The Density-Depth Model by Spectral Fractal Dimension Index

Important note: we use only cross-platform Open Source Software and Open Datasets for our publications. You can free download the used software and data.

There are a lot of ways to estimate fractal dimension but we need someone which is robust and has a well-known relation to density. Spectral density estimator is robust because it's based on power spectral density and we know the relation between it and density too (Miranda et al., 2015) For some reasons we prefer to analyze discrete spatial spectrum produced by a set of band-pass filters and for this case this estimator is easy to define as a standard deviation on the filtered rasters.

Also we use scale factor between wavelength and depth which is equal to 0.707. This one is well-known as some magic constant (Sokolov, 2009):

The ratio between the source depth and characteristic anomaly size should be (very approximately) 1 : 3.

It's not strict enough definition because "characteristic anomaly size" could be defined by very different ways. For circular anomaly the anomaly size is equal to anomaly diameter and in this case depth/diameter ~= 1/3 or depth ~= diameter/3 = (2/3) * radius ~= 0.67*radius. Actually, this approximation is very good because (as you could check in my previous articles) the exact value is 0.707 and the error is only 5% (0.67/0.707 ~= 0.95)The exact scale factor is applicable only for one wavelength but not for wide waveband and it's the main point for this approach. We assume the anomaly radius is equal to wavelength but it can be also equated to the anomaly diameter (Sokolov, 2009). Fortunately, this duality is not a problem because as we checked before (see my previous Linkedin articles) that depends only of the data processing technique.

These two limitations should be satisfied together to get an accurate estimation:

1. Study area should be small enough to be geological homogeneous. See below how it works for a region of Antarctica which is geological homogeneous.
2. Study area should be large enough for deep estimation because maximal depth proportional to wavelength with scale factor 0.707 (see my previous articles if you need to know what is this coefficient). For example wavelength=50 km corresponds to depth=35 km. Hereafter we will illustrate the limitation for an area in the Gulf of Patras region.

We will consider the same Antarctica region below as in my previous article:

Spectral Coherence between Gravity and Bathymetry Grids

The region is geological homogeneous up to ~50 km wavelength (area size 100 km x 100 km):

The produced density-depth model should be correct up to ~50 km wavelength (~35 km depth):

The seafloor level is 3.9km but by technical reasons we have the model values from about 4.5km depth for bathymetry based calculation and from about 0.5km for free-air gravity anomalies based calculation (see attached Python3 notebook for details). These are the commonly used densities of crust and water: 2700 kg/m³ and 1035 kg/m³, respectively. And for the Pacific Ocean the crust density should be larger. On the plot above we have seafloor density in range 2573 ... 2944 kg/m³ and it's reasonable estimation for this area. (Note: I have only Russian language papers to approve it. Maybe do you know some international papers about?) The density estimation by free-air gravity anomalies is wrong due to upward continuation of gravity from the seafloor to the sea surface. Here we need to have a straight gravity values instead of the incorrect free air gravity anomalies. (Note: to be honest, that's the wrong way to define a gravity anomalies on a sea surface by upward continuation instead of using the truth satellite altimetry based sea surface gravity and in addition truth bathymetry based sea floor gravity. That's works on land but water is not the same to an air! It's strange that we have to talk about it).

In theory, the upward continuation effect is the same to the picture above:

Amplitude response of upward and downward continuation operators with respect to wavelength for certain heights (z) of continuation (Ravat, 2007).

The Gulf of Patras is an area with very complicated tectonic where are a lot of faults and thrust faults.

The Density-Depth Model by Spectral Fractal Dimension Index for the Gulf of Patras area calculated on Sandwell and Smith Gravity Anomaly grid at 1 arc-minute resolution and GEBCO_2019 bathymetry grid at 15 arc-second resolution.

The region is geological homogeneous up to ~5 km wavelength (area size 10 km x 10 km):

The produced density-depth model should be correct up to ~5 km wavelength (~3.5 km depth):

The bathymetry resolution is better, and it's produce near-seafloor difference between the models. Also, the gravity-based model shifted a little because of the upward continuation of the gravity (see Sandwell and Smith model internals).

The models shift from the upward continuation is easy to calculate as 0.5 km average seafloor level with depth to wavelength scale factor 0.707 and so the average shift is about 0.35 km. The data grids resolution difference produces the most of the models difference while the upward continuation effect is small because of average seafloor depth level is much smaller than the gravity grid spatial resolution.

Gravity model is great to estimate common seafloor density while bathymetry model is better for exploration of faults and high density rock blocks from deep layers. Our estimation of the seafloor density (2587 kg/m³) looks good and it's close to common value 2700 kg/m³.

See the complete processing code and the source data preparation procedure in these Jupyter notebooks (Python3):

https://github.com/mobigroup/gis-snippets/blob/master/Antarctica/GEBCO_Freeair_fractal.Antarctica.ipynb

https://github.com/mobigroup/gis-snippets/blob/master/Patraikos/GEBCO_Freeair_fractal.Patraikos.ipynb

NB. Pictures for the Gulf of Patras replaced by new ones for changed subarea to illustrate more complex behavior of the Fractal Dimention Index. See the original pictures in the last notebook.

Berke, J., 2007. MEASURING OF SPECTRAL FRACTAL DIMENSION. New Mathematics and Natural Computation (NMNC) 03, 409–418. https://doi.org/10.1142/S1793005707000872

Caratori Tontini, F., Graziano, F., Cocchi, L., Carmisciano, C., Stefanelli, P., 2007. Determining the optimal Bouguer density for a gravity data set: implications for the isostatic setting of the Mediterranean Sea. Geophysical Journal International 169, 380–388. https://doi.org/10.1111/j.1365-246X.2007.03340.x

Miranda, S.A., Herrada, A.H., Pacino, M.C., Miranda, S.A., Herrada, A.H., Pacino, M.C., 2015. Fractalness of land gravity data and residual isostatic anomalies map of Argentina, Chile and western Uruguay. Geofísica internacional 54, 315–322.

Ravat, D., 2007. Upward And Downward Continuation. pp. 974–976. https://doi.org/10.1007/978-1-4020-4423-6_311

Sokolov, S.Yu., 2009. Tectonic elements of the Arctic region inferred from small-scale geophysical fields. Geotecton. 43, 18–33. https://doi.org/10.1134/S0016852109010026