To answer your question in a concise manner, even distribution of data points is one of the things to do in exploration geophysics to reduce the problem of ambiguity or non-uniqueness of geophysical solutions. I understand that during real-life surveys, there could be problem of inaccessibility. Availability of aero/airborne data (preferably high spatial resolution airborne data) can help in some measures to overcome this.

The grid is adequate if an added point does not change the contouring.

Dear Subhendu Mondal, Indian Institute of Technology (ISM) Dhanbad.

It is possible to have gravity and magnetic data at random locations when you are in a land acquisition survey, in which each value of the gravity / magnetic point has been collected at a relatively short time and cost. The random locations may be for example the roads, primary roads, or secondary ones, or because you have areas with obstacles, whether natural or artificial, on the ground. Such raw data is commonly referred to as XYZ data because each data point has an (X, Y) location and one or more measured (Z) values. These are your pattern of data collection points.

But you do not use in the interpretation your raw data collection. You use a grid.

Gridding data is the process of spatial interpolation. The process of gridding takes your pattern collection points (XYZ) and interpolates the readings to determine the values at the nodes of a grid in between the data points. The resulting interpolated data set is known as a grid. There are different gridding methods you can use to grid your data as well as fundamental gridding concepts.

The minimum curvature method interpolates the data by fitting a two-dimensional surface to the raw XYZ data in such a way that the curvature of the surface is minimized. This approach to fit a minimum curvature surface to the data points uses a method like that described by Briggs (1974) and Swain (1976) and is ideal when the surface is expected to be relatively smooth or continuous between data points. It is best used when data is randomly distributed, sampled along arbitrary lines, or if you want to include tie lines.

Minimum curvature gridding first estimates grid values at the nodes of a coarse grid (usually 8 times the final grid cell size). This estimate is based upon the inverse distance average of the actual data within a specified search radius. If there is no data within that radius, the average of all data points in the grid is used. An iterative method is then employed to adjust the grid to fit the actual data points nearest the coarse grid nodes. After an acceptable fit is achieved, the coarse cell size is divided by 2. The same process is then repeated using the coarse grid as the starting surface. This is iteratively repeated until the minimum curvature surface is fit at the final grid cell size.

Minimum curvature gridding has the following capabilities:

• Accessing an unlimited number of input observation points

• Adjustable internal tension

• Applying de-aliasing filter

• Applying linear and logarithmic gridding

• Applying projections

• Blanking un-sampled areas

• No limit on the output grid size

• The actual maximum size of the grid that you can use depends upon the available free disk space of your computer.

I trying to explain a minimum curvature because it is the “widespread” method, but there are others such as bi-directional, inverse distance weighted, kriging, triangular irregular, etc.

In synthesis, if you are confidential with the design and acquisition, following the geological and spatial aliasing criteria, the anomaly final map must represent the rock´s contrast properties in the subsurface in your grid*.

PS: the grid cell size should be 1/4 to 1/2 of the interval data sample. By default, 1/4 multiple by the root square of the grid area divided by the number of collection points.

I attached the classical paper "Machine contouring using minimum curvature" by Ian C. Briggs, Geophysics, vol. 39, No 1 (February 1974), pp. 39-48.

Best regards, Mario E. Sigismondi

Perhaps my first response was too concise. Thanks to Mario for a complete scientific explanation. My practical experience is that in 1960 I spent some time on a gravity crew for Mobil Oil Company. There were two land surveyors and one gravity meter operator. The Party Chief was continually contouring the data, which was updated every day, and he routinely had extra points added to remove any ambiguity in contouring. It goes without saying that after the contouring was complete, a grid was overlaid, grid values determined, various derivatives and other functions were then computed and analyzed.

Thank you very much, Dr. Berkman. Let me share, please, an additional comment.

There is an aspect of your query that is more relevant and is alerted by Ernest Berkman's response: the distribution of reading points should be homogeneous in distance since there is a relationship between the wavelength of the anomalies and the distance between points. Therefore, over the whole interpretation progression after the grid's operations.

When you add more grid points in your gridding achievement, it is possible to change the lateral accuracy: More points you add, more frequencies you can resolve tending towards a better spatial solution. Therefore, the solutions you plot are not rigorously comparable. You can see also the radial average power spectrum content in the wavenumber space as a further comparison. In this sense, a larger grid cell size (filter width) correctly smooths the solution. You should compare please in the example that I attached: as the same area, there are two surveys: one regional and another high-resolution. Even the low and high anomaly distribution is similar, there is more confidence in the map or grid with the lower cell size. The Nyquist spatial frequency and wavelength are affected by the number of points by unit area and its relative distribution. That is:

The Nyquist spatial frequency f = 1/(2*grid spacing). Multiply that by the cutoff fraction (e.g., 0.5) then divide 1 by that number to get the resulting wavelength.

More concisely, cutoff wavelength = (2*grid spacing)/(cutoff fraction).

Best regards, Mario E. Sigismondi

Dear Subhendu Mondal:

Following the former example: if the input grid that you will use to process and compute Euler 3D´s solution is smaller (300 m in this particular case), the output apparent depth to the magnetic source derived from Euler’s homogeneity equation will cover more area without “blanks”, and the depth solutions will be more precise (there will be a smaller range of depth uncertainty, Figure 1); also, it will be smoother (red line in the profile, Figure 2) if you compare with a coarse grid (1000 m in this example). Both maps in Figure 1 have been produced with a 1000m grid cell size from Euler´s database.

Best wishes, Mario

Dear Prof. Mario Emilio Sigismondi thank you so much for your kind effort and sharing your observations here. From your figure, it can be concluded that, if the data point spacing is not ideally or scientifically( following the rule of spatial Nyquist aliasing rule) depth values can be over or underestimated from 3D Eular depth solutions.

Both Geo-potential anomaly (Gravity & magnetic) maps contain valuable information that's unknown, unless one has some understanding of the fundamental processes of the data point distributions.