PhD Frédéric Frappart

PhD Fabrice Papa

PhD Andreas Güntner

Werth Susanna

PhD Joecila Santos Da Silva

PhD Javier Tomasella

PhD Frédérique Seyler

PhD; Catherine Prigent

PhD William B Rossow

PhD Stéphane Calmant

PhD Susanna Werth

PhD Marie-Paule Bonnet

Satellite-based estimates of groundwater storage variations in large drainage basins with extensive floodplains

Keywords: groundwater, remote sensing, hydrological modelling

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Ref.: RSE-D-10-00458 Total water storage decomposition and estimates of groundwater variations in the Negro River Basin

Dear Frédéric, Two reviews of your paper follow. The third reviewer has evidently moved and left no forwarding address, so we will not have a review from him and will rely on these two. They are quite positive, but Reviewer #1 recommends some refocus and that the evidence that the modeling is accurate needs to be stronger.

Please carefully consider the comments and recommendations below and make appropriate changes to the paper. Publication depends on revision and/or rebuttal of the criticisms made. Further review and revision may be necessary before a final decision can be made.

When you submit your revised paper, please provide a summary of the changes you have made and your responses to the review comments and recommendations.

To submit a revision, go to http://ees.elsevier.com/rse/ and log in as an Author. You will see a menu item called "Submission Needing Revision." You will find your submission record there. Please remove any items that have changed or are no longer needed before uploading your revised manuscript.

Please upload your original files, not PDF files. If you have any problems or questions when uploading your revised manuscript, please contact Betty Schiefelbein at: rse@umn.edu. I hope that you will undertake the necessary revisions and will look forward to receiving your revised paper.

Sincerely,

Marvin Bauer Editor-in-Chief Remote Sensing of Environment

Dear Marvin Bauer, Editor-in-Chief Remote Sensing of Environment Please find enclosed the revised version of our manuscript now entitled "Satellite-based estimates of groundwater storage variations in large drainage basins with extensive floodplains ". We have taken into account their constructive comments to improve the quality of the manuscript. As suggested by Reviewer 1, we modified the title of the manuscript and added a new figure presenting a) the annual amplitude of the GRACE-based GW seasonal amplitude, b) the hydrologeological of Brazil from the Departamento Nacional da Produção Mineral (1983) to show the reader the similarities between our estimates and the hydrogeological structures in Response to reviews and summary of revisions the Negro River basin. We explained why this comparison is relevant to validate qualitatively our approach. If the seasonal amplitudes of groundwater storage variations will most probably change with the climate forcing of a particular period, i.e., decrease during an El Niño event as in our case, the spatial patterns of variations will persist even for non-average conditions. We also added in the introduction some sentences on the important role of the floodplains in the hydrological cycle. We responded Reviewer 3 concerning the type of approach used to filter the GRACE data and added some information about the characteristics of the Negro River basin. We responded on the scale problem concerning the comparison between the GRACE-derived GW anomalies and the in situ measurements pointed out by the two Reviewers. We hope these modifications will satisfy the Reviewers comments. We are looking forward to hearing from you. Sincerely yours, Frédéric Frappart, Fabrice Papa, Andreas Güntner, Susanna Werth, Joecila Santos da Silva, Javier Tomasella, Frédérique Seyler, Catherine Prigent, William B. Rossow, Stéphane Calmant, Marie-Paule Bonnet

==============================

Comments from the Reviewers:

Reviewer #1:

This study combines a variety of data and models regarding water levels in various components of the water system in the Negro Basin, South America during 2003-2004. The study starts from GRACE measurements of total water storage (TWS) variations during this time. Then these variations are broken up into variations occurring in the surface water (SW), root zone (RZ), and groundwater (GW) reservoirs (as in equation 1). In particular, SW is constrained by satellite measurements and in situ observations, and RZ from a hydrological model. The authors are then able to solve for GW by removing RZ and SW from the TWS measurements of GRACE. To verify the solution for groundwater, the authors compare to hydrological maps to see if the results seem reasonable.

The authors have thus outlined a method for detecting temporal variations in groundwater storage using GRACE satellite measurements and a variety of additional measurements and models to remove the contributions from SW and RZ. Although similar methods have been used to constrain groundwater variations on a basin-scale (for example, the authors cite [START_REF] Yeh | Remote sensing of groundwater storage changes in Illinois using the Gravity Recovery and Climate Experiment (GRACE)[END_REF]Rodell et al, 2009;[START_REF] Leblanc | Basin-scale, integrated observations of the early 21st century multiyear drought in southeast Australia[END_REF], this is the first time this method has been performed in a region dominated by wetlands.

The manuscript is reasonably well-written. However, I am concerned about a few issues that relate to verification of the method and to importance with respect to the broader hydrological community. First, it seems to me that this manuscript describes a method for estimating groundwater variations in a wetland environment, but it is presented as an investigation of the hydrology of the Negro basin. In fact, I don't think the manuscript tells us anything new about the Negro basin, and therefore I think the focus of the paper should be altered. I describe my thoughts on this in more detail in points 1 and 2 below. Second, the method that the authors describe should, in principle, be able to constrain variations in groundwater in a wetland basin (otherwise their method is not useful). However, the authors only compare ground water variations obtained using their method to a hydrogeological map of Brazil (and don't show the map), and to groundwater measurements in a single location (the Asu catchment) for which the fit is not that great (see Fig. 3a). Thus, I think the authors need to do a better job in demonstrating that their method is accurately constraining actual variations in groundwater. I describe these concerns below in points 3-5 below.

If the issues I list below can be addressed, then I think that the manuscript could be published. In the meantime, I am recommending "major revision".

We would like to thank Reviewer 1 for carefully reading our manuscript and for providing us with useful/interesting comments, which helped us to improve our paper.

1. This study was performed in the Negro Basin, South America during 2003-2004, because this is a location and time period for which sufficient data exists. I am not aware of an alternative reason for estimating the groundwater variations in this time and location -if there is some other reason for choosing the Negro Basin (e.g., there is persistent aquifer depletion there, or a drought), then the authors should stress this more clearly. If the authors are not addressing any groundwater issues related to the Negro basin, then they should stress that the point of this study is method development and verification, and they are just using the Negro Basin to test their method. Also, in this case, I think that the title the authors chose is slightly misleading because of its mention of the Negro Basin. Instead, the title should be something like "A method for estimating basin-scale groundwater storage variations in a wetland". The authors could add "A case study from the Negro Basin 2003-2004" at the end.

Reviewer 1 is right: we present here a methodology to estimate groundwater variations in a large river basin covered with extensive floodplains, and to our knowledge, this is the first attempt in a such environment. We now mentioned this, both in the introduction and, in the conclusion. The aim of this paper is not to address any groundwater issues specifically related to the Rio Negro, but we chose the Negro River basin as a case study for our method because we already successfully applied our methodology to estimate surface water volume variations combining information on inundation extent from satellite images and water levels from radar altimetry in this basin [START_REF] Frappart | Floodplain water storage in the Negro River basin estimated from microwave remote sensing of inundation area and water levels[END_REF]2008). As suggested by Reviewer 1, we modified the title of the paper to "Satellite-based estimates of groundwater storage variations in large basins with extensive floodplains".

2. Furthermore, given that others have estimated groundwater storage variations by combining GRACE and surface water measurements, the angle that is new in this manuscript is the application to a drainage basin dominated by wetlands. Thus, I think that the authors need to stress some of the challenges that are presented by the application to wetlands. Why is performing this type of analysis in a wetland different from performing it in a desert or other environment? I think that this is because much more accurate estimates of surface water fluctuations are necessary in a wetland region. This should be stated clearly. For this study to be useful, the minimum requirements for constraints on surface water variations should also be mentioned -how can a reader determine if their constraints on surface water variations are sufficient? Finally, I think that a sentence or two about the need for better constraints on groundwater variations in wetland regions is necessary -what are the major applications of such measurements, and why is remote detection better than in situ measurements (wells)?

This paper follows two previous studies on the estimate of surface water storage in the Negro basin, the first one on the methodology [START_REF] Frappart | Floodplain water storage in the Negro River basin estimated from microwave remote sensing of inundation area and water levels[END_REF], the second one on the monitoring of the surface waters on the basin over 1993-2002[START_REF] Frappart | Interannual variations of river water storage from a multiple satellite approach: A case study for the Rio Negro River basin[END_REF], as mentioned in part 3.1 Monthly water level maps. The datasets used in this paper are very similar to the ones from [START_REF] Frappart | Interannual variations of river water storage from a multiple satellite approach: A case study for the Rio Negro River basin[END_REF]: the same multisatellite inundation product but which has been extended to the period 2003-2004, a denser network of altimetry stations with more accurate water levels as ENVISAT RA-2 measurements are used instead of Topex/Poseidon. In this previous study we found a maximum error of 23% of the annual surface water variations. In this new study, taking into account the different sources of error, the maximum error is reduced to ~ 11%. (See the part on surface water error estimates in 3.1 Water volume variations for details on how this error was estimated (lines 188-200)). We added in the introduction (lines 55-64) a paragraph on the role of floodplains and the interest of using remote sensing information for large river basins instead of in situ as generally for this information is missing for most of the tropical basin, such as the Amazon: "Although wetlands and floodplains cover only 6% of the Earth surface, they have a substantial impact on flood flow alteration, sediment stabilization, water quality, groundwater recharge and discharge [START_REF] Maltby | Wetland management goals: Wise use and conservation[END_REF][START_REF] Bullock | The role of wetlands in the hydrological cycle, Hydroogy and[END_REF]. Moreover, floodplain inundation is an important regulator of river hydrology owing to storage effects along channel reaches. Reliable and timely information about the extent, spatial distribution, and temporal variation of wetlands and floods as well as the amount of water stored is crucial to better understand their relationship with river discharges, and also their influence on regional hydrology and climate. Remote sensing techniques are a unique mean for monitoring large drainage basins climate and hydrology where in situ information is lacking (as, for instance, over floodplains and wetlands or for groundwater monitoring)". Moreover, wells, especially in tropical areas are very sparse and will never provide a complete view of GW variations nor resolve its variations on shorter "weather-like" time scales, which we want to determine in order to investigate processes. Besides, wells will never provide a complete map of GW for large areas nor, unless continuously monitored, resolve its variations on shorter "weather-like" time scales, which we want to determine in order to investigate processes.

I am concerned about the data and method that the authors use to verify their results.

The authors estimate the amplitude of groundwater variations (Fig. 1f) by removing SW and RZ from TWS -this seems to be their main result. How can the authors know if these results are correct? They compare their results to a hydrogeological map of the Negro River Basin (see text, near line 197) and find that the GW variation pattern "perfectly matches" the hydrological map. This is a qualitative result at best, especially since the text in the sentences following line 197 is the only comparison that the authors present. Only gross generalizations of the spatial variations in the seasonal cycle of groundwater are presented and compared to the model predictions. I think that at a minimum, some sort of quantitative measure of the groundwater variations should be presented. For example, do the amplitudes of the estimated variations for 2003-2004 match those that are "predicted" by the hydrogeological map? Better, a reproduction of the hydrogeological map should be included in the paper for direct comparison to Fig. 1f.

The period 2002-2004 was considered as a small El Niño event according to the Multivariate ENSO Index (http://www.esrl.noaa.gov/psd/people/klaus.wolter/MEI/). For the lower and northern parts of the Amazon basin this period was associated with below average precipitation and water storage (e.g., Chen et al. 2010, Xavier et al., 2010). It has to be noted that the hydrogeological map used here for evaluation only shows the spatial distribution of the aquifers, roughly classified by their hydrological importance or yield. The map does not directly give a long-term average representation of groundwater storage variations. Nevertheless, assuming that important aquifers with high yield and important recharge and drainage volumes tend to show larger seasonal storage variations than local and unimportant aquifers with low porosity, we consider the hydrologeological map as a useful albeit qualitative approach to validate the spatial patterns of our estimates. As it can be seen in the Figure 2 of the revised version, the patterns of the annual groundwater storage amplitudes are similar to the spatial distribution of aquifer types of different importance on the hydrogeological map of Brazil. The seasonal amplitudes of groundwater storage variations will most probably change with the climate forcing of a particular period, i.e., decrease during an El Niño event as in our case, but the spatial patterns of variations will persist even for nonaverage conditions. Thus the evaluation approach remains valid. Given the very low availability of direct groundwater observation data in the area, we consider this approach as another helpful piece of evidence that gives more confidence in our results.

4.

Furthermore, how can we be sure that 2003-2004 was not an unusual year? If, for example, anomalous weather patterns produced unusual spatial variations in groundwater, then a match to the long-term average that is presented in a hydrogeological map would indicate a failure of the model. It seems to me that we should expect to be able to detect unusual ground water variations (e.g., patterns that are not on the hydrogeological map) -otherwise this method is not very useful.

The period 2002-2004 was considered as a small El Niño event according to the Multivariate ENSO Index (http://www.esrl.noaa.gov/psd/people/klaus.wolter/MEI/) and can not be considered such a "unusual year" in the area. This is confirmed by a short analysis on the GPCP data which do not show any large anomalous event as the ones in 2005 (drought) or 2009 (flood). Moreover, the hydrological map shows where are located the aquifers and their capacity. Regions with no aquifer or with low capacity to store water, will not store water even if large rainfall occur during large La Niña events. The water will flow and will be stored in the large floodplains and then recharge the aquifers. The spatial patterns will not change. Only the water levels in the aquifer will change, not their spatial pattern as it is conditioned by the storage capacity of the soil. As it can be seen in the Figure 2 of the revised version, the patterns of the annual groundwater storage amplitudes are consistent with the spatial distribution of aquifer types on the hydrogeological map of Brazil.

5.

The authors do compare their results to in situ measurements of groundwater variations from wells at the Asu micro-catchment. This direct comparison (Fig. 3a) is exactly the type of constraints on the method that are needed. Yet, the authors' method shows a very small variation in groundwater during 2003 (about 100 mm) when in fact there was about 600 mm of variation. Although the method did much better in 2004, the failure to predict 1 out of the 2 groundwater cycles does not give the reader a lot of confidence in the method.

Unfortunately, we found only one small area where groundwater time variations are available. The Asu micro-catchment has a drainage area of ~ 7 km² and is not directly connected to the Negro River. As explained in the manuscript, we can not expect a perfect match between this point-measurement and the GRACE encompassing gridpoint with area of ~10,000 km². The interest of this comparison is to see that the timing is similar between the two datasets and that the range of variations is similar. This is what we observe in Figure 4a.

Other points about the paper:

--Line 100 -I think the figure callout should be Fig. 1b.

We added Fig. 1b to the callout and changed the legend of the other panels and the figure caption. We modified the text accordingly. In this study, we did not pretend to have run neither WGHM nor LaD models. We are directly using outputs from these two hydrological models. We suggest the readers to refer to the articles describing these two models to obtain the information concerning the resolution of the water balance equation and the allocation in the different water reservoirs (the references are given in the text). No uncertainty is provided with the hydrological model estimates, however, we followed a similar approach to the one proposed in previous studies such as Yeh et al., Water Resources Research, 2006;Rodell et al., Hydrogeology Journal, 2007;Strassberg et al., Geophysical Research Letters, 2007;Leblanc et al., Water Resources Research, 2009;Rodell et al., Nature, 2009;Sun et al., Geophysical Research Letters, 2010 and maybe some others. The only difference is we use the outputs from two hydrological models instead of a single one. The outputs of these two models exhibit very similar spatial and temporal patterns, and also have similar amplitude differences. We used their extrema in equations ( 1) and (2) to present a mean behaviour and a range of variations.

--Line 137 -I think that the word "bathymetry" is usually used to mean "seafloor topography", and not the "unflooded land surface". I think that "land surface" would be a better term here.

We changed bathymetry into land surface as suggested by Reviewer 1.

--Line 206 (and earlier in the paragraph) -The authors describe the relative "importance" of aquifers. This is a rather unquantitative term -I expect that there are better ways of comparing the groundwater storage variations (see point 3 above)

We also wish we could use more quantitative data to compare our groundwater storage variations. Unfortunately, it is the way the acquifers are mentioned on the one and only hydrogeological map of Brazil which gives the boundaries of the aquifers and their relative importance. We used it to evaluate the spatial patterns of our estimates. See our response to point 3 above.

--Fig. 3-For parts b and c, it is unclear to me from the caption whether the SW and GW estimates are measured or inferred from the GW=TWS-SW-RZ method described here. Furthermore, the authors state in the text that these two quantities should be the same (since the water table is above the surface), but in that case, shouldn't SW already be subtracted from out, and the GW should be zero?

To make it clearer, we modified Figure 4 (former Figure 3) caption as follows: "Figure 4 We would like to thank both Reviewers for carefully reading our manuscript and for providing us with interesting comments, which helped us to improve our paper. GRACE measures the total mass variations of the Earth at monthly or submonthly timescales. This measurement is converted into anomalies of TWS by removing the static gravity field obtained as a multi-year average of the monthly gravity field. We added several times in the paper the term anomaly to make it clearer to the reader. We also added the area of the Negro basin (~ 700,000 km²), which represents 12% of the Amazon basin (line 57).

The authors indicate that the destriped filter with 300 km smoothing provided the best results, but did not indicate relative to what other approaches that were done? [START_REF] Werth | Evaluation of GRACE filter tools from a hydrological perspective[END_REF] evaluated six post-processing filter methods for derivation of regionally averaged water mass variations from GRACE's global gravity field solutions against hydrological model outputs, and, for each filter method, a wide range of values for the parameters that define the degree of smoothing were tested. These filters are:

-the isotropic Gaussian filter [START_REF] Jekeli | Alternative methods to smooth the Earth's gravity field[END_REF], -two degree-order dependent methods [START_REF] Swenson | Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity[END_REF] -a time-dynamic filter [START_REF] Seo | Terrestrial water mass load changes from Gravity Recovery and Climate Experiment (GRACE)[END_REF] -an empirical method know as destriping method (Swen and Whar, 2006) -an anisotropic method [START_REF] Kusche | Approximate decorrelation and non-isotropic smoothing of time-variable GRACE-type gravity field models[END_REF]. For the Negro basin, the best choice according to this methodology was found to be the destriped and smoothed at 300 km post-processing method. We suggest the readers to refer to the [START_REF] Werth | Evaluation of GRACE filter tools from a hydrological perspective[END_REF] paper already mentioned in the reference list. We added to the paragraph: "among six different filtering methods and different parameters (see [START_REF] Werth | Evaluation of GRACE filter tools from a hydrological perspective[END_REF] for the filters employed and the values of the parameters that define the degree of smoothing used)" (lines 156-157).

The groundwater levels are assumed to be below 2 m in the Asu catchment, and to reach the surface in the swamps of Caapiranga and Morro do Agua Preta. We added "in the Asu catchment" in the text to make this point clearer to the reader.

I think comparing GRACE output with groundwater storage in such a small area is a little problematic.

We totally agree with Reviewer 3 comment. The scales are completely different. Nevertheless, it was the only groundwater measurements we have in the Negro basin. The interest of this comparison is to see if the timing is similar between the two datasets and the range of variations is similar. It is what we observe on Figure 4a.

I did not see where the widths of TW, RZ, etc are explained.

We added the following paragraph explaining how were obtained the widths of the different terrestrial water reservoirs (lines 241-245): "The deviations correspond to the extrema values for the different water reservoirs and obtained as the monthly range of variations of the GRACE-derived TWS from CSR, GFZ and JPL, of RZ from LaD and WGHM outputs, the mean surface water volume variations more or less the error computed using (4), the GW extrema by difference of the formers".

Figure 3. The lines could be labeled directly.

As suggested, we added labels on Figure 4.

See attached file for additional comments and suggestions.

All the comments and corrections suggested by Reviewer 3 have been taken into account. The major ones are responded below:

1) The title has been changed to "Satellite-based estimates of groundwater storage variations in large basins with extensive floodplains" according to Reviewer 1 suggest.

2) We indicated in the abstract that WGHM and Lad hydrological models were used (lines 27-28).

3) GRACE measures anomalies of TWS. We added anomalies in the second paragraph of the introduction (lines 50 and 60). 4) Frappart et al. (2006a) and Santos da Silva et al. (2010) showed that the accuracy of Envisat RA-2 derived surface water levels is most of the time from12 to 40 cm, knowing that the distance between the altimetry and the in situ stations can reach several tenths of kilometres (lines 93-95). [START_REF] Prigent | Global inundation dynamics inferred from multiple satellite observations. 1993-2000[END_REF], uncertainties on the multisatellite inundation product was found to be of ~10% (comparison with high resolution SAR data) with some limitations to detect small wetlands fractions (lines 82-83). As mentioned in Frappartet al. ,2008 in the Negro basin, the multisatellite product is not adequately detecting the small floodplains upstream of the Negro and its two major tributaries (see [START_REF] Frappart | Interannual variations of river water storage from a multiple satellite approach: A case study for the Rio Negro River basin[END_REF] for more details). However, all these informations about the uncertainty are used to compute the error bars on the surface water estimates (see above).

5) In

Introduction

The water cycle of large tropical river basins is strongly influenced by seasonal and interannual variability of rainfall and streamflow, affecting all the components of the water balance [START_REF] Ronchail | Interannual rainfall variability in the Amazon basin and sea-surface temperatures in the equatorial Pacific and the tropical Atlantic oceans[END_REF][START_REF] Marengo | Long-term trends and cycles in the hydrometeorology of the Amazon basin since the late 1920s[END_REF]. The Terrestrial Water Storage (TWS), which represents an integrated measurement of the water stored in the different hydrological reservoirs and is the sum of the surface water, root zone soil water, snowpack and groundwater, is a good indicator of the changes that occur in hydrological conditions globally and at basin scales. Nevertheless, TWS is difficult to measure due to the lack of in situ observations of the terrestrial hydrological compartments.

The Gravity Recovery And Climate Experiment (GRACE) mission, launched in 2002, detects tiny changes in the Earth's gravity field which can be related to spatio-temporal variations of TWS at monthly or sub-monthly time-scales [START_REF] Tapley | The Gravity Recovery and Climate Experiment; mission overview and early results[END_REF]. Previous studies provide important information on changes in TWS over the Amazon [START_REF] Crowley | Annual variations in water storage and precipitation in the Amazon Basin[END_REF][START_REF] Chen | 2005 drought event in the Amazon River basin as measured by GRACE and estimated by climate models[END_REF]. Variations in groundwater storage can be separated from the TWS anomalies measured by GRACE using external information on the other hydrological reservoirs such as in situ observations [START_REF] Yeh | Remote sensing of groundwater storage changes in Illinois using the Gravity Recovery and Climate Experiment (GRACE)[END_REF]), model outputs (Rodell et al., 2009), or both [START_REF] Leblanc | Basin-scale, integrated observations of the early 21st century multiyear drought in southeast Australia[END_REF]. No similar studies have been undertaken yet for large river basins characterized by extensive wetlands or floodplains.

Although wetlands and floodplains cover only 6% of the Earth surface, they have a substantial impact on flood flow alteration, sediment stabilization, water quality, groundwater recharge and discharge [START_REF] Maltby | Wetland management goals: Wise use and conservation[END_REF][START_REF] Bullock | The role of wetlands in the hydrological cycle, Hydroogy and[END_REF]. Moreover, floodplain inundation is an important regulator of river hydrology owing to storage effects along channel reaches. Reliable and timely information about the extent, spatial distribution, and temporal variation of wetlands and floods as well as the amount of water stored is crucial to better understand their relationship with river discharges, and also their influence on regional hydrology and climate. Remote sensing techniques are a unique mean for monitoring large drainage basins climate and hydrology where in situ information is lacking (as, for instance, over floodplains and wetlands or for groundwater monitoring).

In this study, a new technique is proposed to derive the spatio-temporal variations of water volume anomalies in the aquifer of the Negro River basin, a large tropical basin dominated by extensive floodplains (see Figure 1a and b for its location). The Negro River basin, with a drainage area of 700,000 km², is indeed the second largest tributary to the Amazon River, covering 12% of the Amazon basin, with a mean annual discharge of 28.400 m 3 .s -1 [START_REF] Richey | Amazon river discharge and climate variability[END_REF][START_REF] Molinier | Régionalisation des débits du basin amazonien[END_REF]. The method is based on the combination of multisatellitederived hydrological products and outputs from global hydrology models. Water storage anomalies in the different hydrological reservoirs are removed from the TWS anomalies measured by GRACE to isolate the groundwater anomaly storage over 2003-2004. Results are both evaluated and validated using a hydrogeological map of Brazil, in situ measurements of groundwater level variations in a micro-catchment, and altimetry-derived water stages for zones where the aquifers reach the land surface.

Datasets 2.1. GRACE-derived land water mass solutions

The Gravity Recovery And Climate Experiment (GRACE) mission, launched in March 2002, provides measurements of the spatio-temporal changes in Earth's gravity field. Several recent studies have shown that GRACE data over the continents can be used to derive the monthly changes of the total land water storage [START_REF] Ramillien | Time variations of the land water storage from an inversion of 2 years of GRACE geoids[END_REF]2008;Schmitt et al., 2008) with an accuracy of ~1.5 cm of equivalent water thickness when averaged over surfaces of a few hundred square-kilometres. We used the Level-2 land water solutions (RL04) produced by GFZ, JPL (for these two first products, January 2003, June 2003 and January 2004 are missing), and CSR (June 2003 and January 2004 are missing) with a spatial resolution of ~333 km, destriped and smoothed by [START_REF] Chambers | Evaluation of New GRACE Time-Variable Gravity Data over the Ocean[END_REF] with an accuracy of 15-20 mm of water thickness. They are available at ftp://podaac.jpl.nasa.gov/tellus/grace/monthly.

The multisatellite inundation extent

This dataset quantifies at global scale the monthly distribution of surface water extent and its variations at ~25 km of resolution. The methodology which captures the extent (with an accuracy of ~10%) of episodic and seasonal inundations, wetlands, rivers, lakes, and irrigated agriculture over more than a decade, 1993-2004, is based on a clustering analysis of a suite of complementary satellites observations, including passive (SSM/I) and active (ERS) microwaves, and visible and near-IR (AVHRR) observations [START_REF] Prigent | Global inundation dynamics inferred from multiple satellite observations. 1993-2000[END_REF][START_REF] Papa | Wetland dynamics using a suite of satellite observations: A case study of application and evaluation for the Indian Subcontinent[END_REF]2008;2010).

Envisat RA-2 radar altimeter-derived water level heights over rivers and wetlands

Silva dos Santos et al. ( 2010) build 140 time series of water levels derived from RA-2 ranges processed using the Ice-1 retracker over the Negro River drainage basin (see Figure 1c for their locations), for the period 2002-2008, as suggested by Frappart et al. (2006a). The uncertainty associated with the water level height ranges between 5-25 cm for high water season to 12-40 cm during low water season (Frappart et al., 2006a;[START_REF] Da Silva | Water levels in the Amazon basin derived from the ERS 2 and ENVISAT radar altimetry missions[END_REF].

In situ surface water levels

We used daily measurements of water stage from eight leveled in situ gauge stations from the Brazilian Water Agency (Agência Nacional de Águas or ANA -http://www.ana.gov.br), see Figure 1c for their location.

In situ groundwater levels

The Asu micro-catchment, with a drainage area of 6.58 km², ~90 km north-northwest of Manaus, was instrumented with dipwells in 2001 (see [START_REF] Tomasella | The water balance of an Amazonian micro-catchment: the effect of interannual variability of rainfall on hydrological behaviour[END_REF] for a complete description of the catchment instrumentation). We used the well measurements to evaluate our estimates of the groundwater storage variations at that location.

Root zone water storage outputs from hydrological models

Hydrological model outputs are widely used to analyze spatio-temporal variations of water storage content at basin and global scales. We used water storage in the root zone from the Land Dynamics (LaD) model [START_REF] Milly | Global modeling of land water and energy balances: 1. The Land Dynamics (LaD) model[END_REF] outputs and from the latest version [START_REF] Hunger | Value of river discharge data for global-sale hydrological modelling[END_REF] of the WaterGAP Global Hydrology Model (WGHM) [START_REF] Döll | A global hydrological model for deriving water availability indicators: Model tuning and validation[END_REF].

Precipitation estimates from the Global Precipitation Climatology Project (GPCP)

These data quantify the distribution of precipitation over the global land surface [START_REF] Adler | The Version 2 Global Precipitation Climatology Project (GPCP) Monthly Precipitation Analysis (1979-Present)[END_REF]. We used the monthly Satellite-Gauge Combined Precipitation Data product Version 2 data, available from January 1997 to present with a spatial resolution of 1° of latitude and longitude. Over land surfaces, uncertainty in rate estimates from GPCP is generally less than over the oceans due to the in situ gauge input (in addition to satellite) from the GPCC (Global Precipitation Climatology Center). Over land, validation experiments have been conducted in a variety of locations worldwide and suggest that while there are known problems in regions of persistent convective precipitation, non precipitating cirrus or regions of complex terrain, the uncertainty estimates range from 10 to 30% [START_REF] Adler | The Version 2 Global Precipitation Climatology Project (GPCP) Monthly Precipitation Analysis (1979-Present)[END_REF].

Hydrogeological map of Brazil

We used a hydrogeological map from the Brazilian Department of Mineral Production (DNPM, 1983) which provides the boundaries and the hydrogeological importance of the aquifers of the whole Brazil. This map, holdings of ISRIC, is made available by the European Commission -Joint Research Centre through the European Digital Archive of Soil Maps (EuDASM) [START_REF] Selvaradjou | European Digital Archive of Soil Maps (EuDASM) -Soil Maps of Latin America and Carribean Islands (DVD-Rom version). EUR 21822 EN[END_REF]: http://eusoils.jrc.ec.europa.eu/esdb_archive/EuDASM/EUDASM.htm

Methods

Monthly water level maps

Monthly maps of water level over the floodplains of the Negro River Basin have been determined by combining the observations from a multi-satellite inundation dataset, RA-2 derived water levels, and the in situ hydrographic stations for the water levels over rivers and floodplains (see Figure 1c for the location of altimetry-based and in situ stations). For a given month during the flood season, water levels were linearly interpolated over the flooded zones of the Negro River Basin. A pixel of 25 km x 25 km is considered inundated when its percentage of inundated area is greater than 0. The elevation of each pixel of the water level maps is given with reference to its minimum computed over the 2003-2004 period. This minimum elevation represents either the land surface or very low water stage of the floodplain. More details about the methodology used here can be found in [START_REF] Frappart | Floodplain water storage in the Negro River basin estimated from microwave remote sensing of inundation area and water levels[END_REF][START_REF] Frappart | Water volume change in the lower Mekong basin from satellite altimetry and imagery data[END_REF][START_REF] Frappart | Interannual variations of river water storage from a multiple satellite approach: A case study for the Rio Negro River basin[END_REF].

GRACE leveling and time-shift

An optimum filter method was developed by analyzing the correspondence of GRACE basinaverage water storage to the ensemble mean of hydrological models (WGHM, LaD) and by analyzing the error budgets (satellite/leakage errors) and amplitude and phase biases for the different filter types. For the Negro River Basin, the destriped filter with 300 km smoothing radius provides the best results among six different filtering methods and different parameters (see [START_REF] Werth | Evaluation of GRACE filter tools from a hydrological perspective[END_REF] for the filters employed and the values of the parameters that define the degree of smoothing used). Only a very small bias in the seasonal phase of storage changes resulted due to filtering. The GRACE products have been rescaled with a factor of 1.061 to account for amplitude smoothing due to filtering determined from smoothed and unsmoothed basin-average model ensemble time series of water storage.

Groundwater storage estimates

The time variations of the TWS expressed as anomalies are the sum of the contributions of the different reservoirs present in a drainage basin:

TWS SW RZ GW        (1)
where SW represents the total surface water storage including lakes, reservoirs, in-channel and floodplains water; RZ is the water contained in the root zone of the soil (representing a depth of 1 or 2 m), GW is the total groundwater storage in the aquifers. These terms are generally expressed in volume (km 3 ) or mm of equivalent water height.

The GW anomaly over 2003-2004 is obtained in (1) by calculating the difference between the TWS anomaly estimated by GRACE and the SW level anomaly maps previously derived from remote sensing and the RZ anomaly derived from hydrological models outputs. The TWS and RZ monthly anomalies are the average anomalies of respectively the Level-2 GRACE CSR, GFZ and JPL destriped and smoothed solutions at 300 km of averaging radius, and the outputs from LaD and WGHM, resepctively.

Water volume variations

For a given month t, the regional water volume of TWS, SW, RZ or GW storage δV(t) in a basin with surface area S, is simply computed from the water heights δh j , with j = 1, 2, . . .

(expressed in mm of equivalent water height) inside S, and the elementary surface R e 2 sinθ j δλδθ (and the percentage of inundation P j for SW):

2 ( ) ( , , )sin j e j j j j jS V t R P h t          (2)
where λ j and θ j are co-latitude and longitude, δλ and δθ are the grid steps in longitude and latitude (generally δλ= δθ), and R e the mean radius of the Earth (6378 km). The surface and total water volume variations are expressed in km 3 .

Error on anomalies of surface water volumes were computed in the Negro basin using (3) :

      n i i i i i h dS h d S dV 1   ( 3
)
where dV is the error on the monthly water volume anomaly (V), S i the i th elementary surface, δh i the i th elementary water level variation between two consecutive months, dS i the error on the i th elementary surface, and dδh i the error on the i th elementary water level variation between two consecutive months.

The error sources include misclassifications, altimetry measurements and the linear interpolation method. The maximum error on the volume variation are monthly estimated as:

) ( max max max max max h S h S V        (4)
where: ΔV max is the maximum error on the water monthly volume anomaly, S max is the maximum monthly flooded surface, δh max is the maximum water level variation between two consecutive months, ΔS max is the maximum error for the flooded surface, and Δ(δh max ) is the maximum error for the water level between two consecutive months.

Results & Discussion

Monthly estimates of water storage in the different hydrological reservoirs are computed for two years (2003)(2004) for which the different datasets overlap in time. Maps of annual amplitudes of TWS, SW, RZ and GW are respectively presented in Figure 1 d to g. They were obtained by fitting simultaneously the temporal trend, the amplitudes of the annual and semiannual cycles by least-square adjustment at each grid point. The amplitude of the annual cycle for TWS is maximum along the Negro River, and the downstream part of the Branco River, and also over the non flooded areas in the northwest of the Branco River (see [START_REF] Frappart | Floodplain water storage in the Negro River basin estimated from microwave remote sensing of inundation area and water levels[END_REF] for a classification of the vegetation and flood extent in the Negro River Basin), reaching 300 mm in the downstream part (Figure 1d). This area corresponds also to the maximum of amplitude of the SW (Figure 1e), clearly related to substantial backwater effects produced at the Negro-Solimões confluence [START_REF] Filizola | The Rio Negro and Rio Solimões confluence pointhydrometric observations during the 2006/2007 cycle[END_REF]. The amplitude of the annual cycle for RZ (Figure 1f) is small except in the upstream part of the Branco River subbasin, where large precipitation occurred without significant flood events. The largest amplitudes of the annual cycle for the GW (Figure 1g) were observed along the Negro River stream, peaking at 250 mm, i.e., ~72% of the TWS, in the downstream part of the basin. In contrast, small amplitudes were obtained in the Branco and Uaupes Basins. The pattern of GW storage variations observed in Figure 2a tends to be similar to the hydrogeological structures of the Negro River Basin (Figure 2b). For important aquifers, higher yield, recharge and drainage volumes and thus larger seasonal storage variations can be expected than for local and unimportant aquifers with low porosity. According to the hydrogeological map of Brazil (DNPM, 1983), the lower part of the basin (longitude  -67° and latitude 0°), where the amplitude of the GRACE-based GW seasonal cycle is the largest, is characterized by continuous aquifers of medium hydrogeological importance. The Uaupes Basin, which only contains local aquifers of relatively small importance, and the Branco Basin, which presents a mixture of local aquifers and small continuous aquifers of relatively small importance, and zones with almost no aquifers, correspond to the smallest amplitudes of the GW seasonal cycle. Note that a secondary maximum of the amplitude of the GW seasonal cycle (66°W, 2°N) can be observed in the upper part of the Negro River which is in good agreement with the presence of two small aquifers of medium importance (DNPM, 1983).

Figure 3a shows the time variations (and deviation at each time step) of the water storage anomalies in the TWS, SW, RZ and GW reservoirs for 2003 and 2004. The deviations correspond to the extrema values for the different water reservoirs and obtained as the monthly range of variations of the GRACE-derived TWS from CSR, GFZ and JPL, of RZ from LaD and WGHM outputs, the mean surface water volume variations more or less the error computed using (4), the GW extrema by difference of the formers. The TWS signal is dominated during high waters (May to July) by SW variations. The RZ varies in phase with both TWS and SW and the amplitude of its variations represents a third of the amplitude of TWS variations, which is similar to what was obtained by Kim et al. [2009] for the whole Amazon basin. The resulting GW variations exhibit a more complex profile with two peaks.

Its time variations follow the bimodal distribution of the precipitation resulting from the geographical location of the basin in both hemispheres (Figure 3b). A large variability, reaching several months, is observed in the timing the extrema across the basin: GW storage is maximum (minimum) in July-August (December-March) in the western part (Uaupes and west of the Negro), in June-July (February to April) in the centre of the basin and the downstream of the Branco, in August-September in the upper part of the Branco, and in May-June (October to December) for the downstream part of the Negro basin. These results are consistent with in situ measurements from sites located in the downstream part of the Negro basin [START_REF] Do Nascimento | Podzolization as a deferraltization process: dynamics and chemistry of ground and surface waters in an Acrisol -Podzol sequence of the upper Amazon basin[END_REF][START_REF] Tomasella | The water balance of an Amazonian micro-catchment: the effect of interannual variability of rainfall on hydrological behaviour[END_REF] and closely related to the timing of GW recharge and soil thickness. In Manaus, the time-lag between the maxima of rainfall and GW is 3 months, which is similar to what is observed with in situ measurements.

Figure 4 compares in situ measurements of GW levels from the Asu catchment and water levels from the Caapiranga and Morro da água preta swamps with the estimated anomalies of GW.

The GW levels in the Asu micro-catchment (below 2m) were converted into GW storage using a specific yield of 0.17 as in [START_REF] Tomasella | The water balance of an Amazonian micro-catchment: the effect of interannual variability of rainfall on hydrological behaviour[END_REF]. Figure 4a shows the [2003][2004] time variations of the GW storage of the Asu catchment and the encompassing GRACE gridcell. They show similar temporal variations. Very good agreement is found between mid 2003 and 2004. Nevertheless, the increase in GW starts later in 2003 for the in situ measurements and the maximum value is three times lower. A less pronounced decrease can also be observed for 2004. Two main factors can account for these differences: the respective sizes (7 km² against 10,000 km²), and the fact that the Asu catchment is not directly connected to the Negro River, so the recharge processes may be different.

The groundwater table permanently reaches the surface in several parts of the Negro River Basin. Two of these regions, the Caapiranga and Morro da água preta swamps (Figure 1c), are flooded and can be monitored using radar altimetry. In these cases, we expect GW to have similar time variations as surface water levels. Time series of SW and corresponding GW anomalies over 2003-2004 are presented in Figures 4b andc for Caapiranga and Morro da água preta respectively. Except for February 2004, where the SW derived from radar altimetry present an abnormally low level (larger errors on altimetry-derived stages during the low water season, due to the presence of dry land or vegetation in the satellite field of view have also been reported by other studies, see for instance Frappart et al., (2006a) or Santos da Silva et al., (in press), both time series agree well (R=0.76 for Caapiranga and 0.73 for Água do Morro Preta) and exhibit similar temporal patterns and amplitudes. The comparisons in Figure 4 give confidence in the groundwater variations derived by the approach presented here.

Conclusion

This study presents the first attempt to estimate time variations of GW anomalies using GRACE-based TWS in combination with other remote sensing measurements and model outputs for a large river basin characterized by extensive inundation. Both spatial and temporal patterns of ground water storage anomalies exhibit realistic behaviour. Comparisons with scarce in situ and satellite information show good agreement, in spite of the difference in spatial scales. This promising study will be soon extended to the entire Amazon basin and for more years as all datasets will soon be available over a longer period of time (2002 to present). Extending this method to characterize the evolution of water storage in other large river basins, especially in semi-arid regions, is also important as it will provide regional estimates of groundwater variations, a key variable for water resource management.

  Luciano Xavier, M. Becker, A. Cazenave, L. Longuevergne, W. Llovel, O.C. Rotunno Filho (2010). Interannual variability in water storage over 2003-2008 in the Amazon Basin from GRACE space gravimetry, in situ river level and precipitation data Original Research Article Remote Sensing of Environment, 114(8), 1629-1637. Chen JL, Wilson CR, Tapley BD (2010). The 2009 exceptional Amazon flood and interannual terrestrial water storage change observed by GRACE, Water Resources Research, 46, W12526..

-

  -Section 2.6. The authors describe how they use a hydrological model to constrain the root zone water storage variations. It is unclear to the reader what inputs go into this model, and what the uncertainty about the outputs -I think additional detail that describes these aspects should be added. This is potentially important because I expect there are tradeoffs between root zone storage and groundwater storage.

  : a) Time variations of the GW storage in the Asu catchment (in situ -grey) and in the corresponding GRACE gridcell (satellite-based -black). b) and c) Time variations of the surface water levels (altimetry-derived -grey) and the groundwater for the corresponding GRACE gridcell (satellite-based -black) in the swamps of Caapiranga and Morro da Água Preta respectively". It is not exactly what is written in the text. In these two areas, the water table reach the surface, so the surface water and the groundwater should present similar variations. So GW=TWS-SW-RZ ~ SW. It is what is observed on Figure 4 b) and c). ========================= Reviewer #3: This is a very interesting study that combines satellite data and modeling analyses to understand temporal variations in water storage in different components of the system. The strength of the paper comes from the multisatellite data and comparison with model results. I hope the following minor comments improve the manuscript. It seems that GRACE measures changes in water storage, I think it is important to indicate this. Throughout the manuscript it often refers to water storage measurements, rather than specifying changes in water storage. I did not see the area of the basin mentioned in the paper. Maybe I missed it.

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1: a) Overview map of South America with the location of the Negro River Basin (b)). c) Map of the Negro River sub-basin extracted from SRTM DEM. Each thin line of black dots represents a ENVISAT track. Yellow dots represent in situ gauge stations, and red dots represent altimetry stations. d), e), f) and g) Maps of amplitude of the annual cycle for TWS, SW, RZ and GW respectively. 476 477 (c)

Acknowledgements

This work was partly supported by the foundation STAE in the framework of the CYMENT project and the ANR CARBAMA. Two authors (F. Papa and W.B. Rossow) are supported by NASA Grant NNXD7AO90G. The authors would like to thank the Centre de Topographie des Océans et de l'Hydrosphère (CTOH) at Laboratoire d'Etudes en Géophysique et Oc´eanographie Spatiales (LEGOS), Observatoire Midi-Pyrénées (OMP), Toulouse, France,

for the provision of the ENVISAT RA-2 GDR dataset. They wish to thank an anonymous reviewer for helping us in editing the manuscript to improve its quality.

Abstract:

This study presents monthly estimates of groundwater anomalies in a large river basin dominated by extensive floodplains, the Negro River basin, based on the synergistic analysis using multisatellite observations and hydrological models. For the period 2003-2004, changes in water stored in the aquifer is isolated from the total water storage measured by GRACE by removing contributions of both the surface reservoir, derived from satellite imagery and radar altimetry, and the root zone reservoir simulated by WGHM and LaD hydrological models.

The groundwater anomalies show a realistic spatial pattern compared with the hydrogeological map of the basin, and similar temporal variations to local in situ groundwater observations and altimetry-derived level height measurements. Results highlight the potential of combining multiple satellite techniques with hydrological modelling to estimate the evolution of groundwater storage. Keywords: groundwater, remote sensing, hydrological modelling


Miroslav Halás

Claude H Moog

A Polynomial Solution to the Model Matching Problem of Nonlinear Time-delay Systems

In this paper the model matching problem is considered for single input single output nonlinear systems with delays. A full characterization for its solvability is provided within a simple class of compensators. The approach is developed by means of the transfer functions of nonlinear time-delay systems. The state elimination problem for the systems given by their state-space representation is discussed as well.

I. INTRODUCTION

An algebraic formalism of differential forms, originally developed for nonlinear systems without delays [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF], [START_REF] Aranda-Bricaire | Linearization of discretetime systems[END_REF], was recently extended to the case of time-delay systems [START_REF] Márquez-Martínez | Note sur l'accessibilité des systèmes non linéaires à retards[END_REF], [START_REF] Márquez-Martínez | Input-Output Feedback Linearization of Time-Delay Systems[END_REF], [START_REF] Márquez-Martínez | The structure of nonlinear time-delay systems[END_REF], [START_REF] Moog | The disturbance decoupling problem for timedelay nonlinear systems[END_REF], [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over noncommutative rings[END_REF] and was shown to be effective in solving control problems like accessibility and observability, disturbance decoupling, feedback linearization and others. On the other side, in the case of systems without delays, there exists, in comparison to the machinery of one-forms, an alternative approach in which the system properties are described by skew polynomials from non-commutative polynomial rings. Such polynomials act as differential [START_REF] Zheng | A polynomial approach to nonlinear system controllability[END_REF], [START_REF] Zheng | Common factors and controllability of nonlinear systems[END_REF] or shift [START_REF] Kotta | Irreducibility conditions for nonlinear input-output difference equations[END_REF] operators on the differentials of the system inputs and outputs. The polynomial approach to nonlinear systems shows a great similarity to methods well known from the linear theory, see for instance [START_REF] Picard | Model Matching for Linear Systems with Delays and 2D Systems[END_REF]. In particular, one can introduce even a notion of a transfer function of a nonlinear system as was recently shown in [START_REF] Halás | An algebraic framework generalizing the concept of transfer functions to nonlinear systems[END_REF], [START_REF] Halás | Symbolic computation for nonlinear systems using quotients over skew polynomial ring[END_REF], [START_REF] Halás | Extension of the concept of transfer function to discrete-time nonlinear control systems[END_REF], [START_REF] Halás | Transfer Functions of Discrete-time Nonlinear Control Systems[END_REF], [START_REF] Zheng | Transfer function description for nonlinear systems[END_REF]. Such a concept is equivalent to that of [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF] for linear timevarying systems which allows us to associate to a nonlinear system the tangent (or variational) linear system, see for instance [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], over Kähler differentials [START_REF] Johnson | Kähler differentials and differential algebra[END_REF] except that now the time-varying coefficients of the polynomials are not necessarily independent [START_REF] Li | Simplifying skew fractions modulo differential and difference relations[END_REF]. The transfer function formalism of nonlinear systems generalizes well known results valid for linear time invariant systems and was, for instance, already employed in [START_REF] Perdon | The pole-zero structure of nonlinear control systems[END_REF] to investigate some structural properties of nonlinear systems, in [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF] to study the nonlinear model matching problem, in [START_REF] Halás | Synthesis of Observers for Nonlinear Systems: A Polynomial Approach[END_REF] to study the observer design and in [START_REF] Halás | Realization problem of SISO nonlinear systems: a transfer function approach[END_REF] to study the realization problem of nonlinear systems. The polynomial approach to nonlinear systems and the notion of a transfer function was recently carried over also to the case of time-delay systems [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF], [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF] where some basic moog@irccyn.ec-nantes.fr properties were discussed. In this paper, the model matching problem of nonlinear time-delay systems is solved within such a transfer function formalism. As it was done in [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF], a more detailed discussion is provided below, concerning also structural properties as the notions of a relative degree and a relative shift. In addition, we also depict a possible solution to the state elimination problem for the nonlinear time-delay case.

II. TRANSFER FUNCTIONS OF NONLINEAR TIME-DELAY

SYSTEMS

We will use the algebraic formalism of [START_REF] Márquez-Martínez | Note sur l'accessibilité des systèmes non linéaires à retards[END_REF], [START_REF] Márquez-Martínez | Input-Output Feedback Linearization of Time-Delay Systems[END_REF], [START_REF] Márquez-Martínez | The structure of nonlinear time-delay systems[END_REF], [START_REF] Moog | The disturbance decoupling problem for timedelay nonlinear systems[END_REF], [START_REF] Xia | Analysis of nonlinear time-delay systems using modules over noncommutative rings[END_REF] which extends the concept of differential oneforms to the case of time-delay systems and of [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF], [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF] which introduces transfer functions of nonlinear time-delay systems.

In this paper we restrict our attention to the SISO nonlinear time-delay systems defined by an input-output equation of the form

y (n) (t) = ϕ({y (k) (t -i), u (l) (t -j)}) (1)
where 0 ≤ k ≤ n -1; 0 ≤ l ≤ n; i, j ≥ 0 and u ∈ R and y ∈ R denote input and output to the system and ϕ is assumed to be an element of the field of meromorphic functions K. Note that it is not restrictive to assume i, j ∈ N as all commensurable delays can be considered as multiples of an elementary delay h [START_REF] Márquez-Martínez | The structure of nonlinear time-delay systems[END_REF]. Remark 1: In the case of systems without delays even if one starts with a state-space representation it is always possible to eliminate the state variables to get an input-output equation, see for instance [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF]. However, it is not completely clear how one can carry over the idea of the state elimination procedure to the time-delay case. A special case is discussed in Appendix I. In the general case, possible drawbacks may be met in the state elimination process for system [START_REF] Halás | An algebraic framework generalizing the concept of transfer functions to nonlinear systems[END_REF]. For the sake of simplicity, here we assume that the system (10) admits an input-output equation of the form (1).

The Ore algebra K[δ, s] of polynomials in s and δ over K with the usual addition, and the (non-commutative) multiplications given by the commutation rules sa(t) = a(t)s + ȧ(t) δa(t) = a(t -1)δ

(2) sδ = δs where a(t) ∈ K, represents the ring of linear differential time-delay operators that act over the vector space of one-forms E = span K {dξ(t); ξ(t) ∈ K} in the following way

  i,j a ij δ j s i   v(t) = i,j a ij v (i) (t -j)
for any v(t) ∈ E. The commutation rules (2) actually represent the rule for differentiating and, respectively, time-delaying.

Proposition 1 (Ore condition): For all non-zero a, b ∈ K[δ, s], there exist non-zero

a 1 , b 1 ∈ K[δ, s] such that a 1 b = b 1 a.
Thus, the ring K[δ, s] can be embedded to the noncommutative quotient field K δ, s by defining quotients [START_REF] Ore | Linear equations in non-commutative fields[END_REF], [START_REF] Ore | Theory of non-commutative polynomials[END_REF] as

a b = b -1 • a
The addition and multiplication in K δ, s are defined as

a 1 b 1 + a 2 b 2 = β 2 a 1 + β 1 a 2 β 2 b 1
where

β 2 b 1 = β 1 b 2 by Ore condition and a 1 b 1 • a 2 b 2 = α 1 a 2 β 2 b 1 (3)
where β 2 a 1 = α 1 b 2 again by Ore condition. Due to the non-commutative multiplications (2) they, of course, differ from the usual rules. In particular, in case of the multiplication (3) we, in general, cannot simply multiply numerators and denominators, nor cancel them in a usual manner. We neither can commute them as the multiplication in K δ, s is non-commutative as well.

Once the fraction of two skew polynomials is defined we can introduce the transfer function of the nonlinear timedelay system (1) as an element F (δ, s) ∈ K δ, s such that dy(t) = F (δ, s)du(t).

After differentiating (1) we get

dy (n) (t) - n-1 k=0 i≥0 ∂φ ∂y (k) (t -i) dy (k) (t -i) = = n-1 k=0 j≥0 ∂φ ∂u (k) (t -j) du (k) (t -j)
or alternatively

a(δ, s)dy(t) = b(δ, s)du(t) (4)
where

a(δ, s) = s n - n-1 k=0,i≥0 ∂φ ∂y (k) (t-i) δ i s k and b(δ, s) = n-1 k=0,j≥0 ∂φ ∂u (k) (t-j) δ j s k are in K[δ, s]. Then F (δ, s) = b(δ, s) a(δ, s) (5)
= b m (δ)s m + • • • + b 1 (δ)s + b 0 (δ) s n + a n-1 (δ)s n-1 + • • • + a 1 (δ)s + a 0 (δ)
where m = max{i ≥ 0; ∂ϕ/∂u (i) = 0}.

Example 1: Consider the system

ÿ(t) = ẏ(t -1)u(t -1)
After differentiating

dÿ(t) -u(t -1)d ẏ(t -1) = ẏ(t -1)du(t -1) (s 2 -u(t -1)δs)dy(t) = ẏ(t -1)δdu(t)
and the transfer function is

F (δ, s) = ẏ(t -1)δ s 2 -u(t -1)δs (6)
The reader is referred to [START_REF] Halás | An algebraic framework generalizing the concept of transfer functions to nonlinear systems[END_REF], [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF], [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF], [START_REF] Halás | Symbolic computation for nonlinear systems using quotients over skew polynomial ring[END_REF], [START_REF] Halás | Extension of the concept of transfer function to discrete-time nonlinear control systems[END_REF], [START_REF] Halás | Transfer Functions of Discrete-time Nonlinear Control Systems[END_REF], [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF] for more detailed description of the transfer function formalism in nonlinear control theory and to [START_REF] Bronstein | An introduction to pseudo-linear algebra[END_REF], [START_REF] Chyzak | Effective algorithms for parametrizing linear control systems over Ore algebras[END_REF], [START_REF] Chyzak | Non-commutative elimination in Ore algebras proves multivariate identities[END_REF], [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF], [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF] for more detailed description of Ore rings and Ore algebras.

III. STRUCTURAL PROPERTIES

Transfer functions of control systems are well known from linear theory. Their use is the most natural for a several control problems, for they give the insight into the system structure. Recently, it was shown that such an algebraic formalism can be generalized also to nonlinear systems without delays [START_REF] Halás | An algebraic framework generalizing the concept of transfer functions to nonlinear systems[END_REF], [START_REF] Halás | Extension of the concept of transfer function to discrete-time nonlinear control systems[END_REF] and even to the time-delay case [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF]. In this section we, in comparison to [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF], discuss additional structural properties of transfer functions of nonlinear timedelay systems.

In dealing with polynomials from K[δ, s] we define their (highest) degrees and lowest degrees in s and in δ as

deg s a(δ, s) = max k ≥ 0; ∂a(δ, s) ∂s k = 0 deg δ a(δ, s) = max k ≥ 0; ∂a(δ, s) ∂δ k = 0 low deg s a(δ, s) = min k ≥ 0; ∂a(δ, s) ∂s k = 0 low deg δ a(δ, s) = min k ≥ 0; ∂a(δ, s) ∂δ k = 0 where 0 = a(δ, s) ∈ K[δ, s]. If a(δ, s) = 0 then define deg s a(δ, s) = deg δ a(δ, s) = -∞ and low deg s a(δ, s) = low deg δ a(δ, s) = ∞.

A. Structural indices

Certain structural indices of a control system play a key role in solving a number of control problems. One of them is represented by the notion of a relative degree, originally introduced for the case of systems without delays. It can be easily extended also to the time-delay case [START_REF] Moog | The disturbance decoupling problem for timedelay nonlinear systems[END_REF]:

Definition 1: The relative degree of the system (10) is defined as

ρ = min k ≥ 0; ∂y (k) (t) ∂u(t -j) = 0 for some j ≥ 0
It is straightforward to show that the relative degree is given as the difference between orders of polynomials in s in denominator and numerator of the transfer function ( 5)

ρ = rel deg s F (δ, s) = deg s a(δ, s) -deg s b(δ, s)
as in the linear case. Another important structural index in the time-delay case is given by the notion of a relative shift [START_REF] Márquez-Martínez | Input-Output Feedback Linearization of Time-Delay Systems[END_REF], [START_REF] Moog | The disturbance decoupling problem for timedelay nonlinear systems[END_REF]:

Definition 2: Assume that the relative degree ρ of the system (10) is finite. Then the relative shift of this system is defined as

µ = min k ≥ 0; ∂y (ρ) (t) ∂u(t -k) = 0
It can be shown that in terms of the transfer function [START_REF] Chyzak | Non-commutative elimination in Ore algebras proves multivariate identities[END_REF] it

means µ = deg δ b m (δ)
where b m (δ) is the leading coefficient of b(δ, s) in s.

Example 2: Consider the system from Example 1 with the transfer function [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF]. Then

ρ = rel deg s F (δ, s) = 2 µ = deg δ ẏ(t -1)δ = 1
Remark 2: Note that in comparison to the relative degree, the relative shift is not the difference between orders of polynomials in δ in denominator and numerator of the transfer function ( 5), µ = rel deg δ F (δ, s) = deg δ a(δ, s)deg δ b(δ, s), which does not play such an important role as the notion of the relative shift.

B. Accessibility

An important notion, concerning additional structural properties of a given system, is the notion of accessibility. It is related mainly to the possibility of controlling the system and can be carried over also to the time-delay case [START_REF] Márquez-Martínez | Note sur l'accessibilité des systèmes non linéaires à retards[END_REF], [START_REF] Márquez-Martínez | Accessibility of nonlinear time-delay systems[END_REF]:

Definition 3: The system (1) is said to be accessible if there does not exist any non-constant autonomous function.

One possibility how to study the accessibility is to introduce the notion of an autonomous element. In the case of system without delays this results in the equivalence of the two following conditions:

• There does not exist any non-constant autonomous function for system. • System does not have any autonomous elements. For linear time-delay systems both definitions are equivalent to the property that the system is torsion free over the ring R[δ, d/dt], [START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF]. However, in the case of nonlinear time-delay systems their equivalence remains an open problem [START_REF] Márquez-Martínez | Accessibility of nonlinear time-delay systems[END_REF]. The accessibility filtration, employing the notion of an autonomous element, was introduced for nonlinear systems without delays in [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF]. Recently, it was extended to the timedelay case as well [START_REF] Márquez-Martínez | Note sur l'accessibilité des systèmes non linéaires à retards[END_REF], [START_REF] Márquez-Martínez | Accessibility of nonlinear time-delay systems[END_REF]. However, it should be noticed that the submodules H k are, in general, infinite dimensional for time-delay systems, which represents a major drawback of their practical computation. Therefore, an alternative way to compute this filtration was studied [START_REF] Márquez-Martínez | Accessibility of nonlinear time-delay systems[END_REF]. However, another alternative accessibility condition can be stated in terms of polynomials a(δ, s) and b(δ, s) in (4).

Proposition 1: The system (1) is accessible if polynomials a(δ, s) and b(δ, s) have no nontrivial common left factors.

Proof: It was shown in [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF] that a(δ, s) and b(δ, s) have no common left factors if and only if system does not have any autonomous elements, following the same line as in [START_REF] Zheng | A polynomial approach to nonlinear system controllability[END_REF].

C. Observability

Following the lines in [START_REF] Perdon | The pole-zero structure of nonlinear control systems[END_REF] also the observability condition can be stated in terms of polynomials.

Proposition 2: The system (10) is observable if and only if deg s a(δ, s) = n Sketch of the proof: If the system is not observable one obtains, by eliminating the state variables in [START_REF] Halás | An algebraic framework generalizing the concept of transfer functions to nonlinear systems[END_REF], see Remark 1 and Appendix I, an input-output equation of the form

y (r) (t) = ϕ({y (k) (t -i), u (l) (t -j); 0 ≤ k ≤ r -1; 0 ≤ l ≤ r; i, j ≥ 0})
where r < n and then deg s a(δ, s) = r.

IV. MODEL MATCHING PROBLEM

The transfer function approach to the model matching problem is the most natural, in comparison with the state space approaches. For the nonlinear system without delays it was recently considered in [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF]. As in linear case, it is stated as the equality of the transfer functions of the model and of the compensated system. Here, we extend some basic ideas of [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF] to the time-delay case. Consider a nonlinear system F and a model G described by the transfer functions

F (δ, s) = b F (δ, s) a F (δ, s) G(δ, s) = b G (δ, s) a G (δ, s)
respectively. Find a (proper) feedforward compensator R, described by the transfer function

R(δ, s)

such that the transfer function of the compensated system coincides with that of the model G

G(δ, s) = F (δ, s) • R(δ, s)
Proposition 3: Given F (δ, s) = 0 and G(δ, s), there exists a feedforward compensator R(δ, s) which solves the model ,s). Proof: By the transfer function algebra [START_REF] Halás | Ore algebras: a polynomial approach to nonlinear timedelay systems[END_REF] we get

matching problem if a R (δ, s)du -b R (δ, s)dv is integrable, where b R (δ,s) a R (δ,s) = F -1 (δ, s) • G(δ
G(δ, s) = F (δ, s) • R(δ, s) Hence, the compensator R(δ, s) = F -1 (δ, s) • G(δ, s) = b R (δ, s) a R (δ, s) (7)
Clearly, the existence of such a compensator is determined by the integrability of the compensator's equation a R (δ, s)du = b R (δ, s)dv.

Example 3:

Given the system F ẏ(t) = u(t -1) + u 2 (t -2)
with the transfer function

F (δ, s) = δs + 2u(t -2)δ 2 s Consider the following three models G(δ, s) = δ s G (δ, s) = δ s + 1 G (δ, s) = δ s + 2y(t)
By ( 7) and ( 3) we get the following transfer functions of the compensators

R(δ, s) = s δs + 2u(t -2)δ 2 • δ s = δ δs + 2u(t -2)δ 2 R (δ, s) = s δs + 2u(t -2)δ 2 • δ s + 1 = δs (s + 1)(δs + 2u(t -2)δ 2 ) R (δ, s) = s δs + 2u(t -2)δ 2 • δ s + 2y(t) = δs -ẏ(t) y(t) δ (s + 2y(t) -ẏ(t) y(t) )(δs + 2u(t -2)δ 2 )
While R(δ, s) and R (δ, s) result in the integrable compensators

R : u(t) = -u 2 (t -1) + v(t) R : ü(t) = -2u(t -1) u(t -1) -u(t)- -u 2 (t -1) + v(t) R (δ, s) does not.
Remark that in multiplying transfer functions one always has to follow the rule (3) which, in general, yields a different result from the standard multiplication, as can be seen for instance in the case of R (δ, s). Remark 3: Note that in contrast to what happens in the linear case, a class of nonlinear systems for which the solution in terms of a feedforward compensator exists is, due to the integrability condition, quite restricted.

A. Properness of the compensator

Obviously, we are interested in finding a solution in a class of proper compensators. However, in comparison to what happens in the case of systems without delays [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF] here, it is not sufficient to restrict just the relative degrees of the model and of the system by the standard inequality

rel deg s G(δ, s) ≥ rel deg s F (δ, s)
even in the linear case.

Example 4: Consider the system

F (δ, s) = δ 2 s -δ and the model G(δ, s) = δ s 2 However, even if rel deg s G(δ, s) = 2 ≥ rel deg s F (δ, s) = 1 the compensator R(δ, s) = F -1 (δ, s) • G(δ, s) = s -δ δ 2 • δ s 2 = s -δ δs 2 is nonproper ü(t -1) = v(t) -v(t -1) ü(t) = v(t + 1) -v(t)
Here, the necessary and sufficient condition, under which the compensator becomes proper, is more involved. Clearly, to satisfy the propernes of the compensator R the highest derivative of its output has to depend only on the past values of its output, input and their derivatives

du (k) (t -l) ∈ span K {du (i1) (t -j), dv (i2) (t -j); 0 ≤ i 1 ≤ k -1; 0 ≤ i 2 ≤ k, j ≥ l} (8)
for some l ≥ 0. Remark 4: Note that in the linear time-delay systems the properness of the compensator ( 7) is satisfied by the requirement that a n R (δ), which is the leading coefficient of a R (δ, s) in s, contains a nonzero constant term [START_REF] Picard | Model Matching for Linear Systems with Delays and 2D Systems[END_REF]. That is, low deg δ a n R (δ) = 0. However, in that case the solution is, in general, given by a compensator from the class of systems of a neutral type. Such solutions are out of the scope of this paper, as we restricted our attention to systems, models and compensators respectively of the form [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF]. From this point of view, the condition (8) means that the leading coefficient contains only a nonzero constant term; that is,

deg δ a n R (δ) = 0.
Clearly, the condition [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and examples[END_REF], in terms of the transfer function [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF], means that in addition to deg s a R (δ, s) ≥ deg s b R (δ, s) also powers of δ in a R (δ, s) and b R (δ, s) have to be taken into account. Namely

deg δ a n R (δ) ≤ min {low deg δ a R (δ, s), low deg δ b R (δ, s)} (9) where a n R (δ) is the leading coefficient of a R (δ, s) in s.
When coming back to the transfer functions F (δ, s) and G(δ, s) one concludes that Proposition 4: R(δ, s) is proper (causal) if and only if

• rel deg s G(δ, s) ≥ rel deg s F (δ, s) • µ F ≤ min {low deg δ b F (δ, s), low deg δ b G (δ, s)
} where µ F is the relative shift of the system F . Proof: For rel deg s G(δ, s) ≥ rel deg s F (δ, s) it follows the same line as in [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF]. For the second part note firstly that µ F = deg δ b m F (δ) where b m F (δ) is the leading coefficient of b F (δ, s) in s. Now, from [START_REF] Fliess | Une interprétation algébrique de la transformation de Laplace et des matrices de transfert[END_REF] it follows that

deg δ a n R (δ) = deg δ b m F (δ) + deg δ a n G (δ) low deg δ a R (δ) = low deg δ b F (δ) + low deg δ a G (δ) low deg δ b R (δ) = low deg δ a F (δ) + low deg δ b G (δ)
where a n G (δ) is the leading coefficient of a G (δ, s) in s. Since both the system and the model are considered to

V. CONCLUSIONS

In this paper, several problems are discussed within the transfer function formalism of nonlinear time-delay systems. Mainly, the structural properties of the transfer functions and the model matching problem. In the model matching problem it was shown that the existence of the compensator requires a restrictive integrability condition and that its properness requires to restrict not only the relative degrees, as in the case without delays [START_REF] Halás | Transfer function approach to the model matching problem of nonlinear systems[END_REF], but the relative shift and a zero structure of both the system and the model as well. A preliminary result relating the state elimination problem for the nonlinear time-delay case was also given. In conclusion, the paper depicts basic ideas in a quite wide range of problems relating the nonlinear time-delay systems and opens the others that are worth to be studied, as for instance finding a solution to the model matching problem of nonlinear time-delay systems allowing the compensator to be a system of a neutral type, or considering the more general class of feedback compensators, etc.

APPENDIX I STATE ELIMINATION FOR SISO NONLINEAR TIME-DELAY

SYSTEMS

Consider the nonlinear time-delay system of the form

ẋ(t) = f ({x(t -i), u(t -j); i, j ≥ 0}) y(t) = g({x(t -i), u(t -j); i, j ≥ 0}) (10)
where x ∈ R n , u ∈ R and y ∈ R.

To find an input-output equation for the system (10) we will try to carry over the idea of the state elimination procedure known for the systems without delays, where an input-output representation is constructed by applying a suitable change of coordinates [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF], also to the time-delay case. However, as the system (10) is the subject of two operators, in comparison to what happens in the case of systems without delays we may not be able to find such a change of coordinates.

Example 5: Consider the system

ẋ(t) = u(t) y(t) = x(t)x(t -1)
Following the lines of [START_REF] Conte | Algebraic Methods for Nonlinear Control Systems[END_REF] ẏ(t) = u(t)x(t -1) + x(t)u(t -1)

However, one cannot go any further, since from y(t) = x(t)x(t -1) one cannot express x(t). Remark 5: It was shown in [START_REF] Anguelova | State elimination and identifiability of the delay parameter for nonlinear time-delay systems[END_REF] that it is possible to derive N independent equations over K in N variables x i (t -j), i = 1, . . . , n, j ≥ 0, for some N which may be greater than n. For instance, in the previous example one considers y(t) = x(t)x(t -1) y(t -1) = x(t -1)x(t -2) ẏ(t -1) = u(t -1)x(t -2) + x(t -1)u(t -2) However, in that case it yields, in general, an input-output equation representing a system of a neutral type, i.e. not being of the form (1). Thus, for our purpose a stronger result is needed.

In what follows we present the state elimination procedure extended to a class of systems with delays. State elimination procedure: Let s denote the minimum nonnegative integer such that

rank K[δ]    c 0 (δ)
. . . is, in fact, an observability matrix and if s < n the system (10) is not observable. Clearly, to eliminate the state variables in y (s) (t) = g (s) (•) we must have that c(δ) ∈ span K[δ] {c 0 (δ), . . . , c s-1 (δ)}.

Then the input-output relation we are looking for can be found. That is y (s) (t) = ϕ({y (k) (t -i), u (l) (t -j)}) [START_REF] Halás | Nonlinear time-delay systems: a polynomial approach using Ore algebras[END_REF] where 0 ≤ k ≤ s -1; 0 ≤ l ≤ s; i, j ≥ 0. Example 6: Consider the system ẋ1 (t) = x 1 (t)x 2 (t -1) ẋ2 (t) = x 2 (t)u(t) y(t) = 1/x 1 (t)

We have ẏ(t) = -x 2 (t -1)/x 1 (t), ÿ(t) = -x 2 (t -1)u(t -1)/x 1 (t) + x 2 2 (t -1)/x 1 (t) and

x 1 (t) = 1/y(t)

x 2 (t) = -ẏ(t + 1)/y(t + 1) Finally ÿ(t) = ẏ(t)u(t -1) -ẏ2 (t)/y(t)

  be of the form (1) note, however, thatdeg δ a n G (δ) = low deg δ a G (δ) = low deg δ a F (δ) = 0 which means that µ F ≤ min {low deg δ b F (δ, s), low deg δ b G (δ, s)} is equivalent to[START_REF] Fliess | Controllability and observability of linear delay systems: an algebraic approach[END_REF].Note that in Example 4 we have µ F = 2, low deg δ b F (δ, s) = 2 and low deg δ b G (δ, s) = 1 and in Example 3 we have µ F = 1, low deg δ b F (δ, s) = 1 and for all G, G and G low deg δ b G (δ, s) = low deg δ b G (δ, s) = low deg δ b G (δ, s) = 1.

  c i (δ) ∈ K 1×n [δ] and d i (δ) ∈ K[δ] are such that dy (i) (t) = c i (δ)dx(t) + d i (δ)du(t)for i = 0, . . . , s. If c 0 (δ) = 0 we define s = 0 and y(t) = g(•) is the input-output equation. Note that

  Halás is with Institute of Control and Industrial Informatics, Fac. of Electrical Engineering and IT, Slovak University of Technology, Ilkovičova 3, 81219 Bratislava, Slovakia. miroslav.halas@stuba.sk Partially supported by the ESF project JPD 3 2005/NP1-047, code No. 13120200115, and the Slovak Research and Development Agency, contract No. LPP-0127-06.

M. C. H. Moog is with IRCCyN, UMR C.N.R.S. 6597, 1 rue de la Noë, BP 92101, 44321 Nantes Cedex 3, France.