stfinvfdleastsquares.h 9.95 KB
 thomas.forbriger committed Oct 23, 2015 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 /*! \file stfinvfdleastsquares.h * \brief least squares in the frequency domain (prototypes) * * ---------------------------------------------------------------------------- * * \author Thomas Forbriger * \date 06/05/2011 * * least squares in the frequency domain (prototypes) * * Copyright (c) 2011 by Thomas Forbriger (BFO Schiltach) * * ---- * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA * ---- * * * REVISIONS and CHANGES * - 06/05/2011 V1.0 Thomas Forbriger * - 30/09/2011 V1.1 implemented handling of additional time series pairs * - 04/10/2011 V1.2 renamed engine  thomas.forbriger committed Oct 23, 2015 34  * - 14/10/2015 V1.3 new end-user usage functions  thomas.forbriger committed Oct 23, 2015 35 36 37 38 39 40 41 42  * * ============================================================================ */ // include guard #ifndef STFINV_STFINVFDLEASTSQUARES_H_VERSION #define STFINV_STFINVFDLEASTSQUARES_H_VERSION \  thomas.forbriger committed Oct 23, 2015 43  "STFINV_STFINVFDLEASTSQUARES_H V1.3"  thomas.forbriger committed Oct 23, 2015 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249  #include namespace stfinv { /*! \brief Fourier domain least squares engine * \ingroup engines * * \par Concept behind this engine * If * - \f$d_{lk}\f$ is the Fourier coefficient of recorded data at Frequency * \f$f_l\f$ and receiver \f$k\f$ at offset \f$r_k\f$, * - \f$s_{lk}\f$ is the Fourier coefficient of the corresponding * synthetics and * - \f$q_l\f$ is that of the sought source tim function, * . * then this engine will minimize the objective function * \f[ * E=\sum\limits_{l,k}\left|w_{lk}\, * \left(d_{lk}-s_{lk}q_l\right) * \right|^2+\sum\limits_{l}\lambda^2\left|q_l\right|^2 * =\chi^2+\psi^2 * \f] * with respect to the real part \f$q_l^\prime\f$ and the * imaginary part \f$q_l^{\prime\prime}\f$ of * \f[ * q_l=q_l^\prime+i\,q_l^{\prime\prime}. * \f] * In the above expression * \f[ * \chi^2=\sum\limits_{l,k}\left|w_{lk}\, * \left(d_{lk}-s_{lk}q_l\right) * \right|^2 * \f] * is the data misfit with weights \f$w_{lk}\f$ and * \f[ * \psi^2=\sum\limits_{l}\lambda^2\left|q_l\right|^2 * \f] * is used for regularization and will introduce a water-level in the * deconvolution. * \f$\lambda\f$ will balance both contributions. * The conditions * \f[ * \frac{\partial E}{\partial q_l^\prime}\stackrel{!}{=}0 * \quad\wedge\quad * \frac{\partial E}{\partial q_l^{\prime\prime}}\stackrel{!}{=}0 * \f] * result in (Forbriger, 2001, appendix A.3) * \f[ * q_l=\frac{ * \eta^2\sum\limits_{k}f_k^2\,s_{kl}^\ast\,d_{kl} * }{ * \lambda^2+\eta^2\sum\limits_{k}f_k^2\,s_{kl}^\ast\,s_{kl} * } * \quad\forall\, l * \f] * where * \f[ * w_{lk}=\eta\,f_k * \f] * and \f$f_k\f$ is a receiver specific weighting factor. * Now \f$\eta\f$ and \f$\lambda\f$ have to be used to balance the * regularization. * We aim to specify a waterlevel as a fraction of synthetic data energy. * * \par Setting up the waterlevel * The misfit equals one if the scaled energy of the residual * \f$d_{lk}-s_{lk}q_l\f$ equals the scaled energy of the synthetics * \f$s_{lk}\f$ and * \f[ * \eta^2=\frac{1}{\sum\limits_k f_k^2\sum\limits_l \left|s_{lk}\right|^2} * \f] * is the reciprocal of the scaled energy of the synthetics. * If we then choose * \f[ * \frac{\lambda^2}{\eta^2}=\frac{\epsilon^2}{N\eta^2}= * \frac{\epsilon^2}{N}\sum\limits_k f_k^2\sum\limits_{l=0}^{N-1} * \left|s_{lk}\right|^2 * \f] * where \f$N\f$ is the number of frequencies, then \f$\epsilon^2\f$ * will specify a waterlevel as a fraction of the scaled energy of the * synthetics. * * \par Using Parceval's Theorem to calculate signal energy * Parceval's Theorem for a signal \f$a(t)\f$ and its Fourier transform * \f$\tilde{a}(\omega)\f$ is * \f[ * \int\limits_{-\infty}^{+\infty}\bigl|a(t)\bigr|^2\,\textrm{d} t= * \int\limits_{-\infty}^{+\infty}\bigl|\tilde{a}(\omega)\bigr|^2\, * \frac{\textrm{d} \omega}{2\pi}. * \f] * If \f$S_{jk}\f$ are the time series samples corresponding to the Fourier * coefficients \f$\tilde{s}_{lk}\f$ and \f$\Delta t\f$ is the sampling * interval then * \f[ * \sum\limits_{k=0}^{M-1}\left|S_{jk}\right|^2\,\Delta t= * \sum\limits_{l=0}^{M-1}\left|\tilde{s}_{lk}\right|^2\,\frac{1}{M\,\Delta t}, * \f] * where \f$M=2N\f$ is the number of samples in the time series. * In the above calculation the energy sum only uses the positive * frequencies and * \f[ * \sum\limits_k f_k^2\sum\limits_{l=0}^{N-1}\left|\tilde{s}_{lk}\right|^2 * = * N\,(\Delta t)^2\, * \sum\limits_k f_k^2 * \sum\limits_{j=0}^{2N-1}\left|S_{jk}\right|^2. * \f] * Fourier coefficients \f$s_{lk}\f$ calculated by the * stfinv::STFFourierDomainEngine are not scaled (see documentation of * libfourierxx and libfftw3), such that * \f[ * \Delta t\,s_{lk}=\tilde{s}_{lk} * \f] * (both, \f$s_{lk}\f$ and \f$\tilde{s}_{lk}\f$ are Fourier coefficients). * Consequently * \f[ * \sum\limits_k f_k^2\sum\limits_{l=0}^{N-1}\left|s_{lk}\right|^2 * = * N\, * \sum\limits_k f_k^2 * \sum\limits_{j=0}^{2N-1}\left|S_{jk}\right|^2. * \f] * * \par Final calculation recipe * The solution to our problem is * \f[ * q_l=\frac{ * \sum\limits_{k}f_k^2\,s_{lk}^\ast\,d_{lk} * }{ * \epsilon^2\,\sum\limits_k f_k^2 * \sum\limits_{j=0}^{2N-1}\left|S_{jk}\right|^2 * +\sum\limits_{k}f_k^2\,s_{lk}^\ast\,s_{lk} * } * \quad\forall\, l, * \f] * where * \f[ * \sum\limits_{j=0}^{2N-1}\left|S_{jk}\right|^2 * \f] * is the sum of the squared sample values \f$S_{jk}\f$ of the synthetic * time series for receiver \f$k\f$, \f$f_k\f$ are the scaling factors * provided by stfinv::STFBaseEngine::weight(), and \f$\epsilon^2\f$ * is the water level parameter passed to STFEngineFDLeastSquares. * * \par References * Forbriger, T., 2001. Inversion flachseismischer Wellenfeldspektern. * PhD thesis, University of Stuttgart. * http://elib.uni-stuttgart.de/opus/volltexte/2001/861/ * * \par Why was it renamed? * This engine was called 'blind deconvolution' engine. * It was renamed to fourier domain least squares engine in October 2011. * * This engine understands the recorded data as being the synthetic data * being convoled with the STF (source correction filter) * and having some noise * added. * The true impulse response of the subsurface could be obtained by * deconvolution of the recorded data with the STF. * Neither the impulse response of the subsurface nor the STF are known. * For this reason the STF has to be found by minimizing an objective * function. * For this reason the it is called 'blind' deconvolution. * * However, the approach of libstfinv is to convolved the synthetics with * the STF to reducde the misfit to the recorded data, not to deconvolve the * recorded data. * For this reason, maybe, we should better call the approach 'blind * convolution'. * With respect to image processing techniques, all engines in libstfinv do * some kind of 'blind convolution', not only the blind deconvolution * engine. * Consequently this engine should better be called 'least-squares engine'. */ class STFEngineFDLeastSquares: public stfinv::STFFourierDomainEngine { public: //! \brief typedef to refer to base class typedef stfinv::STFFourierDomainEngine Tbase; //! \brief ID used to select this engine static const char* const ID; //! \brief short description of this engine static const char* const description; /*! \brief Constructor. */ STFEngineFDLeastSquares(const stfinv::Tvectoroftriples& triples, const stfinv::Waveform& stf, const std::string& parameters) : Tbase(triples, stf, parameters), Mwaterlevel(1.e-3) { this->initialize(); } /*! \brief Constructor. */ STFEngineFDLeastSquares(const stfinv::Tvectoroftriples& triples, const stfinv::Waveform& stf, const stfinv::Tvectorofpairs& pairs, const std::string& parameters) : Tbase(triples, stf, pairs, parameters), Mwaterlevel(1.e-3) { this->initialize(); } //! \brief abstract base requires virtual destructor virtual ~STFEngineFDLeastSquares() { } //! \brief Start engine virtual void exec(); //! \brief print online help virtual void help(std::ostream& os=std::cout) const; //! \brief print online help static void classhelp(std::ostream& os=std::cout);  thomas.forbriger committed Oct 23, 2015 250 251 252 253  //! \brief print detailed description virtual void usage(std::ostream& os=std::cout) const; //! \brief print detailed description static void classusage(std::ostream& os=std::cout);  thomas.forbriger committed Oct 23, 2015 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270  //! \brief return name of engine virtual const char* name() const; private: //! \brief initialize work space void initialize(); // member data private: //! \brief waterlevel double Mwaterlevel; }; // class STFEngineFDLeastSquares } // namespace stfinv #endif // STFINV_STFINVFDLEASTSQUARES_H_VERSION (includeguard) /* ----- END OF stfinvfdleastsquares.h ----- */