River Rhone originates from the Rhone glacier in Switzerland and reaches the Mediterranean Sea in South-eastern France. Of the total discharge of the river Rhone 41% come from the Alps, which is disproportionally high relative to the areal proportion of the Alpine region [14]. The upper part of the catchment of the river Rhone, down to the lake of Geneva, is located in Switzerland and covers 5250 km² of land. 11.5% of this part of the catchment area are currently glaciated, a proportion expected to reach nearly zero by the end of the century [15]. The contribution of glacier storage change to the August runoff over the last century was >40% and nearly 80% in the extreme year 2003 [15]. The study region encompasses nearly the whole Canton of Valais (Figure 1). The topography is very heterogeneous, ranging in elevation from 372 m above sea level (a.s.l.) at the Lake of Geneva to 4634 m a.s.l. at the so-called ‘Dufour-Spitze’. Annual temperature and precipitation exhibit strong vertical gradients as well as variations along the main valley axis. The climate is further characterized by high radiation and low levels of air humidity.

Compared to the 1000 mm of average annual precipitation recorded on the Swiss Plateau (1961–1990, [16]), annual precipitation amounts in the eastern part of the Rhone valley are low, hardly attaining 600 mm (e.g., station Visp, Figure 1). Of this, only about 250 mm fall between April and September (about 300 mm when considering the period 1981–2009, Table 1) when evaporative demand reaches 10 mm day⁻¹ during summer. Thus, the climate in the main agricultural area can be considered semi-arid.

About one fifth of the area (980 km²) is used for agriculture (Figure 1a), with grasslands and pasture representing by far the largest share (833 km²) [17]. The most important activities from an economic perspective are those related to the cultivation of special crops. Grapevine is grown on the South facing slopes along the Rhone to the East and West of Sion, whereas fruit trees are more protected from frost during critical phases on the North facing slopes and on the flat valley bottom [18]. Arable crops, such as maize, benefit from the deeper soils and a less dry climate in the area close the outlet of the catchment. Otherwise, soils are shallow, with a total water holding capacity (TAW) of less than 20 mm (Figure 1b).

Irrigation channels have been in place since the 13th century, mainly in the area between Sion and Visp, deviating streams to give pastures, fields, vineyards and orchards access to the locally missing water. While snow melt peaks in July, ice melt peaks in August, both contributing to the runoff [15]. The total amount of water withdrawn for irrigation has been estimated in [13] as 97 × 10⁶ m³ yr⁻¹, of which 86.5 × 10⁶ m³ yr⁻¹ are supplied to the fields through open channel systems.

The study area appears as a local irrigation hotspot in the GMIA [6], the EIM [3], and the IWRE [8]. The share of irrigated land as given in the GMIA is 5 to 35% for the area downstream of Sion, with a similar picture emerging from the IEM. IWR amount to 100–250 mm yr⁻¹ in the IWRE, which compares favorably with the 100 mm yr⁻¹ irrigation depth estimated by [12] for grasslands.

Seasonal IWR at the catchment scale was inferred assuming a linear dependence upon the atmospheric water budget, whereby the latter represents the difference between precipitation and reference evapotranspiration. To establish the functional form of the relation, the following steps were required:

The weather data used were the same daily gridded fields for precipitation, air temperature, wind speed, vapor pressure, relative humidity, global radiation and sunshine duration as already employed in [12]. They were obtained through spatial interpolation of station data provided by the Swiss Federal Office of Meteorology and Climatology (MeteoSwiss). The interpolation procedure was based on a weighted combination of inverse-distance interpolation and altitude-dependent regression [19,20]. Before interpolation, precipitation data were error-corrected separately for snow and rain using specific correction factors [20].

Geospatial data used for meta-model application (i.e., upscaling, Section 4), including a digital elevation model, a digital land-use map and digital information concerning soil hydraulic characteristics were obtained from various sources identified by [12]. The 74 land-use categories originally defined in [17] were recombined into four categories (Figure 1a), the non-agricultural land being ignored given the focus on irrigation. All spatial data were mapped with a 500 m × 500 m grid cell size.

Daily irrigation requirements were estimated as in [9] by considering the vertical water balance equation for the root zone. Assuming a single soil layer of constant thickness Z_r and total water holding capacity in the root zone TAW = (Ɵ_FC – Ɵ_WP)∙Z_r, where Ɵ_FC and Ɵ_WP are volumetric water contents at field capacity and wilting point, respectively, the water budget equation can be written as: (1) where D_r is the actual soil water depletion; P the daily precipitation; RO the surface runoff; DP the deep percolation; IWR_day the daily irrigation amount and ET_a the actual evapotranspiration. In solving the balance Equation (1), neither rainfall interception, nor lateral water movements, water ponding at the surface, water storage as snow or capillary rise were considered. Infiltration capacity was defined as the difference between soil porosity and actual water content, and all precipitation in excess of the infiltration capacity was removed as surface runoff. Furthermore, all water content in excess of field capacity was removed as deep percolation. Initial soil moisture was set equal to field capacity.

Following Allen et al. [9], actual evapotranspiration ET_a was computed as: (2) where ET₀ is the so-called reference evapotranspiration; K_c the crop coefficient (which will be further described below); and K_s a multiplicative factor accounting for the effects of water stress. According to Allen et al. [9]; ET₀ is defined as the evapotranspiration of a well-watered grass surface with an assumed height of 0.12 m, a fixed surface resistance of 70 s m⁻¹ and an albedo of 0.23. It is computed by means of the Penman-Monteith equation, which reads: (3) where R_n is the net radiation; G the soil heat flux; (e_s − e_a) represents the vapor pressure deficit of the air; ρ_α is the mean air density at constant pressure; C_p is the specific heat of the air; Δ represents the slope of the saturation vapor pressure temperature relationship; γ is the psychrometric constant; and r_s and r_a are the (bulk) surface and aerodynamic resistances. The estimation of r_a and r_s for use in Equation (3) is detailed in [9], while R_n was estimated from global radiation using the approach suggested by [21], and G was set equal to 0.

In Equation (2), the stress factor K_s can be expressed in terms of D_r as: (4) where p is a crop-specific parameter depending on unstressed crop evapotranspiration ET_c = K_c ∙ ET₀: (5) Where P₅ is a prescribed value of p corresponding to an ET_c of 5mm day⁻¹. Values of P₅ for cut and grazed grassland, maize, apple, apricot and grapevine are 0.55, 0.60, 0.55, 0.50, 0.50 and 0.45, respectively.

Given the daily soil moisture depletion, irrigation was assumed to be necessary whenever D_r > D_r,trig, where the trigger value D_r,trig was assumed to correspond to K_s = 0.8. Inserting the latter equality in Equation (4) gives: (6)

For each irrigation event, the daily irrigation requirement (IWR_day) was defined as the amount of water necessary to bring K_s back to 1. From Equation (4) this is can be calculated as: (7)

Simulations of daily irrigation requirements based on Equation (1) through Equation (7) were performed for five sites representing the range of elevation and climate characteristic of the catchment (Figure 1 and Table 1). Daily weather data covering 1981–2009 were extracted for each site from the gridded data described in Section 3.2.

To obtain IWR_day for a wide spectrum of soil and crop conditions, the following soil and crop categories were considered:

For orchards (i.e., apple and apricot) and vineyards K_c was prescribed identically for each year based on data published in [22,23], with resulting growing season mean K_c of 0.82, 0.51 and 0.36, for apple, apricot and vineyards, respectively. For these crops, we further assumed a fixed growing season length of 183 (apple and apricot) and 216 days (vineyards) (Table 2). A minimum K_c of 0.15 was specified outside the growing season, as more than half of the orchards in the study region have a grass cover between rows [22].

For grassland and maize the crop coefficient K_c was diagnosed from the daily leaf area index (LAI). To this aim specific simulations of crop growth were carried out with the grassland simulator PROGRASS [24] and the process-based crop model STICS [25]. The identified relation reads: (8) where for maize LAI can be expressed as a function of the growing degree days (GDD) as: (9) assuming a base temperature of >6°C and setting the harvest date either corresponding to GDD = 2000 °C day⁻¹ or at latest on 1 November. For cut grasslands, we assumed a cutting regime with cuts taking places when LAI = 4 m² m⁻², but not before 1 May and no later than 1 November. Minimum LAI was set as LAI_min = 0.5 m² m⁻², and within each growing cycle, the daily increment ∆LAI was specified as, (10) where T_a is daily mean air temperature, T_B = 2.5 °C the corresponding base temperature, and initial LAI was set to 0.5 m² m⁻². Pastures (grazed grasslands) were modeled in the same way as cut grassland but with a cutting threshold set to LAI = 2.5 m² m⁻².

For all combinations of sites, soil water holding capacities and crops reference values of the seasonal IWR were obtained by integrating the daily irrigation requirements over the relevant growing season. Starting and ending dates of the effective growing season were specified for maize and grassland with respect to K_c (Table 2), and computed for each year as the dates of the up- and downcrossing of the threshold corresponding to the arithmetic mean between minimum and maximum K_c over the study period. This procedure provides more robust estimates than found e.g., with respect to the exceedance of temperature thresholds.

Preliminary data examination indicated that for a given soil and crop type seasonal IWR can be expressed as a bounded, linear function of the seasonal integral of the atmospheric water budget (<P − ET₀>), after scaling IWR estimates as well as <P − ET₀> to a common growing season length of 200 days. Indicating with IWR₂₀₀ and <P − ET₀>₂₀₀ these scaled values, we have: (11)

This relation is illustrated in Figure 2 for cut grassland and TAW = 32.5 mm.

Examination of similar plots for all combinations of crops and soils further suggested that Equation (11) can be generalized provided that A, B and IWR_min are expressed as a function of TAW and K_c: (12) with: (13)

The only exceptions we found, for which the model given by Equation (11) with Equations (12,13) failed to provide an adequate description of seasonal IWR, were shallow soils, viz. soils with a total holding capacity of TAW < 5 mm for vineyards, TAW < 15 mm for maize, and TAW < 9 mm for all other crops. Equations (12,13) are illustrated in Figure 3.

The values of the 12 model parameters g to u were obtained by stepwise fitting Equations (11–13) for all combinations of crop and soil types. Resulting parameter values were: g = −0.109; h = −0.008; i = −0.038; j = −0.019; k = −92.0 mm; l = −25.5 mm; m = 676.0 mm, n = −54.2 mm; r = −73.0 mm; s = −4.5 mm; t = 420 mm and u = −38.2 mm. It should be noted that for each of the three steps involved in fitting Equations (11–13) the coefficient of determination, R², was high, varying between 0.52 and 0.87 for Equation (11) (and equal to 0.84 in the example of Figure 2), between 0.80 and 1 for Equation (12) and between 0.90 and 0.99 for Equation (13).

After parameter fitting, we verified Equation (11) through Equation (13) by comparing estimates of the seasonal IWR with more detailed results from the point-scale daily model for six additional sites (one per crop type) representing contrasting climate and soil characteristics. Again, meteorological data needed to evaluate seasonal P and ET₀ were extracted from the gridded data described in Section 3.2. Growing season length and seasonal mean of daily K_c was computed according to Equations (8–10), and IWR₂₀₀ was eventually rescaled to provide IWR.

With respect to the model assessment, the overall agreement was very good (not shown), with performance statistics computed over the ensemble of verification runs given as R² = 0.64, root-mean square error RMSE = 41 mm and Bias = +2 mm.

Good model performance was also found with respect to year-to-year variations in IWR. As an illustrative example for grasslands, Figure 4 shows a comparison of results obtained from the daily model, on the one hand, and Equations (11–13), on the other hand, at two calibration sites (Aigle humid, and Visp dry) on a shallow soil and two verification sites on deeper soils, located in side Valleys in the South of the Rhone catchment, respectively.

As seen in this figure, Equations (11–13) were able not only to correctly reproduce differences between crops, soils and sites, but also to provide reliable estimates of seasonal IWR for extreme years. For Visp, average (1981–2009) IWR obtained from Equations (11–13) was 334 mm, as compared to 375 mm obtained with daily simulations (Figure 4b), and for 2003 as the year with the highest atmospheric demand, values of IWR at Visp were 479 and 550 mm, respectively, representing relative anomalies of +43% and +46%. Performance statistics relative to the comparison of the two time series at Visp were R² = 0.69, RMSE = 12% and Bias = −11% relative to the average of 375 mm.

For the estimation of IWR from Equation (11) to Equation (13) at the catchment scale, the gridded soil and land use data described in Section 3.2 were used. Start and end of the growing season and seasonal mean K_c were computed using the procedure outlined in Section 3.3.

In Figure 5, spatial patterns of average IWR for 1981–2009 are plotted along with the spatial distribution of <P − ET₀> and the corresponding anomalies for 2003. Positive values of the atmospheric water budget were obtained for the uppermost and lowermost parts of the catchment, reaching over 200 mm per season. Negative values were found in the central part between Sion and Visp, as well as in lateral valleys on the south side of the main valley. Negative seasonal atmospheric budgets varied typically from 0 to −200 mm, with a maximum of −400 mm in the surroundings of Visp (Figure 5a).

Seasonal IWR ranged from 0 to 50 mm in the area around Aigle and at higher elevations, and reached more than 250 mm in the Visp region (Figure 5c). For 59% of the agricultural area, soil water holding capacities were smaller than 5 mm (vineyards), 15 mm (maize) and 9 mm (all other crops). Corresponding grid points were thus excluded from the analysis. The remaining 41% were assumed to represent the area that potentially requires irrigation. This area may include points with IWR of 0 mm depending on climate variability and related crop-specific evaporative demand.

In 2003, a negative precipitation anomaly was already recorded during winter and spring, and again in July [10], resulting in a strong negative anomaly of the seasonal atmospheric water budget throughout the study region (Figure 5b) with values between −200 and more than −400 mm. Corresponding seasonal IWR exceeded 200 mm upstream and 50–100 mm near the outlet of the catchment, i.e., below Aigle (Figure 5d).

Aggregated values of IWR across the entire catchment varied between 22 and 55 × 10⁶ m³, approximating 32 × 10⁶ m³ on average for the period 1981–2009 (Figure 6a).

In 8 out of 29 years, IWR was below 25 × 10⁶ m³, but exceeded 40 × 10⁶ m³ in 4 years. IWR was particularly high toward the end of the time period considered, reaching 53 × 10⁶ m³ in 2003 and 55 × 10⁶ m³ in 2009. On average, half of total IWR was allocated to areas above 1500 m altitude. The relative anomaly in IWR for 2003 is 65% at catchment scale including the effect of a 6% increase in irrigated area.

Half of the area that would potentially require irrigation (that is 20% of total agricultural area, Figure 6b) received less than 50 mm per season and an additional quarter of the area needed between 50 and 100 mm per season. Only an estimated 5% of the total agricultural area appeared to require, on average, more than 200 mm per season, but this fraction doubled in extreme years (1989, 2003 and 2009).