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/***********************************************/
/**
* @file kernel.h
*
* @brief Isotropic harmonic integral kernels.
* E.g. the Abel-Poisson kernel for the continuation of the potential.
* Can also be used as Radial Basis Functions for splines and wavelets.
*
* @author Torsten Mayer-Guerr
* @date 2003-09-03
*
*/
/***********************************************/
#ifndef __GROOPS_KERNEL__
#define __GROOPS_KERNEL__
// Latex documentation
#ifdef DOCSTRING_Kernel
static const char *docstringKernel = R"(
\section{Kernel}\label{kernelType}
Kernel defines harmonic isotropic integral kernels $K$.
\begin{equation}
T(P) = \frac{1}{4\pi}\int_\Omega K(P,Q)\cdot f(Q)\,d\Omega(Q),
\end{equation}
where $T$ is the (disturbance)potential and $f$ is a functional on the spherical surface~$\Omega$.
The Kernel can be exapanded into a series of (fully normalized) legendre polynomials
\begin{equation}\label{eq.kernel}
K(\cos\psi,r,R) = \sum_n \left(\frac{R}{r}\right)^{n+1}
k_n\sqrt{2n+1}\bar{P}_n(\cos\psi).
\end{equation}
On the one hand the kernel defines the type of the functionals~$f$ that are measured
or have to be computed, e.g. gravity anomalies given by the Stokes-kernel.
On the other hand the kernel functions can be used as basis functions to represent
the gravity field, e.g. as spline functions or wavelets.
)";
#endif
/***********************************************/
#include "base/import.h"
#include "config/config.h"
/**
* @defgroup kernelGroup Kernel
* @brief Isotropic harmonic integral kernels.
* @ingroup classesGroup
* The interface is given by @ref Kernel. */
/// @{
/***** TYPES ***********************************/
class Kernel;
typedef std::shared_ptr<Kernel> KernelPtr;
class GravityfieldBase;
/***** CLASS ***********************************/
/** @brief Isotropic harmonic integral kernels.
* E.g. the Abel-Poisson kernel for the continuation of the potential.
* Can also be used as Radial Basis Functions for splines and wavelets.
* An Instance of this class can be created by @ref readConfig. */
class Kernel
{
public:
/// Destructor.
virtual ~Kernel() {}
/** @brief Function value of the Kernel.
* @param p Computational point.
* @param q Source point. */
virtual Double kernel(const Vector3d &p, const Vector3d &q) const;
/** @brief Radial Derivative.
* @f$ \frac{\partial K}{\partial r} @f$.
* Applied to the computational point @a p.
* @param p Computational point.
* @param q Source point. */
virtual Double radialDerivative(const Vector3d &p, const Vector3d &q) const;
/** @brief Gradient of the Kernel.
* @f$ \nabla K @f$.
* Applied to the computational point @a p.
* @param p Computational point.
* @param q Source point. */
virtual Vector3d gradient(const Vector3d &p, const Vector3d &q) const;
/** @brief Gradient of the gradient (Tensor).
* @f$ \nabla \nabla K @f$.
* Applied to the computational point @a p.
* @param p Computational point.
* @param q Source point. */
virtual Tensor3d gradientGradient(const Vector3d &p, const Vector3d &q) const;
/** @brief Apply inverse kernel to a given Kernel.
* Convolution of Kernel of this class with the passed @a kernel.
* Example: Is this class the Stokes kernel, the gravity anomalies of the other kernel is returned:
* @f[ -\frac{\partial K}{\partial r}-\frac{2K}{r} @f]
* @param kernel This will be convoluted.
* @param p Computational point.
* @param q Source point. */
virtual Double inverseKernel(const Vector3d &p, const Vector3d &q, const Kernel &kernel) const;
/** @brief Apply inverse kernel to a gravity field.
* Example: Is this class the stokes kernel,
* gravity anomalies are computed from the given gravity field:
* @f[ -\frac{\partial T}{\partial r}-\frac{2T}{r} @f]
* @param time For time variable gravity fields.
* @param p Computational point.
* @param field Apply kernel to this field. */
virtual Double inverseKernel(const Time &time, const Vector3d &p, const GravityfieldBase &field) const;
/** @brief Legendre coefficients.
* The Kernel can be represented by a series of LegendrePolynomial:
* @f[ K(\cos\psi,r,R) = \sum_n (R/r)^{n+1} k_n \sqrt{2n+1}P_n(\cos\psi) @f]
* This function returns the inverse coefficients @f$ k_n @f$.
* Example: The Stokes kernel gives @f$ r/(n-1) @f$.
* @param p Computational point.
* @param degree maximum degree.
* @return vector with size @a degree+1. */
virtual Vector coefficients(const Vector3d &p, UInt degree) const=0;
/** @brief Legendre coefficients of the inverse Kernel.
* The inverse Kernel can be represented by a series of LegendrePolynomial:
* @f[ K^(\cos\psi,r,R) = \sum_n (R/r)^{n+1} 1/k_n \sqrt{2n+1}P_n(\cos\psi) @f]
* This function returns the inverse coefficients @f$ 1/k_n @f$.
* Example: The Stokes kernel gives @f$ (n-1)/r @f$.
* @param p Computational point.
* @param degree maximum degree.
* @param interior functional is below masses.
* @return vector with size @a degree+1. */
virtual Vector inverseCoefficients(const Vector3d &p, UInt degree, Bool interior=FALSE) const=0;
/** @brief Minimum for bandlimited kernels.
* Otherwise 0 is returned. */
virtual UInt minDegree() const {return 0;}
/** @brief Maximum degree for bandlimited kernels.
* Otherwise INFINITYDEGREE is returned. */
virtual UInt maxDegree() const {return INFINITYDEGREE;}
/** @brief creates an derived instance of this class. */
static KernelPtr create(Config &config, const std::string &name);
protected:
Double kernel (Vector3d const &p, Vector3d const &q, const Vector &kn) const;
Double radialDerivative(Vector3d const &p, Vector3d const &q, const Vector &kn) const;
Vector3d gradient (Vector3d const &p, Vector3d const &q, const Vector &kn) const;
Tensor3d gradientGradient(Vector3d const &p, Vector3d const &q, const Vector &kn) const;
Vector computeFactors (Double r, Double R, const Vector &kn) const;
Vector computeFactorsRadialDerivative (Double r, Double R, const Vector &kn) const;
Vector computeFactorsRadialDerivative2nd(Double r, Double R, const Vector &kn) const;
};
/***** FUNCTIONS *******************************/
/** @brief Creates an instance of the class Kernel.
* Search for a node with @a name in the Config node.
* if @a name is not found the function returns FALSE and @a kernel is untouched.
* @param config The config node which includes the node with the options for this class
* @param name Tag name in the config.
* @param[out] kernel Created class.
* @param mustSet If is MUSTSET and @a name is not found, this function throws an exception instead of returning with FALSE.
* @param defaultValue Ignored at the moment.
* @param annotation Description of the function of this class.
* @relates Kernel */
template<> Bool readConfig(Config &config, const std::string &name, KernelPtr &kernel, Config::Appearance mustSet, const std::string &defaultValue, const std::string &annotation);
/// @}
/***********************************************/
#endif /* __GROOPS__ */
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