## Extended Euler Solutions

Use the **Pdepth > Extended Euler Solutions** (EXEULER GX) to compute the extended Euler depth solutions from magnetic (or gravity) profiles.

### ExEuler - Generate Extended Euler Solutions dialog options

X channel |
Input x coordinate. (Assumes units are the same for X, Y, Elev, and Topo channels). |

Y channel |
Input y coordinate. (Assumes units are the same for X, Y, Elev, and Topo channels). |

Elev. channel |
Flight elevation (+up) relative to sea level. (Assumes units are the same for X, Y, Elev, and Topo channels). |

Mag channel |
Magnetic (or gravity) anomaly channel. |

Horizontal derivative |
Input horizontal derivative channel. The key word "Calculate" forces the Werner to calculate derivative from the Mag Channel values. |

Vertical derivative |
Input vertical derivative channel. The key word "<Calculate>" forces Exeuler to calculate derivative from the Mag Channel values |

Topography |
Topographic elevation (+up) relative to sea level. (Assumes units are the same for X, Y, Elev, and Topo channels. Can be all dummies). |

Minimum depth |
Solutions shallower than minimum depth are discarded. (Relative to flight elevation, + down). |

Maximum depth |
Solutions deeper than maximum depth are discarded. (Relative to flight elevation, + down). |

Window length |
Length (in horizontal distance units) of operator window used to calculate depths. |

Max % error |
Used to filter out solutions that differ in depth by more than this % when calculated by both Euler and Extended Euler calculations. |

"Dike" structural index |
Structual Index (SI) used to calculate solutions output in the "Dike" channel. (Default 1). Restricted to 0<SI<=3. Solutions output in the "Contact" channel are always calculated with SI = 0. |

Relative strike |
Angle between profile direction and anomaly strike in degrees, positive counterclockwise. (Default 90 degrees). |

Field strength |
Earth's magnetic field strength. Used to calculate susceptibility, which will be in same units. |

Inclination |
Earth's magnetic field inclination in degrees. |

Declination |
Earth's magnetic field declination in degrees. |

Output database name |
If a database with this name exists, Exeuler replaces existing lines with the new results but maintains the existing format and structure. |

### Application Notes

#### Methodology

This GX always makes two passes through the data. The first pass always uses a Structural Index (SI) of zero for "Contact" solutions. The second pass uses the SI given by the user in the dialog entry for "Dike" Structural Index. The results of the second pass are always flagged as "dike" solutions, regardless of the SI is used. The solutions are sorted by distance-along-the-line, referenced to the first point in the input profile. The channel "Z_Dikes" contains the "dike" solution depths relative to the flight elevation. The channel "Z_Contacts" contains the contact solution depths relative to the flight elevation. All of the solutions are also in a 3rd channel named "Z_Both". The channel "Depth_sl" contains the depth relative to sea level calculated using the input "Elev" channel if it was non- dummy. The sign of the Z-axis is negative down for all of these channels (Z_Dikes, Z_contacts, Z_Both, Flt_Elev, and Depth_sl) to facilitate convenient profile plots. "Dike0_Cont1" contains a flag identifying the solution as a dike (0) or a contact (1) solution.

Five of the input channels, re-sampled to an even sample interval are copied to the output database: "Elev.", "Mag", "Horizontal Gradient", "Vertical Gradient" and "Topography". The "Elev" and "Topography" profiles are automatically plotted in the top pane; the "Mag", "Horizontal Derivative", and "Vertical Gradient" profiles are automatically plotted in the middle pane; and symbols for the "Z_dikes" and "Z_Contacts" are automatically plotted in the bottom pane.

#### Gravity Applications

To use Extended Euler Deconvolution on gravity profiles, use the vertical derivative of gravity as the input profile rather than total-field magnetics.

If the "Field Strength" is set to "1", "Inclination" set to "90", and the "Declination" set to zero, the "Susc" output channel will be the calculated density contrast. Note that the "Contact" solutions are computed from the 2nd horizontal derivative in the gravity case, so some low-pass filtering is often required.

Length and depth units (e.g., meters or Kilometers) are determined from X, Y, Elev, and Topography input channels. All distance/depth units must be the same.

The input profiles are interpolated to an even sample interval using the standard OASIS spline calls before processing by Exeuler. The sample interval is the total profile length divided by the number of points in the profile. Therefore, profiles with large gaps should be split into multiple lines.

Exeuler uses an fft technique to calculate the horizontal and vertical derivatives if the user does not specify an input gradient channels. For noisy input profiles, the results can be improved significantly by filtering the input anomaly and gradient data.

The extended Euler calculation routine used in this GX was provided by GETECH and is based on the paper by Mushayandebvu and others, 2001. This approach calculates solutions using both the conventional Euler equation (Reid and others, 1990) and the "rotational constraint" equation from extended Euler. Solving both equations jointly (extended Euler) gives distance, depth, dip and susceptibility, assuming there is no remanent magnetization. Using conventional Euler gives a second estimate for distance and depth. If the relative difference in depth for the two estimates is less than the Max % error given by the user, the solution is retained; otherwise it is rejected.

### References

- [1] Martin F. Mushayandebvu, P. van Driel, Alan B. Reid, and James Derek Fairhead, 2001, "Magnetic source parameters of two-dimensional structures using extended Euler deconvolution",
*Geophysics*, vol. 66 ( 3), pp. 814-823. - [2] A. B. Reid, J. M. Allsop, H. Granser, A. J. Millett, and I. W. Somerton, 1990, "Magnetic interpretation in three dimensions using Euler deconvolution",
*Geophysics*, vol 55 (1), pp. 80-91.

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