As already noted, in this work we have analysed two images of the same site acquired from different satellites, Landsat-7 and Ikonos-2. Both are multi-spectral images with several bands that cover several regions of the electromagnetic spectrum in the visible and near-infrared wavelength.

Landsat-7 was put in orbit in April 1999. This satellite follows a sun-synchronous orbit at 705 km of altitude, with an equatorial crossing time of 10:00 LT in the descending node. It requires 98.8 min to circle the Earth, tracing a worldwide reference system (WRS) of just over 230 ground paths. Over at least three decades, Landsat-7 orbits over each of these paths once every 16 days in a repetitive cycle (Mika, 1997).

The main Landsat-7 sensor for Earth observation is the Enhanced Thematic Mapper Plus (ETM+). The ETM+ operates as a whiskbroom scanner and acquires data for seven spectral bands: visible (ETM+#1, from 0.45 to 0.52 µm; ETM+#2, from 0.53 to 0.61 µm; ETM+#3, from 0.63 to 0.69 µm), near infrared (ETM+#4, from 0.78 to 0.9 µm), shortwave infrared (ETM+#5, from 1.55 to 1.75 µm, and ETM+#7, from 2.09 to 2.35 µm) and thermal infrared (ETM+#6, from 10.4 to 12.5 µm). The ETM+ ground sampling distance (pixel size in the images) is 30 m for the six reflective bands and 60 m for the thermal band. The ETM+ also acquires images for a panchromatic band (ETM+#8, from 0.52 to 0.9 µm) with a 15 m ground sampling distance. The radiometric resolution of the Landsat-7 data is 8 bits per pixel or 256 grey levels for the pixel digital value.

Ikonos-2 was launched in September 1999. Its panchromatic sensor, with a resolution of 0.82 m, provided the first very high-resolution images of the Earth's surface from earth observation satellites (EOS). The Ikonos-2 orbiting altitude is approximately 681 km; it is inclined 98.1∘ to the equator and it provides sun-synchronous operation. The equatorial crossing time of Ikonos-2 is 10:30 LT in the descending node. The orbit provides daily access to sites within 45∘ of nadir (Dial et al., 2003).

The multi-spectral sensor simultaneously collects blue (IK#1, from 0.445 to 0.516 µm), green (IK#2, from 0.506 to 0.595 µm), red (IK#3, from 0.632 to 0.698 µm) and near-infrared (IK#4, from 0.757 to 0.853 µm) bands with 3.28 m resolution at nadir. Both images, panchromatic and multi-spectral, have a radiometric resolution of 11 bits per pixel or 2048 grey levels for the pixel digital value.

The Landsat-7 multi-spectral image used in this study was acquired on 6 August 2000 at 10:46 LT. and it corresponds to the scene with WRS coordinates: path and row 201 and 32, respectively. This scene is located in the central region of Spain and it covers a square surface of approximately 180 km side length, located around Madrid. Solar azimuth and elevation angles for this scene are 132.44 and 58.62∘ respectively.

The Ikonos-2 datum used in this study is a multi-spectral image acquired on 8 August 2000 at 11:03 LT. It covers an area of 11 km2 located near Aranjuez, south of Madrid, in the central region of Spain. Solar azimuth and elevation angles for this scene are 139.5 and 60.79∘ respectively. Both images were corrected geometrically to the same cartographic projection: Universal Transverse Mercatorprojection (UTM), zone 30∘ N, by a co-registration process.

The analysis has been carried out on a subset that covers (approximately) the same area in both the Landsat-7 and the Ikonos-2 images, corresponding to a region located north of the town of Aranjuez. The representative elements of the land used in the selected area are irrigation crops, pastures, heaths, unirrigated land cultivations and olive groves. The Landsat-7 subset image is a square of 512×512 pixels with a resolution of 30 m covering a somewhat larger surface than the Ikonos-2 image. The Ikonos-2 image consists of a square subset with 2048×2048 pixels and 4 m resolution.

Vegetation is one of the landscape elements that has received the most attention in the field of image analysis. Therefore, there are many parameters that can be used to obtain information on vegetation from remote sensing imagery.

One of the main parameters is made up of the vegetation indexes. These indexes allow to detect the presence of vegetation in an area and its activity, since its values are related to this activity. For this, we can use the reflectance values corresponding to the different wavelengths, interpreting these in relation to the photosynthetic activity. Of these indexes, the most commonly used is the NDVI.

The NDVI is defined by NDVI=NIR-RNIR+R, where NIR is the pixel value in the near-infrared band and R the pixel value in the red band. The values of this index are within the range (-1, 1) and their positive values are sensitive to the proportion of soil and vegetation in each pixel (Carlson and Ripley, 1997). Pixels with NDVI < 0.2 are considered without vegetation or bare soil. Pixels with NDVI > 0.5 are considered as fully covered by vegetation.

EVI, the other vegetation index, is defined by EVI=2.5NIR-RL+NIR+C1R-C2B, where NIR is the pixel value in the near-infrared band, R the pixel value in the red band and B the pixel value in the blue band. L, C1 and C2 are constants with the values 1, 6 and 7.5 respectively. The main characteristic of this index is that it corrects some distortions caused by the light dispersion from aerosols, as well as the background soil (Huete et al., 2014).

A monofractal object can be measured by counting the number N of δ size boxes needed to cover the object. The measure depends on the box size as N(δ)∝δ-D0, where D0=lim⁡δ→0log⁡N(δ)log⁡1δ is the fractal dimension. D0 is calculated from slope of a log–log plot. However, many examples are found where a single scaling law cannot be applied and it is necessary to do a multi-scaling analysis.

There are several methods for implementing multi-fractal analysis. The universal multi-fractal (UM) model assumes that multi-fractals are generated from a random variable with an exponentiated extreme Levy distribution (Lavallée et al., 1991; Tessier et al., 1993). In UM analysis, the scaling exponent K(q) is highly relevant. This function for the moments q of a cascade conserved process is obtained according to Schertzer and Lovejoy (1987), as follows: Kq=C1qαL-qαL-1ifαL≠1C1qlog⁡qifαL=1, where C1 is the mean intermittency codimension and αL is the Levy index. These are known as the UM parameters.

Other method is the moment method developed by Halsey et al. (1986) and applied to this case study. This method mainly uses three functions: τ(q), known as the mass exponent function; α, the coarse Hölder exponent; and f(α), multi-fractal spectrum (MFS). A measure (or field), defined in two-dimensional image embedding space (n×n pixels) and with values based on grey tones (for 8 bits, from 0 to 255), cannot be considered as a geometrical set and therefore cannot be characterized by a single fractal dimension.

To characterize the scaling property of a variable measured on the spatial domain of the studied area, it divides the image into a number of self-similar boxes. Applying disjoint covering by boxes in an “up-scaling” partitioning process we obtain the partition function χ(q,δ) (Feder, 1989) defined as follows: χ(q,δ)=∑i=1N(δ)μiq(δ)=∑i=1N(δ)miq, where m is the mass of the measure, q is the mass exponent, δ is the length size of the box and N(δ) is the number of boxes in which mi>0. Based on this, the mass exponent function τ(q) shows how the moments of the measure scales with the box size: τ(q)=lim⁡δ→0log⁡<χ(q,δ)>log⁡(δ)=lim⁡δ→0log⁡<∑i=1N(δ)miq>log⁡(δ), where <> represents a statistical moment of the measure μi(δ) defined on a group of non-overlapping boxes of the same size partitioning as the area studied.

The singularity index, α, can be determined by the Legendre transformation of the τ(q) curve (Halsey et al., 1986) as follows: α(q)=dτ(q)dq. The number of cells of size δ with the same α, Nα(δ), is related to the cell size as Nα(δ)∝δ-f(α), where f(α) is a scaling exponent of the cells with common α. Parameter f(α) can be calculated as follows: f(α)=qα(q)-τ(q). MFS, shown as plot of α versus f(α), quantitatively characterizes variability of the measure studied with asymmetry to the right and left indicating domination of small and large values respectively (Evertsz and Mandelbrot, 1992). There are three characteristic values obtained from MFS, the singularity α(q) values for q=0,1,2. The first value (α0) corresponds to the maximum of MFS and it is related to the box-counting dimension of the measure support; the second value is related to information or entropy dimension (α1) and the third with the correlation dimension. The entropy dimension quantifies the degree of disorder present in a distribution. According to Andraud et al. (1994) and Gouyet (1996), a α1 value close to 2.0 characterizes a system uniformly distributed throughout all scales, whereas a α1 close to 0 reflects a subset of the scale in which the irregularities are concentrated. These three values will be shown from each calculation of MFS.

The width of the MF spectrum (Δ) indicates overall variability (Tarquis et al., 2001, 2014) and we have split it in two sections. Section I correspond to values α(q)<α(0) or q>0 and section II to values with α(q)>α(0) or q<0. In section I the amplitude, or semi-width, was calculated with differences Δ+=α(0)-α(+5), and in section II with Δ-=α(-5)-α(0).

To study the asymmetry of the MFS we have chosen the asymmetry index (AI) estimated as follows (Xie et al., 2010): AI=ΔαL-ΔαRΔαL+ΔαRΔαL=α0-αminΔαR=αmax-α0. In our case, α0 is the singularity for q=0 or α(0), αmin is α(+5) and αmax is α(-5). Therefore, we can rewrite the AI as follows: AI=Δ+-Δ-Δ++Δ-. Expressing AI as Eq. (11), we can see that it is a normalized index based on the amplitudes Δ+ and Δ-.

There are several works relating the UM model and the multi-fractal formalism based on τ(q) (Gagnon et al., 2003; Aguado et al., 2014; Morató et al., 2017, among others) through the following equations: fα=E-cγ;α=E-γ,τq=Eq-1-Kq, where E is the Euclidean dimension in which the measure is embedded, in this case E=2, and c(γ) is the codimension of the singularity of the density of the multi-fractal measure γ.

To study the influence of radiometric resolution on Ikonos-2 image information complexity, the original pixel code (11 bits) has been transformed to 8 bits through a rescaling based on minimum and maximum values between 0 and 255, with the aim of preserving the initial histogram shape.

We first discuss the results obtained for the 2048×2048 pixel Ikonos-2 image shown in Fig. 1, in bands combination of false colour (IK#4, IK#3, IK#2 band combination in RGB visualization). In Fig. 2 IK#1, IK#2, IK#3 and IK#4 band histograms are shown. In the right column are histograms with the original radiometric resolution and in the left column the corresponding histograms are rescaled to 8 bits. The histograms present a bimodal structure with a narrow peak of low-value pixels (dark grey) showing a sharp maximum and a wider peak around a second lower maximum. For bands IK#1, IK#2 and IK#3, the narrow peak maximum corresponds to vegetation, mainly irrigation crops, showing strong water absorption. This effect is particularly important in band IK#3. High-value pixels (lighter grey) correspond to ground zones with lower vegetation content. However, as vegetation shows high reflectivity in the near-infrared, IK#4 band histogram shows a predominance of high-value pixels (lighter grey pixels) corresponding to dense vegetation parts. For both radiometric resolutions the shapes of the histograms are very similar, as it was our intention (see Fig. 2).

We cover the image with boxes of size δ=2-n and we change the box size from 2048 to 2 pixels, that is, δ=2048/2n with n=0, 1, 2, …, 10. For each value of the parameter q, from -5 to +5 with increments of 0.5, the partition function (Eq. 6) is computed and log χ(q,δ) versus log⁡δ is plotted in Fig. 3. Each graph contains 11 points and from these a range of scales are selected for the least-square linear fit reaching the maximum possible scales and with a standard error in the slope, the estimated values of τ(q), less than 0.01. Then, using Eqs. (7)–(8), α(q) and f(α) are obtained. Comparing the range of scales used in both radiometric resolutions, the bands using the original data (11 bits) showed a wider range of scales for the linear fit, up to 4 pixels, whereas in the 8-bit radiometric resolution were required up to 32 pixels (see arrows in Fig. 3).

The MF spectra f(α) corresponding to the four bands of multi-spectral Ikonos images are shown in Fig. 4. These differences found in the multi-scaling behaviour of each band are in agreement with previous works (Cheng, 2004; Lovejoy et al., 2008). Just by visual observation, there is a remarkable difference between the bands #3 and #4, and red and NIR, between 8 and 11 bits respectively. Higher radiometric resolution gives a higher range of possible grey values per pixel. Note that this radiometric resolution effect is manifested in both sections of the MF spectra (for q>0 and for q<0).

Some characteristic parameters obtained from these MF spectra are shown in Table 1 and Table 2. As expected, in both radiometric resolutions and in each band the α0 is practically 2, as the measure is defined in the entire plane and it has an Euclidean dimension of 2. With respect to the α1 value, certain differences are found. Comparing the bands in 8 bits to the same ones in 11 bits, the entropy dimension was always higher. However, considering the standard errors, only IK#1 (B) and IK#2 (G) bands were significantly different, with the blue band showing the highest difference. Meanwhile, red and NIR bands are not significantly different. This shows that a more spatially uniform distribution for the bands of Ikonos-2 8 bits than in 11 bits. The same behaviour is observed in the α2.

The amplitudes calculated (Δ+ and Δ-) in Ikonos-2 11-bit bands present opposite trends (Table 1). Note that amplitude Δ+ decreases as band wavelength grows, whereas the other amplitude Δ- diminishes. Observing these parameters in Ikonos-2 8-bit bands (Table 2) a different trend and behaviour are found. In this case both Δ+ and Δ- increase as the wavelength increases for the three visible bands, but decrease for the near-infrared band (IK#4).

The AI estimated on these MFS amplitudes on each radiometric resolution are shown in the last column of Table 1 and Table 2. Comparing the bands in 8 bits to the same ones in 11 bits, the behaviour is similar: there is a decreasing trend from IK#1 to IK#4, although the range of values is different. At a resolution of 11 bits from a positive AI=0.240 at blue band goes to a negative AI=-0.237 at NIR band. On the other hand, at a resolution of 8 bits, an AI=0.092 goes to a negative AI=-0.347. The more symmetric MFS are found in green and red bands at a resolution of 11 bits and in blue and green bands at a resolution of 8 bits.

Doing the same study for the vegetation indexes we found the following. The bi-log plot of the partition function (χ(q,δ)) versus δ is plotted in Fig. 5 for both vegetation indexes at both radiometric resolutions. Each graph contains 11 points as the bands from where they were estimated. The linear fit was done with the same methodology as that for the four bands. In this case only the EVI at 8 bits shows a better linear trend in a wider range of scales. However, to better compare both vegetation indexes, from both radiometric resolutions, a range achieving 32 pixels (128 m) was selected as shown by arrows in Fig. 5.

The MF spectra f(α) of EVI and NDVI estimated for both radiometric resolutions of Ikonos images are shown in Fig. 4. Both vegetation indexes show differences due to the transformation from 11 to 8 bits. However, NDVI shows higher differences in the MFS, mainly in the part corresponding to q negative values (right side). Even EVI presents changes; its MFS is closer at both radiometric resolutions. Comparing the range of f(α) values in the vegetation indexes to the range obtained in the four bands (left column in Fig. 4), there is a remarkable contrast. Meanwhile the NIR band of 8 bits achieves a f(α) value close to 0.5; EVI and NDVI achieve values closer to 0.2. These differences are higher in the 11-bit image; the red band achieves a f(α) value close to 0.9 and vegetation indexes again achieve values of ∼0.2. The same characteristic parameters obtained from the band MF spectra were calculated for the vegetation indexes and are shown in Table 1 and Table 2.

With respect to the α1 values, certain differences are found between the vegetation indexes. Comparing the NDVI in 8 bits to the same ones in 11 bits, the entropy dimension was always higher than it was found to be in the bands. However, EVI shows the contrary: entropy values of the 11-bit image are higher than the 8-bit image, although the differences are not significant. Therefore, the radiometric resolution affects NDVI more than EVI. The former presents a more uniform space distribution in 8 bits than in the 11-bit image. The same behaviour is observed in α2.

The amplitudes calculated (Δ+ and Δ-) in Ikonos-2 11-bit vegetation indexes present a similar situation (Table 1). The amplitude Δ+ is lower than amplitude Δ- and therefore the AI estimated is negative. This is visually perceived in Fig. 4 (right column). Observing these parameters in Ikonos-2 8 bits vegetation indexes (Table 2), similar situations are found but the values are lower. In both images (11 and 8 bits) NDVI shows higher values for both amplitudes, Δ+ and Δ-.

All the AI estimated for both vegetation indexes on each radiometric resolution are negative (Table 1 and Table 2), indicating a high asymmetry on the right part of the MFS, as shown in Fig. 4. Comparing the AI values in 8 bits to the same ones in 11 bits, they are similar which shows that the shape of the MFS is similar as this index is a normalized index. However, the values of the amplitudes mark a higher change in NDVI than in EVI.

A comparison is made between Landsat, with an original pixel code of 8 bits, and the rescaled histograms from Ikonos, with an original pixel code of 11 bits. In this section, we discuss the results obtained in the MF analysis on the 512×512 pixel Landsat-7 image shown in Fig. 6, in bands combination of false colour (ETM+#3, ETM+#2 and ETM+#1 band combination in RGB visualization).

In the right column of Fig. 6, the histograms of the Landsat-7 image for the first four bands are shown. The histograms present a bimodal structure, except for ETM+#4 (NIR), in which there is only one peak. Comparing these histograms with those obtained for Ikonos-2 8-bit image (Fig. 2), the peaks are not so abrupt and narrow. At the same time, ETM+#1, ETM+#2 and ETM+#3 bands show the absolute maximum peak at high-value pixels (light grey) and a second one at lower-value pixels (dark grey). These bands are more centred and do not show a shift to the left as the Ikonos-2 8-bit bands (IK#1, IK#2 and IK#3. In the case of the NIR band, Landsat-7 and Ikonos-2 8 bits are quite similar except for the absence of a second peak.

In the calculations, box sizes range from 512 to 2 pixels, that is, δ=512/2n with n=0, 1, …, 8. For each value of the parameters q, from -5 to +5 with increments of 0.5, we compute the partition function, and the bi-log χ(q,δ) versus δ is plotted in Fig. 7. In this case each linear fits contains only 9 points as the size of the image is 512×512 pixels. The same method was applied to select the range of scales used in the linear fit, achieving a scale of 4 pixels. Changing from pixels to metres, the scale achieved, used in Landsat-7 in the MF analysis, was ∼120 m. In the case of the Ikonos-2 8 bits the scale was 32 pixels or 128 m, very close to Landsat-7.

The MF spectra, f(α), corresponding to the first four bands of multi-spectral Landsat-7 images are shown in Fig. 8. From a comparison of Figs. 4 and 8 we see that Landsat-7 image MF spectra are always located inside the corresponding Ikonos-2 MF spectra. For a given value of Hölder exponent α, the relation fLandsat(α)≤fIkonos(α) is always satisfied. This result means that Landsat-7 images show lower complexity than Ikonos-2 8-bit images. As stated in Sect. 2.1 Ikonos-2 satellite data are coded in 11 bits in contrast with Landsat-7 8-bit-coded data. To compare both sensors, with different spatial resolution, we pass Ikonos-2 from 11 to 8 bits, observing that the latter shows more complexity than Landsat.

The MF-spectra parameters from Landsat-7 are shown in Table 3. In this section we will compare the MF spectra and the vegetation indexes of the Ikonos-2 8 bits (Table 2). The α1 values from the four bands of Landsat-7 are higher than the ones presented by Ikonos-2 8 bits, indicating a more uniform space distribution. Comparing between the bands, there are not significant differences, contrary to the trend we observed among them in Ikonos-2 8 bits. The α2 shows the same behaviour.

The amplitudes calculated (Δ+ and Δ-) in Landsat-7 bands present few variations (Table 3). The amplitude Δ+ decreases from ETM+#1 to ETM+#3 and then presents an increase in ETM+#4 (NIR) whereas the other amplitude Δ- remains practically constant. Observing these parameters in Ikonos-2 8-bit bands (Table 2), there are variations in value and behaviour for the four bands. In this case, both Δ+ and Δ- increase as the wavelength increases for the three visible bands, but decrease for the NIR band (IK#4).

The AIs estimated on these MFS amplitudes on each Landsat-7 bands are positive, except for ETM+#3 (red band). For the green band (ETM+#3) the symmetry of the MFS is complete. The band that shows certain asymmetry is the blue band (ETM+#1).

Regarding the vegetation indexes, estimated on Landsat-7 bands, we found the following. The bi-log plot of the partition function (χ(q,δ)) versus δ is plotted in Fig. 9 for both vegetation indexes. Each graph contains 9 points corresponding to the bands from which they were estimated. The linear fit was done with the same methodology as that for the four bands. EVI and NDVI show the same behaviour, and the same range of scale was selected achieving 8 pixels, as shown by arrows in Fig. 9.

The MF spectra f(α) of EVI and NDVI, estimated based on the Landsat-7 image, are shown in Fig. 8. Both vegetation indexes show differences mainly in the right side of the MFS (for q negative values). Comparing the range of f(α) values in the vegetation indexes to the range obtained in the four bands (left column in Fig. 8), there is a remarkable contrast. Meanwhile the NIR band of 8 bits achieves f(α) value close to 1.6, EVI and NDVI achieve values closer to 1. A similar situation was found with both images of Ikonos-2.

We are going to study the parameters obtained from the MF spectra for the vegetation indexes (Table 3). The results are quite similar to those found for the Landsat-7 bands, showing even higher values: 1.996 in NDVI and 1.997 in EVI.

The amplitude Δ+ is quite low compared with the bands and to the vegetation indexes of Ikonos-2 8 bits. On the other hand, the amplitude Δ- is higher than Landsat-7 bands but by only a third of the values shown by Ikonos-2 8-bit vegetation indexes (Table 2). The AI estimated for both vegetation indexes are negative, indicating a high asymmetry on the right part of the MFS, as shown in Fig. 8. Comparing the AI values of Landsat-7 vegetation indexes with the ones of both Ikonos-2 images, these are the highest, indicating that the most unbalanced MFS shifted totally on the right side of the spectrum.