simpeg/SimPEG/Examples/DC.py

from SimPEG import *



class DCData(Data.BaseData):
    """
        **DCData**

        Geophysical DC resistivity data.

    """

    def __init__(self, mesh, model, **kwargs):
        problem.BaseProblem.__init__(self, mesh, model)
        self.mesh.setCellGradBC('neumann')
        Utils.setKwargs(self, **kwargs)

    def reshapeFields(self, u):
        if len(u.shape) == 1:
            u = u.reshape([-1, self.RHS.shape[1]], order='F')
        return u

    def dpred(self, m, u=None):
        """
            Predicted data.

            .. math::
                d_\\text{pred} = Pu(m)
        """
        if u is None:
            u = self.field(m)

        u = self.reshapeFields(u)

        return Utils.mkvc(self.P*u)



class DCProblem(Problem.BaseProblem):
    """
        **DCProblem**

        Geophysical DC resistivity problem.

    """

    dataPair = DCData

    def __init__(self, mesh, model, **kwargs):
        problem.BaseProblem.__init__(self, mesh, model)
        self.mesh.setCellGradBC('neumann')
        Utils.setKwargs(self, **kwargs)


    def createMatrix(self, m):
        """
            Makes the matrix A(m) for the DC resistivity problem.

            :param numpy.array m: model
            :rtype: scipy.csc_matrix
            :return: A(m)

            .. math::
                c(m,u) = A(m)u - q = G\\text{sdiag}(M(mT(m)))Du - q = 0

            Where M() is the mass matrix and mT is the model transform.
        """
        D = self.mesh.faceDiv
        G = self.mesh.cellGrad
        sigma = self.model.transform(m)
        Msig = self.mesh.getFaceMass(sigma)
        A = D*Msig*G
        return A.tocsc()

    def field(self, m):
        A = self.createMatrix(m)
        solve = Solver(A)
        phi = solve.solve(self.RHS)
        return Utils.mkvc(phi)

    def J(self, m, v, u=None):
        """
            :param numpy.array m: model
            :param numpy.array v: vector to multiply
            :param numpy.array u: fields
            :rtype: numpy.array
            :return: Jv

            .. math::
                c(m,u) = A(m)u - q = G\\text{sdiag}(M(mT(m)))Du - q = 0

                \\nabla_u (A(m)u - q) = A(m)

                \\nabla_m (A(m)u - q) = G\\text{sdiag}(Du)\\nabla_m(M(mT(m)))

            Where M() is the mass matrix and mT is the model transform.

            .. math::
                J = - P \left( \\nabla_u c(m, u) \\right)^{-1} \\nabla_m c(m, u)

                J(v) = - P ( A(m)^{-1} ( G\\text{sdiag}(Du)\\nabla_m(M(mT(m))) v ) )
        """
        if u is None:
            u = self.field(m)

        u = self.reshapeFields(u)

        P = self.P
        D = self.mesh.faceDiv
        G = self.mesh.cellGrad
        A = self.createMatrix(m)
        Av_dm = self.mesh.getFaceMassDeriv()
        mT_dm = self.model.transformDeriv(m)

        dCdu = A

        dCdm = np.empty_like(u)
        for i, ui in enumerate(u.T):  # loop over each column
            dCdm[:, i] = D * ( Utils.sdiag( G * ui ) * ( Av_dm * ( mT_dm * v ) ) )

        solve = Solver(dCdu)
        Jv = - P * solve.solve(dCdm)
        return Utils.mkvc(Jv)

    def Jt(self, m, v, u=None):
        """Takes data, turns it into a model..ish"""

        if u is None:
            u = self.field(m)

        u = self.reshapeFields(u)
        v = self.reshapeFields(v)

        P = self.P
        D = self.mesh.faceDiv
        G = self.mesh.cellGrad
        A = self.createMatrix(m)
        Av_dm = self.mesh.getFaceMassDeriv()
        mT_dm = self.model.transformDeriv(m)

        dCdu = A.T
        solve = Solver(dCdu)

        w = solve.solve(P.T*v)

        Jtv = 0
        for i, ui in enumerate(u.T):  # loop over each column
            Jtv += Utils.sdiag( G * ui ) * ( D.T * w[:,i] )

        Jtv = - mT_dm.T * ( Av_dm.T * Jtv )
        return Jtv



def genTxRxmat(nelec, spacelec, surfloc, elecini, mesh):
    """ Generate projection matrix (Q) and """
    elecend = 0.5+spacelec*(nelec-1)
    elecLocR = np.linspace(elecini, elecend, nelec)
    elecLocT = elecLocR+1
    nrx = nelec-1
    ntx = nelec-1
    q = np.zeros((mesh.nC, ntx))
    Q = np.zeros((mesh.nC, nrx))

    for i in range(nrx):

        rxind1 = np.argwhere((mesh.gridCC[:,0]==surfloc) & (mesh.gridCC[:,1]==elecLocR[i]))
        rxind2 = np.argwhere((mesh.gridCC[:,0]==surfloc) & (mesh.gridCC[:,1]==elecLocR[i+1]))

        txind1 = np.argwhere((mesh.gridCC[:,0]==surfloc) & (mesh.gridCC[:,1]==elecLocT[i]))
        txind2 = np.argwhere((mesh.gridCC[:,0]==surfloc) & (mesh.gridCC[:,1]==elecLocT[i+1]))

        q[txind1,i] = 1
        q[txind2,i] = -1
        Q[rxind1,i] = 1
        Q[rxind2,i] = -1

    Q = sp.csr_matrix(Q)
    rxmidLoc = (elecLocR[0:nelec-1]+elecLocR[1:nelec])*0.5
    return q, Q, rxmidLoc


if __name__ == '__main__':
    import matplotlib.pyplot as plt

    # Create the mesh
    h1 = np.ones(20)
    h2 = np.ones(100)
    M = mesh.TensorMesh([h1,h2])

    # Create some parameters for the model
    sig1 = np.log(1)
    sig2 = np.log(0.01)

    # Create a synthetic model from a block in a half-space
    p0 = [5, 10]
    p1 = [15, 50]
    condVals = [sig1, sig2]
    mSynth = Utils.ModelBuilder.defineBlockConductivity(p0,p1,M.gridCC,condVals)
    plt.colorbar(M.plotImage(mSynth))
    plt.show()

    # Set up the projection
    nelec = 50
    spacelec = 2
    surfloc = 0.5
    elecini = 0.5
    elecend = 0.5+spacelec*(nelec-1)
    elecLocR = np.linspace(elecini, elecend, nelec)
    rxmidLoc = (elecLocR[0:nelec-1]+elecLocR[1:nelec])*0.5
    q, Q, rxmidloc = genTxRxmat(nelec, spacelec, surfloc, elecini, M)
    P = Q.T

    # Create some data
    problem = DCProblem(M)
    problem.P = P
    problem.RHS = q
    data = problem.createSyntheticData(mSynth, std=0.05)

    u = problem.field(mSynth)
    u = problem.reshapeFields(u)
    M.plotImage(u[:,10])
    # plt.show()

    # Now set up the problem to do some minimization
    # problem.dobs = dobs
    # problem.std = dobs*0 + 0.05
    m0 = M.gridCC[:,0]*0+sig2

    opt = inverse.InexactGaussNewton(maxIterLS=20, maxIter=3, tolF=1e-6, tolX=1e-6, tolG=1e-6, maxIterCG=6)
    reg = inverse.Regularization(M)
    inv = inverse.Inversion(problem, reg, opt, data, beta0=1e4)

    # Check Derivative
    derChk = lambda m: [inv.dataObj(m), inv.dataObjDeriv(m)]
    tests.checkDerivative(derChk, mSynth)



    print inv.dataObj(m0)
    print inv.dataObj(mSynth)

    m = inv.run(m0)

    plt.colorbar(M.plotImage(m))
    print m
    plt.show()