In this study, we use multifractal analysis, through generalized
dimensions (

The

The four variables studied presented a weak multifractal character with a
low variation in

Soils exhibit spatial variation operating over several scales. This observation points to “variability” as a key soil attribute that should be studied (Burrough et al., 1994). Soil variability has often been considered to consist of “functional” (explained) variations plus random fluctuations or noise (Goovaerts, 1997, 1998). However, the distinction between these two components is scale-dependent because increasing the scale of observation almost always reveals structure in the noise (Logsdon et al., 2008). Geostatistical methods and, more recently, multifractal/wavelet techniques have been used to characterize the scaling and heterogeneity of soil properties, among other approaches coming from complexity science (de Bartolo et al., 2011). These methods study the structure of the property measured in the sense that compares the probability distribution at each scale and among scales.

Multifractal formalism, first proposed by Mandelbrot (1982), is suitable for variables with self-similar distribution on a spatial domain (Kravchenko et al., 2002). Multifractal analysis can provide insight into spatial variability of crop or soil parameters (Kravchenko et al., 2002, 2003; Vereecken et al., 2007). This technique has been used to characterize the scaling properties of a variable measured along a transect as a mass distribution of a statistical measure on a spatial domain of the studied field (Zeleke and Si, 2004; López de Herrera et al., 2016). To do this, it divides the transect into a number of self-similar segments. It identifies the differences among the subsets by using a wide range of statistical moments.

Wavelets were developed in the 1980s for signal processing and later introduced to soil science by Lark and Webster (1999). The wavelet transform decomposes a series; whether this be a time series (Whitcher, 1998; Percival and Walden, 2000), or as in our case a series of measurements made along a transect; or into components (wavelet coefficients) which describe local variation in the series at different scale (or frequency) intervals, giving up only some resolution in space (Lark et al., 2003). Wavelet coefficients can be used to estimate scale-specific components of variation and correlation. This allows us to see which scales contribute most to signal variation, or to see at which scales signals are most correlated (Lark et al., 2004). This can give us an insight into the dominant processes.

An alternative to both of the above methods has been described recently. Relative entropy and increments in relative entropy have been applied in soil images (Bird et al., 2006) and in soil transect data (Tarquis et al., 2008) to study scale effects localized in scale and provide the information that is complementary to the information about scale dependencies found across a range of scales. We will use them in this work to describe the spatial scaling properties of a set of data measured on a common 80 m transect across a wheat crop field. This is an indirect way to study the N variability left in the soil by the previous crop.

Nitrogen fertilizer inputs for intensive production of irrigated crops can
contribute to elevated NO

The data discussed in this paper result from two consecutive experiments
performed near two hydrological units (UH) protected by the government of
Castilla-La Mancha concerning the protection of waters against pollution
caused by nitrates from agricultural sources. These two units, Mancha
Occidental (UH04.04, 6.953 km

In this study, we have analysed the transect data for nitrogen content and the weight of the grain and of the whole plant of the wheat crop. First, correlations between these four variables and the different nitrogen application doses in the previous crop were estimated, without considering spatial structure. Then, multifractal and relative entropy analyses were applied to investigate the structure among the scales. This work is the first application of both types of analysis to the same data set.

A croquis of the experimental melon crop layout. The nine subplots
of the melon crop experiment through which the wheat transect ran are shown.
The wheat transect is shown by the dark green line. The fertilizer levels are
shown in the figure: N0, N1 and N2 represent 0, 150 and 300 kg N ha

Field trials were conducted in La Entresierra field station of Ciudad Real
in the central region of Spain (3

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 (

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).

The treatments applied to the melon crop, total irrigation (applied irrigation, taking initial establishment irrigation into account, in the different treatments: 60 ETc (W1), 100 ETc (W2) and 140 % ETc (W3); 15 to 104 DAT) and applied nitrogen information. From Milne et al. (2010) with permission.

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

The fertilizer treatments consisted of different N doses: 0 (N0), 150 (N1)
and 300 (N2) kg ha

Monthly precipitation and irrigation applied, in millimetres, for melon and wheat crop.

The plots were fertilized with 120 kg of P

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).

Original data of the four variables studied, including the nitrogen
doses applied in the melon crop along the transect:

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

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

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

A simple analysis, regardless of spatial position, was applied to the data
collected. The correlation (

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

The aim of a multifractal analysis (MFA) is to study how a normalized
probability distribution of a variable (

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

We consider a transect of length

Descriptive statistics of variables studied: grain N content (GN), grain weight (GW), wheat N content (PN) and wheat weight (PW).

Correlations with nitrogen applied (N

Here we consider some special cases. When we increase the resolution by a
factor of 2, we observe that

Further, if

Effect of N applied in a previous melon crop on

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 (N

Statistical trend significance between the variables studied and the distance in the transect (see Fig. 3): grain N content (GN), grain weight (GW), wheat N content (PN) and wheat weight (PW).

s.e.: standard error; ns: not significant.

Multifractal analysis of the four variables studied:

Entropy study:

Relative entropy (

However, we can observe that at each of the N

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.

Increment of relative entropy (

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

Multifractal analysis was applied to the four variables. In all cases, a

Calculating the difference of

To compare the spatial scaling behaviour of these four variables with the
N

We have plotted each variable (Fig. 8)

The increments of

All the values of

Comparing these results with those published by Milne et al. (2010), we
found agreement on N

Four variables, the biomass and nitrogen content of wheat and grain, have been studied on transect data selected from a set of experimental plots where different fertigation treatments were applied to a previous melon crop.

First, classical statistics were applied without considering the spatial arrangement to study these variables. None presented extreme values of kurtosis and asymmetry, but comparing the values showed a difference between variables related to nitrogen content and variables related to weight. In addition, the coefficient of variation was lower in the nitrogen-related variables.

Then, the relationships between the variables and with the nitrogen applied to the previous crop were studied. The positive effect of N addition to the melon experiment was observed through increased grain weight (GW), wheat N content (PN) and wheat weight (PW), but even these correlations present a high volatility, and it is not clear whether a first- or second-order regression could fit better. However, GW versus PN and PW versus PN presented a clear logarithmic relation tending to a maximum.

Considering the spatial arrangement of the variables' values, we have conducted a multifractal analysis on transect data as we checked that there was a non-significant trend along the transect. The Dq obtained indicates a non-strong multiscale structure in the four variables studied, but different strength was nonetheless observed between variables related to nitrogen content (GN and PN) and variables related to weight (GW and PW). In this case, the generalized dimensions did not give us the relevant information we expected on multiscale heterogeneity but did discriminate between the two types of variables, as in the classical statistics.

A relative entropy analysis was used to identify local maxima within the data
structure. Grain and plant weight (GW and PW respectively) present a
maximum structure at a scale of 5 m that corresponds to N

The proposed approach provides information about scale dependencies related to factors that created spatial variability and is complementary to multiscale analysis and descriptive statistics.

Data are available by email request to the corresponding author.

The authors declare that they have no conflict of interest.

This project has been partially supported by INIA-RTA04-111-C3 and by the Ministerio de Economía y Competitividad (MINECO) under contract nos. MTM2015-63914-P and CICYT PCIN-2014-080. Edited by: A. Biswas Reviewed by: three anonymous referees