An efficient workflow to accurately compute groundwater recharge for the study of rainfall-triggered deep-seated landslides , application to the Séchilienne unstable slope ( western Alps )

Pore water pressure build-up by recharge of underground hydrosystems is one of the main triggering factors of deep-seated landslides. In most deep-seated landslides, pore water pressure data are not available since piezometers, if any, have a very short lifespan because of slope movements. As a consequence, indirect parameters, such as the calculated recharge, are the only data which enable understanding landslide hydrodynamic behaviour. However, in landslide studies, methods and recharge-area parameters used to determine the groundwater recharge are rarely detailed. In this study, the groundwater recharge is estimated with a soil-water balance based on characterisation of evapotranspiration and parameters characterising the recharge area (soil available water capacity, runoff and vegetation coefficient). A workflow to compute daily groundwater recharge is developed. This workflow requires the records of precipitation, air temperature, relative humidity, solar radiation and wind speed within or close to the landslide area. The determination of the parameters of the recharge area is based on a spatial analysis requiring field observations and spatial data sets (digital elevation models, aerial photographs and geological maps). This study demonstrates that the performance of the correlation with landslide displacement velocity data is significantly improved using the recharge estimated with the proposed workflow. The coefficient of determination obtained with the recharge estimated with the proposed workflow is 78 % higher on average than that obtained with precipitation, and is 38 % higher on average than that obtained with recharge computed with a commonly used simplification in landslide studies (recharge = precipitation minus non-calibrated evapotranspiration method).


Introduction
Pore water pressure build-up by recharge of aquifers is one of the main triggering factors of destabilisation of deep-seated landslides (Noverraz et al., 1998;Van Asch et al., 1999;Guglielmi et al., 2005;Bogaard et al., 2007;Bonzanigo et al., 2007).In most deep-seated landslides, pore water pressure data are not available since piezometers, if any, have a very short lifespan because of slope movements.In addition, landslides show heterogeneous, anisotropic and discontinuous properties (Cappa et al., 2004;Binet et al., 2007a) and local measurements are rarely representative of the overall behaviour of the landslide aquifers.In the absence of piezometric measurements, the groundwater recharge is used as the most relevant parameter to characterise the pore water pressure of the landslide aquifers.Groundwater recharge (hereafter recharge), also referred to as deep percolation, is the part of the precipitation which recharges the saturated zones (aquifers).
Landslide studies involve a wide range of specialities (subsurface geophysics, structural geology, modelling, geotechnics, and geomechanics).Scientists or engineers in charge of landslides may not have the required hydrology knowledge to accurately estimate the recharge.In most cases, deep-seated landslide studies devoted to characterise the rainfall-destabilisation relationships do not take into account recharge with enough accuracy.In particular, some studies estimate the recharge without calibration of the evapotranspiration estimation methods and without soil-water balance (Canuti et al., 1985;Alfonsi, 1997;Hong et al., 2005;Binet et al., 2007b;Durville et al., 2009;Pisani et al., 2010;Prokešová et al., 2013).Lastly, several studies use precipita-tion data instead of the recharge (Rochet et al., 1994;Zêzere et al., 2005;Meric et al., 2006;Helmstetter and Garambois, 2010;Belle et al., 2014).These approaches can overestimate the groundwater recharge and can thus bias the characterisation of the relationship between rainfall and destabilisation.A more accurate estimation of the groundwater recharge signal can improve the accuracy of these studies.So far, no computation workflow has been proposed to estimate simply and accurately the recharge in the context of landslide studies.Patwardhan et al. (1990) showed that the soil-water balance method is an accurate way to estimate groundwater recharge.Recharge computation with a soil-water balance depends mainly on the surface runoff, the soil available water capacity (SAWC) and the specific vegetation (so-called crop) evapotranspiration (ET c , also referred to as potential evapotranspiration), itself being deduced from reference vegetation evapotranspiration (ET 0 ) with a vegetation coefficient (K c ).The Penman-Monteith method (Eq.A6 in Appendix A), hereafter referred to as the ET 0 standard equation or FAO-56 PM, developed in the paper FAO-56 (Food and Agriculture Organization of the United Nations) is considered by the scientific community as a global standard method to estimate ET 0 worldwide (Jensen et al., 1990;Allen et al., 1998).This method requires the knowledge of the air relative humidity, the air temperature, the wind speed and the solar radiation.However, most weather stations in landslide areas record only air temperature and rainfall.Unlike the FAO-56 PM method, methods based only on air temperature and solar radiation (R S ) allow for a simpler expression of ET 0 (Tabari et al., 2013).Besides, R S can also be estimated only from air temperature (Almorox, 2011), thus allowing ET 0 to be obtained only from air temperature records.These reducedset methods are developed under specific site conditions and must be calibrated in order to improve accuracy (Allen et al., 1994;Shahidian et al., 2012).
The objective of this study is to develop a parsimonious, yet robust, guideline workflow to calculate time series of groundwater recharge at the scale of the recharge area, time series that can subsequently be used as a deterministic variable in landslide studies.To maximise the accessibility to various user groups, we strive to develop an efficient method, balancing technical accuracy with operational simplicity.The proposed workflow is applied on the deepseated Séchilienne landslide.To test its reliability, a correlation analysis is used to evaluate whether the calculated groundwater recharge is more strongly correlated with measured land mass displacement velocities than with precipitation or with recharge estimated with a common simplification in landslide studies (recharge = precipitation minus non-calibrated ET 0 (Canuti et al., 1985;Binet et al., 2007b;Pisani et al., 2010;Prokešová et al., 2013).The significance of the correlations is assessed with bootstrap tests.The proposed study aims at showing that an accurate estimation of the recharge can significantly improve the results of rainfalldisplacement studies.

General workflow
In the case of deep-seated landslides triggered by deep watersaturated zones, the impact of a multiday cumulative rainfall is far more significant than rainfall duration or intensity (Van Asch et al., 1999;Guzzetti et al., 2008).For these reasons, the workflow is developed to compute daily groundwater recharge.Similarly, this study is based on displacement recorded at a daily time step.For the sake of simplicity, the daily displacement, equivalent to a velocity measurement in millimetres per day, is hereafter referred to as displacement.The groundwater recharge is estimated with a soilwater balance based on characterisation of ET 0 and parameters characterising the recharge area (SAWC, runoff and K c ).The computation workflow (Fig. 1), hereafter referred to as LRIW (Landslide Recharge Input Workflow), includes four steps.
The estimation of the ET 0 requires the records of air temperature within the landslide area and relative humidity, solar radiation and wind speed within or close to the landslide area.In the case of a landslide-located weather station recording only the temperature, the first step (detailed in Sect.2.2) consists of a regional calibration of ET 0 and R S reduced-set equations (equations detailed in Appendix A).The calibrated methods then allow estimating evapotranspiration based only on temperature records.In the case of a landslide weather station recording the full set of parameters, the first step can be skipped and the FAO-56 PM method can then be used to estimate ET 0 .The second step (detailed in Sect.2.3) consists in estimating the recharge-area parameters (surface runoff, SAWC and K c ) using a GIS (geographic information systems) composite method requiring field observations and spatial data sets (digital elevation models (DEMs), aerial photographs and geological maps).The third step (detailed in Sect.2.4) uses a soil-water balance to estimate the recharge with the estimated ET 0 and the estimation of the recharge-area parameters.The fourth step (detailed in Sect.2.5) consists of a sensitivity analysis based on a recharge-displacement velocity correlation and is performed in order to refine the estimations of SAWC and runoff coefficient.

2.2
Step 1: regional calibration of ET 0 and R S methods ET 0 reduced-set and R S temperature methods were initially developed for given regions or sites with their own climatic conditions and must be calibrated to take into account the weather conditions of the study site.Details about calibration can be found in the literature (Allen et al., 1994;Itenfisu et al., 2003;Lu et al., 2005;Alkaeed et al., 2006;Alexandris et al., 2008;Shahidian et al., 2012;Tabari et al., 2013).
The regional calibration method (step 1; Fig. 1) is performed using the records of nearby weather stations (here- Step 1: calibration of standard ET 0 and R S methods. Step 2: estimation of recharge-area parameters required for the soil-water balance (R coeff , K c and SAWC) and the infiltration structures.
Step 3: computation of the recharge with the soil-water balance.* In the case of a landslide-located weather station recording the full set of parameters, the first step can be skipped and the ET 0 of step 3 can be estimated directly at the study site with the standard ET 0 method (FAO-56 PM method).
after referred to as reference weather stations) having similar climatic conditions as the study site and recording the required meteorological parameters.The calibration of R S and ET 0 methods are performed for each reference weather station (local scale).The local adjustment coefficients of the reference stations are then averaged in order to define a regional calibration.The user has to maintain a balance between the number of selected reference stations and the necessity for these stations to be located in areas with climatic conditions similar to those of the study site.For sites with a sparse weather station network, one reference station can be sufficient for the calibration, provided that this station has the same weather conditions as those of the studied site.
The performance assessment of regional-scale calibrated methods is based on the comparison between observed measurements and calibrated estimates for R S and between FAO-56 PM estimates and calibrated estimates for ET 0 for each reference weather station.Performance indicators are the coefficient of determination (R 2 ), the slope and the intercept from linear regression (independent variable: estimated pa-rameter; dependant variable: reference parameter), and the root mean square error (RMSE).Bristow and Campbell (1984) and Hargreaves and Samani (1985) proposed methods to compute R S based solely on the air temperature measurement (Eqs.A1 and A2 in Appendix A).Castellvi (2001) demonstrated that both methods show good results for daily frequencies.The coefficients of the Bristow-Campbell method have to be evaluated.The coefficients of the Hargreaves-Samani method have default values.However, Trajkovic (2007) showed that the regional calibration of the Hargreaves-Samani method is significantly improved by an adjustment of the coefficients rather than by a linear regression.Therefore, all the HS mod R s coefficients are adjusted.In this study, modified forms of the Bristow-Campbell method (Eq.A3) and Hargreaves-Samani method (Eq.A4) are used.For the R S equations, the adjustment of the local calibration coefficients is non-linear.To adjust the calibration coefficients, a grid search iterative algorithm is used to maximise the R 2 value while minimising the RMSE at each reference weather station.

Evapotranspiration methods
ET 0 is the evapotranspiration from a reference grass surface and is used as a standard from which ET c is deduced as follows (Allen et al., 1998): where K c is the vegetation coefficient.Several ET 0 methods using a reduced data set in comparison to the FAO-56 PM method have been developed worldwide.Only a few methods are commonly used.This is the case with the five ET 0 methods selected for this study, which have shown good performance when using daily to weekly frequencies (Trajkovic, 2005;Yoder et al., 2005;Alexandris et al., 2008;Shahidian et al., 2012;Tabari et al., 2013).The five selected ET 0 methods, namely the methods of Hargreaves and Samani (1985), Makkink (1957), Turc (1961), Priestley and Taylor (1972), and the Penman-Monteith reduced-set method (Allen et al., 1998), require records of R S and temperature (Eqs.A7-A12 in Appendix A).As R S can be estimated with a calibrated R S temperature-based method, ET 0 can thus be obtained with temperature records only.
ET 0 is calculated using data collected at each reference weather station (independent ET 0 estimates).These calculations follow the FAO-56 PM method outlined in the FAO-56 document (Allen et al., 1998).These independent ET 0 estimates are then used as pseudo-standards for the purpose of calibrating the regional-scale ET 0 methods.A linear regression is performed for each of the evapotranspiration methods and for each reference weather station (Eq.2).The slope a and the intercept b of the best-fit regression line are used as local calibration coefficients.
where ET 0FAO-56PM is the ET 0 estimated with the standard method and ET 0method is the ET 0 obtained by any of the five methods tested in this study.The linear regression method has been widely used to calibrate ET 0 methods (Allen et al., 1994;Trajkovic, 2005;Shahidian et al., 2012).

2.3
Step 2: estimation of the parameters of the recharge area The estimation of the recharge with the soil-water balance (step 3; Sect.2.4) requires the calculation, at the scale of the recharge area, of three parameters which are SAWC, runoff coefficient R coeff , and K c .These three parameters are controlled by one or several factors which are, in this study, the slope gradient, the geological nature of the substratum and the type of vegetation cover.Moreover, at the scale of the recharge area, the controlling factors are commonly heterogeneous and thus the recharge-area parameters cannot be readily computed.For each of the controlling factors, the recharge area is divided into subareas (hereafter referred to as factor subareas) characterised by homogenous factor properties.Factor subareas can be either continuous or discontinuous, and their number and shape can differ, depending of the spatial distribution of the factors.Relevant factor subareas are in turn used to define parameter subareas.For a given parameter subarea, the value of the parameter is estimated from either field measurements or from the literature.The parameter values at the scale of the recharge area are then calculated by taking into account the relative surface of the parameter subareas (step 2; Fig. 1).Lastly, if preferential infiltration structures (hereafter referred to as infiltration structures) such as sinkholes, cracks, reverse slope areas, bare ground or any topographical depression which can collect the surface runoff are present in the recharge area, the above-mentioned parameters have to be adjusted.For such areas, the SAWC and R coeff , being very low, will be set at 0 in the calculations.
Similarly, for such areas, ET 0 is negligible and therefore the surface of these areas is disregarded for the K c computation.The parameter values are afterwards refined by a sensitivity analysis (step 4; Sect.2.5) in order to find the optimal set of recharge-area parameters.
The K c parameter takes into account four key characteristics (vegetation height, albedo, canopy resistance and evaporation from soil) that distinguish the vegetation type of a given subarea from the reference grass used to estimate ET 0 (Allen et al., 1998).The K c subareas are defined according to the type of vegetation (e.g.meadows and forests) obtained from aerial photographs.The dominant vegetation species assigned to each vegetation type can be obtained from the literature (e.g.forest agency data) or from field observations.Since the K c parameter depends on the stage of development  (Verstraeten et al., 2005).
The SAWC parameter refers to the difference between a maximum water content above which all free water is drained through gravity (field capacity) and a minimum moisture content below which plant roots cannot extract any water (permanent wilting point).The SAWC is mainly affected by soil texture and thickness, both depending primarily on the geological substratum and the vegetation.The SAWC subareas are defined according to the type of vegetation (obtained from aerial photographs) and to the geological substratum (obtained from geological maps).SAWC values can be either calculated with pedotransfer functions (Bruand et al., 2004;Pachepsky and Rawls, 2004) from soil properties (type of horizon, texture and bulk density) and thickness or obtained directly from the literature.Soil properties and thickness can be obtained from the literature (e.g.pedological maps), from morphological description or laboratory measurements of auger hole cores.
The method used to estimate the surface runoff is similar to the commonly used "runoff rational method".The R coeff parameter depends mainly on topography and vegetation.The R coeff subareas are defined according to the vegetation (obtained from aerial photographs).An average slope gradient obtained from the DEM is assigned to each vegetation subarea.The R coeff values can then be calculated from vegetation cover and slope gradient through the use of charts such as the Sautier chart (Musy and Higy, 2011).
Infiltration structures are first located through examination of aerial photographs (lineament analysis) and geological maps, and then inspected in the field.

Step 3: recharge computation with soil-water balance
The soil-water balance workflow used to estimate the recharge at a daily frequency is detailed in Fig. 2. All terms required for the soil-water balance estimation are expressed in water amount (millimetres), except for R coeff expressed in percentage.The soil-water balance is based on ET c , SAWC, K c and R coeff .The precipitation (P ) is the amount of liquid (rain) or solid (snow) water which falls on the recharge area.The precipitation will be taken here as the sum of snowmelt and rainfall.A part of this water amount is intercepted by the vegetative canopy (interception; Fig. 2a).The remainder of precipitation reaches the ground surface and forms (i) the runoff (Rf), which is the water joining the surface drainage network and (ii) the infiltration (I ) into the soil layer which supplies the SAWC.The remaining part of the precipitation not taken-up by evapotranspiration and runoff and not stored in the SAWC is called the recharge (R).It corresponds to deep percolation and is the component of the precipitation which recharges the saturated zone (Fig. 2a).
The ET c is a lumped parameter including potential transpiration, potential soil evaporation and canopy interception evaporation (Verstraeten et al., 2005).In the proposed computation diagram workflow (Fig. 2b) the interception component is therefore integrated in the ET c component.The ET c is the water evapotranspired without any other restrictions than the atmospheric demand (assuming unlimited soil water availability).However, field conditions do not always fulfil these requirements, particularly during low rainfall periods when water supplies are inadequate to support vegetation uptakes.The actual evapotranspiration (ET a ) corresponds to the actual amount of evapotranspired water.
Runoff takes place when the intensity of a precipitation event exceeds the soil infiltration capacity.The use of a daily measurement frequency for precipitation does not allow for an accurate estimation of rainfall intensity.Instead, a R coeff is applied only for days when precipitation is greater than the average.Such days are considered as high intensity rainfall days.The R coeff is applied only to excess precipitation, after the demands of evapotranspiration and SAWC are met, i.e. when SAWC is fulfilled (Fig. 2b).

Step 4: sensitivity analysis of the recharge-area parameters
In the landslide recharge area, recharge can be considered as spatially heterogeneous.Indeed, in fractured rocks, the groundwater flow is mainly driven by an anisotropic fracture network.The proportion of infiltrated water which flows toward the landslide aquifer can significantly differ between two zones of the recharge area.Nevertheless, the GIS composite method considers that any part of the recharge area has the same weight with respect to the groundwater which flows toward the landslide aquifer.This homogeneous recharge assumption can lead to biased estimations of the recharge-area parameters.However, uncertainties in the delimitation of the recharge area can also lead to biased estimations.A sensitivity analysis evaluates the possible overestimation or underestimation of the set of recharge-area parameters.The infiltration-structure subareas are used as fitting factors (varying from 0 to 100 % of the recharge area surface) to adjust the estimation of the set of recharge-area parameters.A variation of the infiltration structure percentage corresponds to a variation of the contribution weight of the infiltration structures to the recharge of the landslide aquifer.Consequently, a variation of the infiltration structure percentage does not affect the relative proportion of the other subarea surfaces but only their contribution weights.The sensitivity analysis is based on the performance of a linear correlation between daily time series of recharge and displacement.The landslide displacement triggered by pore water pressure is therefore related to the hydrodynamic variations P j >= SAWC max -SAWC j-1 + ETc j & P j > avg(P) of the landslide aquifers.For this reason, the performance of the correlation between recharge and displacement informs whether the recharge-area parameters are satisfactorily estimated.The sensitivity analysis allows determining the optimal set of recharge-area parameters which maximise the performance of the correlation.

Antecedent cumulative sum
The correlation between water input and displacement requires measurements of landslide displacements at the same temporal frequency (daily frequency in this study) as the measurements of water input (precipitation or recharge).The groundwater hydrodynamic processes in aquifers are nonlinear.A former rainfall event displays less impact (though not negligible) than a recent one on the aquifer hydrodynamic fluctuations (Canuti et al., 1985;Crozier, 1986;Diodato et al., 2014).The daily precipitation/recharge time series cannot therefore be used without appropriate corrections.An an-tecedent cumulative sum of precipitation/recharge weighted by a factor α is applied as a moving window to the daily precipitation/recharge time series (Eq.3).The antecedent cumulative sum allows approximating the daily triggering impact of the aquifer (ATI) on the landslide destabilisation.In order to take into account the groundwater transit time, a β time-lag factor is introduced.This factor can shift the moving window from the target date t.
where ATI t is the aquifer triggering impact (in mm) at the date t; β is the time shift of the moving window (in days); i is the ith day from the date t (i = t + β: start of the moving window and i = t + β + n: end of the moving window); n is the length of the moving window of the cumulative period (in days); W i water input, i.e. precipitation or recharge at the ith day (in mm), and α is the weighting factor.An iterative grid search algorithm is used to find the optimal set of parameters of the antecedent cumulative sum.
The optimal set of parameters is the set that maximises the correlation performance itself based on the R 2 indicator.The grid search algorithm investigates the following parameter ranges: n from 1 to 250 days (increment: 1 day), α from 0 to 0.5 (increment: 0.0001) and β from 1 to 10 days (increment: 1 day).

Significance of the water input-displacement correlation
The bootstrap method, which is an inference statistical resampling method, is used to estimate the confidence interval (CI) of estimated parameters and to perform statistical hypothesis tests (Chernick, 2008).The bootstrap method uses resampling with replacement and preserves the pair-wise relationship.However, for interdependent data (such as time series), the structure of the data set has to be preserved during the resampling.The moving block bootstrap is a variant of the bootstrap method.It divides data into blocks for which the structure is kept (Cordeiro and Neves, 2006).The moving block bootstrap method is performed with a 90-day block size (season) and 50 000 iterations for each run.
To estimate the significance of the linear regression, the lower bound of the confidence interval (LBCI) of R 2 is used at the level of confidence of 90 % (equivalent to a one-tailed test at the significance level of 5 %).An LBCI value greater than 0 means that the relationship is significant.Particular to statistical hypothesis tests is the definition of the tested null hypothesis which is often a default position opposite to the aim of the test, i.e. by stating that "there is no relationship between the two considered quantities".The null hypothesis is assumed to be true until it is rejected by statistical evidence in favour of the alternative opposite hypothesis.The recharge estimated with the LRIW workflow is hereafter called R LRIW .The recharge estimated by subtracting a noncalibrated ET 0 from precipitation is hereafter called R PMNE , PMNE standing for precipitation minus non-calibrated ET 0 .
To estimate whether the R PMNE -displacement correlation R 2 is significantly better than the precipitation-displacement correlation R 2 value, the Null Hypothesis 1 (NH1) is tested.The NH1 states that the R PMNE -displacement correlation R 2 value is not significantly greater than the R 2 value obtained from precipitation.In other words, the NH1 statistic test is the difference between the R PMNE R 2 value and the precipitation R 2 value, expected to be 0 if no difference.Similarly, the Null Hypothesis 2 (NH2) and the Null Hypothesis 3 (NH3) are tested.NH2 estimates whether the R LRIW -displacement correlation R 2 is significantly better than the precipitationdisplacement correlation R 2 value.NH3 estimates whether the R LRIW -displacement correlation R 2 is significantly better than the R PMNE -displacement correlation R 2 value.
To estimate whether the best precipitation-R LRIWdisplacement correlation R 2 value computed from the sensitivity analysis is significantly better than the other R 2 values obtained, the Null Hypothesis 4 (NH4) is tested.The NH4 states that the best R 2 value is not significantly greater than the ones obtained with all the remaining combinations.In other words, the NH4 statistic test is the difference between the best R 2 value and the R 2 values obtained with the remaining combinations, expected to be 0 if no difference.
For all null hypotheses, the decision of rejection is made by determining how much of the bootstrap distribution (among 50 000 iterations) falls below 0 by using the LBCI at the level of confidence of 90 %, equivalent to a one-tailed test at the significance level of 5 %.An LBCI value greater than 0 allows rejecting the null hypotheses.
3 Application to the Séchilienne landslide

Geological settings and rainfall triggering
The Séchilienne landslide is located in the French Alps on the right bank of the Romanche River, on the southern slope of the Mont Sec Massif (Fig. 3).The climate is mountainous with a mean annual precipitation of 1200 mm.The geological nature of the area is composed of vertical N-S foliated mica schists unconformably covered by carboniferous to Liassic sedimentary deposits along the massif ridge line above the unstable zone.Quaternary glacio-fluvial deposits are also present.The Séchilienne landslide is limited eastwards by a N-S fault scarp and northwards by a major head scarp of several hundred metres wide and tens of metres high below the Mont Sec.The slope is cut by a dense network of two sets of near-vertical open fractures trending N110-N120 and N70 (Le Roux et al., 2011).
The Séchilienne landslide is characterised by a deep progressive deformation controlled by the network of faults and fractures.A particularity of the Séchilienne landslide is the absence of a well-defined basal sliding surface.The landslide is affected by a deeply rooted (about 100-150 m) toppling movement of the 50-70 • N slabs to the valley (accumulation zone) coupled with the sagging of the upper slope (depletion zone) beneath the Mont Sec (Vengeon, 1998;Durville et al., 2009;Lebrouc et al., 2013).A very active moving zone is distinguishable from the unstable slope where high displacement velocities can be 10 times higher than the rest of the landslide.
The landslide shows a higher hydraulic conductivity than the underlying stable bedrock (Vengeon, 1998;Meric et al., 2005;Le Roux et al., 2011), thus leading to a landslide perched aquifer (Guglielmi et al., 2002).The recharge of the landslide perched aquifer is essentially local, enhanced by the trenches and the counterscarps which tend to limit the runoff and to facilitate groundwater infiltration in the landslide area.However, the hydrochemical analyses of Guglielmi et al. (2002) show that the sedimentary deposits distributed above the landslide hold a perched aquifer which can recharge the landslide perched aquifer.The fractured metamorphic bedrock beneath the landslide contains a deep saturated zone at the base of the slope and an overlying vadose zone.The groundwater flow of the entire massif is mainly controlled by the network of fractures with high flow velocities (up to a few kilometres per day; Mudry and Etievant, 2007).The hydromechanical study of Cappa et al. (2014) shows that the deep aquifer can also trigger the Séchilienne landslide destabilisation as a result of stress transfer and frictional weakening.Thus, the Séchilienne landslide destabilisation is likely triggered by a twolayer hydrosystem: the landslide perched aquifer and the deep aquifer.The Séchilienne landslide behaviour is characterised by a good correlation between precipitations and displacement velocities (Rochet et al., 1994;Alfonsi, 1997;Durville et al., 2009;Chanut et al., 2013).The seasonal variations of the daily displacements are clearly linked to the seasonal variations of the recharge (high displacements during high flow periods and low displacements during low flow periods).

Method implementation
The recharge computation uses the daily rainfall recorded at the weather station located at Mont Sec, a few hundred metres above the top of the landslide (Table 1, Fig. 3).This station is equipped with rain and snow gauges and a temperature sensor.However, the temperature measurements at the Mont Sec station are considered unreliable because of a non-standard setting of the temperature sensor and numerous missing data.Consequently, the temperature at the Mont Sec station has to be estimated in order to estimate the evapotranspiration at the landslide site (see details about the computation in Appendix B).
Since the Mont Sec station does not record the full set of parameters (relative humidity, temperature, wind speed and solar radiation), a regional calibration of ET 0 and R S reduced-set methods is required.Three weather stations located at less than 60 km from the studied site are used as reference weather stations: Grenoble-Saint-Geoirs, Saint-Jean-Saint-Nicolas and Saint-Michel-Maur (Table 1, Fig. 3).The Saint-Michel-Maur weather station does not measure R S , which is estimated with the Angström formula (Eq.A5 in  Appendix A) using sunshine duration data recorded at the station.The Angström formula empirical default coefficients are tuned with the two other weather stations (a S = 0.232 and b S = 0.574).
The delimitation of the recharge area of the two-layer hydrosystem (Fig. 3) of the Séchilienne landslide is based on the geological and hydrochemical studies of Vengeon (1998), Guglielmi et al. (2002) and Mudry and Etievant (2007).The recharge area is delimited by the spatial extent of the sedimentary cover of which the hosting perched aquifer recharges the two-layer hydrosystem.Groundwater flow of the entire Mont Sec Massif is controlled by faults and fractures.The N20 fault bordering the sedimentary cover to the east as well as the N-S fault zone bordering the landslide to the east are structures which delimitate the recharge area.The scarcity of information does not allow accurately defining the actual extent of the recharge area.The sensitivity analysis mentioned in Sect.2.5 allows compensating for the possible biases introduced by this uncertainty.The following spatial data sets are used for the estimation of the parameters of the recharge area.The aerial photographs (0.5 m resolution) and a DEM of 25 m resolution are provided by the Institut National de l'Information Géographique et Forestière (IGN) and geological maps are provided by the French Geological Survey (BRGM).
The Séchilienne landslide is permanently monitored by a dense network of displacement stations managed by the CEREMA Lyon (Duranthon et al., 2003).In this study, one infrared station (1101) and three extensometer stations (A16, A13 and G5) are used.Stations 1101, A13 and A16 are representative of the most active zone (median displacements of 2.5, 1.75 and 2.98 mm day −1 , respectively), while G5 is located on a much less active zone (median displacement of 0.05 mm day −1 ; Fig. 3, Table 2).
The sensitivity analysis is performed on the A16 extensometer on the period from 1 May 1994 to 1 January 2012, period during which both A16 extensometer and recharge data sets are available.The performance test of the LRIW workflow against precipitation and R PMNE is performed on the four displacements station in the period from 1 January 2001 to 1 January 2012, period during which the four stations and recharge data sets are available.The R PMNE is estimated with the non-calibrated Turc equation (Eq.A8) which is the most appropriate ET 0 reduced-set equation for the Séchilienne site.Indeed, the Turc equation was developed initially for the climate of France.The Turc equation requires the estimation of R S which is performed with the non-calibrated Hargreaves-Samani equation (Eq.A2).

Displacement data detrending
The long-term displacement monitoring shows that displacement rate and amplitude exponentially increased with time as illustrated by the records of extensometer A16 (Fig. 4a).
The rainfall data series does not show any trend over the year, meaning that the displacement trend is independent of the recharge amount.Consequently, on the Séchilienne landslide, for the same amount of rainfall, the displacement rate and magnitude responses increase steadily with time.The observed trend is the consequence of a progressive weakening of the landslide due to long-term repetitive stresses.The accumulating deformation can be assimilated to long-term creep (Brückl, 2001;Bonzanigo et al., 2007) and can be explained by a decrease of the slope shear strength (Rutqvist and Stephansson, 2003).As shown by the detrended displacement, the Séchilienne landslide is constantly moving and shows large daily to seasonal variations which seem to be the landslide response to the precipitation trigger.Consequently, the precipitation-displacement correlation is performed on the detrended displacement.
The exponential trend is removed with the statistical multiplicative method (y t = T t S t I t ) where the time series (y t ) is composed of three components (Madsen, 2007;Cowpertwait and Metcalfe, 2009;Aragon, 2011): trend (T t ), seasonal (S t ) and irregular (I t ).In this study, the irregular and seasonal components are both assumed to be linked to the rainfall triggering factor (y t = T t D t with D t = S t I t ).The trend is determined by curve fitting of a fourth-order polynomial (parametric detrending).The result is a detrended unitless time series (D t ) with both variance and mean trend removed.The time series decomposition process is illustrated with the A16 extensometer in Fig. 4. The two calibrated R S methods show good results with respect to R S measured at the reference weather stations.
The BC mod R S method is selected as it shows a better performance (R 2 = 0.864; RMSE = 1.567) than the HS mod R S method (R 2 = 0.847; RMSE = 1.625).Equation ( 4) presents the calibrated BC R S method with all the calibrated coefficients.
The cloud cover adjustment factor α is either equal to 0.79 (cloud impact) or to 1.All the equation terms are described in the Appendix A. The BC mod R S calibrated method is then used to compute R S input data of the five ET 0 reduced-set methods.
Overall, all of the ET 0 methods tested show good results for regional calibration and are all suitable for the Séchilienne site (Table 3).Among the ET 0 methods tested, the PM red ET 0 method shows the best performance (R 2 = 0.932; RMSE = 0.505) and requires only a low regional adjustment (a = 0.994 and b = 0.013).Therefore, the PM red ET 0 method is selected to compute ET 0 for the Séchilienne site (hereafter referred to as ET 0Séch ). Figure 5 displays the estimated ET 0Séch versus the FAO-56 PM computation for each reference weather station.
Equation ( 5) is the final calibrated PM red ET 0 method with all the calibrated coefficients.The input R n term is deduced from the calibrated BC mod R S method (Eq.4).

Recharge-area parameters
Subareas are expressed in percentages of the whole recharge area (Table 4, Fig. 6).Two types of vegetation cover, pasture and forest, are defined using aerial photographs, with proportions of 23 and 53 %, respectively.The Séchilienne forest is mainly composed of beeches (Fagus sylvatica) and conifers (Picea excelsa), which are associated occasionally with ashes (Fraxinus) and sweet chestnuts (Castanea sativa).Three main geology subareas, mica schist bedrock (15 %), sedimentary cover (20 %) and superficial formations (41 %), are defined through examination of the geological map and field investigations.Infiltration structures are centred on the major faults identified on the geological map, on lineaments deduced from aerial-photograph analysis and on geomorphological features (sinkholes, cracks, etc.).A 50 m wide influence zone is added to the identified objects, leading to an infiltration-structure subarea representing 24 % of the recharge area.For K c estimation, the proportion of beeches and conifers is assumed to be identical for the Séchilienne forest (each 50 % of forest subarea) and other species are ignored.K c values are set to 0.71 and 0.97 for conifers, and to 0.78 and 0.9 Table 4. Estimation of K c , SAWC and runoff for the recharge area of the Séchilienne landslide.Geology and vegetation are the subarea factors identified and expressed in relative proportion of the recharge area.The average slope gradient is the slope gradient for each identified vegetation subarea factor.K c , R coeff and SAWC columns are the estimated values for each subarea factor.K c RA, SAWC RA and R coeff RA columns are the contribution of each subarea parameter at the scale of the recharge area.The recharge area (bottom row) stands for the estimation at the scale of the recharge area.for beeches according to Verstraeten et al. (2005).Most pastures are anthropogenic and consist of grass.K c values are set to 0.85 and 1 according to Allen et al. (1998).Infiltration structure subareas are not taken into account in the K c estimation, so the relative proportions of pasture and forest become 30 and 70 %, respectively.The contribution of each subarea (column K c RA, Table 4) allows determining the recharge area K c values at the scale of the recharge area (0.777 to 0.955).
The combination of geology and vegetation subareas results in six types of SAWC subareas (Table 4).For each SAWC subarea, at least one auger hole was drilled.For each soil auger core, the soil texture, the stoniness and the organic-matter content are estimated by morphological description (Baize and Jabiol, 2011).Based on these estimations, the SAWC is then computed using the pedotransfer functions of Jamagne et al. (1977) and Bruand et al. (2004).The average estimation of SAWC at the recharge area scale is 106 ± 10 mm (rounded to 105 mm).
To estimate R coeff , an average slope gradient is computed from slope gradient analysis of the DEM and is assigned to each vegetation subarea.Pasture and forest subareas show an average slope gradient of 14 and 20.6 • , respectively.R coeff values of 22 % for pasture and 15 % for forest are deduced from the Sautier chart (Musy and Higy, 2011).This chart was developed for Switzerland where environmental conditions are similar to the French Alps.A 12.8 % runoff coefficient is then estimated at the recharge area scale, according to the respective proportions of vegetation subareas (Table 4).

Sensitivity analysis of the parameters of the recharge area
Sensitivity analysis is performed for SAWC ranging from 0 (100 % of infiltration structures corresponding to precipitation) to 145 mm of SAWC (0 % infiltration structures +10 mm of SAWC uncertainties measurement) with increments of 10 mm.The coupled surface R coeff ranges from 0 to 16.3 % (with increments of about 1 %).For each combination, recharge is computed according to the soil-water balance (step 3; Figs. 1, 2) with (i) the temperature estimated for the recharge area (Appendix B), (ii) the precipitation recorded at Mont Sec weather station, and (iii) the parameters of the recharge area.
All the best computations have a 1-day lag, with periods ranging from 56 to 104 days (Fig. 7a, Table 5).The best R 2 obtained from recharge is obtained with both the estimated recharge-area parameters (SAWC = 105 mm, R 2 = 0.618) and the recharge-area parameters for SAWC adjusted from 75 (R 2 = 0.616) to 115 mm (R 2 = 0.617; Fig. 7b, Table 5).One of the best correlation performances is obtained for the estimated recharge-area parameters.This shows that the delimitation of the recharge area properly reflects the actual field conditions.The best correlation performance is assumed to be obtained with the estimated recharge-area parameters for NH4, i.e. testing R 2 obtained with the estimated recharge-area set (SAWC = 105 mm) minus R 2 obtained with each of the other adjusted rechargeparameter sets of the sensitivity analysis (Table 5).
For all the recharge combinations tested, the LBCI values from bootstrap testing of NH2 are greater than 0, allowing for the rejection of NH2 (Fig. 7c).In other words, it shows that the R 2 obtained with recharge is always significantly higher than the one computed with precipitation (R 2 = 0.311) even for a SAWC of 5 mm (R 2 = 0.426; Table 5).For the adjusted recharge-area parameter scenarios having SAWC values above 45 mm, the LBCI values from bootstrap testing of NH4 are lower than 0, not allowing for the rejection of NH4 (Table 5, Fig. 7d).In other words, it shows that the R 2 obtained with a SAWC of 105 mm is not significantly higher than the ones obtained from SAWC above 45 mm.Rechargedisplacement correlations for SAWC values ranging from 75 (runoff = 9 %) to 115 mm (runoff = 13.9 %) show (i) a cumu-lative period computation (n) below 101 days that is within the third quartile, (ii) an R 2 greater than 0.616 that is within the third quartile, (iii) LBCI values of NH2 greater than 0, and (iv) LBCI values of NH4 lower than 0 (Table 5, Fig. 7).These SAWC and runoff values seem to statistically reflect the recharge area properties of the landslide and are suggested for further work on the Séchilienne landslide.

Estimation of the recharge for the Séchilienne landslide
For the remaining part of this paper, R LRIW is based on the estimated recharge-area parameters (infiltration structures = 24 %, SAWC = 105 mm, and R coeff = 12.8 %).Indeed, among all solutions giving satisfying performances in the sensitivity analysis, these parameters arise from actual field data.R LRIW is compared with the precipitation signal in Fig. 8.The R LRIW signal differs significantly from the precipitation signals, marked by a high seasonal contrast.This is especially true during summer when ET c is important.Indeed, the first rainfall events after a dry period do not reach the aquifer until the SAWC is exceeded.Figure 9 shows the best correlation results for precipitation and R LRIW , together with A16 detrended daily displacements.The cumulative recharge signal reproduces well the displacement acceleration and deceleration phases, and especially the dry summers where displacement dramatically dropped (summers 1997, 1998, 2003, 2004and 2009;Fig. 9b).On the contrary, the cumulative precipitation signal is more contrasted and more noisy, and does not manage to reproduce several peaks (in width as well as in intensity) of the detrended displacement signal (winters 1997, 2000, 2004, 2005 and 2010).In addition, the cumulative precipitation signal shows a weak correlation with displacement deceleration phases (summers 1998, 1999, 2000 2006, 2009 and 2010).

Relevance of the LRIW method
Figure 10 summarises the comparison of the performances between the precipitation, the R PMNE and the R LRIW based on the NH1, NH2 and NH3 tests for the four displacement stations.LBCI values from bootstrap testing of NH1 are lower than 0 for 1101, A13 and A16 stations and greater than 0 for the G5 station.NH1 cannot be rejected, meaning that the R 2 values obtained with R PMNE are not significantly higher than those computed with precipitation, except for the G5 station.All LBCI values from bootstrap testing of NH2 and NH3 are greater than 0, allowing for the rejection of these two null hypotheses for the four stations (Fig. 10a).Rejection of NH2 shows that the R 2 values obtained with R LRIW are significantly higher than those computed with precipitation.Similarly, rejection of NH3 shows that R 2 values  obtained with R LRIW are significantly higher than those computed with R PMNE .R 2 values vary from 0.0006 to 0.343 for precipitation, from 0.076 to 0.444 for R PMNE and from 0.243 to 0.586 for R LRIW , for G5 and A16 extensometer, respectively (Table 2).On average, R PMNE allows increasing the R 2 value by 29 % relative to precipitation, while R LRIW allows increasing the R 2 by 78 % (Fig. 10b).The R 2 values obtained with R LRIW are 38 % higher on average than those obtained with R PMNE .
These results are confirmed by the LBCI and by the observed values of the NH2 test which are always greater than those from the NH1 test as well as by the positive LBCI values of the NH3 test (Fig. 10).The correlation performance for the recharge estimated with the LRIW method signif-1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008  icantly exceeds the performance of the two other signals, making the LRIW method particularly appropriate to be used in landslide studies.A discussion about the benefit of this study for the understanding of the rainfall-displacement relationship in the case of the Séchilienne landslide can be found in Appendix C.

Applicability of the LRIW method to other landslides
Several studies have shown the relevance of the recharge signal for various landslide types: coastal landslides (Maquaire, 2000;Bogaard et al., 2013), unstable embankment slope landslides (Cartier and Pouget, 1987;Delmas et al., 1987;Matichard and Pouget, 1988) and deep-seated earth flow landslides (Malet et al., 2003;Godt et al., 2006).In addition, destabilisation of shallow landslides is known to be influenced by antecedent soil moisture and precipitation (Brocca et al., 2012;Garel et al., 2012;Ponziani et al., 2012).Recharge, which implicitly combines antecedent soil moisture and precipitation, can be a significant parameter to consider.
Although the method proposed in this study has not yet been tested at other sites, there are several arguments which suggest its applicability elsewhere.First, the FAO Penman-Monteith method used in this study is considered worldwide as the evapotranspiration method standard (Allen et al., 1998;Shahidian et al., 2012).Several evapotranspiration methods were developed locally and many of them can be calibrated against reference methods in other contexts (Hargreaves and Allen, 2003;Yoder et al., 2005;Alkaeed et al., 2006;Igbadun et al., 2006;Trajkovic, 2007;Alexandris et al., 2008;López-Moreno et al., 2009;Sivaprakasam et al., 2011;Tabari and Talaee, 2011;Shahidian et al., 2012;Tabari et al., 2013).Otherwise, the Penman-Monteith or Hargreaves-Samani methods are recommended (Allen et al., 1998).Several solar radi-   LBCI is the lower bound of the confidence interval.G5 station is disregarded in the the performance average variation calculation since the R 2 value obtained at G5 from precipitation is close to 0, therefore leading to a non-representative variation.
ation methods were developed and can be applied worldwide if locally calibrated, allowing for the estimation of evapotranspiration from temperature alone (Allen et al., 1998;Almorox, 2011).Recharge-area parameters can be estimated locally or with local or global literature reference values.The use of global values will increase recharge estimation uncertainties.However, the implementation of a sensitivity analysis allows refining the recharge-area parameters in order to compensate for the lack of site-specific data.Pachepsky and Rawls (2004) developed pedotransfer functions to estimate SAWC for various regions of the world.R coeff values from the widely used rational method can be applied, as well as most of the runoff coefficients from the literature (McCuen, 2005;Musy and Higy, 2011).In addition, pedotransfer functions can also be used for runoff estimation.Cuenca, 2000;Verstraeten et al., 2005;Hou et al., 2010), but can also be found in the literature for many species (Allen et al., 1998).

Conclusion and perspectives
A method based on a soil-water balance, named LRIW, is developed to compute recharge on a daily interval, requiring the characterisation of evapotranspiration and parameters characterising the recharge area (soil available water capacity and runoff).A workflow is developed to compute daily groundwater recharge and requires the records of precipitation, air temperature, relative humidity, solar radiation and wind speed within or close to the landslide.The determination of the parameters of the recharge area is based on a spatial analysis requiring field observations and spatial data sets (digital elevation models, aerial photographs and geological maps).Once determined, the parameters are refined with a sensitivity analysis.
The method has been tested on the Séchilienne landslide.The tests demonstrate that the performance of the correlation with landslide displacement velocity data is significantly enhanced using the LRIW estimated recharge.The R 2 values obtained with the LRIW recharge are 78 % higher on average than those obtained with precipitation and are 38 % higher on average than those obtained with recharge computed with a commonly used simplification in several landslide studies (recharge = precipitation minus non-calibrated ET 0 ).The sensitivity analysis of the LRIW workflow appears to be an appropriate alternative to estimate or to refine soilwater balance parameters of the recharge area, especially in the case of insufficient field investigations or in the absence of the necessary spatial data set.
The LRIW workflow is developed to be as universal as possible in order to be applied to other landslides.The workflow is developed in order to be sufficiently simple to guide any non-hydrogeology specialist who intends to estimate the recharge signal in the case of rainfall-landslide displacement studies.Within this scope, a software is planned to be developed in the near future in order to provide a user-friendly tool for recharge estimation.In addition, the LRIW workflow also enables the reconstruction of retrospective time series for sites recently equipped with weather stations designed to measure a full set of parameters.A further step will have to account for the spatial and temporal variabilities of precipitation and recharge area properties, thus providing a better estimation of the recharge.In addition, taking recharge into account can assist in determining a warning rainfall threshold for the deep-seated slope movements.

Figure 1 .
Figure 1.LRIW diagram.Step 1: calibration of standard ET 0 and R S methods.Step 2: estimation of recharge-area parameters required for the soil-water balance (R coeff , K c and SAWC) and the infiltration structures.Step 3: computation of the recharge with the soil-water balance.*In the case of a landslide-located weather station recording the full set of parameters, the first step can be skipped and the ET 0 of step 3 can be estimated directly at the study site with the standard ET 0 method (FAO-56 PM method).

Figure 2 .
Figure 2. Soil-water balance: (a) soil-water balance conceptual representation and (b) soil-water balance diagram used for recharge computation on a daily frequency.SAWC: soil available water capacity; SAWC max : SAWC threshold (possible maximum); P : precipitation (rainfall + snowmelt); avg (P ): precipitation average of the entire record; I : part of precipitation which infiltrates the soil; Rf: surface runoff; R coeff : runoff coefficient; ET c : specific vegetation evapotranspiration; ET a : actual vegetation evapotranspiration; R: recharge.Units: millimetres of water, except R coeff in percentage.Subscript j is the computation day and subscript j − 1 is the day before.TRUE and FALSE are the answers of the conditional inequality statements.

Figure 3 .
Figure 3. Location map of the Séchilienne landslide.(a) Map of the Séchilienne unstable slope and recharge area with the Mont Sec weather station.(b) Enlarged map of the most active area showing displacement stations.(c) Map showing the weather stations used for the temperature estimation at Mont Sec.(d) Map showing the weather stations used for evapotranspiration and solar radiation method calibration.

Figure 4 .
Figure 4. Trend removal of A16 extensometer displacement data.(a) A16 displacement data and the fourth-order polynomial curve fitting considered as the displacement trend; (b) A16 detrended data (unitless) corresponding to A16 displacement data for which the trend is removed by a multiplicative method.

Figure 5 .Figure 6 .
Figure 5. ET 0 regional calibration results at the three reference weather stations (Grenoble-Saint-Geoirs, Saint-Jean-Saint-Nicolas and Saint-Michel-Maur).(a) ET 0Séch and FAO-56 PM ET 0 as a function of time.(b) Linear regression between ET 0Séch (x axis) and FAO-56 PM ET 0 (y axis).ET 0Séch stands for ET 0 computed with the combination of calibrated ET 0 Penman-Monteith reduced-set method and calibrated R S modified Bristow-Campbell method.

Figure 7 .
Figure 7. Results of the sensitivity analysis relative to SAWC for (a) the computation period, (b) the R 2 and the LBCI of R 2 , (c) the LBCI of NH2 and (d) the LBCI of NH4.

Figure 8 .
Figure 8. Recharge computation with the LRIW method at Séchilienne with an SAWC of 105 mm and a runoff coefficient of 12.8 %.ET c : specific vegetation evapotranspiration; ET a : actual vegetation evapotranspiration, SAWC: soil available water capacity.

Figure 9 .
Figure 9. Best linear correlations for precipitation and recharge computed with the LRIW method.IS is for infiltration structures.SAWC soil available water capacity.Cumulative period (n) and shift factor (β) are the terms of Eq. (3).(a) Linear regression between precipitation/R LRIW and A16 detrended displacement.(b) Correlation between precipitation/R LRIW and A16 detrended displacement as a function of time.

Figure 10 .
Figure 10.Performance of the LRIW workflow.(a) Bootstrap distribution of NH1, NH2 and NH3 tests for four displacement recording stations.(b) R 2 values for the four displacement recording stations obtained with the precipitation, recharge PMNE, and recharge LRIW.LBCI is the lower bound of the confidence interval.G5 station is disregarded in the the performance average variation calculation since the R 2 value obtained at G5 from precipitation is close to 0, therefore leading to a non-representative variation.

Table 1 .
Summary of weather data sets with parameters used (X) at the various locations.Distance is measured from the Séchilienne landslide, R S is the solar radiation, N is the sunshine duration, W is the wind speed, H is the humidity, T is the temperature and P is the precipitation depth.

Table 2 .
Statistics of the displacement records and results of the best linear correlation between precipitation/R LRIW and displacement records for four displacement stations (1101, A13, A16 and G5).The displacement column indicates basic statistics of the displacement records: first quartile (Q 1 ), median and third quartile (Q 3 ).Cumulative period (n), shift factor (β) and weighting factor (α) are the terms of Eq. (3).P stands for precipitation, R 1 stands for R PMNE and R 2 stands for R LRIW .

Table 3 .
Calibration and performance of the five tested ET 0 methods, in relation to the FAO-56 PM ET 0 standard (Penman-Monteith method defined in the FAO-56 paper).All the ET 0 methods are detailed in Appendix A. a, b and R 2 are the results of linear regression between FAO-56 PM ET 0 and tested ET 0 methods.RMSE is the root mean square error.

Table 5 .
Sensitivity analysis results of the best correlation between precipitation/R LRIW and A16 extensometer detrended displacement.IS is for infiltration structures.SAWC is the soil available water capacity.LBCI is the lower bound of the confidence interval.R 2 row is the R 2 computed from recharge-area parameters indicated in each table row.Cumulative period (n), shift factor (β) and weighting factor (α) are the terms of Eq. (3).NH2 test: R 2