source: issm/trunk/src/c/classes/kriging/Observations.cpp

/*
 * \file Observations.cpp
 * \brief: Implementation of Observations class, derived from DataSet class.
 */

/*Headers: {{{*/
#ifdef HAVE_CONFIG_H
        #include <config.h>
#else
#error "Cannot compile with HAVE_CONFIG_H symbol! run configure first!"
#endif

#include <vector>
#include <functional>
#include <algorithm>
#include <iostream>

#include "../Options/Options.h"
#include "./Observations.h"
#include "./Observation.h"
#include "../../datastructures/datastructures.h"
#include "../../shared/shared.h"

#include "./Quadtree.h"
#include "./Covertree.h"
#include "./Variogram.h"
#include "../../toolkits/toolkits.h"

using namespace std;
/*}}}*/

/*Object constructors and destructor*/
Observations::Observations(){/*{{{*/
        this->treetype  = 0;
        this->quadtree  = NULL;
        this->covertree = NULL;
        return;
}
/*}}}*/
Observations::Observations(IssmPDouble* observations_list,IssmPDouble* x,IssmPDouble* y,int n,Options* options){/*{{{*/

        /*Check that there are observations*/
        if(n<=0) _error_("No observation found");

        /*Get tree type (FIXME)*/
        double dtree = 0.;
        options->Get(&dtree,"treetype",1.);
        this->treetype = reCast<int>(dtree);
        switch(this->treetype){
                case 1:
                        this->covertree = NULL;
                        this->InitQuadtree(observations_list,x,y,n,options);
                        break;
                case 2:
                        this->quadtree = NULL;
                        this->InitCovertree(observations_list,x,y,n,options);
                        break;
                default:
                        _error_("Tree type "<<this->treetype<<" not supported yet (1: quadtree, 2: covertree)");
        }
}
/*}}}*/
Observations::~Observations(){/*{{{*/
        switch(this->treetype){
                case 1:
                        delete this->quadtree;
                        break;
                case 2:
                        delete this->covertree;
                        break;
                default:
                        _printf_("Tree type "<<this->treetype<<" not supported yet (1: quadtree, 2: covertree)");
        }
        return;
}
/*}}}*/

/*Initialize data structures*/
void Observations::InitCovertree(IssmPDouble* observations_list,IssmPDouble* x,IssmPDouble* y,int n,Options* options){/*{{{*/

        /*Intermediaries*/
         IssmPDouble  minspacing,mintrimming,maxtrimming;

        /*Checks*/
        _assert_(n);

        /*Get trimming limits*/
        options->Get(&mintrimming,"mintrimming",-1.e+21);
        options->Get(&maxtrimming,"maxtrimming",+1.e+21);
        options->Get(&minspacing,"minspacing",0.01);
        if(minspacing<=0) _error_("minspacing must > 0");

        /*Get maximum distance between 2 points
         *  maxDist should be the maximum distance that any two points
         *  can have between each other. IE p.distance(q) < maxDist for all
         *  p,q that you will ever try to insert. The cover tree may be invalid
         *  if an inaccurate maxDist is given.*/
        IssmPDouble xmin = x[0];
        IssmPDouble xmax = x[0];
        IssmPDouble ymin = y[0];
        IssmPDouble ymax = y[0];
        for(int i=1;i<n;i++){
                if(x[i]<xmin) xmin=x[i];
                if(x[i]>xmax) xmax=x[i];
                if(y[i]<ymin) ymin=y[i];
                if(y[i]>ymax) ymax=y[i];
        }
        IssmPDouble maxDist = sqrt(pow(xmax-xmin,2)+pow(ymax-ymin,2));
        IssmPDouble base    = 2.;
        int         maxdepth = ceilf(log(maxDist)/log(base));

         _printf0_("Generating covertree with a maximum depth " <<  maxdepth <<"... ");
    this->covertree=new Covertree(maxdepth);

    for(int i=0;i<n;i++){

                /*First check limits*/
                if(observations_list[i]>maxtrimming) continue;
                if(observations_list[i]<mintrimming) continue;

                /*Second, check that this observation is not too close from another one*/
                Observation newobs = Observation(x[i],y[i],observations_list[i]);
                if(i>0 && this->covertree->getRoot()){
                        /*Get closest obs and see if it is too close*/
                        std::vector<Observation> kNN=(this->covertree->kNearestNeighbors(newobs,1));
                        Observation oldobs = (*kNN.begin());
                        if(oldobs.distance(newobs)<minspacing) continue;
                }

                this->covertree->insert(newobs);
    }
         _printf0_("done\n");
}
/*}}}*/
void Observations::InitQuadtree(IssmPDouble* observations_list,IssmPDouble* x,IssmPDouble* y,int n,Options* options){/*{{{*/

        /*Intermediaries*/
        int          i,maxdepth,level,counter,index;
        int          xi,yi;
        IssmPDouble  xmin,xmax,ymin,ymax;
        IssmPDouble  offset,minlength,minspacing,mintrimming,maxtrimming;
        Observation *observation = NULL;

        /*Checks*/
        _assert_(n);

        /*Get extrema*/
        xmin=x[0]; ymin=y[0];
        xmax=x[0]; ymax=y[0];
        for(i=1;i<n;i++){
                xmin=min(xmin,x[i]); ymin=min(ymin,y[i]);
                xmax=max(xmax,x[i]); ymax=max(ymax,y[i]);
        }
        offset=0.05*(xmax-xmin); xmin-=offset; xmax+=offset;
        offset=0.05*(ymax-ymin); ymin-=offset; ymax+=offset;

        /*Get trimming limits*/
        options->Get(&mintrimming,"mintrimming",-1.e+21);
        options->Get(&maxtrimming,"maxtrimming",+1.e+21);
        options->Get(&minspacing,"minspacing",0.01);
        if(minspacing<=0) _error_("minspacing must > 0");

        /*Get Minimum box size*/
        if(options->GetOption("boxlength")){
                options->Get(&minlength,"boxlength");
                if(minlength<=0)_error_("boxlength should be a positive number");
                maxdepth=reCast<int,IssmPDouble>(log(max(xmax-xmin,ymax-ymin)/minlength +1)/log(2.0));
        }
        else{
                maxdepth = 30;
                minlength=max(xmax-xmin,ymax-ymin)/IssmPDouble((1L<<maxdepth)-1);
        }

        /*Initialize Quadtree*/
        _printf0_("Generating quadtree with a maximum box size " << minlength << " (depth=" << maxdepth << ")... ");
        this->quadtree = new Quadtree(xmin,xmax,ymin,ymax,maxdepth);

        /*Add observations one by one*/
        counter = 0;
        for(i=0;i<n;i++){

                /*First check limits*/
                if(observations_list[i]>maxtrimming) continue;
                if(observations_list[i]<mintrimming) continue;

                /*Second, check that this observation is not too close from another one*/
                this->quadtree->ClosestObs(&index,x[i],y[i]);
                if(index>=0){
                        observation=xDynamicCast<Observation*>(this->GetObjectByOffset(index));
                        if(pow(observation->x-x[i],2)+pow(observation->y-y[i],2) < minspacing) continue;
                }

                this->quadtree->IntergerCoordinates(&xi,&yi,x[i],y[i]);
                this->quadtree->QuadtreeDepth2(&level,xi,yi);
                if((int)level <= maxdepth){
                        observation = new Observation(x[i],y[i],xi,yi,counter++,observations_list[i]);
                        this->quadtree->Add(observation);
                        this->AddObject(observation);
                }
                else{
                        /*We need to average with the current observations*/
                        this->quadtree->AddAndAverage(x[i],y[i],observations_list[i]);
                }
        }
        _printf0_("done\n");
        _printf0_("Initial number of observations: " << n << "\n");
        _printf0_("  Final number of observations: " << this->quadtree->NbObs << "\n");
}
/*}}}*/

/*Methods*/
void Observations::ClosestObservation(IssmPDouble *px,IssmPDouble *py,IssmPDouble *pobs,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius){/*{{{*/

        switch(this->treetype){
                case 1:
                        this->ClosestObservationQuadtree(px,py,pobs,x_interp,y_interp,radius);
                        break;
                case 2:
                        this->ClosestObservationCovertree(px,py,pobs,x_interp,y_interp,radius);
                        break;
                default:
                        _error_("Tree type "<<this->treetype<<" not supported yet (1: quadtree, 2: covertree)");
        }

}/*}}}*/
void Observations::ClosestObservationCovertree(IssmPDouble *px,IssmPDouble *py,IssmPDouble *pobs,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius){/*{{{*/

        IssmPDouble hmin  = UNDEF;

        if(this->covertree->getRoot()){
                /*Get closest obs and see if it is too close*/
                Observation newobs = Observation(x_interp,y_interp,0.);
                std::vector<Observation> kNN=(this->covertree->kNearestNeighbors(newobs,1));
                Observation observation = (*kNN.begin());
                hmin = observation.distance(newobs);
                if(hmin<=radius){
                        *px   = observation.x;
                        *py   = observation.y;
                        *pobs = observation.value;
                        return;
                }
        }

        *px   = UNDEF;
        *py   = UNDEF;
        *pobs = UNDEF;
}/*}}}*/
void Observations::ClosestObservationQuadtree(IssmPDouble *px,IssmPDouble *py,IssmPDouble *pobs,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius){/*{{{*/

        /*Output and Intermediaries*/
        int          nobs,i,index;
        IssmPDouble  hmin,h2,hmin2;
        int         *indices      = NULL;
        Observation *observation  = NULL;

        /*If radius is not provided or is 0, return all observations*/
        if(radius==0) radius=this->quadtree->root->length;

        /*For CPPcheck*/
        hmin = 2*radius;

        /*First, find closest point in Quadtree (fast but might not be the true closest obs)*/
        this->quadtree->ClosestObs(&index,x_interp,y_interp);
        if(index>=0){
                observation=xDynamicCast<Observation*>(this->GetObjectByOffset(index));
                hmin = sqrt((observation->x-x_interp)*(observation->x-x_interp) + (observation->y-y_interp)*(observation->y-y_interp));
                if(hmin<radius) radius=hmin;
        }

        /*Find all observations that are in radius*/
        this->quadtree->RangeSearch(&indices,&nobs,x_interp,y_interp,radius);
        for (i=0;i<nobs;i++){
                observation=xDynamicCast<Observation*>(this->GetObjectByOffset(indices[i]));
                h2 = (observation->x-x_interp)*(observation->x-x_interp) + (observation->y-y_interp)*(observation->y-y_interp);
                if(i==0){
                        hmin2 = h2;
                        index = indices[i];
                }
                else{
                        if(h2<hmin2){
                                hmin2 = h2;
                                index = indices[i];
                        }
                }
        }

        /*Assign output pointer*/
        if(nobs || hmin==radius){
                observation=xDynamicCast<Observation*>(this->GetObjectByOffset(index));
                *px   = observation->x;
                *py   = observation->y;
                *pobs = observation->value;
        }
        else{
                *px   = UNDEF;
                *py   = UNDEF;
                *pobs = UNDEF;
        }
        xDelete<int>(indices);

}/*}}}*/
void Observations::Distances(IssmPDouble* distances,IssmPDouble *x,IssmPDouble *y,int n,IssmPDouble radius){/*{{{*/

        IssmPDouble xi,yi,obs;

        for(int i=0;i<n;i++){
                this->ClosestObservation(&xi,&yi,&obs,x[i],y[i],radius);
                if(xi==UNDEF && yi==UNDEF){
                 distances[i]=UNDEF;
                }
                else{
                 distances[i]=sqrt( (x[i]-xi)*(x[i]-xi) + (y[i]-yi)*(y[i]-yi) );
                }
        }
}/*}}}*/
void Observations::ObservationList(IssmPDouble **px,IssmPDouble **py,IssmPDouble **pobs,int* pnobs){/*{{{*/

        /*Output and Intermediaries*/
        int          nobs;
        IssmPDouble *x            = NULL;
        IssmPDouble *y            = NULL;
        IssmPDouble *obs          = NULL;
        Observation *observation  = NULL;

        nobs = this->Size();

        if(nobs){
                x   = xNew<IssmPDouble>(nobs);
                y   = xNew<IssmPDouble>(nobs);
                obs = xNew<IssmPDouble>(nobs);
                int i=0;
                for(Object* & object: this->objects){
                        observation=xDynamicCast<Observation*>(object);
                        observation->WriteXYObs(&x[i],&y[i],&obs[i]);
                        i++;
                }
        }

        /*Assign output pointer*/
        *px=x;
        *py=y;
        *pobs=obs;
        *pnobs=nobs;
}/*}}}*/
void Observations::ObservationList(IssmPDouble **px,IssmPDouble **py,IssmPDouble **pobs,int* pnobs,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius,int maxdata){/*{{{*/

        switch(this->treetype){
                case 1:
                        this->ObservationListQuadtree(px,py,pobs,pnobs,x_interp,y_interp,radius,maxdata);
                        break;
                case 2:
                        this->ObservationListCovertree(px,py,pobs,pnobs,x_interp,y_interp,radius,maxdata);
                        break;
                default:
                        _error_("Tree type "<<this->treetype<<" not supported yet (1: quadtree, 2: covertree)");
        }
}/*}}}*/
void Observations::ObservationListCovertree(double **px,double **py,double **pobs,int* pnobs,double x_interp,double y_interp,double radius,int maxdata){/*{{{*/

        double *x            = NULL;
        double *y            = NULL;
        double *obs          = NULL;
        Observation observation=Observation(x_interp,y_interp,0.);
        std::vector<Observation> kNN;

        kNN=(this->covertree->kNearestNeighbors(observation, maxdata));
        //cout << "kNN's size: " << kNN.size() << " (maxdata = " <<maxdata<<")"<<endl;

        //kNN is sort from closest to farthest neighbor
        //searches for the first neighbor that is out of radius
        //deletes and resizes the kNN vector
        vector<Observation>::iterator it;
        if(radius>0.){
                for (it = kNN.begin(); it != kNN.end(); ++it) {
                        //(*it).print();
                        //cout << "\n" << (*it).distance(observation) << endl;
                        if ((*it).distance(observation) > radius) {
                                break;
                        }
                }
                kNN.erase(it, kNN.end());
        }

        /*Allocate vectors*/
        x   = new double[kNN.size()];
        y   = new double[kNN.size()];
        obs = new double[kNN.size()];

        /*Loop over all observations and fill in x, y and obs*/
        int i = 0;
        for(it = kNN.begin(); it != kNN.end(); ++it) {
                (*it).WriteXYObs((*it), &x[i], &y[i], &obs[i]);
                i++;
        }

        *px=x;
        *py=y;
        *pobs=obs;
        *pnobs = kNN.size();
}/*}}}*/
void Observations::ObservationListQuadtree(IssmPDouble **px,IssmPDouble **py,IssmPDouble **pobs,int* pnobs,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius,int maxdata){/*{{{*/

        /*Output and Intermediaries*/
        bool         stop;
        int          nobs,tempnobs,i,j,k,n,counter;
        IssmPDouble  h2,radius2;
        int         *indices      = NULL;
        int         *tempindices  = NULL;
        IssmPDouble *dists        = NULL;
        IssmPDouble *x            = NULL;
        IssmPDouble *y            = NULL;
        IssmPDouble *obs          = NULL;
        Observation *observation  = NULL;

        /*If radius is not provided or is 0, return all observations*/
        if(radius==0.) radius=this->quadtree->root->length*2.;

        /*Compute radius square*/
        radius2 = radius*radius;

        /*Find all observations that are in radius*/
        this->quadtree->RangeSearch(&tempindices,&tempnobs,x_interp,y_interp,radius);
        if(tempnobs){
                indices = xNew<int>(tempnobs);
                dists   = xNew<IssmPDouble>(tempnobs);
        }
        nobs = 0;
        for(i=0;i<tempnobs;i++){
                observation=xDynamicCast<Observation*>(this->GetObjectByOffset(tempindices[i]));
                h2 = (observation->x-x_interp)*(observation->x-x_interp) + (observation->y-y_interp)*(observation->y-y_interp);

                if(nobs==maxdata && h2>radius2) continue;
                if(nobs<maxdata){
                        indices[nobs]   = tempindices[i];
                        dists[nobs]     = h2;
                        nobs++;
                }
                if(nobs==1) continue;

                /*Sort all dists up to now*/
                n=nobs-1;
                stop = false;
                for(k=0;k<n-1;k++){
                        if(h2<dists[k]){
                                counter=1;
                                for(int jj=k;jj<n;jj++){
                                        j  = n-counter;
                                        dists[j+1]   = dists[j];
                                        indices[j+1] = indices[j];
                                        counter++;
                                }
                                dists[k]   = h2;
                                indices[k] = tempindices[i];
                                stop = true;
                                break;
                        }
                        if(stop) break;
                }
        }
        xDelete<IssmPDouble>(dists);
        xDelete<int>(tempindices);

        if(nobs){
                /*Allocate vectors*/
                x   = xNew<IssmPDouble>(nobs);
                y   = xNew<IssmPDouble>(nobs);
                obs = xNew<IssmPDouble>(nobs);

                /*Loop over all observations and fill in x, y and obs*/
                for(i=0;i<nobs;i++){
                        observation=xDynamicCast<Observation*>(this->GetObjectByOffset(indices[i]));
                        observation->WriteXYObs(&x[i],&y[i],&obs[i]);
                }
        }

        /*Assign output pointer*/
        xDelete<int>(indices);
        *px=x;
        *py=y;
        *pobs=obs;
        *pnobs=nobs;
}/*}}}*/
void Observations::InterpolationIDW(IssmPDouble *pprediction,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius,int mindata,int maxdata,IssmPDouble power){/*{{{*/

        /*Intermediaries*/
        int         i,n_obs;
        IssmPDouble prediction;
        IssmPDouble numerator,denominator,h,weight;
        IssmPDouble *x   = NULL;
        IssmPDouble *y   = NULL;
        IssmPDouble *obs = NULL;

        /*Some checks*/
        _assert_(maxdata>0);
        _assert_(pprediction);
        _assert_(power>0);

        /*Get list of observations for current point*/
        this->ObservationList(&x,&y,&obs,&n_obs,x_interp,y_interp,radius,maxdata);

        /*If we have less observations than mindata, return UNDEF*/
        if(n_obs<mindata){
                prediction = UNDEF;
        }
        else{
                numerator   = 0.;
                denominator = 0.;
                for(i=0;i<n_obs;i++){
                        h = sqrt( (x[i]-x_interp)*(x[i]-x_interp) + (y[i]-y_interp)*(y[i]-y_interp));
                        if (h<0.0000001){
                                numerator   = obs[i];
                                denominator = 1.;
                                break;
                        }
                        weight = 1./pow(h,power);
                        numerator   += weight*obs[i];
                        denominator += weight;
                }
                prediction = numerator/denominator;
        }

        /*clean-up*/
        *pprediction = prediction;
        xDelete<IssmPDouble>(x);
        xDelete<IssmPDouble>(y);
        xDelete<IssmPDouble>(obs);
}/*}}}*/
void Observations::InterpolationKriging(IssmPDouble *pprediction,IssmPDouble *perror,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius,int mindata,int maxdata,Variogram* variogram){/*{{{*/

        /*Intermediaries*/
        int           i,j,n_obs;
        IssmPDouble   prediction,error;
        IssmPDouble  *x      = NULL;
        IssmPDouble  *y      = NULL;
        IssmPDouble  *obs    = NULL;
        IssmPDouble  *Lambda = NULL;

        /*Some checks*/
        _assert_(mindata>0 && maxdata>0);
        _assert_(pprediction && perror);

        /*Get list of observations for current point*/
        this->ObservationList(&x,&y,&obs,&n_obs,x_interp,y_interp,radius,maxdata);

        /*If we have less observations than mindata, return UNDEF*/
        if(n_obs<mindata){
                *pprediction = -999.0;
                *perror      = -999.0;
                return;
        }

        /*Allocate intermediary matrix and vectors*/
        IssmPDouble* A = xNew<IssmPDouble>((n_obs+1)*(n_obs+1));
        IssmPDouble* B = xNew<IssmPDouble>(n_obs+1);

        IssmPDouble unbias = variogram->Covariance(0.,0.);
        /*First: Create semivariogram matrix for observations*/
        for(i=0;i<n_obs;i++){
                //printf("%g %g ==> %g\n",x[i],y[i],sqrt(pow(x[i]-x_interp,2)+pow(y[i]-y_interp,2)));
                for(j=0;j<=i;j++){
                        A[i*(n_obs+1)+j] = variogram->Covariance(x[i]-x[j],y[i]-y[j]);
                        A[j*(n_obs+1)+i] = A[i*(n_obs+1)+j];
                }
                A[i*(n_obs+1)+n_obs] = unbias;
                //A[i*(n_obs+1)+n_obs] = 1.;
        }
        for(i=0;i<n_obs;i++) A[n_obs*(n_obs+1)+i]=unbias;
        //for(i=0;i<n_obs;i++) A[n_obs*(n_obs+1)+i]=1.;
        A[n_obs*(n_obs+1)+n_obs] = 0.;

        /*Get semivariogram vector associated to this location*/
        for(i=0;i<n_obs;i++) B[i] = variogram->Covariance(x[i]-x_interp,y[i]-y_interp);
        B[n_obs] = unbias;
        //B[n_obs] = 1.;

        /*Solve the three linear systems*/
#if _HAVE_GSL_
        DenseGslSolve(&Lambda,A,B,n_obs+1);    // Gamma^-1 Z
#else
        _error_("GSL is required");
#endif

        /*Compute predictor*/
        prediction = 0.;
        for(i=0;i<n_obs;i++) prediction += Lambda[i]*obs[i];

        /*Compute error (GSLIB p15 eq II.14)*/
        error = variogram->Covariance(0.,0.)*(1. - Lambda[n_obs]);;
        for(i=0;i<n_obs;i++) error += -Lambda[i]*B[i];

        /*clean-up*/
        *pprediction = prediction;
        *perror = error;
        xDelete<IssmPDouble>(x);
        xDelete<IssmPDouble>(y);
        xDelete<IssmPDouble>(obs);
        xDelete<IssmPDouble>(A);
        xDelete<IssmPDouble>(B);
        xDelete<IssmPDouble>(Lambda);
}/*}}}*/
void Observations::InterpolationNearestNeighbor(IssmPDouble *pprediction,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius){/*{{{*/

        /*Intermediaries*/
        IssmPDouble x,y,obs;

        /*Get clostest observation*/
        this->ClosestObservation(&x,&y,&obs,x_interp,y_interp,radius);

        /*Assign output pointer*/
        *pprediction = obs;
}/*}}}*/
void Observations::InterpolationV4(IssmPDouble *pprediction,IssmPDouble x_interp,IssmPDouble y_interp,IssmPDouble radius,int mindata,int maxdata){/*{{{*/
        /* Reference:  David T. Sandwell, Biharmonic spline interpolation of GEOS-3
         * and SEASAT altimeter data, Geophysical Research Letters, 2, 139-142,
         * 1987.  Describes interpolation using value or gradient of value in any
         * dimension.*/

        /*Intermediaries*/
        int         i,j,n_obs;
        IssmPDouble prediction,h;
        IssmPDouble *x       = NULL;
        IssmPDouble *y       = NULL;
        IssmPDouble *obs     = NULL;
        IssmPDouble *Green   = NULL;
        IssmPDouble *weights = NULL;
        IssmPDouble *g       = NULL;

        /*Some checks*/
        _assert_(maxdata>0);
        _assert_(pprediction);

        /*Get list of observations for current point*/
        this->ObservationList(&x,&y,&obs,&n_obs,x_interp,y_interp,radius,maxdata);

        /*If we have less observations than mindata, return UNDEF*/
        if(n_obs<mindata || n_obs<2){
                prediction = UNDEF;
        }
        else{

                /*Allocate intermediary matrix and vectors*/
                Green = xNew<IssmPDouble>(n_obs*n_obs);
                g     = xNew<IssmPDouble>(n_obs);

                /*First: distance vector*/
                for(i=0;i<n_obs;i++){
                        h = sqrt( (x[i]-x_interp)*(x[i]-x_interp) + (y[i]-y_interp)*(y[i]-y_interp) );
                        if(h>0){
                                g[i] = h*h*(log(h)-1.);
                        }
                        else{
                                g[i] = 0.;
                        }
                }

                /*Build Green function matrix*/
                for(i=0;i<n_obs;i++){
                        for(j=0;j<=i;j++){
                                h = sqrt( (x[i]-x[j])*(x[i]-x[j]) + (y[i]-y[j])*(y[i]-y[j]) );
                                if(h>0){
                                        Green[j*n_obs+i] = h*h*(log(h)-1.);
                                }
                                else{
                                        Green[j*n_obs+i] = 0.;
                                }
                                Green[i*n_obs+j] = Green[j*n_obs+i];
                        }
                        /*Zero diagonal (should be done already, but just in case)*/
                        Green[i*n_obs+i] = 0.;
                }

                /*Compute weights*/
#if _HAVE_GSL_
                DenseGslSolve(&weights,Green,obs,n_obs); // Green^-1 obs
#else
                _error_("GSL is required");
#endif

                /*Interpolate*/
                prediction = 0;
                for(i=0;i<n_obs;i++) prediction += weights[i]*g[i];

        }

        /*clean-up*/
        *pprediction = prediction;
        xDelete<IssmPDouble>(x);
        xDelete<IssmPDouble>(y);
        xDelete<IssmPDouble>(obs);
        xDelete<IssmPDouble>(Green);
        xDelete<IssmPDouble>(g);
        xDelete<IssmPDouble>(weights);
}/*}}}*/
void Observations::QuadtreeColoring(IssmPDouble* A,IssmPDouble *x,IssmPDouble *y,int n){/*{{{*/

        if(this->treetype!=1) _error_("Tree type is not quadtree");
        int xi,yi,level;

        for(int i=0;i<n;i++){
                this->quadtree->IntergerCoordinates(&xi,&yi,x[i],y[i]);
                this->quadtree->QuadtreeDepth(&level,xi,yi);
                A[i]=(IssmPDouble)level;
        }

}/*}}}*/
void Observations::Variomap(IssmPDouble* gamma,IssmPDouble *x,int n){/*{{{*/

        /*Output and Intermediaries*/
        int          i,j,k;
        IssmPDouble  distance;
        Observation *observation1 = NULL;
        Observation *observation2 = NULL;

        IssmPDouble *counter = xNew<IssmPDouble>(n);
        for(j=0;j<n;j++) counter[j] = 0.0;
        for(j=0;j<n;j++) gamma[j]   = 0.0;

        for(Object* & object : this->objects){
                observation1=xDynamicCast<Observation*>(object);

                for(Object* & object : this->objects){
                        observation2=xDynamicCast<Observation*>(object);

                        distance=sqrt(pow(observation1->x - observation2->x,2) + pow(observation1->y - observation2->y,2));
                        if(distance>x[n-1]) continue;

                        int index = int(distance/(x[1]-x[0]));
                        if(index>n-1) index = n-1;
                        if(index<0)   index = 0;

                        gamma[index]   += 1./2.*pow(observation1->value - observation2->value,2);
                        counter[index] += 1.;
                }
        }

        /*Normalize semivariogram*/
        gamma[0]=0.;
        for(k=0;k<n;k++){
                if(counter[k]) gamma[k] = gamma[k]/counter[k];
        }

        /*Assign output pointer*/
        xDelete<IssmPDouble>(counter);
}/*}}}*/