The experiment was conducted over two consecutive growing seasons (2011–2012) in a 0.2 ha vineyard (Vitis vinifera L.) planted with cultivar Albariño, located in the experimental farm of the Estación de Viticultura e Enoloxía de Galicia (EVEGA), in Leiro (42∘21.6′ N, 8∘7.02′ W; elevation 115 m), Ourense, Spain (Fig. 1). Vines were grafted in 1998 on 196-17C rootstock and trained to a vertical trellis on a single cordon system (10–12 buds per vine). Rows were east–west oriented, spacings between vines and between rows were 1.25 and 2.4 m, respectively (3333 vines ha-1). The soil at the site was sandy textured (64 % sand, 16 % silt, 20 % clay), slightly acidic (pH 6.3), medium fertility (2.7 % organic matter), and with a rather shallow profile (≈ 1.2 m). The climate of the studied site is temperate, humid with cool nights (Fraga et al., 2014).

The reference evapotranspiration (ET0) per week for the site was calculated from weather variables recorded at a station located 150 m away from the experimental vineyard using the Penman–Monteith equation (Allen et al., 1998). The ET0 was then used, along with a constant crop coefficient (Kc = 0.8) to compute the amount of water required by the vines (Trigo-Córdoba et al., 2015). Precipitation was subtracted from ETc each week. The calculated amount of water was applied the following week.

Treatments consisted of a rain-fed control and an irrigation to the 50 % of ETc. Irrigation was applied from late June to early July (after bloom) till mid-August, approximately 2 weeks prior to harvest through two pressure-compensated emitters of 4 L h-1 located 25 cm on either side of the vine. Irrigation water was of good quality, with a pH of 6.35, electrical conductivity of 163.4 µS cm-1, and 0.4 mg L-1 of suspended solids. The water amount applied each season was 40 and 50 mm for 2011 and 2012, respectively (Table S1 in the Supplement).

The volumetric soil water content was continuously monitored through the soil profile in two spots of the experimental vineyard (one in the rain-fed treatment and another in the irrigated treatment) using two capacitance probes (EnviroSCAN, Sentek, Australia), based on the frequency domain reflectometry (FDR) technique. Each probe was equipped with three sensors installed on an access tube at 20, 40, and 60 cm depth and connected to a data logger. The probes were properly maintained for recording soil water content at half-hour intervals over the 2011 and 2012 seasons. Here, data from the irrigation period (mid-June to late August) are reported.

In each treatment, the probe was located within two vines (Fig. 1), avoiding proximity to the emitters (25 cm from the emitter and 50 cm from the vine trunk, approximately). The equation provided by the manufacturer was used for transforming permittivity data registered by the probes into soil water content since we only wanted to compare relative contents between these two irrigation regimes. Previous work suggests that soil type greatly affects the FDR readings, but the default equation is valid for differential measurements (Paraskovas et al., 2012).

The concepts of multifractals and their estimation methods that were used in the current study are next summarized. For detailed descriptions, further information can be found in Chhabra et al. (1989) and Everstz and Mandelbrot (1992).

To implement the multifractal analysis of one-dimensional soil water content time distributions supported on a given interval I = [a, b], a set of non-overlapping subintervals of I with equal length is required. A common choice is to consider dyadic downscaling (Everstz and Mandelbrot, 1992; Caniego et al., 2005), which means successive partitions of I in k stages (k = 1, 2, 3 …). Hence, at each scale, d, a number of segments, N(δ) = 2k, are obtained with characteristic time resolution, δ = L × 2-k, covering the whole extent of I.

A multifractal approach applied to time series has already been described (Jiménez-Hornero et al., 2010), hence, we only summarize the technique used in the current study. The time interval of soil water content data series, L, varied from half an hour to 2 months and the minimum time resolution, δini, was chosen accounting for containing at least one half-hourly averaged soil moisture datum, θini, at every initial interval. According to this, the probability mass distribution, pi(δ), at time resolution δ was estimated as pi(δ)=θi(δ)∑jniniθinij, where θi is the water content of the ith interval and nini is the number of initial intervals with mean soil water content θini.

The method of the moments was used (Chhabra et al., 1989) to analyse the multifractal spectrum of the probability mass function, pi(δ). The partition function χ(q, δ) was estimated as χ(q,δ)=∑i=1npi(δ)q, where moment q is a real number between -∞ and +∞.

A log–log plot of the partition function vs. δ for different values of q yields χ(q,δ)∝δ-τ(q), where τ(q) is the mass scaling function of order q. The functions f(α) and α can be obtained by Legendre transformation of the mass exponent, τ(q), as f(α) = α(q)–τ(q) and α(q) = dτ(q)/dq, respectively. Log–log plots of χq(δ) vs. δ, however, typically exhibit linearity across a limited scale range (e.g. Posadas et al., 2003), which results in drawbacks when using the moment method to obtain the singularity spectrum.

The direct method (Chhabra and Jensen, 1989) avoids inaccuracies associated with the estimation of α(q) by Legendre transformation. This method is based on the calculation of the contributions of individual segments, μi(q, δ), to the partition function, which are defined as μi(q,δ)=μiq(δ)/∑1N(δ)μiq(δ). Then, using a set of real numbers, q, (-∞ < q < -∞), the relationships applied to calculate f(α) and α, can be expressed as f(α(q))∝∑i=1N(δ)μi(q,δ)log⁡μi(q,δ)log⁡(δ), and α(q)∝∑i=1N(δ)μi(q,δ)log⁡μi(δ)log⁡(δ). The f(α)–α spectrum is reduced to a point for monofractal scaling type. The minimum scaling exponent (αmin) corresponds to the most concentrated region of the measure, and the maximum exponent (αmax) corresponds to the rarefied regions of the measure. A plot of f(α) vs. α is called multifractal spectrum. It is a downward function with a maximum at q = 0. The width of the multifractal spectrum (w = αmax–αmin) indicates overall variability (Moreno et al., 2008) similar to the nugget effects in geostatistics. For each data series, we calculated multifractal spectrum with q from -10 to +10 in steps of 0.5, fine enough to show the multifractal behaviour in the studied moment range.

Multifractal measures can also be characterized on the basis of the generalized dimension, Dq, of the moment of order q of a distribution, defined by Grassberger and Procaccioa (1983), based on the work of Rényi (1955). The Dq of a multifractal measure is calculated as Dq=τ(q)q-1=1q-1lim⁡δ→0log⁡χq(δ)log⁡δ,q≠1, and D1≈lim⁡δ→0∑i=1n(δ)μi(δ)log⁡μi(δ)log⁡δ,q=1. Equation (6a) shows that τ(q) is also related to the generalized fractal dimension, Dq. In fact, the concept of generalized dimension, Dq, corresponds to the scaling exponent for the qth moment of the measure. Using Eq.  (6a), D1 becomes indeterminate. Therefore, for the particular case that q = 1, Eq. (6b) was employed.

For a monofractal, Dq is a constant function of q. However, for multifractal measures, the relationship between Dq and q is described by a S-shaped curve. In this case, the most frequently used generalized dimensions are D0 for q = 0, D1 for q = 1, and D2 for q = 2, which are referred to as capacity, information (or Shannon entropy), and correlation dimension, respectively. The information dimension, D1, provides insight about the degree of heterogeneity in the distribution of the measure. The correlation dimension, D2, is associated to the uniformity of the measure among intervals and describes the average distribution density of the measure. In general, the generalized dimension, Dq, is more useful for the comprehensive study of multifractals. Differences between Dq allow comparison of the complexity between measured soil water content data series. In homogeneous structures Dq are close, whereas in a monofractal they are equal.

Temperatures for the two studied growing seasons were similar on average (Table S1); however, rainfall and evapotranspiration were higher in 2012. Harvest date was almost the same in both years. Nevertheless, the temporal evolution of rainfall and ETc differed from year to year (Fig. 2), being greater during 2012, especially at the beginning of the study period. This fact caused a different scheduling of irrigation between years.

Soil water content decreased over the growing season under rain-fed conditions in both years (Fig. 3). However, when irrigation was initiated, soil water content became more stable in the irrigated treatment (Fig. 3). The magnitude of the soil water loss was more evident in the layers of 20 and 40 cm depth, and less important in the 60 cm layer, which may indicate the depth of the active root zone as well as the intensity of root water uptake at each soil layer, as reported for other cultivars and crops (Intrigliolo and Castel, 2009; Mestas-Valero et al., 2011), and proved that FDR probes can be successfully used for irrigation scheduling (Goldhamer et al., 1999), calibrating them with established indicators such as midday stem water potential (Mirás-Avalos et al., 2014) and soil evaporation.

Soil water content time series obeyed power-law scaling, as shown by the double log plots (Fig. S1 in the Supplement). These plots allow to identify the range of moments needed to describe the scale variation of the studied parameter (Vidal Vázquez et al., 2010).

Figure S1 shows the partition functions for rain-fed and irrigation conditions at 20 cm depth in 2011. Visually, a slight departure from the straight line model was observed for moments q < -1 (Fig. S1). In general, higher deviations from linearity were found for the highest q moments in the data series from the irrigation treatment, when compared to those from the rain-fed treatment, especially in 2012. Nevertheless, determination coefficients, R2, were greater than 0.9 for statistical moments in the range from q = -10 to q = 10, in all the studied data sets. Consequently, scalings are adequately defined. Similar results were found by Mestas-Valero et al. (2011) for soil water content under rain-fed grassland.

The τ(q) functions were different from a monofractal type of scaling for all series analysed, especially under irrigation conditions (Fig. S2), similar to results obtained by Biswas et al. (2012b) for soil water storage. In fact, the heterogeneity of the soil water content data series from the irrigated treatment was greater than that of the rain-fed treatment (Fig. S2).

The value of D1 is a good indicator of the heterogeneity degree in temporal distributions of a given variable. The closer the D1 value to D0, the more homogeneous is the distribution of the variable. In our case, rain-fed series were more homogeneous than the irrigated ones. In general, soil water content recorded at 60 cm depth presented the lower differences between D1 and D0 (Table 1), thus being more homogeneous both under rain-fed and irrigation conditions. Moreover, the 2012 data series presented a higher heterogeneity than those from 2011 (Table 1) for both treatments, caused by the greater rainfall amount collected in 2012.

A monofractal would be characterized by D0 = D1 = D2 (Evertsz and Mandelbrot, 1992). In all the studied data series D0 > D1 > D2 (Table 1), indicating that soil water content had a tendency to behave as a multifractal. However, differences (D0–D1) ranged from 0.051 to 0.222 and (D1–D2) oscillated between 0.053 and 0.168, which suggests a different degree in the homogeneity/heterogeneity of soil water content depending on the treatment imposed and the depth in the soil profile. In general, data series from the irrigation treatment showed greater differences between D0, D1, and D2 than the series from the rain-fed treatment for both growing seasons. Moreover, the 60 cm depth layer presented smaller differences than the 20 and 40 cm layers (Table 1). The width of the Dq spectra, determined by indicators such as D0–D10, showed different degrees of heterogeneity, with a trend to decrease in depth and under rain-fed conditions when compared with the irrigation treatment (Table 1). This is caused by the spiky nature of soil water content and indicates a multiple scaling nature at shallow depths. Moreover, the width of the Dq spectra increased from 2011 to 2012 in both treatments, mainly in the 20 and 40 cm depths.

Generalized dimensions, or Rényi spectra, calculated for the range between q = -10 and q = 10 for soil water content data series at three depths under rain-fed and irrigation conditions are displayed in Fig. 4. All the data series studied showed Rényi spectra as asymmetric sigma-shaped curves with more curvature for the negative values of q than for positive ones (Fig. 4). The left part of the curves is concave down and it changes to concave up on the right of the vertical axis. In the case of the soil water content series from the rain-fed treatment, the most curved spectra corresponded to the 40 cm depth data series, whereas for the irrigation treatment, the most curved one was the 20 cm depth data series (Fig. 4). When compared between treatments, Rényi spectra were more curved under irrigation conditions and the estimation errors were also greater under this treatment (Fig. 4). These results confirmed the higher heterogeneity (multifractality) of the data series from the irrigation treatment when compared to those from rain-fed treatment.

Mestas-Valero et al. (2011) obtained monofractal distributions of soil water content time series under grassland when measured at depths greater than 40 cm, in contrast with our results. This disagreement is likely casued by the greater depth reached by grapevine roots when compared to grass roots. Therefore, grapevines may uptake water from deeper soil layers than grasses.

Determination coefficients, R2, were highest for moments q = 0 and q = 1 and diminished for the other |q| moments. In the case of q = 10, R2 was greater than 0.97 and 0.95 in the rain-fed and irrigated data sets, respectively. For q = -10, R2 values for rain-fed and irrigated data series were greater than 0.99 and 0.91, respectively (data not shown). Standard errors of Dq values increased with increasing |q| moments and they were much lower for the right (q > 0) than for the left (q < 0) branch of the Rényi spectra (Fig. 4).

Parameter α0 from the singularity spectra ranged from 1.056 to 1.146 in the rain-fed treatment and from 1.075 to 1.187 in the irrigated treatment (Table 2). The singularity spectrum allows for analysing similarity or difference between the scaling properties of the measures as well as assessing the local scaling properties of soil water content measurements. The wider the spectrum is (i.e. the largest αq-–αq+ value), the higher the heterogeneity in the scaling indices and vice versa (Vidal Vázquez et al., 2010). Moreover, the f(α) spectrum branch length gives insight about the abundance of the measure. Hence, small f(α) values at the end of a long branch correspond to rare events. Our results showed that the width of the singularity spectra increased in both treatments from 2011 to 2012 (Table 2).

Singularity spectra are characterized by a concave down shape (Fig. 5), showing an asymmetrical curve with wider but shorter right side. Rain-fed data series showed a shorter f(α) spectrum in both years, confirming their low degree of multifractality when compared to the irrigated data series (Fig. 5).

Differences (αq-–α0 and α0–αq+) indicate the deviation of the spectrum from its maximum value (q = 0) towards the right side (q < 0) and the left side (q > 0), respectively (Vidal Vázquez et al., 2010). Usually, soil water content data series from the rain-fed treatment showed lower α0–αq+ values than those from the irrigated treatment (Table 2). Moreover, the highest values for this multifractal parameter were observed at 40 cm depth in both treatments and years (Table 2). This may indicate that higher soil water contents were more frequent under irrigation, with the greatest differences observed at 40 cm depth in 2012. In contrast, the right branch (αq-–α0) of the spectrum was usually wider for rain-fed conditions (Table 2). These results confirm the differential homogeneity/heterogeneity pattern between treatments evidenced by the generalized dimension, Dq, analysis (Table 1, Fig. 4).