Bathymetry from satellite images: a proposal for adapting the band ratio approach to IKONOS data

The acquisition of bathymetric data in shallower waters is difficult to attain using traditional survey methods because the areas to investigate may not be accessible to hydrographic vessels, due to the risk of grounding. For this reason, the use of satellite detection of depth data (satellite-derived bathymetry, SDB) constitutes a particularly useful and also economically advantageous alternative. In fact, this approach based on analytical modelling of light penetration through the water column in different multispectral bands allows to cover a big area against relatively low investment in time and resources. Particularly, the empirical method named band ratio method (BRM) is based on the degrees of absorption at different bands. The accuracy of the SDB is not comparable with that of traditional surveys, but we can certainly improve it by choosing satellite images with high geometric resolution. This article aims to investigate BRM applied to high geometric resolution images, IKONOS-2, concerning the Bay of Pozzuoli (Italy), and improve the accuracy of results performing the determination of the relation between band ratio and depth. Two non-linear functions such as the exponential function and the 3rd degree polynomial (3DP) are proposed, instead of regression line, to approximate the relationship between the values of the reflectance ratios and the true depth values collected in measured points. Those are derived from an Electronic Navigational Chart produced by the Italian Hydrographic Office. The results demonstrate that the adopted approach allows to enhance the accuracy of the SDB, specifically, 3DP supplies the most performing bathymetric model derived by multispectral IKONOS-2 images.


Introduction
The determination of the bathymetry (water depth) is of fundamental importance to understand the topography of the seabed, river and lake. It is essential to analyse the dynamics of the marine environment, both in terms of sediment transport, in relation to the prediction of tides, currents and waves, to detect foreign objects present in the seabed and to produce nautical charts supporting navigation. Such information is essential for continuous monitoring of coastal areas (Shah 2020;Gao 2009) which are particularly sensitive, both for the high number of activities that take place there and for the risk of environmental pollution. For centuries, large amounts of waste and pollutants have been thrown into the seas, e.g. solid waste, sewage sludge, boat wastewater and oil: the interaction of coastal and submarine morphology with the hydrodynamics regime conditions the dispersion of sediments and potential pollutants existing in the area (Pippo et al. 2002), so the exact knowledge of bathymetry is crucial. In addition, the seabed can be subject to change due to bradyseismic motions, the gradual uplift (positive bradyseism) or descent (negative bradyseism) of part of the Earth's surface generally caused by the filling or emptying of an underground magma chamber and/or hydrothermal activity (Scafetta and Mazzarella 2021), so the morphology of the coast can vary rapidly and therefore the entire coastal ecosystem changes (Mattei et al. 2020).
Due to the continuous variation in shape of the seabed, bathymetric maps are fast becoming outdated for precise navigation; therefore, it is necessary to perform periodical depth measurement to keep them updated on a current basis (Specht et al. 2017).
In classical bathymetry, depth measurements are carried out by means of oceanographic campaigns with specific instruments, such as echo sounders mounted on special vessels for the purpose. These techniques use the principle of acoustic waves to sound the bottom and determine the depth (Amoroso and Parente 2021), but they are often constrained by inefficiency, expense and inaccessibility. In fact, in some remote and difficult areas, e.g. unseen reefs, creeks and estuaries, hydrographic surveys are so complex to carry out due to risk of life of men and loss of materials; in addition, alternative technologies like remotely operated vehicles (ROVs) and autonomous underwater vehicles (AUVs) are also very expensive for their high acquiring and maintenance costs (Ashphaq et al. 2021).
Remote sensing permits to override these limits by providing medium and high geometric resolution images that are suitable to obtain the bathymetry. In fact, this technique is a powerful tool to map the bathymetry of the seabed because of its extensive coverage of the area, low cost and repetitiveness (Jagalingam et al. 2015). The satellite-derived bathymetry (SDB) method is suitable for bathymetric survey of shallow coastal areas with clear water, approximately to the depth of 2 Secchi disc depth (Leder et al. 2020). In any case, the accuracy of SDB does not meet current International Hydrographic Organization (IHO) S-44 standards (Edition. 2008), so this approach at present can be used to plan hydrographic survey of marine areas not surveyed or areas with old data (Pe'eri et al. 2013); (Leder et al. 2019).
To increase the accuracy of SDB, high-resolution images are to prefer. In the last years, different space missions have been lunched to provide high-resolution images; among them, we find the IKONOS-2 mission (Kramer 2002); (ESA European Space Agency 2000-2021) that achieves a good combination between a high-spatial resolution (1 m in the panchromatic) and a high-spectral resolution (four multispectral bands and one panchromatic band). SDB uses optical remote sensing data for depth determination. In fact, it is based on analytical modelling of light penetration through the water column in visible and infrared bands: the solar radiation is partly absorbed by the water and partly diffused, whereby a portion is dispersed while through the water and the residual energy is backscattered, intercepted by the sensor and recorded in the remote sensing images (Leder et al. 2019). In the literature, there are several SDB methods. These methods can be grouped into two categories: empirical methods, which use direct observations of the depth of water in the study area to calibrate reflectance as a function of depth, and physical methods that use radiative transfer models (the interaction of light with matter) to derive the depth of water without in situ calibration data (Lyons 2011). One of the most important empirical methods is the band ratio method (BRM) (Stumpf et al. 2003) that is based on the degrees of absorption at different bands (with different wavelengths). Using this method, a regression line must be developed between the values of the reflectance ratios and the true depth values. However, relationship is not evermore of the linear type, because it depends on various aspects related to the interaction of sunlight with the marine environment, so the analysis of the specific situation is necessary to identify an empirical modelling of the examined area.
This article aims to analyse and test BRM applied to IKONOS-2 high-resolution images and improve the accuracy of results performing the determination of the relation between depth and band ratio.
For the application of the SDB methods, the free and open-source software Quantum GIS (QGIS) version 3.16 was used. It is a software designed to acquire, store, manipulate, analyse, manage and present spatial or geographical data. In other terms, it has the typical GIS software characteristics, allowing users to create interactive queries, analyse spatial information, modify data in maps and present the results of all these operations (Clarke 1986).
The paper is organized as follows. The "Data and methods" section illustrates the main characteristics of the study area and the dataset including IKONOS-2 images; in addition, this section also presents the BMR as proposed by Stumpf et al. and our proposal for increasing the accuracy of the results. The "Results and discussion" section introduces, compares and discusses the results obtained by traditional application of BRM as well as by its adaptation. The "Conclusion" section presents our conclusions remarking the importance of the work and suggesting the potential applications and extensions for future studies.

Study area
The study area chosen for this work concerns the Bay of Pozzuoli that it is located west of Naples, in the Tyrrhenian Sea. This area includes the famous submerged archaeological park of Baiae that is a protected marine area, known as the "Underwater Pompeii." The study area is shown in the next two figures. Particularly, Fig. 1 geolocalizes this area in the Italian territory using WGS84 ellipsoidal coordinates and equirectangular cartographic representation, while Fig. 2 supplies a RGBcoloured view of the considered zone extending between the following UTM/WGS84 -Zone 33 N plane coordinates: E 1 = 421,100 m; N 1 = 4,518,600 m; E 2 = 423,100 m; N 2 = 4,520,000 m.
The Bay of Pozzuoli is included in the sector called "Campi Flegrei" (Phlegraean Fields), a large volcanic area that is dominated by one large caldera collapse structure consisting of 24 craters and volcanic edifices; most of them lie under water (Scandone et al. 1991). Hydrothermal activity can be observed at Lucrino, Agnano and the town of Pozzuoli; there are also effusive gaseous manifestations in the Solfatara crater, the mythological home of the Roman god of fire, Vulcan (ESA). The whole area (including the emerged part and the submerged part) is subject to slow soil deformation known as bradyseism (literally slow movement of the soil) which occurs in different ways over time, leading to both the uplift and subsidence of the affected area (Passaro 2013); therefore, the sea level has changed over the ages. Pioneer geologists used the pillars of the Roman market remains in Pozzuoli as a paleotide gauge (Lyell 1830); recently, radiocarbon analysis of biological indicators on these remains had shown three 7-m relative sea-level highstands during the fifth century A.D., the early Middle Ages Fig. 1 Territorial framework of the study area related to WGS84 ellipsoidal coordinates and before the 1538 eruption of Monte Nuovo (Morhange et al. 2006). In modern times (since 1905), the techniques of geodetic levelling and in recent decades also the measurements using GPS and interferometric data, allow to evaluate in real time the variations of the soil and therefore to monitor the phenomenon of Phlegraean bradyseism (Usai et al. 1999); (Minet et al. 2012). In the 1970s and 1980s of the last century the Phlegraean area, and the inhabited area of Pozzuoli in particular, were affected by a sudden uplift of the soil which brought the latter to an overall higher level of about 3.5 m and caused numerous earthquakes, with severe damage to buildings. Later, the Campi Flegrei caldera was characterized by general subsidence for about twenty years until 2005, the year in which a period of uplift that is currently underway began. The areal distribution of the recent uplift reaches the maximum value in the area of the Port of Pozzuoli, with a radial decrease towards the edges of the caldera.
These repeated uplift and subsidence cycles indicate that the volcanic activity is still ongoing; in fact, the last eruption was in 1538 AD (Morhange 2006), but a recent study had interpreted the ground deformation, observed through satellite interferometry and GPS measurements, as the effect of the intrusion at shallow depth (3090 ± 138 m), 500 m south from the port of Pozzuoli, of 0.0042 ± 0.0002 km 3 of magma within a sill; this phenomenon interrupts about 28 years of dominant hydrothermal activity and occurs in the context of an unrest phase which began in 2005 and within a more general ground uplift that goes on since 1950 (D' Auria et al. 2015). Also, other studies ascribe uplift to repeated injections of magma at shallow depth (Troise et al. 2019); particularly, the recurrent seismic events that hit the area, especially in recent times, are interpreted as being related to repeated injections of high temperature magmatic fluids into the hydrothermal system feeding the fumaroles of Solfatara (Chiodini et al. 2017). The dynamic nature of the Phlegraean Fields confirms the necessity to continuous update 3D model not only for the emerged part but also for the submerged part of this area, so SDB and possibility to increase the accuracy of the resulting depth data are crucial.

IKONOS-2 and ENC data
The multispectral images of the IKONOS-2 satellite (as shown in Fig. 3) were used in this work for bathymetry data extraction. This satellite was launched into space in September 1999 and was the first satellite in the world (commercially available) with high geometric resolution, in fact imagery exceeding 1 m of resolution, since pixel dimensions is 82 cm in the panchromatic and 3.28 m in the multispectral. The 4 multispectral bands available are blue (445-516 nm), green (506-595 nm), red (632-698 nm) and near infrared/ NIR (757-853 nm) while panchromatic sensor acquires in the range of 450-900 nm. The images have a radiometric resolution (dynamic range) in 11 bits.
Even today, it orbits our planet at 681 km, and until 2007, IKONOS-2 was considered the industry most agile satellite with the ability to acquire images of a large portion of the earth every 3 days; the satellite has not collected data since December 2014 (Apollo Mapping 2022). The images of the study area were acquired on 2005-11-04 at 10:09 GMT. Metadata contained in a separate file added to the IKONOS-2 images provide further information, such as info regarding the sensors and orbital parameters. The main information for our dataset is shown in Table 1: Regarding bathymetric information, we extracted depth data from an Electronic Navigational Chart (ENC) produced by the Istituto Idrografico della Marina Militare (IIMM), the Italian Hydrographic Office, in scale 1:7500, identified as n° IT50083D: ENC is a digital navigational chart in vector format that complies specific standard established by the International Hydrographic Organization (IHO) and is currently used in Electronic Chart Display and Information Systems (ECDISs) to support maritime navigation (Brčić et al. 2015). As reported in Fig. 4, three vector files extracted from ENC were used concerning, respectively, to the coastline, the depth points distributed in the affected sea area and finally the contour lines relating to the depths 2 m, 5 m, 10 m, 20 m, 30 m and 50 m.

Data pre-processing
Satellite image: from digital number to reflectance As well known, a sensor transforms the reflected or emitted electromagnetic energy, which it detects, into an electrical signal that is converted into a digital value. Therefore, it is necessary to convert the raw image data (digital numbers, DNs) in radiance, performing more operations that, according to (Fleming 2001), can be summarized for IKONOS as follows.
The spectral radiance of the IKONOS-2 images can be expressed by the following relationship provided by Space Imaging Document Number SE-REF-016 n.d:  The final formula to obtain the reflectance values is: where d expresses the distance between the Earth and the Sun on the day of the acquisition, i.e. 4/11/2005 which is equal to 0.99177 astronomical units, while ϴs is the zenith angle of the sun equal to 56.89°.
Finally, the Esun values (Table 4) express the average solar spectral radiations of the band (W m −2 μm).

Glint correction
In accordance with the pre-processing workflow present in the literature (Traganos 2018;Jagaligam 2015), we applied sun glint correction. Since glint is due to the fact that sun reflects on the sea surface, this contribution distorts the values of reflectance. Whereby the sensor acquires an extra quantity which for the final evaluation misrepresents the   bathymetric models. The method developed by (Hedley et al. 2005) permits to do this correction.
Once the DN values have been transformed into reflectance values, a sea area is selected where the sun glint is not present (i.e. in deep water), determining the minimum NIR value.
A regression line is established between the NIR values (x-axis) and the values of the band whose sun glint is to be corrected (y-axis); generally, this correction is applied to the visible bands. The slope (b i ) of this line is calculated. Finally, the following formula is applied to obtain corrected value of the considered visible band: where R i is the initial value of the band of interest, b i is the slope of the regression line, R NIR is the value of NIR band and Min NIR is the minimum value of NIR band registered in the selected area.

3D bathymetric model from ENC
Starting from the bathymetric values of the ENC, we applied the linear spatial interpolation of the TIN (Triangulated Irregular Network) to obtain a continuous model of the bathymetry in the study area (Floriani 1989); Yang 2005).
Even if TIN is a vector 3D Model, QGIS software gives the rasterization of it with a cell size established by the user: considering the scale of the ENC (1: 7,500) and the geometric resolution of the multispectral images of the IKONOS-2 satellite, in this application, the dimension of the grid pitch is equal to 4 m. This value is in accordance with the formulas proposed by (Hengl 2006) for finding the right pixel size p (in metres) for the 3D model in relation to the scale factor SN of the used cartographic data: The resulting bathymetric model derived from the ENC (ENCBM) is shown in Fig. 5.

Delimitation of the zone of interest
The depth up to which SDB is effective depends on different factors, i.e. the clarity of the water, the bottom type and the method used. The linear transform proposed by (Lyzenga 1978) does not distinguish depths > 15 m and is more subject to variability across the studied atolls; the ratio transform can, in clear water, retrieve depths in 25 m of water (Stumpf et al. 2003  We noticed that the coastline present in the ENC did not coincide with that evident in the image (Fig. 4): in Bathymetric grid resulting from the ENC correspondence with the port of Pozzuoli, new infrastructures had been added, after the date of acquisition of the image, so they were present in the ENC but not in the IKONOS. Furthermore, in the image, there were boats moored at the old piers that occupied an area by changing its natural reflectance as water. For consequence, we decided to apply the Normalized Difference Water Index (NDWI) (McFeeters 1996); (Costantino et al. 2020); (Alcaras et al. 2022) to the IKONOS dataset for distinguishing two classes (water/no water) and extracting coastline in automatic way. The normalized difference between blue and NIR reflectance values allows to highlight the pixels containing water. In fact, the spectral signature of water, that is the plot of the percent reflectance values for landscape feature across any range of wavelengths, presents a reflectance peak in the blue bands (0.4-0.5 μm) and an absorbance peak in the near infrared. Subsequently, NDWI based on those two bands produces an image with lighter pixels of water and darker pixels of soil and vegetation. Therefore, we proceeded with the typical approach of supervised classification (Alcaras et al. 2020), using training sites and the maximum likelihood classifier (Bruzzone and Prieto 2001) to obtain an image where the pixels of water are distinguished from the context. Therefore, the polygonization permits to obtain the boundaries separating water from no water (such as bare soil, vegetation and ships). To establish the area with depths no more than 15 m, the contour line extracted from the ENCBM was used. Finally, a sea polygon including all water pixel within the study area presenting depths in the range (0 m, − 15 m) was used as a mask for a first clipping of the raster files to be processed.

Interaction between electromagnetic radiation and sea water
The penetration of sunlight into the sea is defined as the depth above which 90% of the diffuse-reflected radiation originates (Gordon 1975). To extract the depths of the water from satellite imagery, it is important to understand the interaction of light with the sea. The total radiance (L t ) registered by the sensor is the sum of four radiance components according to the formula: where: L p is the path radiance given by the effects of atmospheric diffusion and the contribution of areas adjacent to the water; (7) L t = L p + L s + L v + L b L s is the radiance due to the descending solar radiation arriving at the air-water interface (it gives information about qualities of the water that are close to the sea surface); L v is subsurface volumetric radiance as the part of radiance going under the surface of the water and coming into contact with the natural / inorganic constituents present; L b is the part of total radiance that arrives at the bottom of the sea giving depth information (Bukata et al. 2018;Legleiter and Roberts 2005). Figure 6 shows the components of total radiance registered by the sensor: The light is attenuated exponentially with the depth in the water column. Therefore, the maximum depth of penetration is the inverse of the absorption coefficient of the medium expressed by Beer's law (Parente and Pepe 2018). This law describes the phenomena of absorption of electromagnetic radiation. The absorbance (a) is expressed through the relationship: where t expresses the transmittance (the fraction of incident light which it transmits).
Beer's law concerns absorbance and states that this magnitude is directly proportional to the concentration of the solution: ε indicates the molar absorptivity or molar absorption coefficient, which is expressed in M −1 cm −1 , l is the optical path or the thickness of the solution and finally c expresses the concentration of the solution (Stavn 1988).

Band ratio method (BRM)
BRM for SDB compares band ratios within situ data to obtain a relation. It exploits different reflectance bands based on the intrinsic optical properties of water, as the blue band and green band. In fact, in clear waters, the blue band will have a greater peak of reflectance than the green band. While, in the coastal area, organic and inorganic elements are abundant, and therefore, the green band will have a greater peak than the blue band. These different responses provide useful information for determining bathymetric trend of the study area, so depth (Z) can be calculated using the formula: where m 1 is the constant to scale the depth ratio; n is the fixed constant for the whole area to make the logarithm positive; m 0 is the depth offset where z = 0; ρ W is the reflectance of water and λ i and λ j are two different bands. The advantages of this method are: 1. it works on all types of bottoms even when they are of a variable type; 2. it requires only two coefficients when compared to other methods; 3. it uses an easy-to-apply algorithm; 4. it penetrates better in coastal waters when compared to other methods.
Using the BRM, a regression line must be developed between the values of the reflectance ratios and the real depth values (e.g. obtained by in situ measurement). A method of determining the quality of the regression line is the coefficient of determination also called the correlation coefficient (R 2 ). When its value tends to 1, it means that there is a strong correlation between the values, while 0 means maximum uncorrelation (Stumpf et al. 2003).

BRM application
Once the multispectral imagery was processed to obtain the reflectance values and the zone of interest was outlined, the BRM was applied. The application of this technique initially involved the construction of the map resulting from the ratio between blue and green band as recommended for this method in literature (e.g. Chénier et al. 2018;Shah 2020). Afterwards, low pass filter 3 × 3 was applied to the previously obtained map to remove noise and decrease local variation.
The following steps of BRM were applied: 1. Choice of transects for training; 2. Extraction of the data of the band ratio and of the depth data obtained from the grid along the transects; 3. Plot of previously obtained values and determination of the regression line; 4. Application of the equation of the regression line; 5. Calculation of the goodness of the data.
In the first phase of the elaboration, three transects were chosen, reaching the maximum depth of 15 m. The three transects taken into consideration are reported (white lines) on map in Fig. 7.
In the second phase, using the QGIS tools, the band ratio values and the depth values obtained from the ENCBM were extracted for all pixels covered by the transects.
In the third phase, the data were plotted in an Excel sheet and the regression line was obtained.
The graph below (Fig. 8) shows an example of a linear relationship (RL) between the depth and the band ratio values (note that the measured depths are reported as positive values).
In the fourth phase of processing, the equation obtained from each regression line was applied using the QGIS tool named "raster calculator" to derive the bathymetric data.
In this way, one model containing the depth values was obtained as bathymetric grids.
In the last phase, it was necessary to evaluate the validity of the application. To do this, we considered both the Fig. 6 Representation of total radiance adaptation from (Campbell and Wynne 1996) ENC depth data (including points and contour line vertices) and the ENCBM, as resulting in the range (0 m, − 15 m): the difference between depth values of each initial dataset (ENC or ENCBM) and each raster obtained by the BRM application provided residuals that were used as indicators of the accuracy of the new produced model. Each of the two dataset used for controlling the accuracy of the BRM products presents limits and advantages: ENC depth data are more accurate but not uniformly distributed, ENCBM are affected by errors introduced with the interpolation process but numerous and regularly located. The results are shown and discussed in the "Results and discussion" section.

Proposal for increasing BRM performance
Using a linear model to interpret the relation between band ratio and depth, the classic method defined by Stumpf et al. allows to determine the bathymetry of the entire image. However, this relationship is not always linear; in fact, the propagation of light in water depends on various aspects related to the interaction of sunlight with the marine environment. This aspect is particularly evident in this study as Figs. 9, 10 and 11 show.
For consequence, we tried to approximate the plotted points using non-linear functions such as the exponential function and the 3rd degree polynomial The applications were carried out for all the transects previously shown, also in this case for a maximum depth of 15 m. The resulting graphs are reported below with an indication of the equations of the interpolating functions. Figures 9, 10 and 11 show exponential regression function (ERF), applied respectively to transects A, B and C. Fig. 9 Relationship between depth and band ratio-plotting of the points and relative exponential regression line for transect A Fig. 10 Relationship between depth and band ratio-plotting of the points and relative exponential regression line for transect B Fig. 11 Relationship between depth and band ratio-plotting of the points and relative exponential regression line for transect C Figures 12, 13 and 14 display third-order polynomial regression (TPR) to interpolate the points.
In accordance with the steps of the procedure reported in the previous section, the new equations were applied to the band ratio image using "raster calculator." Analogously to the previous applications, all the resulting maps were tested using ENC depth data and the ENCBM, in the range (0 m, − 15 m).

Results and discussion
Considering that BRM method and its adjustments (using ERF, TPR) were applied to each transect, we obtained 6 bathymetric models as reported in Figs. 15,16 and 17. In relation to the 4th step of the procedure, the statistical parameter values (i.e. minimum, maximum, average, standard deviation and root mean square error) relating Fig. 12 Relationship between depth and band ratio-plotting of the points and relative thirdorder polynomial regression for transect A Fig. 13 Relationship between depth and band ratio-plotting of the points and relative thirdorder polynomial regression for transect B Fig. 14 Relationship between depth and band ratio-plotting of the points and relative thirdorder polynomial regression for transect C to the residuals between each model and the ENC depth data as well as the ENCBM, were calculated. Those values are shown below, so as to provide synoptic frameworks that facilitate the comparison. Particularly, Table 5 shows statistics of residuals calculated for each model compared with the ENC depth data. The first column indicates the data source for the equation determination (transects A, B, C), the second column the method used (ERF, TPR), the following columns the statistic values (mean, standard deviation, RMSE, maximum, minimum), the last two the correlation (R 2 ) and the equation of the adopted model respectively.
The results reveal that the level of accuracy varies according to the transect used and the method applied. Generally, a high level of correlation between depth values and band ratio values represented by a specific equation on a transect provides an accurate model. In fact, TPR founded on the transect C presents the highest value of R 2 (0.98) as well as the lowest RMSE (1.88 m). RL never produces the bestperforming model. The ERF can be considered unstable as it presents different results that are not in agreement with the fact that high level of correlation corresponds to low value of RMSE. TPR always gives the best result also in terms of RMSE, so we conclude that this method is preferred for SDB based on the results of comparing the different models with the ENC depth data. Table 6 shows statistics of residuals calculated for each model compared with the ENCBM. The table structure is the same of the previous one (we omitted to report the correlation and the equation of the adopted model since they are the same shown in Table 5).
Despite the change in the comparison term, the results are fully confirmed: the level of accuracy varies according to the transect used as well as the method applied; the higher the level of correlation between depth values and band ratio values, the smaller the RMSE. In addition, TPR founded on the transect C confirms the best-performing method (RMSE = 1.724 m, the smallest of all). Ultimately, TPR is confirmed as the method to prefer for SDB according to the results of the comparison of the different models with the ENC points. Later, we wanted to carry out a further check; therefore, analysing the values are tabulated below (Table 7), on 67.305 points distributed over the study area: in the case of the regression line, we have at best 24.22% of points with a residual greater than 4 m (equal to the pixel size). In the other elaborations, this percentage drops sharply reaching 2.7% as regards the TPR.
Finally, we present the results of the residual analysis carried out on the bathymetric models compared with the ENCBM, this time in relation to three different classes of depth. Specifically, the residual statistics are shown in: Table 8 for the depth range (0 m, − 5 m), Table 9 for the depth range (− 5 m, − 10 m) and Table 10 for the depth range (− 10 m, − 15 m). In all cases, the good performance of TPR was confirmed. It is also evident that the results are more accurate in shallower waters.

Conclusion
BRM is a powerful tool for SDB also when applied to VHR satellite images such as IKONOS. Nevertheless, its performance can be increased analysing the correlation between the band ratio values and depth values. This study showed that each time it is necessary to analyse the specific situation and define for each single transect the mathematical model able to better approximate the correlation between the two types of data. The experiments conducted on the area of the Pozzuoli bay proved that for the three transects chosen in no case the best mathematical model was the linear one: ERF and TPR supplied higher value of R 2 and generated most suitable SDB maps.
To establish the performance of each method, two terms of comparison were chosen: ENC depth values and ENCBM, the first more accurate but not characterized by uniformly distributed points, the second affected by errors introduced with the interpolation process but including numerous and regularly located points. For both of them, TPR was able to provide the most accurate results in terms of RMSE. In fact, it produces only values below sea level and supplies the lowest RMSE, so it can be considered as the best-performing method for SDB compared with RL and ERF.
Comparing the bathymetric models with the ENCBM and analysing the residuals in relation to three different classes of depth, i.e. 0 m, − 5 m, − 5 m, − 10 m and − 10 m, − 15 m, the good performance of TPR was confirmed. The experiments remarked that the results of BRM, also in the case of our improvements introduced with TPR remain more accurate in shallower waters.