Basic Geophysics FWI – Application on Different Scales

What have seismic and medical imaging in common? Very much in deed. Even though human tissue and the Earth subsurface are different in their structure, both can be analysed using acoustic waves. In medical diagnostics ultrasound waves with wavelengths in the submillimetre to centimetre range are used to examine unborn babies in their mother’s womb as well as inner organs, or they are used for cancer prevention. In Geophysics seismic waves with wavelengths in a wide range between centimetres and kilometres give insights into the Earth’s structures and help to understand tectonic processes and prospect for natural resources. The analysis of the full wavefield by Full Waveform Inversion allows to model both structures in detail to give valuable insights in high resolution. Hello and welcome. In this video you will learn about the application of the Full Waveform Inversion method to such different fields like ultrasound imaging for detecting breast cancer and seismic imaging of gas hydrates in the Black Sea. These are applications on very different scales and you will learn what this means regarding wavelength, frequency, wave velocity, and the quantity of propagated wavelengths. The two applications are exemplary studies by Fabian Kühn and Laura Gassner, who did their master’s and PhD-thesis on those topics. Medical ultrasound uses frequencies of up to 18 MHz. That is around a factor of thousand higher than sound waves, which can be heard by humans from around 16 Hz to 16 kHz. Longitudinal seismic P-waves however, travel through the Earth with frequencies up to a factor of one thousand lower than audible signals. They reach from millihertz to some kilohertz. Frequency is related to wavelength lambda, which is equal to propagation velocity v divided by frequency f. With typical values for v, wavelengths from micrometres to hundreds of kilometres result. This determines the spatial dimensions of the study object that can be analysed accurately. The number of propagated wavelengths N_lambda is the ratio of the characteristic size of an object L to the wavelength lambda used. In heterogeneous media it is a measure for the complexity of the wavefield. A high number of propagated wavelengths increases the computational costs and the risk of non-convergence of the Full Wavefrom Inversion. Hence, you see that scale matters. Full Waveform Inversion – FWI for short – can be applied on different scales and allows the investigation of different structures such as human tissue or the Earth’s subsurface. The spatial resolution of such analyses depends on the wavelength used and must be adapted to the objects of interest. The spatial resolution of FWI is around half the analysed wavelength. Two examples will demonstrate this: first, from medical imaging. Breast cancer is one of the most common types of cancer. The typical approach for breast cancer screening today is X-Ray mammography. An alternative approach is ultrasound tomography. It has the advantage that it allows the non-invasive examination of the undeformed breast tissue, which may provide more reproducible results than conventional mammography, especially for dense tissues. A prototype for automatised 3D ultrasound tomography was developed at the KIT Institute for Data Processing and Electronics. The devise consists of a water-filled ellipsoidal half-sphere embedded in a patient’s bed. It is equipped with 157 array systems, each of which consists of nine receiver and four source elements. Those are so-called transducers that transform electrical to mechanical signals and vice versa. Fabian Kühn analysed the performance of FWI on 2D synthetic data. For this purpose, he forward calculated the wavefield through a realistic visco-acoustic model of a human breast. Visco-acoustic modelling allows to calculate wave propagation including effects of anelastic attenuation. Around the model 16 equally spaced sources and 256 receivers were placed. He inverted those synthetic data using FWI to reconstruct the original model. This animation shows the wave propagation in the numerical breast model. First, you see the direct wave, which is refracted and reflected at the interfaces of skin and tissue. Finally, you see the head wave as a bend wave at the bottom of the model. The model has a diameter of 15 centimetres and in medical ultrasound typically frequencies of 1 to 3 MHz are used. With a velocity of around 1500 metres per second in human tissue this corresponds to wavelengths of around 0.5 to 1.5 millimetres. This results in propagated wavelengths of 100 to 300, which is relatively high and indicates a fairly complex model space. Here, I show you the final reconstructed model for ultrasound velocities by Fabian Kühn. You clearly see that the reconstruction on the right matches the input model on the left quite well. For comparison you see the tomography result in the central figure, which only uses traveltime data and therefore gives a rather smooth picture of the structure. Inversion results for density and seismic attenuation also provide high quality reconstructions but they are not shown here. Fabian Kühn found out that medical ultrasound imaging using 2D visco-acoustic FWI is able to significantly exceed the resolution limit of standard raybased methods. And as computational power continues to increase rapidly, FWI is expected to become more important for clinical applications in the future. This could help to detect small cancers earlier and thus improve the survival rate of patients. From this synthetic and small-scale application of FWI, I come to the second example with data observed at a larger study object – the Earth. At depths of around 400 metres beneath the ocean floor, gas hydrates are assumed to be stored in fine grained layers of organic origin. Those gas hydrates are a potential source for future energy supply and geophysicist Laura Gassner applied FWI on seismic data to explore the subsurface for those natural resources. Gas hydrates are crystalline solids, which contain gases in a hydrogen structure. Especially methane hydrates have a great potential to enhance the global reserves of fossil fuels. Typically, they are stored in marine environments. Laura Gassner analysed data from the Black Sea, which has extensive shelf regions. Their edges are promising for finding gas hydrates. At the bottom of the hosting sedimentary layers so-called bottom-simulating reflectors – BSR for short – exist. They result from a contrast in the P-wave velocity of sediments hosting hydrate compared to water- or gas-saturated sediments. To find those reflectors Laura Gassner analysed seismic signals recorded in the Black Sea using Full Waveform Inversion. Seismic waves were emitted by air guns with a shot spacing of about 8 to 10 metres. The signals were recorded by ocean bottom seismometers. In the figure they are indicated as white circles. Lines show the track of the vessel. At first, Laura Gassner performed synthetic tests showing that FWI can robustly reconstruct P-wave velocity distributions similar to an actual BSR. S-wave velocities and densities however, could not be recovered because of the little sensitivity of FWI to these parameters in the discussed setting. FWI iteratively inverts observed seismic data based on an initial model. Here, the initial model is taken from an interpolated ray-based traveltime tomography and shows a smoothed 2D structure of increasing P-wave velocities with depth. Seismometers again are indicated as white circles. With an average P-wave velocity of 1800 m/s and seismic waveforms recorded with frequencies higher than 3 Hz a wavelength lambda of up to 600 metres results. With that and an effective horizontal extension of around 12 kilometres, the number of propagated wavelengths can be calculated. Hence, N_lambda equals at least 20, indicating a rather low complexity of the modelled wavefield in a heterogeneous medium. Using this starting model, seismic P-waves could be inverted using FWI for the detailed subsurface structure shown here. Clearly visible is the gas zone with decreased P-wave velocities just below the BSR. Anyhow, around the ocean bottom seismometers circular artefacts occur. Laura Gassner found out that these are due to the 3D distribution of the ocean bottom seismometers that cannot be accounted for by the 2D model. Since those waves do not map the BSR, she applied a time window to suppress them and primary reflections as well. She showed that this is very beneficial for the inversion, because strong artefacts near the ocean bottom seismometers are reduced and the BSR can be clearly resolved. Another example, that FWI can improve the spatial resolution obtained by traveltime tomography. In this video you learned that FWI can be applied on various fields, independent of the spatial scale. In both examples shown a heterogeneous structure could be recovered in great detail: with synthetic data for medical imaging and with field data for seismic imaging. The propagated wavelength allows estimating the complexity of the modelled wavefield in heterogeneous media. In our examples, the propagated wavelength varies between 20 for the seismic application and up to 300 for medical imaging, showing the high wavefield complexity and the high computational power necessary. Both applications of FWI improve the resolution compared to conventional tomography and thus show the potential of inverting full waveforms for many different applications. Future studies may include the inversion of even lower frequency waves radiated by earthquakes as well as higher frequencies in non-destructive testing.

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