Cagniard–De Hoop method

{{Short description|Mathematical technique in seismology}}

In the mathematical modeling of seismic waves, the Cagniard–De Hoop method is a sophisticated mathematical tool for solving a large class of wave and diffusive problems in horizontally layered media. The method is based on the combination of a unilateral Laplace transformation with the real-valued and positive transform parameter and the slowness field representation. It is named after Louis Cagniard and Adrianus de Hoop; Cagniard published his method in 1939, and De Hoop published an ingenious improvement on it in 1960.{{cite journal | url=https://doi.org/10.1007%2FBF02920068 | doi=10.1007/BF02920068 | title=A modification of cagniard's method for solving seismic pulse problems | date=1960 | last1=De Hoop | first1=A. T. | journal=Applied Scientific Research, Section B | volume=8 | pages=349–356 | s2cid=121665626 }}

Initially, the Cagniard–De Hoop technique was of interest to the seismology community only. Thanks to its versatility, however, the technique has become popular in other disciplines and is nowadays widely accepted as the benchmark for the computation of wavefields in layered media. In its applications to calculating wavefields in general N-layered stratified media, the Cagniard–De Hoop technique is also known as the generalized ray theory. The complete generalized-ray theory, including the pertaining wave-matrix formalism for the layered medium with arbitrary point sources, has been developed by De Hoop (with his students) for acoustics waves,{{Cite journal |last=de Hoop |first=A. T. |year=1988 |title=Acoustic radiation from impulsive sources in a layered fluid |url=https://zbmath.org/0676.76071 |journal=Nieuw Archief voor Wiskunde |volume=6 |issue=1–2 |pages=111–129}} elastic wavesVan Der Hijden, J. H. M. T., "Propagation of Transient Elastic Waves in Stratified Anisotropic Medium," North-Holland, Amsterdam, 1987. and electromagnetic waves.{{cite journal | url=https://ieeexplore.ieee.org/document/6420895 | doi=10.1109/TAP.2013.2242835 | title=Generalized Ray Theory for Time-Domain Electromagnetic Fields in Horizontally Layered Media | date=2013 | last1=Stumpf | first1=Martin | last2=De Hoop | first2=Adrianus T. | last3=Vandenbosch | first3=Guy A. E. | journal=IEEE Transactions on Antennas and Propagation | volume=61 | issue=5 | pages=2676–2687 | bibcode=2013ITAP...61.2676S | s2cid=44047405 }}

Early applications of the Cagniard-DeHoop technique were limited to the wavefield propagation in piecewise homogeneous, loss-free layered media.{{cite journal | url=https://link.springer.com/article/10.1007%2FBF02920070 | doi=10.1007/BF02920070 | title=Radiation of pulses generated by a vertical electric dipole above a plane, non-conducting, earth | date=1960 | last1=De Hoop | first1=A. T. | last2=Frankena | first2=H. J. | journal=Applied Scientific Research, Section B | volume=8 | pages=369–377 | s2cid=18578107 }} To circumvent the limitations, a number of extensions enabling the incorporation of arbitrary dissipation and loss mechanisms{{cite journal | url=https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/96RS02191 | doi=10.1029/96RS02191 | title=The electromagnetic field emitted by a pulsed current point source above the interface of a nonperfectly conducting Earth | date=1996 | last1=Kooij | first1=B. J. | journal=Radio Science | volume=31 | issue=6 | pages=1345–1360 | bibcode=1996RaSc...31.1345K }}{{cite journal | url=https://ieeexplore.ieee.org/document/6595020?section=abstract | doi=10.1109/TAP.2013.2281376 | title=On the Limitations of the Time-Domain Impedance Boundary Condition | date=2013 | last1=Stumpf | first1=Martin | last2=Vandenbosch | first2=Guy A. E. | journal=IEEE Transactions on Antennas and Propagation | volume=61 | issue=12 | pages=6094–6099 | bibcode=2013ITAP...61.6094S | s2cid=35360524 }} and continuously-layered media{{cite journal | url=https://asa.scitation.org/doi/pdf/10.1121/1.400080 | doi=10.1121/1.400080 | title=Acoustic radiation from an impulsive point source in a continuously layered fluid—An analysis based on the Cagniard method | date=1990 | last1=De Hoop | first1=Adrianus T. | journal=The Journal of the Acoustical Society of America | volume=88 | issue=5 | pages=2376–2388 | bibcode=1990ASAJ...88.2376D }}{{Cite journal |doi=10.1111/j.1365-246X.1990.tb05684.x |doi-access=free |title=Determination of seismic wavefields in arbitrarily continuously layered media using the modified Cagniard method |date=1990 |last1=Verweij |first1=M. D. |last2=Hoop |first2=A. T. |journal=Geophysical Journal International |volume=103 |issue=3 |pages=731–754 |bibcode=1990GeoJI.103..731V }} were introduced. More recently, the Cagniard–De Hoop technique has been employed to put forward a fundamentally new time-domain integral-equation technique in computational electromagnetics, the so-called Cagniard–De Hoop Method of Moments (CdH-MoM), for time-domain modeling of wire and planar antennas.{{Cite book |last=Stumpf |first=Martin |url=https://www.wiley.com/en-us/Time+Domain+Electromagnetic+Reciprocity+in+Antenna+Modeling-p-9781119612377 |title=Time-Domain Electromagnetic Reciprocity in Antenna Modeling |date=September 2019 |publisher=Wiley-IEEE Press |isbn=978-1-119-61237-7 |language=en-us}}{{Cite book |last=S̆tumpf |first=Martin |title=Metasurface electromagnetics: The cagniard-dehoop time-domain approach |date=2022 |publisher=The Institution of Engineering and Technology |isbn=978-1-83953-613-7 |series=The ACES series on computational and numerical modelling in electrical engineering |location=London}}