Irrigation Quantification Through Backscatter Data Assimilation With a Buddy Check Approach

Irrigation is an important component of the terrestrial water cycle, but it is often poorly accounted for in models. Recent studies have attempted to integrate satellite data and land surface models via data assimilation (DA) to (a) detect and quantify irrigation, and (b) better estimate the related land surface variables such as soil moisture, vegetation, and evapotranspiration. In this study, different synthetic DA experiments are tested to advance satellite DA for the estimation of irrigation. We assimilate synthetic Sentinel‐1 backscatter observations into the Noah‐MP model coupled with an irrigation scheme. When updating soil moisture, we found that the DA sets better initial conditions to trigger irrigation in the model. However, DA updates to wetter conditions can inhibit irrigation simulation. Building on this limitation, we propose an improved DA algorithm using a buddy check approach. The method still updates the land surface, but now the irrigation trigger is not primarily based on the evolution of soil moisture, but on an adaptive innovation (observation minus forecast) outlier detection. The new method was found to be optimal for more temperate climates where irrigation events are less frequent and characterized by higher application rates. It was found that the DA outperforms the model‐only 14‐day irrigation estimates by about 20% in terms of root‐mean‐squared differences, when frequent (daily or every other day) observations are available. With fewer observations or high levels of noise, the system strongly underestimates the irrigation amounts. The method is flexible and can be expanded to other DA systems, also real‐world cases.


Introduction
Irrigation represents more than 70% of freshwater withdrawals (Gleick et al., 2009) making it the most important human activity impacting the terrestrial water cycle.Over the last decades, irrigated areas have expanded almost sixfold (Siebert et al., 2015), and contributed significantly to the increase in global crop production over the same time period (Foley et al., 2011).Under a growing population, food demand will continue to rise, which will inevitably lead to a further expansion and intensification of irrigated agriculture (Foley et al., 2011).In parallel, climate change will impact the irrigation water needs as a result of the expected rising temperatures and drier periods in many regions (Busschaert et al., 2022;Döll, 2002;Fischer et al., 2007).Conversely, irrigation also plays an important role in weather and climate dynamics (Bonfils & Lobell, 2007;Hirsch et al., 2017;Mahmood et al., 2014;Thiery et al., 2017Thiery et al., , 2020)), but it is still not or poorly included in Earth system models (Cook et al., 2015;Gormley-Gallagher et al., 2022;Valmassoi & Keller, 2022).Not only is there a call to monitor irrigation in order to ensure that the available water meets the future irrigation demands, but future climate-related research, and Earth system models in general, could significantly benefit from large-scale irrigation estimates.
In the last years, several methods have been developed to map and quantify irrigation by making use of satellite remote sensing data (Massari et al., 2021).These observations (optical, microwave and gravimetric measurements) are used alone, combined with each other, or with models.Optical (visible or thermal) observations were first used to map irrigation relying on the difference in spectral responses between irrigated and non-irrigated areas (e.g., Ozdogan & Gutman, 2008;Pervez et al., 2014;Salmon et al., 2015;Xie & Lark, 2021;L. Zhang et al., 2022), and more recently using machine learning-based methods (e.g., Jin et al., 2016;Magidi et al., 2021;Nagaraj et al., 2021;C. Zhang et al., 2022).Optical data have further been used to quantify irrigation amounts, mostly using estimates of actual evapotranspiration (ET) based on vegetation indices, sometimes also including models (land surface, water and energy balance), or combining visible and thermal bands (e.g., Bretreger et al., 2022;Brombacher et al., 2022;Droogers et al., 2010;Le Page et al., 2012;Maselli et al., 2020;Olivera-Guerra et al., 2020;van Eekelen et al., 2015;Vogels et al., 2020).Furthermore, satellite-based leaf area index (LAI) and ET products have been assimilated into the Noah-MP land surface model (LSM; Niu et al., 2011) with a focus on improving irrigation estimations by updating the land surface (Nie et al., 2022;J. Zhang et al., 2023).
While irrigation detection methods based on visible and thermal data have progressed and shown some promising results, they typically rely on proxies, and are limited by cloud cover.By contrast, long microwave signals can directly be related to water, and are less limited by atmospheric conditions.Despite their coarse resolutions, soil moisture retrievals from passive L-band radiometers or from active C-band scatterometers can detect wetter moisture when large-scale irrigation water is applied.The first microwave-based irrigation estimates were derived by inverting the soil water balance (SM2RAIN algorithm; Brocca et al., 2014) using several surface soil moisture (SSM) products (Soil Moisture Active Passive [SMAP], Soil Moisture Ocean Salinity [SMOS], Advanced SCATterometer [ASCAT], Advanced Microwave Scanning Radiometer 2 [AMSR2]) at a 25-km resolution (Brocca et al., 2018).They found satisfactory results in terms of irrigation quantification, but the outcome strongly depended on the revisit and the uncertainty of the SSM retrievals.Following the same approach, Dari et al. (2020) achieved finer resolution quantification by downscaling SMAP and SMOS data using the Disaggregation based on Physical and Theoretical scale Change algorithm (DisPATCh; Merlin et al., 2008).Jalilvand et al. (2019) applied this method in a more arid climate (Iran).SSM retrievals (containing irrigation in the signal) were also contrasted against LSM simulations (without irrigation), in order to estimate the amounts of water applied (Zaussinger et al., 2019;Zohaib & Choi, 2020).Despite these methodological advances toward irrigation estimation, the currently most accurate microwave-based satellite SSM retrievals are, for the time being, only available at resolutions coarser than most irrigated fields in Europe.
C-band synthetic aperture radar (CSAR) observations provide data at finer (field-scale) resolutions, and they are also sensitive to soil moisture, albeit with less penetration depth than L-band observations.The Sentinel-1 (S1; Torres et al., 2012) mission from the European Space Agency (ESA) offers the opportunity for frequent (∼2-3 days revisit in Europe) and fine-scale (10 m) observations, which are required for irrigation detection and quantification purposes.The S1 mission comprises a constellation of two satellites (S1-A and S1-B) sensing in two polarizations over land: co-polarized VV (vertically transmitted, vertically received), and cross-polarized VH (vertically transmitted, horizontally received).The S1 CSAR instruments on board of S1-A and S1-B have respective radiometric accuracies (error standard deviation) of 0.25 and 0.32 dB (varying with the acquisition mode and polarization; Miranda et al., 2017).In December 2021, S1-B became unresponsive, resulting in fewer observations from that time onwards.High-resolution SSM estimates retrieved from S1 backscatter have been developed in the last years.Zappa et al. (2021) used the TU Wien S1 SSM product (Bauer-Marschallinger et al., 2019) to detect and quantify irrigation at a local scale, based on spatiotemporal variations in SSM.The method showed promising results in terms of detection and correlation, but systematic underestimations of the irrigation water amounts were found when the observation interval was longer than 1 day (in a follow-up synthetic experiment; Zappa et al., 2022).The first regional datasets of high resolution irrigation water use, based on S1 data, have been released by Dari et al. (2023) using the soil moisture-based inversion approach.While the backscatter itself has already been used in irrigation mapping and timing studies at the local scale (Bazzi, Baghdadi, Fayad, Charron, et al., 2020;Bazzi, Baghdadi, Fayad, Zribi, et al., 2020), the direct use of S1 data to quantify irrigation is only in its infancy, as changes in backscatter are affected by the water in the topsoil, but also by the vegetation (water, volume, density, and geometry), and terrain roughness (McNairn & Shang, 2016).
The most optimal and spatio-temporally complete estimates of irrigation could theoretically be expected to result from a combination of observations (e.g., microwave observations) with models through data assimilation (DA; De Lannoy et al., 2022).Abolafia-Rosenzweig et al. ( 2019) performed SMAP SSM DA into the variable infiltration capacity (VIC) LSM, using a particle batch smoother.With the intent of going to a finer resolution, Jalilvand et al. (2023) used a similar approach using the S1-SMAP SSM product (Das et al., 2019).Ouaadi et al. (2021) assimilated S1-derived SSM data into the FAO-56 (Allen et al., 1998) model with a particle filter.In the three aforementioned studies, irrigation was treated as a model input and not explicitly simulated.A series of DA synthetic experiments were carried out to evaluate the impact of for example, time interval between the assimilated observations and their level of error.They concluded that the proposed techniques could accurately estimate irrigation amounts and timing (only for Ouaadi et al., 2021) but that small errors (levels of noise) and frequent observations are crucial.
The above studies assimilated SSM products, but retrieval assimilation requires rescaling to remove the bias between the forecast (modeled) and observed soil moisture to achieve an optimal DA system.Some of these rescaling approaches can remove irrigation from the signal (Kwon et al., 2022).Moreover, retrievals can introduce errors and inconsistencies in the DA system (De Lannoy et al., 2022), and might suppress irrigation signals.Indeed, microwave-based retrievals often rely on ancillary data, and on empirical change detection algorithms in the case of active measurements.Based on these limitations, Modanesi et al. (2022) decided to directly assimilate the S1 backscatter signal into the Noah-MP LSM (Niu et al., 2011) equipped with a sprinkler irrigation scheme (Ozdogan et al., 2010), where irrigation is dynamically modeled and triggered based on a soil moisture deficit approach.The DA updated SSM and LAI using an Ensemble Kalman Filter (EnKF) and a calibrated Water Cloud Model (WCM; Attema & Ulaby, 1978;Modanesi et al., 2021) as observation operator to map between SSM, LAI and backscatter.The idea is to provide the model with a better initial state, in terms of soil moisture and vegetation, to improve the triggering and estimation of irrigation.However, the method has also shown several limitations related to the model (soil texture, crop type, irrigation parametrization), and the DA system itself.An important problem is that irrigation events could be missed when the DA updates soil moisture to wetter conditions, thereby preventing irrigation simulation.
We set up synthetic experiments based on the system of Modanesi et al. (2022) with the goal to investigate the exact benefits and shortcomings of S1 backscatter DA and to address the main shortcomings (Section 2).In this context, synthetic backscatter observations are generated from a nature run (also called "truth") with a calibrated WCM as observation operator (Modanesi et al., 2021).These observations are then assimilated into erroneous model simulations for which the forcings are altered compared to the reference nature run.We then propose and test a novel method based on an innovation (observation minus forecast) buddy check approach.The land surface state is still updated to have better initial conditions to estimate irrigation, but anomalous high backscatter observations are not assimilated, and used to flag an unmodeled process and trigger irrigation instead.This new method is evaluated for three different sites, under different forcing errors, observation intervals and observation errors in Section 3. Finally, we discuss (Section 4) the possible future developments of the method, along with the opportunity to bring this system to a real world experiment.
The irrigation scheme, coupled to the Noah-MP LSM, was initially developed by Ozdogan et al. (2010) and based on a soil moisture deficit approach.In case of a fully irrigated pixel, the irrigation scheme depends on two conditions for irrigation to be triggered: (a) the day must fall within the growing season, and (b) the rootzone soil moisture must reach a certain depletion.First, the growing season is defined by a greenness vegetation fraction (GVF [ ]) threshold, GVF irr , as suggested by Ozdogan et al. (2010): In this study, GVF is based on a monthly climatology, and GVF min and GVF max are respectively the minimum and maximum monthly GVF.Second, the soil must be dry enough, meaning that the rootzone soil moisture has to reach a certain depletion (MA irr ).In the irrigation scheme, the depletion is defined by the moisture availability (MA [ ]) as follows: where θ l [m 3 m 3 ] and RD l [m] are the soil moisture content and rooting depth (RD) of the l th soil layer, and θ WP [m 3 m 3 ] and θ FC [m 3 m 3 ] are the water contents at wilting point and field capacity of the corresponding soil texture.lroot is the number of considered soil layers in the computation of MA.This number varies over the growing season given that RD i (sum of RD l at day i) directly depends on the GVF i , that is, in which RD max [m] is the maximum rooting depth (a vegetation parameter), and GVF is based on a monthly climatology (as in Equation 1).When both conditions (growing season and dry soil) are fulfilled, irrigation is triggered and the amount of water required brings the rootzone soil moisture back to field capacity.The irrigation rate (Irr rate [mm s 1 ]) is then defined as follows: where the Irr time (in seconds) corresponds to the period when irrigation is allowed.This time frame is set to 06:00 to 10:00 LT following Ozdogan et al. (2010).The Irr rate is then added to the precipitation at each model time step.
In our study, we report the total irrigation amount per day, which is effectively applied at each model time step within the 4-hr irrigation period.
To simulate realistic Irr rate at the field scale, the θ WP and θ FC were chosen to be in line with the Cosby et al. (1984) soil parameters, that is, they are derived with the Campbell (1974) relation: where θ [m 3 m 3 ] is the volumetric soil water content for a defined pressure head h [m H 2 O], and the water content at saturation θ s , the air entry pressure h s [m H 2 O], and the shape parameter b are taken from the Noah-MP v3.6 parameter table.For all soil classes, θ WP is derived for a pressure head of 150 m (pF 4.2).For field capacity, we used the water content at pF 2.5, which is a common reference, as introduced by Colman (1947).Note that the resulting θ WP and θ FC values differ from those in the default Noah-MP v3.6 parameter table created by Chen and Dudhia (2001).In their work, θ WP and θ FC were based on previous literature but then further adapted to intentionally artificially increase the total available water (TAW).This was motivated to indirectly account for the effect of subgrid soil moisture variability on ET dynamics at large-scale simulations.However, we noticed that those adapted values lead to unrealistic irrigation amounts per event compared to the field reference data.The default and updated parameters are presented in Appendix A. Note that the choice was made to use the Community Land Model (CLM) type soil hydraulic scheme (Oleson et al., 2004;Yang & Dickinson, 1996) to simulate the stomatal response to soil moisture as this scheme is not a function of the parameters θ WP and θ FC but solely depends on soil matric potential parameters (Li et al., 2021), taken from the Noah-MP v3.6 parameter table.

Study Areas
Experiments are performed on three sites with different climates and soil textures, which consequently introduce variation in the irrigation volumes and number of applications per season.Table 1 presents the location of the study sites with their main characteristics (Köppen climate class, soil texture, and irrigation characteristics).The MA threshold was chosen to simulate realistic irrigation events compared to benchmark data at these sites.For instance, the measured sprinkler irrigation applications at the Italian site are typically around ∼20 mm/day.The simulated irrigation with a MA threshold of 0.70 [ ] resulted in similar volumes per application.

Nature Run and Synthetic Observations
For all sites, the Noah-MP with irrigation module was used to create a nature run (also called the "truth") that provides reference data of soil moisture, LAI, and irrigation, along with all other variables.The model was run at a temporal resolution of 15 min and at a 0.01°× 0.01°lat-lon spatial resolution.The setup can be readily expanded to other domains.Meteorological data to force the nature run were extracted from the Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA2; Gelaro et al., 2017), which were remapped from a spatial resolution of 0.5°× 0.625°to the resolution of this study by bilinear interpolation.Soil texture parameters were taken from the 1-km Harmonized Soil World Database (HWSD v1.21).Irrigation was triggered when the MA reached the site-specific threshold MA irr (Table 1).The nature simulation ran over the period from 2010 through 2019 after a model spin-up starting on 1 January 2000.The growing season was defined based on the GVF climatology (0.144°spatial resolution; Gutman & Ignatov, 1998) and is specified in Table 1.
Based on these Noah-MP simulations, synthetic γ 0 VV observations were generated daily (at 06:00 LT), by propagating the SSM and LAI estimates through a WCM.The WCM describes the local soil and vegetation scattering processes through semi-empirical formulas (Attema & Ulaby, 1978), using a simple linear relationship between SSM and backscatter (Ulaby et al., 1978).The WCM calibration was done for each site separately, based on Noah-MP SSM and LAI simulations and real S1 γ 0 VV observations, following Modanesi et al. (2021Modanesi et al. ( , 2022)).The synthetic observations are assimilated after perturbing them with different levels of Gaussian white noise, with standard deviations ranging from 0 to 0.7 dB (see Section 2.4).Such observation errors could partly reflect sensor error and mild white noise errors in the observation operator (WCM), incl.roughness and incidence angle effects (for different orbits).Future sensitivity studies could, for example, explicitly consider differences between the "truth" and the assumed LSM and WCM.

Model-Only Run
An overview of all experiments is given in Figure 1.Model-only runs, also called open loop (OL), were performed with the same settings and inputs as the nature run, but with an introduction of forcing error.Specifically, the meteorological forcings were altered in two different ways: (a) all forcings were kept identical to the nature run (MERRA2) except for precipitation which was shifted in time (using 2000-2009, instead of 2010-2019), referred to as OL H (high forcing error); and (b) all MERRA2 forcings were replaced with the European Center for Medium-Range Weather Forecasts (ECMWF) Reanalysis version 5 (ERA5; Hersbach et al., 2020), referred to as OL M (mild forcing error).For the first case, shifting the MERRA2 precipitation (also performed by Girotto et al., 2021) introduces long-term errors since the interannual variability of the precipitation deviates from the truth.By contrast, the experiments with mild forcing errors will use monthly and annual precipitation patterns that are closer to the truth and the errors mostly represent short-term deviations.The OL experiments served as a reference to assess the skill gain of the DA experiments.

Default DA Experiments
In the default DA experiments (DA def ), daily synthetic γ 0 VV , without any addition of noise, are assimilated to update the SSM at all time steps.An additional experiment assimilating an observation every other day with 0.3 dB of white noise is performed for all sites.The irrigation scheme is primarily triggered by the (updated) soil moisture deficit, similar to the OL.In line with the OL simulations, the DA def experiments were also run for a high (DA def,H ) and mild (DA def,M ) forcing error.The aim of DA def is to identify the strengths and weaknesses of the approach by Modanesi et al. (2022).

Buddy Check DA Experiments
The last series of experiments aim at testing our new buddy check approach (DA BC ) described in Section 2.5.2.Again, the DA was tested for two setups of forcing error, that is, DA BC,H and DA BC,M .The method was tested for daily perfect (no white noise) observations, as well as for different overpass intervals (one observation every 1, 2, 3, and 7 days), and different white noise levels in the assimilated observations.White noise is added to the observations in time through a Gaussian distribution of mean zero and different standard deviations (σ): 0, 0.3, 0.5, and 0.7 dB.This range in total observation error (measurement + representativeness error) was chosen considering the radiometric accuracies of the S1 CSAR instruments and the fact that the observation operator is assumed to be perfectly calibrated (i.e., the calibrated WCM is the truth), which limits the representativeness error (van Leeuwen, 2015).When white noise was added to the signal, experiments were run for three different seeds of random noise.Note that all observation intervals and noise combinations are only tested for the German site.

Ensembles
For all experiments (OL H , OL M , DA def,H , DA def,M , DA BC,H , and DA BC,M ), a total of 24 ensemble members were used to estimate the forecast uncertainty.The ensembles were generated by perturbing model forcings (rainfall, incident longwave radiation, incident shortwave radiation) and the SSM state variable (only, i.e. the LAI is not additionally perturbed), with the same perturbation parameters for all experiments.For further details on the perturbation parameters, the reader can refer to Modanesi et al. (2022).It should be noted that in contrast to the setup of Modanesi et al. (2022), a perturbation bias correction method was applied in this study.This adjustment was proposed by Ryu et al. (2009) to avoid unintended biases in the forecast of soil moisture.To be able to use this option with the soil moisture deficit irrigation approach (OL and DA def ), the conditions for which irrigation is triggered required slight modifications.In the Noah-MP v3.6 LSM, irrigation is triggered by considering the MA of each ensemble member individually.This is not compatible with the perturbation bias correction option as the correction can bring the soil moisture of several ensemble members sooner to an irrigation state when some other members have not reached the MA threshold yet.Therefore, in this study and for all experiments, irrigation was triggered based on the ensemble mean MA and the same amount of irrigation water (also calculated from the ensemble mean) is applied to each ensemble member.This was already corrected in the irrigation module of the latest Noah-MP version (4.0.1) implemented in LIS.The observation error standard deviation was set to 1 dB, as in Modanesi et al. (2022).

Synthetic γ 0
VV observations were assimilated to update the SSM, and all other variables via model propagation.An ensemble Kalman filter (EnKF) was employed to ingest γ 0 VV observations into an erroneous version of the Noah-MP LSM (i.e., different from that of the nature run, see Section 2.4).The "true" calibrated WCM was used as observation operator to produce observation predictions based on the erroneous LSM simulations of SSM and LAI.The update equation of the EnKF can be written as follows: for which x+ i is an ensemble of the updated model states at time step i, x i is the ensemble forecast state, y obs,i is the assimilated observation γ 0 VV ) , K i is the Kalman gain, and h i (.) is the WCM observation operator.The innovation at time i (innov i ) is defined as the residual between the observed and the forecast γ 0 VV and is expressed in decibels (dB): Even though the observation predictions use both SSM and LAI as input, the update is limited to SSM here for simplicity (unlike Modanesi et al., 2022).

Buddy Check Approach
Because Modanesi et al. (2022) reported that DA updates to a wetter soil moisture could possibly lead to missed simulated irrigation events, we tested a novel approach in this study, illustrated in Figure 2 for a case with daily observations.We still update SSM as in Equation 6.However, the triggering of irrigation simulation is now not merely based on a modeled soil moisture deficit, but instead always requires a high positive difference between the observed and forecast γ 0 VV (innovation; Equation 7).In other words, the timing of the irrigation is now primarily observation-based.The new method builds on a buddy check approach, commonly used in atmospheric DA (e.g., Dee et al., 2001), avoiding the assimilation of outlier observations.In this case, outliers are detected when the γ 0 VV innovation is suddenly large and positive (highlighted blue dots in Figure 2a).These sudden "jumps" in the innovations can be detected by looking at the difference between two successive innovation, that is, Δinnov i is defined as: where T is the overpass time interval [days].For days in the growing season, irrigation is triggered when: An observation is detected as an outlier, (a) if Δinnov i exceeds a multiple of the standard deviation of the innovations (SD innov,n , always positive) computed on the antecedent n days (excluding the outliers), and (b) when the moisture conditions m are likely to support irrigation.
By definition, the expected error standard deviation of Δinnov is ̅̅ ̅ 2 with a moving window to adaptively account for the natural variability of the model and observation errors.In this study, windows of 30 (SD innov,30 ) and 60 days (SD innov,60 ) were considered.
A strong positive Δinnov i (outlier) hints to an unmodeled process, that is, irrigation.To ensure that irrigation is limited to realistic conditions and to avoid over-irrigation, the dynamic threshold is modulated by a conditional factor m, which is a function of the MA: This rescaling factor gives a higher chance to an irrigation event when the soil is dry than when the rootzone moisture is close to field capacity (MA = 1).A fixed MA threshold for irrigation is thus avoided and replaced by an observation-based trigger, for example, a farmer can irrigate before the uncertain model reaches a critical MA.
Furthermore, unlike the nature (or OL or DA def case), the DA BC approach allows the MA to decrease freely, mimicking the reality that farmers might sometimes irrigate later than expected.The MA is illustrated in Figure 2c.For each irrigation event, the rootzone soil moisture is increased to field capacity, corresponding to MA = 1.However, this is not visible in MA time series (Figure 2c) as the plotted MA is computed based on the daily averaged soil moisture contents.Likewise, m is computed on the daily average MA of the antecedent day.
In short, an outlier innovation is not used to update the soil moisture state, but to correct the model input by adding water as irrigation.The amount of irrigation water is computed by the irrigation scheme coupled to the Noah-MP LSM described in Section 2.1, and it benefits from the updated soil moisture prior to the irrigation trigger.This method was implemented in the LIS framework itself, allowing an online modeling of irrigation with this approach.However, since irrigation starts at 6:00 LT in the irrigation scheme, and the synthetic γ 0 VV observations were produced at the same time of the day, irrigation is not yet part of the observations when they are checked for assimilation (and either assimilated or flagged as outlier): the irrigation of day i is only visible in the observation of day i + 1 (see Figure 2).Hence, with this buddy check approach, irrigation events are always delayed by a day compared the truth, and a negative innovation can be expected on the day following an irrigation event.This technical detail is tied to observation and irrigation times, and could be overcome via postprocessing or model rewinding in future work.

Evaluation Metrics
The experiments were evaluated in terms of irrigation, soil moisture (through the MA), ET, and LAI.The metrics used in this study to evaluate these variables are the Pearson correlation (R), the percentage bias (PBIAS), and the root-mean-square difference (RMSD), and are defined as follows: where x is the value of the simulated land surface variable from the OL or DA experiment, y is the reference value (from the nature run), and N are the number of reference data in time (n = 1, …, N). x and ȳ represent the temporal mean values.The land surface variables are evaluated using a 3-day smoothing window to account for the technical 1-day delay of irrigation.MA is evaluated for all the growing seasons over the years 2010 through 2019 (10 years).LAI and ET are evaluated over the entire 10 years and the R is computed on the anomalies (anomR), because these land surface variables have a clear climatological pattern and naturally result in high R values.
The normalized information contribution in R (NIC R ) and RMSD (NIC RMSD ) are commonly used to describe the improvement or degradation of the estimates compared to a model only (OL) run: Positive NIC values correspond to an improvement while a negative NIC indicates poorer estimations than those of the OL.
Irrigation is evaluated for the growing season only with the same metrics, considering different levels of smoothing where the antecedent n daily irrigation amounts are averaged.Smoothing windows of different lengths (n days) are considered to better grasp for which time intervals (e.g., daily, weekly, monthly) the irrigation events can be accurately simulated.Additionally, binary metrics are considered to assess the performance to detect (in terms of timing) the irrigation events.The probability of detection (POD) and the false alarm ratio (FAR) are computed on a daily basis and were defined by Roebber (2009) as follows: where TP, FN, and FP are the true positive (detected), false negative (missed), and false positive (false) irrigation events.Both metrics range from 0 to 1 and should be equal to 1 and 0 for the POD and FAR, respectively, in an ideal case.Note that POD and FAR were computed on the daily irrigation estimates ±1 day, therefore accounting for the technical delay of irrigation with the buddy check approach (see Section 2.5.2).

Results
First, the performance of the default DA def and new DA BC in terms of irrigation and MA is shown for the 3 different sites with daily assimilation of perfect observations, and assimilation every other day with noisy observations.The impact of both DA approaches on LAI and ET is also presented.Second, the detailed sensitivity analysis of the results to the observation noise and frequency, and to the smoothing level, is done for the German site only (Sections 3.2 and 3.3).

Irrigation and Soil Moisture
In Figure 3, irrigation and MA estimates are evaluated for the three sites and for four different experiments with high model errors: DA def,H and DA BC,H , both assimilating daily perfect observations (1 day 0 dB), and assimilating observations every 2 days containing some white noise (2 days 0.3 dB).DA BC experiments used a 30-day window for the outlier detection threshold (SD innov,30 ).The results are presented in terms of NIC R and NIC RMSD , that is, relative to the corresponding model only run (OL H ) of that site, and for two levels of smoothing (3 days, 14 days).
For irrigation (Figures 3a-3d), the DA BC outperforms the DA def for 2 of the 3 sites, when daily perfect observations are available.DA def shows the best short-term performance at the Italian site, where the irrigation application volumes are smaller (Table 1).When averaged over 14 days, the performance for DA def increases in the scenario with high model errors.When noisy observations are used every other day in DA BC , the NICs become lower and even negative, with a very poor performance in Spain.In this drier region with frequent (weekly) irrigation, the events become harder to detect when only a limited number of observations are available between two consecutive irrigation applications.When an irrigation event is missed, the soil can quickly dry out leading to severe dry biases, conditions in which outliers become hard to detect since the innovations remain large and positive.
For the MA (Figures 3e-3h), the NIC R values for the DA BC experiments are mostly larger than for the irrigation.This can be explained by the fact that the DA BC also catches the true rainfall events that were not in the meteorological forcings of the DA BC .The water signal contained in the synthetic observation can be attributed to rainfall or irrigation, and therefore corrects the MA but does not necessarily improve the irrigation estimates.Less and noisy data assimilation again limits the added value of the satellite data.
The effect of the model error is summarized in Table 2 and Appendix B. Compared to DA def,H , Appendix B shows that DA def performs relatively worse under mild model errors, so that DA BC always outperforms DA def when assimilating daily perfect observations.The irrigation detection skill (without considering volumes) of DA BC assimilating daily observations without noise is shown in Table 2 in terms of POD and FAR, for high and mild model error.Whereas DA def does not consistently improve the POD for all sites, DA BC significantly increases the POD for all sites relative to the OL.Overall, the POD is higher under a mild model error and the FAR is decreased.Falsely detected irrigation events are most common in Germany.In this region, the rainfall is typically larger compared to the other drier sites and the total number of irrigation events over the 10-year experiment is significantly smaller than for the other sites (56 vs. >100), allowing the proportion of falsely detected irrigated events to increase more rapidly.

Impact on Other Land Surface Variables
The quality of irrigation estimates has an impact on the other land surface variables.Figure 4 shows the anomR values between 3-daily smoothed results for selected OL and DA experiments, relative to the truth.The DA BC results are shown for an outlier threshold based on SD innov,30 .First, the difference between the two forcing errors is reflected in the lower anomR values for OL H (Figures 4a and 4c) than for OL M (Figures 4b and 4d), suggesting less room for improvement in the latter case.DA BC using daily observations without noise (DA BC 1 day, 0 dB) is generally superior or equivalent to DA def (with daily, perfect observations).The anomR values for DA BC with less frequent and noisy observations vary across sites with a good performance for Germany and poorer or equivalent anomR values than the OL for Italy and Spain.LAI is less impacted by irrigation over the German site due to the choice of the soil parameters and the model for the stomatal response to water stress (see Section 2.1).On this silt loam site, the vegetation is less sensitive to water and MA levels are generally kept at higher levels.The LAI is more responsive to missed irrigation events in (a) a site presenting for example, a sandy loam texture (such as the Italian site), and (b) in drier and warmer regions, where crop growth mainly relies on irrigation, in contrast to more temperate regions, such as Germany, where irrigation is used to supplement the precipitation.

DA def : Soil Moisture Updating Can Limit Irrigation Estimation
In the DA def experiments, irrigation is primarily triggered when the modeled (or analysis) soil moisture deficit exceeds a threshold.The limitation of this method is obvious at the German site.For this site, Figure 5a illustrates  First, it can be seen that the γ 0 VV assimilation brings the soil moisture, and consequently the MA, to a state that is closer to the nature run, sometimes correctly moving irrigation events closer to the nature run compared to the OL (e.g., in early July).In contrast, another true irrigation event (in June) is delayed in the DA def,H compared to the OL.The MA of DA def,H does not reach the 0.60 threshold before the true event and the irrigation simulation is consequently prevented by updates to wetter soil moisture conditions as a result of large positive innovations (Figure 5).This effect can also clearly be observed in August, where none of the two irrigation events are simulated based on improved initial conditions, but positive soil moisture increments are applied instead.More generally, over the 10-year experiment, the DA irrigation events that are not estimated before the true irrigation are typically delayed or skipped, resulting in daily irrigation estimates ±1 day with low POD values (Table 2).
The missed or delayed irrigation events can be identified in the innovation time series.Large innovations occur on true irrigation days (Figure 5a), highlighting that a process is missed by the model.Instead of avoiding or delaying the event in a DA run, DA BC can identify and trigger such irrigation events, as described next.

Assimilation of Daily Perfect Backscatter Observations
The results of the new buddy check approach are first presented for daily assimilation of perfect observations and using a 30-day window to compute the outlier threshold (SD innov,30 ) at the German site.Figure 6 shows the innovations and irrigation results of DA BC,H for the period 2014-2016 (panel a), with the associated MA time series for 2015 in panel b.The DA BC,H irrigation estimates (dashed blue bars) capture all of the true irrigation events (full black lines) over these 3 years.The falsely detected irrigation events correspond to true rainfall events, leading to an occasional large positive innovation and irrigation is consequently triggered (if the MA is not too high).This proves that when the forcings are erroneous (i.e., rainfall is missed in the DA run), irrigation cannot be dissociated from rainfall in the γ 0 VV signal.By capturing these true rainfall events, the irrigation estimation R may deteriorate, but in turn causes the soil moisture to follow more closely the nature run (here represented by the MA in Figure 6b).Note that the false irrigation events can sometimes lead to a slight wet bias in the MA.Over the whole 10-year experiment, irrigation estimates are strongly improved by DA BC , significantly increasing the POD of the irrigation estimates, and also decreasing the FAR (Table 2).values are mostly positive but improvements remain limited.When the model error is high (DA def,H ), NIC R values increase with smoothing (peak from bimonthly estimates onwards, Figure 7a).However, for the NIC RMSD , even yearly irrigation estimates remain poor (Figure 7b).When the model error is mild (DA def,M ), NICs stay below 0.2 for all smoothing levels.This can be explained by the difference in forcings.The ERA5 meteorology follows the seasonal patterns of the MERRA2 meteorology used for the truth, therefore the seasonal amount of irrigation from a model-only OL M run is close to the nature, as indicated by the high R and low RMSD values, especially for the longer smoothing windows in Figures 7c and 7d.

Effect of Irrigation Smoothing
For mild model errors, DA BC,M is superior to DA def,M for all smoothing intervals.When model errors are high, DA def,H shows a better performance for longer smoothing intervals, likely due to the detection of true rainfall events in DA BC,H (expressed in the high FAR value).Between a 3-day and monthly smoothing interval, R and RMSD values are improved by more than 30% and 10%, respectively, for all forcing errors.The poor effect on irrigation quantification at a daily scale can be attributed to the timing of the observation and irrigation application (see technical 1-day delay, Section 2.5.2).

Effect of Observation Interval and White Noise
DA BC was tested for different observation intervals (1, 2, 3, and 7 days) and observation noise levels (0, 0.3, 0.5, and 0.7 dB) at the German site, to assess which observation configuration would be ideally suited for irrigation estimation.The 14-day smoothed daily DA BC irrigation estimates are evaluated relative to the OL through the NIC R and NIC RMSD .The PBIAS (difference between the daily simulated and nature irrigation relative to the nature irrigation, in %) is also assessed to indicate if there is a general over-or underestimation of the irrigation amounts.The results for the DA BC,H and DA BC,M experiments are shown in Figure 8. Two thresholds were tested to trigger irrigation, using two different window sizes for the calculation of SD innov : 30 days (SD innov,30 ), and 60 days (SD innov,60 ).Note that the experiments assimilating observations with a 7-day interval could only be performed with a 60-day window in order to compute a standard deviation with enough data points.This longest observation interval is also reason why 14-day irrigation estimates are shown, and not estimates smoothed for shorter intervals, which resulted in slightly higher NICs (Figure 7).All experiments with white noise in the observation signal were performed for three random seeds of added noise and the average metric is presented.
The performance of the buddy check approach degrades with longer observation intervals, to reach negative NICs for the assimilation of weekly observations (Figures 8a, 8b, 8d, and 8e).As already shown in Figure 3, the DA BC does not always result in a higher performance compared to the DA def .For the Italian site (shown in Figure 8), DA def already outperforms DA BC when observations are not assimilated on a daily basis.With frequent observations, there is a slight positive bias (Figures 8c and 8f), meaning that more irrigation is simulated compared to the nature run.This effect can be attributed to the detection of true rainfall events in addition to the irrigation events.Less frequent observations lead to stronger underestimations of the irrigation amounts, as shown by the negative PBIAS (Figure 8f).This has consequences for the LAI (and other land surface variables), as already demonstrated for the Spanish and Italian sites in Section 3.1.2.The largest underestimation is found for DA BC,M with a 7-day observation interval and noise.Counter-intuitively, underestimations are less severe for DA BC,H likely due to the 14-day smoothing window, in which false irrigation events (more frequent for DA BC,H ) compensate for the missed true events.This compensation results in a larger MA improvement for DA BC,H than for DA BC,M (shown in Appendix C).
The white noise in the signal strongly affects the performance under all observation intervals by increasing the number of missed irrigation events, as shown by the decreasing PBIAS when noise is added (Figures 8c and 8f).and (c, d) PBIAS [%] of 14-day irrigation amount estimates for the different DA BC , for 1, 2, 3, and 7-day observation intervals, for the German site.The colors correspond to the level of Gaussian white noise added to the signal (σ, dB), and for all experiments with noise, the mean metric is taken from the three runs.The rows are associated with the window size taken to compute the standard deviation for the irrigation threshold, where (a-c) are based on a 30-day window (SD innov,30 ), and (d-f) relate to experiments with a 60-day window (SD innov,60 ).Dots correspond to DA BC,H and crosses to DA BC,M .
Sorted from the most to least important factor, the added noise tends to degrade the ability to detect an outlier (and hence an irrigation event) by: (a) increasing the SD innov,n , (b) occasionally decreasing the irrigation signal in the observation affecting the Δinnov, and (c) resulting in biased MA at a certain time step (wetter or dryer) affecting the rescaling of the model error (2 * SD innov ).The threshold for the outlier is strongly affected, increasing this threshold (2 * SD innov,30 * m) from 1 dB for perfect observations to 1.7 dB for observations with 0.7 dB white noise for daily DA BC,H .For reasonable levels of noise and frequent observations (≤0.3 dB), the RMSD is reduced by at least 15% for both forcing errors compared to the OL, and the R is increased by at least 25% (Figures 8d  and 8e).
Increasing the window size for the computation of the SD innov did not significantly alter the NIC values.However, the PBIAS is overall decreased, meaning that the irrigation overestimation is more limited under shorter observation intervals (1 and 2 days) but this also results in more severe underestimations under sparse observations (Figure 8e).More irrigation (or true rainfall) events remain undetected because the SD innov considers innovations up to 60 days before the event.This is problematic, especially at the beginning of the season as the natural variation in the innovations is larger in the late winter or spring (wet season), which are then considered in the computation of the SD innov,60 .A window size of 30 days seems more appropriate, in the sense that this SD innov should capture the natural variation of the innovations which is mainly induced by forcing errors but also vegetation.

Novel Approach to Estimate Irrigation in a DA System
The new method shows good performance over the three sites when daily observations are available.However, when the observations become sparse in time, the skill of the buddy check approach is highly influenced by the irrigation frequency and volumes (mainly defined by the climate, the soil texture, and the irrigation threshold in this synthetic setup).The poor irrigation estimation skill over the sites in Italy and Spain was mainly related to the high frequency of irrigation events (up to one application every 3 days).In these regions, the system could greatly benefit from model rewinding to avoid missing irrigation events when the previous estimation is delayed.Another determining factor is the level of white noise in the observations affecting the irrigation detection skill of the presented method, especially where irrigation application volumes are small (∼20 mm), such as for the Italian site.In this region, DA def showed large improvements (Figure 3), likely related to the fact that positive innovations following an irrigation event are smaller, limiting the shortcoming of DA def (explained in Section 3.2).
In Germany, the buddy check approach still works reasonably when observations are available every 2 or 3 days, corresponding to the initial revisit interval of the S1 constellation over Europe.Weekly observations would not be sufficient to guarantee the irrigation detection skill and lead to severe underestimations of irrigation water.The failure of S1-B halved the number of available observations (one every ∼4 days in Europe, 12 days elsewhere), making the buddy check approach (used within a S1 DA system) unsuitable outside of Europe until the launch of the next satellite (S1-C, expected in the near future).Other synthetic studies assimilating S1-related SSM products (Abolafia-Rosenzweig et al., 2019;Jalilvand et al., 2023;Ouaadi et al., 2021;Zappa et al., 2022) also highlighted the importance of frequent observations.Zappa et al. (2022) reported large irrigation underestimations when observations are too sparse in time.Similar to our study using Δinnov to detect outliers, their approach is based on observed differences in soil moisture (ΔSM).Both methods are observation-based, making them sensitive to observation noise, and to underestimation of the irrigation amounts with less frequent observations, because the irrigation signal fades away in the observations with time.In short, our buddy check approach underestimates irrigation more for infrequent observations or when a large time window is used to compute the Δinnov threshold (SD innov,60 ), but the remaining detected irrigation events are identified accurately in time and with the correct amounts of irrigation water, or they compensate for missed rainfall.precipitation RMSD values of around 5 mm day 1 and 2 mm day 1 , respectively (also for the growing season).For daily observations, good performances were found across the three sites without introducing a new locationdependent parameter.Second, using the MA to rescale the outlier threshold is more realistic than the use of a fixed threshold.For irrigation to be triggered, the MA can be off from the truth and still detect an irrigation event if the signal is strong enough in the innovations.This is more in line with a real world situation where irrigation is not necessarily determined by a fixed soil moisture deficit value, but depends on the agricultural practices and more generally on the water availability (Nie et al., 2021).

Limitations and Opportunities
The main limitation of the buddy check approach is the missing of irrigation events, esp.for longer observation intervals or drier regions where the soil moisture dries out rapidly and irrigation events are frequent.A first improvement would be to implement a model rewind system to minimize the technical 1-day delay of irrigation events.However, even with this development, irrigation events will still be missed if satellite observations are not frequent enough.We could then consider a hybrid DA system, where the buddy check approach is supplemented with a pure MA-based irrigation model trigger, if the observation interval exceeds the surface soil memory of an irrigation event.The missed irrigation events have a strong impact on vegetation in drier regions, where LAI strongly declines.This issue can be tackled by jointly updating SSM and LAI, as done by Modanesi et al. (2022).Vegetation updating would ask for the assimilation of backscatter in cross-polarization (γ 0 VH , or γ 0 VH /γ 0 VV ) as this signal has shown to be more affected by vegetation (Patel et al., 2006;Vreugdenhil et al., 2018).A joint assimilation of γ 0 VV and γ 0 VH would require a combination of both innovations in our buddy check method approach.
Future system developments could involve leveraging the ensembles within the EnKF.The ensemble spread before assimilation can serve as basis to determine whether irrigation should be triggered in the system.For example, if the observation plus its associated uncertainty falls outside the forecasted ensemble, irrigation could be applied.An alternative development entails refraining from triggering irrigation for all ensemble members simultaneously but rather applying irrigation to each member individually (similarly to Modanesi et al., 2022), allowing for irrigation uncertainty estimates.However, this alternative would not be compatible with the perturbation bias correction option proposed by Ryu et al. (2009), as described in Section 2.4.4.In general, the use of ensemble information to trigger or estimate irrigation will require a careful optimization of the ensemble perturbations.
The newly proposed buddy check approach could also be used to estimate irrigation with other (e.g., particle) filters or other observations than backscatter data.High resolution L-band soil moisture data would be interesting to guide the estimation of irrigation amounts.Such data can be obtained from for example, downscaled SMOS or AMSR-E with DisPATCh (Malbéteau et al., 2016;Merlin et al., 2013) or future missions such as the Copernicus ROSE-L (Davidson & Furnell, 2021) and the SMOS-HR (Rodríguez-Fernández et al., 2019).Nevertheless, the outcome would strongly depend on the quality of the retrievals, and an appropriate bias treatment is needed to avoid the attenuation of the irrigation in the signal (Kumar et al., 2015;Kwon et al., 2022).Instead of changing the type of assimilated observations, the buddy check method could also be used in systems with other models, possibly crop models that are originally designed for agriculture, offering new opportunities in such fields of application.

Future Real World Experiment
The success of a real world DA experiment with the buddy check approach will depend on the observability of irrigation and model-related limitations.First, satellite observations need to be available at high spatial and temporal resolution, and the actual type of irrigation method needs to be detectable.There is a chance that irrigation is applied for consecutive days over different fractions of the observed satellite footprint (one or a few fields receive irrigation per day).In that case, "jumps" in the innovations will not be detected and the backscatter signal is likely to remain high for these consecutive days.Future research is necessary to counter this limitation, or higher resolution input data and observations are needed.Similarly, some types of irrigation will be easy to detect, whereas others not.Punctual large sprinkler events, as simulated in this study, are more easily detectable than for example, drip irrigation, which is typically applied in smaller amounts and more frequently.
Second, the LSM and WCM are assumed to be perfect in our synthetic study, but the model-related limitations already mentioned in Modanesi et al. (2022) will be important when going to a real world experiment.Concerning the LSM, the quality of the input data is crucial.Erroneous crop rooting depth, soil texture, or irrigation fraction, would automatically lead to a bias in the irrigation amounts since these factors directly influence the volumes of irrigation water (see LSM equations in Section 2.1).Though more flexible than a rigid soil moisture threshold to trigger irrigation, the MA irr parameter, used to rescale the outlier threshold (Equation 10), will need some calibration, as this value varies across regions.Likely, the MA irr parameter will determine the sensitivity of the model to true rainfall events as the latter cannot be distinguished from true irrigation events (if they are of the same magnitude).However, this shortcoming benefits the DA BC in its ability to improve the MA and consequently the water balance (when frequent observations are available).The observation operator (e.g., WCM) could also pose a limitation, when directly assimilating microwave signals.Rather than calibrating an empirical model, novel machine-learning based observation operators could improve the system (de Roos et al., 2023;Rains et al., 2022), but in both cases, the observation operator training might suffer from an inaccurate match between irrigation simulation (and the effect on soil moisture) and irrigation observed in the satellite signals.
In short, a controlled field-scale experiment will be needed to bring the buddy check approach to the real world and consider further developments that were discussed above.Compared to DA def , the new method relies on observations to trigger irrigation, which has the potential to improve field-scale irrigation estimates by estimating the timing and volumes more accurately but is likely not suited for coarser resolution applications, or when the interval between two observations becomes too large.Even if DA def presents a strong shortcoming, it will likely be able to better estimate irrigation at the regional level when irrigation amounts are aggregated over biweekly or monthly time scales, or when observations are not frequently available, as compared to the irrigation frequency.

Conclusions
Irrigation detection and quantification are major challenges.New methods based on remote sensing data are now emerging, including the use of microwave observations in combination with models through data assimilation (DA).Modanesi et al. (2022) assimilated Sentinel-1 backscatter observations into the Noah-MP version 3.6, coupled to a sprinkler irrigation scheme.The soil moisture and vegetation state were updated to set better initial conditions to trigger irrigation simulation, but the system also had limitations, esp.when large updates to wetter conditions delayed or completely inhibited the process-based modeling of irrigation events.
In this study, we conducted synthetic experiments for the assimilation of backscatter observations γ 0 VV ) to update soil moisture in a system with erroneous meteorological forcings.After illustrating the shortcoming of blindly assimilating all data for state updating, a new method was developed based on a buddy check approach, in which unexpected changes in innovations (observation minus forecast) are detected and not assimilated.The method still updates the land surface to guarantee the best possible initial conditions to estimate irrigation amounts, but when an outlier in the Δinnov (difference between two consecutive innovations) is detected, an unmodeled process is assumed and the large innovation is not assimilated.Consequently, the "missed" irrigation is triggered, if the rootzone soil moisture is dry and it is a day in the growing season.The new method is now primarily observationbased, and better adapts to the timing of real irrigation events.The threshold value to identify outlier innovations was made dependent on the locally and temporally varying errors in the system.The method was tested on three different sites (in Germany, Italy, and Spain) with different climates, soil texture, and irrigation thresholds.A detailed evaluation was then performed for the German site where the method was tested for several observation intervals and noise.The main results can be summarized as follows: 1.When daily observations are available with reasonable levels of noise (≤0.3 dB), the method shows good performances for all three study sites.The probability of irrigation detection more than doubles for two sites, when assimilating perfect daily observations.When the observations become sparser in time, or when they contain larger noise levels, the performance decreases rapidly for regions where the irrigation events are very frequent (weekly or less), or when application rates are lower.2. For biweekly aggregated irrigation estimates in Germany, and compared to a model-only run, the new DA method reaches about 40% and 20% of improvement in terms of Pearson R and RMSD, respectively, when frequent observations (daily or every other day) are assimilated.From a 3-day observation interval onward, the performance degrades but remains reasonable (NICs > 10%), and for weekly observations, there is no improvement compared to a model-only run (NICs close to zero or negative).

Figure 1 .
Figure 1.Overview of the experiments.All experiments are repeated for high forcing error (OL H , DA def,H , DA BC,H ) and mild forcing error (OL M , DA def,M , DA BC,M ).All combinations of overpass and white noise for DA BC are tested for the German site only.For the DA BC over the other sites and DA def , two experiments are performed: (1) assimilating daily perfect γ 0 VV

Figure 2 .
Figure 2. Illustration of the buddy check approach for the Spanish site under a mild model error (DA BC,M ).(a) Irrigation time series [mm day 1 ] of the nature run and the DA run with buddy check approach with the innovations [dB] in gray.(b) Δinnov [dB] time series with the dynamic threshold (based on a 30-day window SD innov,30 ).(c) Moisture availability (MA [ ]) of the antecedent day of the nature run (and corresponding MA irr threshold in gray) and the DA buddy check.The shaded blue stripes highlight the irrigation events.The green arrows illustrate (1) the rescaling of the threshold by the MA and (2) the outlier that triggers irrigation.The blue arrow (3) links the irrigation event to the increase in MA.

Figure 3 .
Figure 3. NIC R and NIC RMSD for irrigation [mm day 1 ] (a-d) and moisture availability [ ] (e-h) smoothed over two time windows (3 and 14 days; columns).DA def,H and DA BC,H experiments are presented in green and blue, respectively, with full bars corresponding to the assimilation of daily observations without noise (1 day 0 dB), and the stippled bars present experiments assimilating observations every 2 days with 0.3 dB of white noise (2 days 0.3 dB).For the experiments with white noise, the bar represents the mean of the metric across the three seeds and the whiskers extend to the minimum and maximum NIC.The metrics (R and RMSD) of the OL H are shown in the plot for each domain.
some irrigation events of the nature run ("truth"), OL H , and DA def,H along with the γ 0 VV innovations.The corresponding MA time series are shown in Figure 5b with the MA Irr threshold of 0.60 [ ].

Figure 4 .
Figure 4. anomR [ ] for 3-daily LAI [ ] (a, b) and ET [mm day 1 ] (c, d) for the entire simulation period.The columns correspond to the model error (high or mild).DA def and DA BC experiments are presented in green and blue, respectively, with full bars corresponding to the assimilation of daily observations without noise (1 day 0 dB), and the stippled bar presents experiments assimilating observations every 2 days with 0.3 dB of white noise (2 days 0.3 dB).For the experiments with white noise, the bar represents the mean of the metric across the three seeds and the whiskers extend to the minimum and maximum anomR.

Figure 7
Figure 7 further compares the performance of daily DA BC and DA def with perfect observations, as a function of model error and temporal smoothing at the German site.The NIC R and NIC RMSD values are shown and after smoothing the daily irrigation estimates with various time windows.For the DA def experiments (green), NIC

Figure 6 .
Figure 6.(a) Time series for the German site of the irrigation [mm day 1 ] of the nature run, DA def,H , and DA BC,H for daily observations without noise and an outlier threshold based on SD innov,30 .Innovations of DA BC,H are shown in the background in gray.(b) Time series for the growing season of 2015 (shaded in blue in a) of the moisture availability (MA [ ]) of the nature run, DA def,H , and DA BC,H for the irrigation months (April through October).In (b), the full line corresponds to the threshold for the nature run and DA def,H (0.60 [ ]).

Figure 7 .
Figure 7. (a) NIC R [ ] and (b) NIC RMSD [ ] for irrigation smoothed with different window sizes for the German site.(c) Pearson R [ ] and (d) RMSD [mm day 1 ] of the OL.The dots and the crosses correspond to the high and mild forcing error experiments, respectively.DA BC were performed by assimilating daily perfect observations.All metrics were computed on the irrigation months only (April through October) over the 10-year experiment.

Figure 8 .
Figure 8. (a, d) NIC R [ ], (b, e) NIC RMSD [ ],and (c, d) PBIAS [%] of 14-day irrigation amount estimates for the different DA BC , for 1, 2, 3, and 7-day observation intervals, for the German site.The colors correspond to the level of Gaussian white noise added to the signal (σ, dB), and for all experiments with noise, the mean metric is taken from the three runs.The rows are associated with the window size taken to compute the standard deviation for the irrigation threshold, where (a-c) are based on a 30-day window (SD innov,30 ), and (d-f) relate to experiments with a 60-day window (SD innov,60 ).Dots correspond to DA BC,H and crosses to DA BC,M .
The main advantages of the new buddy check approach are (a) the flexibility of the outlier detection method to different situations (and different errors), and (b) the substitution of the strict model-based soil moisture threshold for irrigation by a rescaling of the outlier threshold, making the irrigation estimates more in line with what happens in reality (as observed by the satellite).First, the standard deviation of the γ 0 VV innovations for the DA H and DA M experiments reach average values of 0.8 and 0.6 dB, respectively, during the growing season for the German site.These values are in line with the magnitude of the expected high and mild forcing errors, with Journal of Advances in Modeling Earth Systems 10.1029/2023MS003661 BUSSCHAERT ET AL.

Figure C1 .
Figure C1.(a, d) NIC R [ ], (b, e) NIC RMSD [ ],and (c, d) PBIAS [%] of 14-day MA estimates for the different DA BC , for 1, 2, 3, and 7-day observation intervals, for the German site.The colors correspond to the level of Gaussian white noise added to the signal (σ, dB), and for all experiments with noise, the mean metric is taken from the three runs.The rows are associated with the window size taken to compute the standard deviation for the irrigation threshold, where (a-c) are based on a 30-day window (SD innov,30 ), and (d-f) relate to experiments with a 60-day window (SD innov,60 ).Dots correspond to DA BC,H and crosses to DA BC,M .

Table 1
Coordinates (Lat-Lon) of the Different Sites With the Corresponding Köppen Climate Class, Soil Texture, Chosen Irrigation Threshold (MA irr [ ]), Average Irrigation Interval [Days] Over the Summer Months (June, July, August), and Average Irrigation Application (Irr rate [mm day 1 ]) Over the Growing Season, Based on the Nature Run

Table 2
POD and FAR for Daily Irrigation Estimates ±1 Day for High Model Error (OL H , DA def,H , DA BC,H ) and Mild Model Error (OL M , DA def,M , DA BC,M ) Experiments POD OL H DA def,H DA BC,H OL M DA def,M DA BC,M BUSSCHAERT ET AL.