions can evolve over time, due to the possible formation of neighboring bubbles on the electrode surface. In such a case, the formation of a high pressure region between the bubbles acts like an external force to push them apart [START_REF] Liu | Electrode-normal magnetic field facilitating neighbouring electrochemical bubble release from hydrophobic islets[END_REF]. Furthermore, the influence of the micro-MHD flow can be fine-tuned by changing the low pressure regions, either by using porous [START_REF] Liu | Porous electrode improving energy efficiency under electrode-normal magnetic field in water electrolysis[END_REF] or fringed electrodes [START_REF] Takagi | Nanobubble formation from ionic vacancies in an electrode reaction on a fringed disk electrode under a uniform vertical magnetic field -1. Formation process in a vertical magnetohydrodynamic (MHD) flow[END_REF][START_REF] Takagi | Nanobubble formation from ionic vacancies in an electrode reaction on a fringed disk electrode under a uniform vertical magnetic field -2. Measurement of the angular velocity of a vertical magnetohydrodynamic (MHD) flow by the microbubbles originating from ionic vacancies[END_REF] and arrangements of multiple magnets [START_REF] Weier | The effect of a Lorentz-force-driven rotating flow on the detachment of gas bubbles from the electrode surface[END_REF]. The hydrodynamic convection driven by the induced Lorentz force, impacts the overall performance of an electrocatalytic reaction in three different ways: (i) by reducing the ohmic voltage drop, (ii) by decreasing the activation potential, due to the continuous release of bubbles that otherwise would inhibit the active surface of the electrode, and (iii) by partially decreasing the thickness of the diffusion layer, which leads to an enhanced mass transport. Such advantages have been explored to enhance the performance of alternative electrocatalytic systems. For example, Kenis et al., observed a decrease of the overall cell potential during the electrolysis of CO2, in the presence of an external magnetic field [START_REF] Bhargave | Decreasing the Energy Consumption of the CO2 Electrolysis Process Using a Magnetic Field[END_REF]. Furthermore, Wilmer et al., demonstrated the possible enhancement of the conversion rates of bioelectrocatalytic reactions. [14,15] A 3-fold increase of the bioelectrochemical conversion rate of oxygen, lactate and glucose, in the presence of a constant magnetic field, was observed.

Redox magnetohydrodynamic microfluidics

A second and less obvious field of research using the MHD effect is the design of microfluidic systems for directional pumping and mixing, which has gained considerable attention due to potential applications ranging from biochemical analysis to chemical synthesis [START_REF] Xia | Nonlinear microfluidics: device physics, functions, and applications[END_REF]. Microfluidics is an interesting field of research since it allows the handling of fluids in small quantities in an automated manner. This leads to different advantages such as the production of less waste, reduced energy consumption, portability, and enhanced sensitivity. Different well-stablished microfluidic pumping methods have been developed i.e., electrokinetic, mechanical, centrifugal and redox-MHD [START_REF] Xia | Nonlinear microfluidics: device physics, functions, and applications[END_REF][START_REF] Weston | Magnetic Fields for Fluid Motion[END_REF]. In particular, the latter presents the unique opportunity to control microfluidics, that means to stop, start, reverse and fine-tune the flow without the need of an external valve or redesigning the device. It is important to highlight that magnetic forces can be use in a wide range of microfluidics applications, i.e. mixing, transport and trapping or for responsive valves. However, for such sophisticated applications, hybrid ferromagnetic materials are required [START_REF] Pamme | Magnetism and microfluidics[END_REF].

The redox-MHD microfluidic approach takes advantage of the synergy between the ionic flux associated to a faradaic process and a magnetic field orthogonal to the electrode surface [START_REF] Anderson | Investigations of Redox Magnetohydrodynamic Fluid Flow At Microelectrode Arrays Using Microbeads[END_REF][START_REF] Weston | Redox-Magnetohydrodynamic Microfluidics Without Channels and Compatible with Electrochemical Detection Under Immunoassay Conditions[END_REF][START_REF] Sikes | Characterization of Nanoparticles in Diverse Mixtures Using Localized Surface Plasmon Resonance and Nanoparticle Tracking by Dark-Field Microscopy with Redox Magnetohydrodynamics Microfluidics[END_REF]. In this approach, the redox reactions are triggered by applying a relatively low potential difference between two electrodes (of the order of millivolts), using 1:1 molar ratio solutions of the corresponding electroactive species. Under these conditions, electrodissolution of the electrodes and bubble formation at their surface can be avoided. As stated above, the magnitude of the induced Lorentz force is given by equation 1, thus in a first order approximation, under the influence of a constant B , relatively low faradic currents may already induce a MHD-flux at the surface of the electrodes. Different reversible redox probes with relatively low electrode potentials and good electron transfer kinetics can be used. Various attempts to visualize the redox MHD flow patterns triggered by the oxidation/reduction of classic redox couples, i.e., Fe(CN)64-/3-and Ru(NH3)62+/3+ or by using the ionic exchange involved in the charging/discharging process of π-conjugated polymers, have been developed [START_REF] Sahore | Flat Flow Profiles Achieved with Microfluidics Generated by Redox-Magnetohydrodynamics[END_REF][START_REF] Khan | Chip-Scale Electrodeposition and Analysis of Poly(3,4-ethylenedioxythiophene) (PEDOT) Films for Enhanced and Sustained Microfluidics Using DC-Redox-Magnetohydrodynamics[END_REF][START_REF] Khan | Redox-Magnetohydrodynamically Controlled Fluid Flow with Poly(3,4-ethylenedioxythiophene) Coupled to an Epitaxial Light Sheet Confocal Microscope for Image Cytometry Applications[END_REF]. In the majority of the studies, the redox-MHD microfluidic device is based on a microarrangement of parallel band electrodes (Figure 3a). Furthermore, coupling redox MHD with microscopy techniques, such as dark-field microscopy or light sheet microscopy [START_REF] Sikes | Characterization of Nanoparticles in Diverse Mixtures Using Localized Surface Plasmon Resonance and Nanoparticle Tracking by Dark-Field Microscopy with Redox Magnetohydrodynamics Microfluidics[END_REF][START_REF] Khan | Redox-Magnetohydrodynamically Controlled Fluid Flow with Poly(3,4-ethylenedioxythiophene) Coupled to an Epitaxial Light Sheet Confocal Microscope for Image Cytometry Applications[END_REF], facilitates single-particle tracking at the nanoscale range, which allows quantifying the fluid velocity produced by relatively low currents (Figure 3b). Commonly, it has been demonstrated that the MHD flow speed, between the band electrodes, is in the 5 -50 µm/s range [START_REF] Gao | 3D Imaging of Flow Patterns in an Internally-Pumped Microfluidic Device: Redox Magnetohydrodynamics and Electrochemically-Generated Density Gradients[END_REF]. It is important to highlight that the intensity of (F_L ) and its concomitant fluid velocity depends on additional experimental factors, such as the viscosity of the solution, the border effect, the distance between the electrodes and counteracting flows or shear stress. However, the most outstanding feature of the redox MHD microfluidic approach is the possible control of the directionality of the flux, by changing the orientation of the magnetic or the electric field. For example, Fritsch et al., took advantage of such a direction control in order to develop a semicontinuous, unidirectional microfluidic pumping device [START_REF] Khan | Sustaining redox-magnetohydrodynamics (R-MHD) microfluidics by switching oppositely-polarized permanent magnets: Synchronized activation and automation[END_REF]. In this work, a multi-pulse galvanostatic perturbation scheme was efficiently coupled to an automatic switching of the B orientation to produce a semi-continuous unidirectional flux. In addition, the flux velocity produced by B , which is orthogonal to the microfluidic device, can be influenced by the presence of magnetic field gradients ((∇B) ), positioned close to the electrodes [42]. The force produced by (∇B) is able to generate a crossflow that overlaps with the classic MHD flux, causing a slight redirection of the main flow. This is associated to a localized change of the magnetic flux density, which induces an additional field gradient force (F_B ) . Figure 3. (a) Schematic illustration of the electrochemical processes involved in a RMHD device with a representation of the microarrangement of parallel band electrodes, the redox reactions, the ionic current, and the magnetic field which produces the MHD force and the fluid flow. (b) Trajectory plots of mixed NP suspensions at a constant applied current (50 μA) in the absence (top) and presence of a magnetic field (bottom). Adapted from reference [START_REF] Sikes | Characterization of Nanoparticles in Diverse Mixtures Using Localized Surface Plasmon Resonance and Nanoparticle Tracking by Dark-Field Microscopy with Redox Magnetohydrodynamics Microfluidics[END_REF].

Lorentz force-driven swimmers

A complementary approach with respect to the one described in the previous paragraph is to generate MHD not in a stationary device like a microfluidic channel, but around mobile objects, in order to induce a reciprocal effect. This should greatly affect their dynamic features. A family of objects, which is interesting in this context, are macro-, micro-and nano swimmers. These are devices that harvest energy from an external input and transform it into motion. Such systems have gained considerable attention due to their numerous potential applications ranging from sensing to biomedicine [START_REF] Yuan | Janus Micromotors Coated with 2D Nanomaterials as Dynamic Interfaces for (Bio)-Sensing[END_REF][START_REF] Ou | Micro-/Nanomotors toward Biomedical Applications: The Recent Progress in Biocompatibility[END_REF][START_REF] Salinas | Recent progress in enzyme-driven micro/nanoswimmers: From fundamentals to potential applications[END_REF]. Different propulsion mechanisms based on asymmetric release of bubbles, selfgeneration of local chemical gradients and electric fields (self-diffusiophoresis and selfelectrophoresis, respectively) or applying external stimuli i.e., light, sound and electric or magnetic fields, have been extensively studied [START_REF] Salinas | Chemo-and Magnetotaxis of Self-Propelled Light-Emitting Chemoelectronic Swimmers[END_REF][START_REF] Arnaboldi | Direct dynamic readout of molecular chirality with autonomous enzyme driven swimmers[END_REF][START_REF] Palagi | Light-Controlled Micromotors and Soft Microrobots[END_REF][START_REF] Fernández-Medina | Recent advances in nano-and micromotors[END_REF][START_REF] Han | Engineering of Self-Propelling Microbots and Microdevices Powered by Magnetic and Electric Fields[END_REF][START_REF] Zhou | Magnetically Driven Micro and Nanorobots[END_REF]. However, developing novel propulsion mechanisms that enhance the swimmer speed and enable trajectory control is still a challenge. For example, recently self-propelled magnetotactic swimmers, where motion is triggered by a physico-chemical processes and the trajectory is guided by a magnetic field, have been designed [START_REF] Zhou | Magnetically Driven Micro and Nanorobots[END_REF]. However, such devices require the presence of ferromagnetic components, which makes their design more complicated and limits the degree of freedom in terms of materials choice. In this context, the MHD effect is an interesting alternative to boost the dynamic behavior of these moving particles. Theoretical studies show that the MHD acceleration of fluid surrounding an object enables propulsion, since the produced thrust is comparable to the weight of the swimmer [START_REF] Zharov | Magnetohydrodynamic Self-propulsion of Active Matter Agents[END_REF]. Recently, the synergy between the self-electrophoretic mechanism and an external magnetic field was explored. In this work, an external B acts on the ionic fluxes produced by the spontaneous oxidation of Mg and the reduction of H+ at the extremities of a Janus Mg/Pt swimmer (Figure 4a) [START_REF] Salinas | Lorentz Force-Driven Autonomous Janus Swimmers[END_REF]. Such reactions cause a continuous flux of ions towards or away from its surface, due to the consumption of electroactive species, which produces, by placing a magnetic field orthogonal to the surface of the self-electrophoretic object, a MHD vortex at each extremity of the device. Under these conditions two well-defined macro-MHD flow patterns along the edges of the electrodes were observed. In addition to the MHD vortices, an ion flux parallel to the surface of the device is formed to compensate the electron flow from the anode to the cathode. Such a horizontal ionic current, in the presence of the external B , generates an additional Lorentz force, inducing a torque that causes a predictable clockwise or anticlockwise motion as a function of the orientation of the magnetic field (Figure 4b). As stated above, the generated hydrodynamic flow improves the reaction kinetics due to a better mixing of the solution in the vicinity of the swimmer surface, and by removing efficiently gas bubbles. Thus, this novel propulsion mechanism, based on the synergy between the magnetic field and ionic flux, is an interesting alternative to boost the propulsion speed by up to two orders of magnitude. Finally, it is important to point out, since the ionic flux is triggered by a selfelectrophoretic mechanism, that this concept is generic. Therefore, it is possible to envision the use of this approach for the design of a vast variety of Lorentz force-driven swimmer with alternative functionalities.

Conclusions and perspectives

This short review allows a comprehensive in-depth look into the principles governing electrochemically triggered MHD convection. We specifically aim to draw attention to current and unconventional applications of the MHD effect, which illustrate its potential interest for optimizing electrocatalysis, redox magnetohydrodynamic microfluidics, and Lorentz force-triggered motion in microswimmers. The MHD effect can be strategically used to control with precision the flow of ionic currents, by adjusting the strength and spatial position of the magnetic field. In addition, the Lorentz force promotes fluid circulation and ultimately diminishes the impeding influences of the diffusion layer and bubble formation. Furthermore, this concept allows also a precise control of microswimmer propulsion, in terms of trajectory and speed of the object. The described Mg/Pt hybrid microswimmers enable bypassing the integration of ferromagnetic components, and this could inspire the development of alternative Lorentz force-driven Janus swimmers composed of a wide range of materials. Although the MHD effect is very well-known, its application to alternative electrochemical research topics, such as electroanalysis, water remediation and electrosynthesis, is promising and merits further exploration. For instance, the straightforward optimization of the intensity and spatial position of the magnetic field, allows fine-tuning the MHD convection, and thus maximizing the efficiency of the electrochemical systems. In addition, since the induced Lorentz force is a phenomenon localized close to the surface, it can be influenced by the physico-chemical properties of the working electrode, allowing eventually the on purpose generation of micro-MHD vortices. For example, the use of highly ordered macroporous electrodes or well-positioned nanoparticles can lead to the formation of original localized micro-MHD vortices that may enhance the overall reaction kinetics (Figure 5). This type of hydrodynamic convection at very small scales may lead to a decrease of mass transfer resistance at such well-structured surfaces, which is not possible by conventional convection approaches (rotating disk electrodes or classic stirring). Finally, it is important to highlight that the interdisciplinarity of the concept and the resulting applications of the MHD effect should stimulate collaborative approaches to further study its universal impact on electrochemistry and also on more technologically orientated innovations.

Figure 1 .

 1 Figure 1. Illustration of the mechanism of the mass transport of electroactive species (a) in the absence and (b) in the presence of an applied magnetic field. J is the diffusional flux of the electroactive species, Cel is the concentration at the electrode surface, δD and δ0 are the Nernst diffusion layer and the hydrodynamic boundary layer thickness, respectively, and U0 is the fluid velocity at the outer edge of the hydrodynamic boundary layer. (c) Illustration of the MHD flow under the influence of a homogeneous magnetic field when B J and (d) when B // J.

Figure 2 .

 2 Figure 2. (a) Illustration of the micro-MHD flux generated around bubbles when B // J. (b) Illustration of the forces involved in the bubble release mechanism under the influence of a homogeneous magnetic field when FL and Fb are oriented in the same direction and (c) in the opposite direction. Adapted from reference [19].

Figure 4 .

 4 Figure 4. (a) Illustration of the Janus Mg/Pt swimmer with a representation of the associated chemical reactions, the spontaneous ionic currents, the magnetic field and the resulting Lorentz force. (b) Trajectory plots of two micro-Janus swimmers as a function of the magnetic field orientation. Inset: optical pictures of the Mg rods before and after asymmetrical modification. Adapted from reference [52].

Figure 5 .

 5 Figure 5. Illustration of micro-MHD vortices under the influence of an orthogonal homogeneous magnetic field; (a) inside the pores of a macroporous electrode, (b) around a metallic nanoparticle.

Acknowledgements

The work has been funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement n° 741251, ERC Advanced grant ELECTRA).

References and recommended reading

•Paper of special interest.

••Paper of outstanding interest.