diff --git a/CMakeLists.txt b/CMakeLists.txt
index e3781979..0ab796e8 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -115,3 +115,25 @@ target_link_libraries(Test_ordering
     Forward_obj
     Inversion_obj
     )
+
+add_executable(Synthetic_data1
+    src/Synthetic_data1.cpp)
+target_link_libraries(Synthetic_data1
+    PRIVATE
+    Forward_obj
+    Inversion_obj
+    )
+add_executable(test_wavelet_forward1
+    src/test_wavelet_forward1.cpp)
+target_link_libraries(test_wavelet_forward1
+    PRIVATE
+    Forward_obj
+    Inversion_obj
+    )
+add_executable(test_wavelet_forward2
+    src/test_wavelet_forward2.cpp)
+target_link_libraries(test_wavelet_forward2
+    PRIVATE
+    Forward_obj
+    Inversion_obj
+    )
diff --git a/src/Forward_modelling/Fwd.cpp b/src/Forward_modelling/Fwd.cpp
index 00108f99..cb908b11 100755
--- a/src/Forward_modelling/Fwd.cpp
+++ b/src/Forward_modelling/Fwd.cpp
@@ -192,7 +192,6 @@ void Fwd::compute_G_wavelet()
       for (int k = 0; k < n_fields; k++)
       {
         sensitivity_data[k][j] = field[basic_field_index[k]];
-        // G(i + k * N_obs, j) = field[basic_field_index[k]];
       }
     }
 
@@ -201,17 +200,8 @@ void Fwd::compute_G_wavelet()
       this->n_dy_for_wavelet = compress_vec(
           sensitivity_data[k], comp_col_ids[i + k * N_obs],
           comp_G_coeffs[i + k * N_obs], this->compression_threshold);
+      // sensitivity_data[k] has been resized in the compress_vec function
       assert(n_dy_for_wavelet == sensitivity_data[k].size());
-      // comp_col_ids[i + k * N_obs].clear();
-      // comp_G_coeffs[i + k * N_obs].clear();
-      // for (int j = 0; j < n_dy_for_wavelet; ++j) {
-      //    if (std::fabs(sensitivity_data[k][j]) > 1e-15) {
-      //        n_non_zero++;
-      //        comp_col_ids[i + k * N_obs].push_back(j);
-      //        comp_G_coeffs[i + k * N_obs].push_back(
-      //            sensitivity_data[k][j]);
-      //    }
-      //}
     }
   }
   int n_non_zero = 0;
diff --git a/src/GraInvRect.o b/src/GraInvRect.o
deleted file mode 100644
index 2d89b5e7..00000000
Binary files a/src/GraInvRect.o and /dev/null differ
diff --git a/src/test_wavelet_forward1.cpp b/src/test_wavelet_forward1.cpp
new file mode 100644
index 00000000..1a491aab
--- /dev/null
+++ b/src/test_wavelet_forward1.cpp
@@ -0,0 +1,205 @@
+#include <chrono>
+#include <fstream>
+#include <random>
+// #include<function>
+
+#include "GaussNewtonInversion.h"
+#include "timer.h"
+void func1();
+void func2(double);
+int main(int argc, char **argv)
+{
+    if (argc != 2)
+    {
+        cout << "A thresholding parameter should follow the program name. "
+                "Usage:\n";
+        cout << "program_name [relative threshold]" << endl;
+        return 1;
+    }
+    double relative_threshold = atof(argv[1]);
+    func1();
+    func2(relative_threshold);
+    return 0;
+}
+void func1()
+{
+    double x_lim[2] = {0, 2000};
+    double y_lim[2] = {0, 2000};
+    double z_lim[2] = {0, 1000};
+
+    Mesh mesh;
+    mesh.generate_regular_mesh(x_lim, 40, y_lim, 40, z_lim, 20);
+
+    double block_z[2] = {200, 500};
+    double block_x[2] = {600, 900};
+    double block_y[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x, block_y, block_z, 500);
+
+    double block_z2[2] = {200, 500};
+    double block_x2[2] = {1200, 1500};
+    double block_y2[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x2, block_y2, block_z2, -500);
+
+    double block_z3[2] = {200, 500};
+    double block_x3[2] = {900, 1500};
+    double block_y3[2] = {1300, 1500};
+    mesh.set_parameter_in_a_region(block_x3, block_y3, block_z3, 500);
+
+    mesh.out_model_vtk("test_model.vtk");
+#ifdef USE_NETCDF
+    mesh.out_model_netcdf("test_model.nc");
+#endif
+    mesh.out_model_txt("test_model.txt");
+
+    /******************Foward modelling*****************/
+    VectorXd rho;
+    mesh.get_model_parameter_from_mesh(rho, 0);
+    Observation ob;
+
+    int nx = 41;
+    int ny = 41;
+    ob.generate(nx, x_lim[0], x_lim[1], ny, y_lim[0], y_lim[1], -0.1);
+    ofstream os("sites");
+    os << ob;
+
+    Fwd forward(mesh, ob, Compute_g_z);
+    // Fwd forward(mesh, ob,
+    // Compute_T_rr|Compute_T_rtheta|Compute_T_rphi|Compute_T_thetatheta|Compute_T_thetaphi|Compute_T_phiphi,8);
+    // Fwd forward(&mesh, &ob, Compute_T_rr|Compute_T_rtheta, 8);
+
+    Timer timer;
+    timer.start();
+    cout << "Generating synthetic data..." << endl;
+    forward.compute_G();
+    const Eigen::MatrixXd &G = forward.get_G();
+    timer.stop();
+    cout << "Time for calculating sensitivity: " << timer.getElapsedTimeInSec()
+         << " s" << endl;
+    timer.start();
+    Eigen::VectorXd d_obs = G * rho;
+    cout << "Calculation of synthetic data completed" << endl;
+    timer.stop();
+    cout << "Multiplication time: " << timer.getElapsedTimeInSec() << " s"
+         << endl;
+
+    VectorXd test_vec(d_obs.size());
+    for (int i = 0; i < test_vec.size(); ++i)
+    {
+        test_vec(i) = 1;
+    }
+    timer.start();
+    VectorXd GTv1 = G.transpose() * test_vec;
+    timer.stop();
+    cout << "Multiplication time of GT*v: " << timer.getElapsedTimeInSec() << "s"
+         << endl;
+
+    ofstream ofs("GTv1.txt");
+    Mesh m = forward.get_mesh();
+    for (int i = 0; i < GTv1.size(); i++)
+    {
+        int id = m.get_reordered_id(i);
+        ofs << setw(23) << scientific << GTv1(id);
+        ofs << endl;
+    }
+
+    // cout << "||d_obs||=" << d_obs.norm() << endl;
+    // cout << "0.001*d_obs.norm()=" << 0.001 * d_obs.norm() << endl;
+
+    // GaussNewtonInversion inv(mesh, ob,
+    // Compute_T_rr|Compute_T_rtheta|Compute_T_rphi|Compute_T_thetatheta|Compute_T_thetaphi|Compute_T_phiphi);
+    GaussNewtonInversion inv(mesh, ob, Compute_g_z);
+
+    inv.set_dobs(d_obs);
+    inv.output_obs_data("dobs");
+}
+
+void func2(double relative_threshold)
+{
+    double x_lim[2] = {0, 2000};
+    double y_lim[2] = {0, 2000};
+    double z_lim[2] = {0, 1000};
+    double reference_surface = 6378137;
+
+    Mesh mesh;
+    mesh.generate_regular_mesh(x_lim, 40, y_lim, 40, z_lim, 20);
+
+    double block_z[2] = {200, 500};
+    double block_x[2] = {600, 900};
+    double block_y[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x, block_y, block_z, 500);
+
+    double block_z2[2] = {200, 500};
+    double block_x2[2] = {1200, 1500};
+    double block_y2[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x2, block_y2, block_z2, -500);
+
+    double block_z3[2] = {200, 500};
+    double block_x3[2] = {900, 1500};
+    double block_y3[2] = {1300, 1500};
+    mesh.set_parameter_in_a_region(block_x3, block_y3, block_z3, 500);
+
+    mesh.out_model_vtk("test_model.vtk");
+#ifdef USE_NETCDF
+    mesh.out_model_netcdf("test_model.nc");
+#endif
+    mesh.out_model_txt("test_model.txt");
+
+    VectorXd rho;
+    mesh.get_model_parameter_from_mesh(rho, 0);
+    Observation ob;
+
+    int nx = 41;
+    int ny = 41;
+    ob.generate(nx, x_lim[0], x_lim[1], ny, y_lim[0], y_lim[1], -0.1);
+    ofstream os("sites");
+    os << ob;
+
+    //***********************************WAVELET
+    Fwd forward_wl(mesh, ob, Compute_g_z);
+    forward_wl.set_use_wavelet(true);
+    Timer timer2;
+    timer2.start();
+    forward_wl.set_compression_threshold(relative_threshold);
+    cout << endl;
+    cout
+        << "=====================Using wavelet transfrom======================="
+        << endl;
+    cout << "Using wavelet transform\nGenerating synthetic data..." << endl;
+    forward_wl.compute_G_wavelet();
+    timer2.stop();
+    cout << "Time for calculating sensitivity: " << timer2.getElapsedTimeInSec()
+         << " s" << endl;
+
+    VectorXd d_obs_wl;
+    timer2.start();
+    // cout << rho << endl;
+    forward_wl.G_vec_mul(rho, d_obs_wl);
+    timer2.stop();
+    cout << "Multiplication time: " << timer2.getElapsedTimeInSec() << " s"
+         << endl;
+
+    VectorXd test_vec(ob.get_n_obs());
+    for (int i = 0; i < test_vec.size(); ++i)
+    {
+        test_vec(i) = 1;
+    }
+    timer2.start();
+    VectorXd GTv2;
+    forward_wl.GT_vec_mul(test_vec, GTv2);
+    timer2.stop();
+    cout << "Multiplication time of GT*v using wavelet compression: "
+         << timer2.getElapsedTimeInSec() << endl;
+
+    ofstream ofs("GTv2.txt");
+    Mesh m = forward_wl.get_mesh();
+    for (int i = 0; i < GTv2.size(); i++)
+    {
+        int id = m.get_reordered_id(i);
+        ofs << setw(23) << scientific << GTv2(id);
+        ofs << endl;
+    }
+
+    GaussNewtonInversion inv2(mesh, ob, Compute_g_z);
+    inv2.set_dobs(d_obs_wl);
+    inv2.output_obs_data("dobs_wavelet");
+}
diff --git a/src/test_wavelet_forward2.cpp b/src/test_wavelet_forward2.cpp
new file mode 100644
index 00000000..53621764
--- /dev/null
+++ b/src/test_wavelet_forward2.cpp
@@ -0,0 +1,230 @@
+#include <chrono>
+#include <fstream>
+#include <random>
+// #include<function>
+
+#include "GaussNewtonInversion.h"
+#include "timer.h"
+void func1();
+void func2(double);
+int main(int argc, char **argv)
+{
+    if (argc != 2)
+    {
+        cout << "A thresholding parameter should follow the program name. "
+                "Usage:\n";
+        cout << "program_name [relative threshold]" << endl;
+        return 1;
+    }
+    double relative_threshold = atof(argv[1]);
+    func1();
+    func2(relative_threshold);
+    return 0;
+}
+void func1()
+{
+    double x_lim[2] = {0, 2000};
+    double y_lim[2] = {0, 2000};
+    double z_lim[2] = {0, 1000};
+
+    Mesh mesh;
+    mesh.generate_regular_mesh(x_lim, 40, y_lim, 40, z_lim, 20);
+
+    double block_z[2] = {200, 500};
+    double block_x[2] = {600, 900};
+    double block_y[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x, block_y, block_z, 500);
+
+    double block_z2[2] = {200, 500};
+    double block_x2[2] = {1200, 1500};
+    double block_y2[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x2, block_y2, block_z2, -500);
+
+    double block_z3[2] = {200, 500};
+    double block_x3[2] = {900, 1500};
+    double block_y3[2] = {1300, 1500};
+    mesh.set_parameter_in_a_region(block_x3, block_y3, block_z3, 500);
+
+    //     mesh.out_model_vtk("test_model.vtk");
+    // #ifdef USE_NETCDF
+    //     mesh.out_model_netcdf("test_model.nc");
+    // #endif
+    //     mesh.out_model_txt("test_model.txt");
+
+    /******************Foward modelling*****************/
+    VectorXd rho;
+    mesh.get_model_parameter_from_mesh(rho, 0);
+    Observation ob;
+
+    int nx = 41;
+    int ny = 41;
+    ob.generate(nx, x_lim[0], x_lim[1], ny, y_lim[0], y_lim[1], -0.1);
+    ofstream os("sites");
+    os << ob;
+
+    GaussNewtonInversion temp_var(mesh, ob, Compute_g_z);
+    Mesh &temp_mesh0 = temp_var.get_mesh();
+    temp_var.compute_G();
+    temp_var.set_m(rho);
+    temp_var.set_density_to_mesh();
+    temp_var.refine_mesh(0.05);
+    // temp_var.out_model_vtk("test_model.vtk");
+    Mesh &mesh1 = temp_var.get_mesh();
+    mesh1.out_model_vtk("test_model.vtk");
+
+    mesh1.get_model_parameter_from_mesh(rho, 0);
+    // temp_mesh.show_ordering();
+
+    Fwd forward(mesh1, ob, Compute_g_z);
+    // Fwd forward(mesh, ob,
+    // Compute_T_rr|Compute_T_rtheta|Compute_T_rphi|Compute_T_thetatheta|Compute_T_thetaphi|Compute_T_phiphi,8);
+    // Fwd forward(&mesh, &ob, Compute_T_rr|Compute_T_rtheta, 8);
+
+    Timer timer;
+    timer.start();
+    cout << "Generating synthetic data..." << endl;
+    forward.compute_G();
+    const Eigen::MatrixXd &G = forward.get_G();
+    timer.stop();
+    cout << "Time for calculating sensitivity: " << timer.getElapsedTimeInSec()
+         << " s" << endl;
+    timer.start();
+    Eigen::VectorXd d_obs = G * rho;
+    cout << "Calculation of synthetic data completed" << endl;
+    timer.stop();
+    cout << "Multiplication time: " << timer.getElapsedTimeInSec() << " s"
+         << endl;
+
+    VectorXd test_vec(d_obs.size());
+    for (int i = 0; i < test_vec.size(); ++i)
+    {
+        test_vec(i) = 1;
+    }
+    timer.start();
+    VectorXd GTv1 = G.transpose() * test_vec;
+    timer.stop();
+    cout << "Multiplication time of GT*v: " << timer.getElapsedTimeInSec() << "s"
+         << endl;
+
+    ofstream ofs("GTv1.txt");
+    Mesh m = forward.get_mesh();
+    for (int i = 0; i < GTv1.size(); i++)
+    {
+        int id = m.get_reordered_id(i);
+        ofs << setw(23) << scientific << GTv1(id);
+        ofs << endl;
+    }
+
+    // cout << "||d_obs||=" << d_obs.norm() << endl;
+    // cout << "0.001*d_obs.norm()=" << 0.001 * d_obs.norm() << endl;
+
+    // GaussNewtonInversion inv(mesh, ob,
+    // Compute_T_rr|Compute_T_rtheta|Compute_T_rphi|Compute_T_thetatheta|Compute_T_thetaphi|Compute_T_phiphi);
+    GaussNewtonInversion inv(mesh1, ob, Compute_g_z);
+
+    inv.set_dobs(d_obs);
+    inv.output_obs_data("dobs");
+}
+
+void func2(double relative_threshold)
+{
+    double x_lim[2] = {0, 2000};
+    double y_lim[2] = {0, 2000};
+    double z_lim[2] = {0, 1000};
+    double reference_surface = 6378137;
+
+    Mesh mesh;
+    mesh.generate_regular_mesh(x_lim, 40, y_lim, 40, z_lim, 20);
+
+    double block_z[2] = {200, 500};
+    double block_x[2] = {600, 900};
+    double block_y[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x, block_y, block_z, 500);
+
+    double block_z2[2] = {200, 500};
+    double block_x2[2] = {1200, 1500};
+    double block_y2[2] = {550, 850};
+    mesh.set_parameter_in_a_region(block_x2, block_y2, block_z2, -500);
+
+    double block_z3[2] = {200, 500};
+    double block_x3[2] = {900, 1500};
+    double block_y3[2] = {1300, 1500};
+    mesh.set_parameter_in_a_region(block_x3, block_y3, block_z3, 500);
+
+    //     mesh.out_model_vtk("test_model.vtk");
+    // #ifdef USE_NETCDF
+    //     mesh.out_model_netcdf("test_model.nc");
+    // #endif
+    //     mesh.out_model_txt("test_model.txt");
+
+    VectorXd rho;
+    mesh.get_model_parameter_from_mesh(rho, 0);
+    Observation ob;
+
+    int nx = 41;
+    int ny = 41;
+    ob.generate(nx, x_lim[0], x_lim[1], ny, y_lim[0], y_lim[1], -0.1);
+    ofstream os("sites");
+    os << ob;
+
+    //***********************************WAVELET
+    GaussNewtonInversion temp_var(mesh, ob, Compute_g_z);
+    Mesh &temp_mesh0 = temp_var.get_mesh();
+    temp_var.compute_G();
+    temp_var.set_m(rho);
+    temp_var.set_density_to_mesh();
+    temp_var.refine_mesh(0.05);
+    // temp_var.out_model_vtk("test_model.vtk");
+    Mesh &mesh1 = temp_var.get_mesh();
+    mesh1.out_model_vtk("test_model.vtk");
+
+    mesh1.get_model_parameter_from_mesh(rho, 0);
+
+    Fwd forward_wl(mesh1, ob, Compute_g_z);
+    forward_wl.set_use_wavelet(true);
+    Timer timer2;
+    timer2.start();
+    forward_wl.set_compression_threshold(relative_threshold);
+    cout << endl;
+    cout
+        << "=====================Using wavelet transfrom======================="
+        << endl;
+    cout << "Using wavelet transform\nGenerating synthetic data..." << endl;
+    forward_wl.compute_G_wavelet();
+    timer2.stop();
+    cout << "Time for calculating sensitivity: " << timer2.getElapsedTimeInSec()
+         << " s" << endl;
+
+    VectorXd d_obs_wl;
+    timer2.start();
+    // cout << rho << endl;
+    forward_wl.G_vec_mul(rho, d_obs_wl);
+    timer2.stop();
+    cout << "Multiplication time: " << timer2.getElapsedTimeInSec() << " s"
+         << endl;
+
+    VectorXd test_vec(ob.get_n_obs());
+    for (int i = 0; i < test_vec.size(); ++i)
+    {
+        test_vec(i) = 1;
+    }
+    timer2.start();
+    VectorXd GTv2;
+    forward_wl.GT_vec_mul(test_vec, GTv2);
+    timer2.stop();
+    cout << "Multiplication time of GT*v using wavelet compression: "
+         << timer2.getElapsedTimeInSec() << endl;
+
+    ofstream ofs("GTv2.txt");
+    Mesh m = forward_wl.get_mesh();
+    for (int i = 0; i < GTv2.size(); i++)
+    {
+        int id = m.get_reordered_id(i);
+        ofs << setw(23) << scientific << GTv2(id);
+        ofs << endl;
+    }
+
+    GaussNewtonInversion inv2(mesh, ob, Compute_g_z);
+    inv2.set_dobs(d_obs_wl);
+    inv2.output_obs_data("dobs_wavelet");
+}