Field trials were conducted in La Entresierra field station of Ciudad Real in the central region of Spain (3∘56′ W, 39∘0′ N; 640 m of altitude) during May 2006 to June 2007. The soil of the experimental site, classified as Petrocalcic Palexeralfs in the USDA system (Soil Survey Staff, 2010), presented very low vertical variability up to a depth of 60 cm, from which one finds a discontinuous and fragmented petrocalcic horizon. The soil was sandy loam in texture, moderately basic (pH 7.9), with a medium level of organic matter (2.2 %), rich in potassium (0.9–1.0 meq L-1, ammonium acetate) and with a medium level of phosphorous (16.4 to 19.4 ppm, Olsen) with ECw. 0.1–0.2 dS m-1.

The area is characterized by a continental Mediterranean climate with widely fluctuating daily temperatures (for more details, see Castellanos et al., 2010).

During the 3 years prior to this experiment, the plots did not receive any organic or fertilizer amendments and were used to grow non-irrigated winter wheat (Triticum aestivum L.).

In this experiment, a randomized complete block design was used, with three nitrogen treatments and three irrigations. The irrigation treatment was applied at the main plot level, and N rates were replicated in the subplots. Each treatment was replicated four times in subplots measuring between 7.5 and 16.5 m in width and 12 m in length. The subplot widths ranged in size for practical reasons. The plots were arranged on a 4 by 9 grid (Fig. 1). Each subplot had 5, 7 or 11 rows of melons, according to its width (see Fig. 1).

Each crop row was drip irrigated from a line with emitters spaced at 0.5 m, which dripped water at a rate of 2 L h-1. Initially, to facilitate crop establishment, all plots received 30 mm of water. The irrigation schedule was calculated from 8 June to 6 September, with a single daily irrigation of 60 (W1), 100 (W2) or 140 % (W3) of melon crop evapotranspiration (ETc) depending on the irrigation treatment. Crop evapotranspiration (ETc) was calculated daily following the Food and Agriculture Organization of the United Nations (FAO) method (Doorenbos and Pruitt, 1977) as follows: ETc=Kc×ETo, where Kc is the crop coefficient, which was obtained in the same area for the melon crop in earlier years (Ribas et al., 1995), and ETo is the reference evapotranspiration calculated by the FAO Penman–Monteith method (Allen et al., 2002) using daily data from a meteorological station sited near the experimental field. The rainfall was negligible during the crop experiment, so the water applied was calculated as the ratio between the ETc of the previous week and the efficiency of the system, which considers the salt tolerance of the crop, the quality of the irrigation, the soil texture and the homogeneity of the irrigation system (Rincón and Giménez, 1989), estimated as 0.81 under the study conditions (more details in Castellanos et al., 2010). The irrigation calculated in this form was the theoretical irrigation and was divided by the number of days to obtain daily irrigation requirements. The total irrigation applied was registered on the water meter. The ETo during the irrigation schedule was 572 mm, the ETc was 419 mm, and the total irrigation applied was 343, 553 and 756 mm for W1, W2 and W3 respectively (Table 1). The irrigation water quality was measured weekly through a chemical analysis to estimate the nitrogen content of the water (Nw; Table 1).

The fertilizer treatments consisted of different N doses: 0 (N0), 150 (N1) and 300 (N2) kg ha-1. The N fertilizer was applied in the form of ammonium nitrate from 9 June to 18 August, from a single pool at one end of the field where irrigation water was mixed with the respective doses of N (Table 1). The total amount of N applied was the sum of the N fertilizer and N in the irrigation water, so all the treatments appear in Table 1.

The plots were fertilized with 120 kg of P2O5 ha-1 (phosphoric acid) for the season, added to the irrigation water and injected daily from 8 June to 30 August.

Melons were harvested when there was a significant amount of ripe fruit in the field from 26 July to 7 September, with a total of seven harvests.

The duration of the melon experiment was from 24 May to 7 September, and it is described more fully in Castellanos et al. (2010).

Winter wheat (cv. Soissons) was grown on the same experimental sites where the melon crop was before (Fig. 2). It was sown on 20 December 2006 in rows spaced 0.15 m apart at a population of 400 seeds m-2. Post-emergence herbicides were used to control weeds. No fertilizer or organic amendments were used for the cereal crop. Wheat crop was harvested on 6 June 2007.

At this time a transect was selected in the field that went through several plot treatments as shown in Fig. 1. Each 0.5 m a frame of 0.5 × 0.5 m2 was placed on the soil and the wheat plants captured were harvested and placed in labelled samples. A total of 160 samples were collected, traversing a length of 80 m.

Sub-samples of the dry plants and wheat grain were ground to a fine powder to determine the N content using the Kjeldahl method (Association of Official Analytical Chemists, 1990). The N uptake by the plant (PN) and by the grain (GN) was obtained as a product between N concentration and biomass (PW and GW respectively). The resulting data are shown in Fig. 3a and c.

In each sample, the wheat grain was placed apart from the rest of the plant to obtain the dry weight of each sample separately. The grain dry weight (GW) and plant dry biomass were determined by oven drying at 80 ∘C to constant weight. The plant dry weight (PW) was the sum of the GW and plant biomass. The data are shown in Fig. 3b and d.

A simple analysis, regardless of spatial position, was applied to the data collected. The correlation (r) and the determination coefficient (R2) between the nitrogen applied during the melon crop (Napp) and each variable (PW, GW, PN and GN) were estimated and plotted.

At the same time, the relations between nitrogen content and weight were studied for the grain (GW versus GN) and the whole plant (PW versus PN) as well as GW versus PN to compare with other studies performed on wheat crops.

Finally, a statistical test was applied for each variable to determine whether there was any significant trend with distance that would not allow the application of a straight multifractal analysis to the original data. The measure used was the coefficient of the slope of the regression line along the distance. This coefficient is derived using the least squares method and then compared to zero using the Student t test. If the t value is less than a critical t value at the 95 % level for the degrees of freedom, then the slope is considered to be zero.

The aim of a multifractal analysis (MFA) is to study how a normalized probability distribution of a variable (μi) varies with scale, as it is one way to study the structure of a measure. In this sense, the density levels of these probabilities are evaluated through the behaviour of a range of statistical moments of the partition function (χ(q,δ)). Let us consider a grid segment of length δ covering a part of the transect, with total length L. The measure of the ith segment is defined as Mi(δ). The probability is μi(q,δ)=Miq(δ)∑j=1N(δ)Mjq(δ). For a multifractal measure, χ(q,δ) will have scaling properties (Evertsz and Mandelbrot, 1992), namely χ(q,δ)∼δτ(q) being χ(q,δ)=∑j=1N(δ)μjq(δ) where τ(q) is a nonlinear function of q (Feder, 1989). For each q, τ(q) may be obtained as the slope of a log-log plot of χ(q,δ) against δ. A generalized dimension function Dq is then derived as (Hentschel and Procaccia, 1983) Dq=τ(q)/(1-q) for q≠1. The case D1 is defined as the limit D1=lim⁡q→1Dq. This leads to the scaling relation of entropy given by S(δ)=-∑i=1n(δ)μiln⁡(μi)∼D1ln⁡(δ). The dimension D1, known as the entropy dimension, can then be extracted from a plot of entropy against ln⁡(δ).

Given these definitions and the behaviour to expect in case of a multifractal measure, we are going to focus on the scaling properties of entropy as a tool to quantify the heterogeneity of a coarse-grained measure μi(δ), or signal, derived from the transect data as it has been applied previously to black and white soil thin sections (Bird et al., 2006).

We consider a transect of length L for a bin size δ; the entropy (S(δ)) is defined by Eq. (6). We use here a relative entropy (E(δ)) in order to establish what difference exists from the entropy of a uniform measure, given by E(δ)=∑iμi(δ)ln⁡μi(δ)-ln⁡δL, where the second term is the entropy of the uniform measure. Plotting this against the resolution of observation δ then reveals how heterogeneity in the signal evolves with increasing resolution, being another way to study the structure of the measure or variable (Tarquis et al., 2008). We may use this simple procedure to identify multiscale signals arising from the superposition of structure at different scales and assess the degree of this scale-dependent structure.

Here we consider some special cases. When we increase the resolution by a factor of 2, we observe that E(δ/2)=E(δ)+∑iμi(piln⁡pi+qiln⁡qi), where p and q control the distribution of the measure in the finer partition and p+q=1. Then ΔE(δ)=E(δ/2)-E(δ)=∑iμi(piln⁡pi+qiln⁡qi)+ln⁡2. If p and q are independent of i, then ΔE(δ)=(piln⁡pi+qiln⁡qi)+ln⁡2. This increases as the difference between p and q increases and more structure is observed in the data at this scale. If p=q=0.5, namely there is no structure revealed on increasing the resolution, then ΔE(δ)=0.

Further, if p and q are independent of δ, then we arrive at a binomial cascade. This is a multifractal measure and relative entropy scales logarithmically as E(δ)=(D1-1)ln⁡(δ/L).

Classical statistical analyses were performed on each of the variables to study their first statistical moments (Table 2). We could observe that the average and median present differences for each variable, in contrast to a normal distribution where both coincide. However, kurtosis and asymmetry do not present values higher than the unit in absolute terms. GW and PW present the highest kurtosis (0.82 and 0.78) and are negative. On the other hand, GN and PN have the highest asymmetry and are positive. The coefficient of variation is higher in variables related to nitrogen content (GN and PN) and lower in variables related to weight (GW and PW).

To study the relationships of GW, PW, GN and PN with the nitrogen applied during the melon crop season (Napp), we have plotted these variables without considering any spatial factors (Fig. 4). All of them show a tendency, as we expected, to increase in value as Napp increases. The correlation coefficient (r) for the four variables ranges from 0.66 (the GN case) up to 0.77 (the PN case), demonstrating that there are statistically significant correlations with the N application in the melon crop experiment (Napp), as the wheat crop did not receive any N directly. For this reason, the relationship that we can observe could be considered linear, as the range we are studying is suboptimal and not as in other studies (e.g. Hawkesford, 2014). However, a quadratic relation can be fitted to all the variables with a similar R2 (results not shown).

However, we can observe that at each of the Napp values, the variables show variability. This result is a consequence of a set of processes occurring from melon fertigation to wheat harvest, such as nitrogen uptake by the melon crop, organic soil nitrogen mineralization, nitrogen leaching, horizontal diffusion of soluble nitrogen forms and nitrogen uptake by the wheat crop (Milne et al., 2010).

The positive effect of increasing grain weight together with the additional benefit of increasing wheat N content with increasing N application is shown in Fig. 5a. Moreover, the same positive effect of N addition was observed, increasing wheat weight together with increasing wheat N content (Fig. 5b). Closer inspection of Fig. 4 reveals that the variability was much higher when the N application was higher. Barraclough et al. (2010), in an experiment with N fertilization applied homogenously directly to the wheat crop, found that much of the additional N taken up by the plant (PN) is manifested in higher yield (GW), although we remark again that in this work, the N application was performed in the melon crop experiment, through fertigation on crop lines, and the wheat crop did not receive any N fertilization and was not irrigated.

This positive effect of N addition has been observed in numerous studies (Barraclough et al., 2010, and references therein). Several works determine the N optimum in the wheat crop, but in this study, the optimal N dose was not obtained because we sought to study the variability and the effect of the residual N resulting from N application to a previous melon crop months before.

Before applying the multifractal analysis, a statistical test was applied to each variable to determine whether it presented a significant trend with distance. The results are shown in Table 3, where the estimated t was always lower than the critical t value, implying that no spatial trend was significant.

Multifractal analysis was applied to the four variables. In all cases, a τ(q) function reflected a hierarchical structure from one scale to the other with values of q= 1, τ(1)= 0, indicating the conservative character of the variables (Fig. 6a). Therefore, we estimated the Dq in an interval of q=±4 (Fig. 6b). The results show a weak variation in the values near 1, highlighting the difficulty in characterizing the multiscale heterogeneity in this type of analysis. In this case, the scale dependency found across a range of scales is not strong enough to show a high variation in Dq versus q, and τ(q) presents an almost linear trend. There are several works on soil transect data that present similar results (Caniego et al., 2005; Zeleke and Si, 2006).

Calculating the difference of D-4 and D4 can provide an estimate of the variation of Dq for each variable. A higher difference implies a stronger multifractal character. The variables related to nitrogen content (GN and PN) show a higher variation in Dq values (0.151 and 0.150 respectively) than the variables related to weight (0.088 for GW and 0.092 for PW), highlighting a different multifractal character of the two types of variables. In this sense, GW and PW behave very similarly, as do GN and PN. This information is complementary to the descriptive statistics performed in Sect. 3.1, in which the spatial factor was not considered.

To compare the spatial scaling behaviour of these four variables with the Napp behaviour, E(δ) was calculated, and the results are shown in Figs. 7a and 8. The translation from the number of data points (δ= 1, 2, 5, 10, 20, 40, 80 and 160) to the distance in metres is marked in Fig 7a. The trend in each case is not log-linear, as we would expect for a pure multifractal measure. In the case of Napp, the range of values reached -0.20 (Fig. 7a), and in the rest, they approach -0.06 (GW and PW) or -0.11 (GN and PN; Fig. 8).

We have plotted each variable (Fig. 8) E(δ) calculated at each δ following Eq. (7) and based on D1 estimated in the above section using Eq. (11). At certain scales, both present the same value, but most of the scales show variations (Fig. 8). Comparing the straight line slopes (see Fig. 8), which derived from the D1 values, higher and very similar values are found in GN and PN. On the other hand, GW and PW present lower values and are very similar to each other.

The increments of E(δ) (ΔE(δ)), between two consecutive scales, calculated for Napp and the four variables, are shown in Figs. 7b and 9 respectively. PN, GW and PW present a similar scaling trend, with a maximum structure revealed at scale δ= 10, corresponding to a distance of 5 m. This behaviour is the same found in Napp in the melon crop. In the case of N, the maximum structure is found at δ= 20 (10 m), indicating that the interaction of other factors influences this variation, and that Napp is not the main one.

All the values of ΔE(δ) at the smallest scales, δ= 5, 2 and 1 (2.5, 1 and 0.5 m respectively), show an increase, giving the second maximum value for GN, GW and PW. This result suggests that at those scales, the variation is mainly due to the melon cropping lines, as the uptake of the applied nitrogen by this crop left a lower amount of available nitrogen for the wheat crop. In the case of PN, the second maximum was found at δ= 20 (10 m) followed by the one at the smallest scales, δ= 2 and 1 (1 and 0.5 m), as in the other variables.

Comparing these results with those published by Milne et al. (2010), we found agreement on Napp as the main factor affecting PW change in structure and a noticeable influence at the smallest scales, highlighting the importance of crop melon space arrangement.