the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 06 Jan 2025
Spatial error constraints reduce overfitting for potential field geophysical inversion
Abstract. Geophysical inversion is an important tool for characterising the structure of the Earth. The utility of geophysical inversion has led to widespread adoption by resource explorers, and used to adapt gravity, magnetic, seismic and electrical datasets into petrophysical models that can be used for targeting. However, inherent ambiguity means that an infinite number of petrophysical models exist that can explain the geophysical data, so constraints such as geological models and petrophysical data have been employed to reduce the solution space. The constraints, like the data, are subject to noise and error resulting in uncertainty propagating to the final model. This is because inversion is designed to use the algorithm and constraints to find the ‘best’ solution by optimising the lowest misfit between the data and model. If the data is uncertain, the model fit to that data is likewise uncertain, and misrepresentative. Optimising misfit also means that inversion is subject to overfitting. Overfitting is when the lowest misfit values are attained by fitting the model to data noise. Overfitting inversion can create anomalies in the near-surface that can be mistakenly identified as legitimate targets for exploration rather than possible model artefacts. This contribution describes the use of spatial error constraints calculated from geophysical data to reduce overfitting for geophysical inversion. The spatial error estimate is derived from a geostatistical model calculated using Integrated Nested Laplacian Approximation (INLA). A region in the East Kimberley, northern Western Australia, is subject to gravity inversion using Tomofast-x, an open-source inversion platform. Inversion using different percentiles from the geophysical model explores whether the extrema of gravimetry values should be considered to explore the model space. Examination of inversion using and not using spatial error constraints shows that overfitting reduction can be achieved while using different percentiles as the observed field has lesser benefits.
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RC1: 'Comment on egusphere-2024-3754', Anonymous Referee #1, 05 Mar 2025
This paper adresses an important issue in geophysical inversion namely how to estimate uncertainties for data where such estimates are not readily available. The authors use gravity as an example and it is common practice to assume a constant error for all data points. This is clearly a strong simplification and I was hoping for a thorough geostatistical treatment of the topic to inform my own inversion practice. Unfortunately, in its current form I do not think the paper achieves this and I have doubts about the fundamental premise of the study. I think all aspects have to be reconsidered and the presentation has to be improved significantly. In its current form I have significant trouble to understand what the authors are doing. For this reason I recommend to reject the manuscript in its current form. I will explain my concerns in more detail below.
1. I find the writing style difficult to follow. The description switches between a tutorial style (lines 78-91) and compressing complex technical issues into a couple of sentences, e.g. the description of INLA in lines 286-294 (the previous paragraph makes only very general statements) or the discussion of previous geostatistical approaches in Lines 142-150. I also find many sentences very hard to follow, e.g. Line 207 "The second approach is to focus parameter changes in parts of the petrophysical model which are supported by uncertain regions interpolated grid." Line 613 "That these small anomalies only feature in one of our three slices would suggest that they are unlikely given they are produced from a gravity realisation that sits in the tails of the probability distribution, but may not be simply noise." I think the mansucript needs a thorough revision and rebalancing.
2. Many figures are poor quality and contain elements that cannot be read even at high zoom levels of the pdf (e.g. Figures 7, 8). The different panels in Figure 8 use different color scales for gravity (which can only been seen when zooming in close). I cannot draw much useful information from the misfit plots in Figure 11. The authors should think of a different way of presenting this information, maybe zoom into a representative region that patterns (or lack thereof) can be better understood. The caption for Figure 12 states "The selection of anomalies is made for visualisation and bears no particular importance." So why are these features included in the visualization. The model in Figure 9b from the previous study should be visualized in a style suitable for detailed comparison. The multi-physics presentation with two simultaneous color gradients for two different physical properties makes it hard to impossible for the reader to compare.
3. The spatial error estimation is based on the INLA method, but the description of this method is inadequate. Are there any user determined parameters that need to be specified? How are these selected and what is the impact on the result? What are the latent variables in this context? Is the Gaussian approximation produced by INLA adequate for the data used here? How can this be identified?
4. I would challenge the assertion that " grids [are] the format typically used." Line 172. If this is true in some contexts, then I question the appropriateness of such an approach outside traditional FFT based analysis of potential field data. For physical modelling based inversions (such as Tomofast-X and equivalent codes), I see no reason why would perform such an interpolation. Looking at Figure 2 the authors have generated 40,000 "measurements" out of 700 actual readings, i.e. 98% of the interpolated dataset are not based on reality. Furthermore, the interpolation is purely mathematical and thus does not obey the physics of potential fields. As a consequence the interpolated points will also violate potential field physics and do not contain any significant information. The authors rectify this problem by introducing their spatially varying error estimate, but I would argue that you should not interpolate to begin with. I would also like to note that the paper by Lelievre et al. 2009 given as a reference for the use of gridded data does not state that they use gridded data. In fact they state "direct observations should be considered more reliable than interpolations" and the other reference (Fullagar et al. 2000) is not part of the reference list. All modern potential field inversion codes I am aware of can deal with gravity data in irregular locations and use actual measurements, so I cannot see the value in interpolating and then penalizing.
5. As explained in 4. I doubt the application of the spatial uncertainty estimation on gridded data is very useful as we first create artificial data only to then penalize it with high uncertainty to exclude it from the inversion. A more useful and very interesting application of this method would be to use a relatively dense gravity survey (e.g. airborne) and see if spatial uncertainties can be derived from such a dataset without interpolation. The fundamental issue dealt with here is significant: Traditionally we would ascribe the same error to all measurements but for various reasons (corrections, measurement issues) this might not be the case. Thus any progress in this direction would be very helpful. I hope my comments help to achieve this.
6. A more minor comment: The paper is written very much from a mineral exploration perspective. For Solid Earth a more broad view would invite a broader readership.
Citation: https://doi.org/10.5194/egusphere-2024-3754-RC1 -
RC2: 'Comment on egusphere-2024-3754', Anonymous Referee #2, 03 Jun 2025
In this paper, the authors are tackling the important issue of estimating data uncertainties in the context of potential data inversion when sparse datasets are processed. In general, constant/same errors for all data are taken into account in the inversion processes, but in fact this can not be the case in all cases due to spatially lacks of data measurments. Therefore, the authors propose to estimate and add error covariance matrices to counterbalance the uncertainties in some spatial areas where data distributions are sparse. And this is one of my main concerns. Why do the authors interpolate data and then penalize them ? I do not see why the authors are doing this. Firstly, because there is an error in the measurments and secondly because additional errors are introduced by using interpolated data. It would be more interesting to consider only the real data where they actually are and adapt the density model grid (different SRTM topography grid resolutions for instance) to the actual location of the data points. Furthermore, the computational resources used to invert interpolated data can be much more important than when using spatially sparse data distributions. Besides, depending on the degree of sparsity of the data (dense or sparse data regions), the wavelengths that can be solved are different. So, what can be interesting is to perform a first FFT analysis to estimate and separate the different wavelengths that can be solved by inversion etc ...
- Besides, it seems that the authors are trying to provide a protocol of finding the best models for interpolated or not interpolated data on grids. However, the authors are not providing a real mathematical or physical criterion to discriminate those best models. It seems that only a visual discrimination procedure is depicted. For instance authors are claiming this at lines 429-433 in the first paragraph of section 3.4 "Inversion Visualisation" and particularly in the sentence "The thresholds for the relative density values were chosen mainly to aid visualisation of interesting geobodies rather than by some sophisticated statistical measure. Note while the range of values for each inversion ....". Furthermore part of everything presented after this paragraph is based on this procedure, which is not a mathematical objective criterion. All the discussion and results are mainly based on trial and error tests without substantial rigorous statistical and mathematical procedures.
- The results and discussions detailed in sections 3 and 4 (please replace "1. Results" by "3. results" in page 10, and "2. Discussion" by "4. Discussion") should be better presented. Sometimes it is very confusing to understand in which case [ i.e for interpolated data or not, and with or without uncertainties (spatial error constraints) taken into account in the inversions ] the solutions are the best. More particularly what are the objective (i. not subjective) mathematical criteria providing the highest probability of having a coherent and physical solution distribution or not ? An effort of synthesis of the results for the different cases should be done. For instance, a table summarizing/synthetizing them would be a suited way to clarify all the long discussions of the results in sections "3.4 Inversion Visualisation”, "3.5 Misfit” and “4. Discussion”. Besides, many results of the models obtained after inversion are discussed on the basis of visual arguments. This looks like somewhat sloppy. The authors should propose a suitable metric to estimate which model would have the highest probability to be acceptable physically. From a mathematical and physical point of view this will give more consistency to the results obtained.
- In the abstract, the introduction and also in the text body the INLA method is used. However, the method is never described and the parametrization of this method for the special geophysical case studied here is never provided. So, how the results could be reproduced by different users ? Furthermore, the authors are not clearly explaining the choice of this method nor justifying the Gaussian approximation applied to the data used here.
- What do authors mean exactly by "extrema of these alternatives" at line 224 page 6 ? Authors should state more clearly what this term (i.e "extrema") mean since it appears several times in the rest of the manuscript. This term should be clarified everywhere in the text.
- Lines 294 page 9 : What do "lower limit" and "upper limit" grids mean in practice ? Is this related to small or huge grids in size with large or small spatial discretization steps for data or density model grids (or for both data and density model grids) ? This should be better explained because the reader can not find out what those terms are covering exactly.
- At lines 538-542, authors are saying that boundary effects are immediately recognizable and can be ignored. So, if those artefacts are present, do they have an impact on the solutions in the most inner part of the computational domain or not ? What is the impact on the solutions and what is the criterion used to justify that these artefacts can be ignored ? This claim is very weird. This should be clarified.
- In the discussion section (by the way this should be section 4 and not section 2 !!!!), at lines 574-579, authors write :
" The next step is to establish the impact this has on the use of geophysics for mineral exploration. From a practical perspective, one may not care that the inversion has overfitted, especially since visual inversion results are quite similar (Figure 8) and the data cost for the overfitted results are very low. For some, this argument is valid, but it depends on where and what they are interested in."
So, if "from a practical perspective one may not care the inversion has overfitted, especially since visual inversion results are quite similar", why do the authors are performing all this present study ? It does not make any sense.
- Besides, the legends are too tiny and very hard to read in almost all the figures.
- In Figure 11-a, there are red and black (contour) lines. What do those colors represent ? In particular, what is the red line crossing the Figure. This is never mentioned. The units of the misfits in 11b and 11-c are not provided, please add them to the legends.
For all those reasons, and before any publication of this study, I suggest that major revisions of the manuscript have to be made and further efforts in terms of mathematical formalism and clarifications must be provided.
Best regards
The reviewer
Citation: https://doi.org/10.5194/egusphere-2024-3754-RC2
Status: closed
-
RC1: 'Comment on egusphere-2024-3754', Anonymous Referee #1, 05 Mar 2025
This paper adresses an important issue in geophysical inversion namely how to estimate uncertainties for data where such estimates are not readily available. The authors use gravity as an example and it is common practice to assume a constant error for all data points. This is clearly a strong simplification and I was hoping for a thorough geostatistical treatment of the topic to inform my own inversion practice. Unfortunately, in its current form I do not think the paper achieves this and I have doubts about the fundamental premise of the study. I think all aspects have to be reconsidered and the presentation has to be improved significantly. In its current form I have significant trouble to understand what the authors are doing. For this reason I recommend to reject the manuscript in its current form. I will explain my concerns in more detail below.
1. I find the writing style difficult to follow. The description switches between a tutorial style (lines 78-91) and compressing complex technical issues into a couple of sentences, e.g. the description of INLA in lines 286-294 (the previous paragraph makes only very general statements) or the discussion of previous geostatistical approaches in Lines 142-150. I also find many sentences very hard to follow, e.g. Line 207 "The second approach is to focus parameter changes in parts of the petrophysical model which are supported by uncertain regions interpolated grid." Line 613 "That these small anomalies only feature in one of our three slices would suggest that they are unlikely given they are produced from a gravity realisation that sits in the tails of the probability distribution, but may not be simply noise." I think the mansucript needs a thorough revision and rebalancing.
2. Many figures are poor quality and contain elements that cannot be read even at high zoom levels of the pdf (e.g. Figures 7, 8). The different panels in Figure 8 use different color scales for gravity (which can only been seen when zooming in close). I cannot draw much useful information from the misfit plots in Figure 11. The authors should think of a different way of presenting this information, maybe zoom into a representative region that patterns (or lack thereof) can be better understood. The caption for Figure 12 states "The selection of anomalies is made for visualisation and bears no particular importance." So why are these features included in the visualization. The model in Figure 9b from the previous study should be visualized in a style suitable for detailed comparison. The multi-physics presentation with two simultaneous color gradients for two different physical properties makes it hard to impossible for the reader to compare.
3. The spatial error estimation is based on the INLA method, but the description of this method is inadequate. Are there any user determined parameters that need to be specified? How are these selected and what is the impact on the result? What are the latent variables in this context? Is the Gaussian approximation produced by INLA adequate for the data used here? How can this be identified?
4. I would challenge the assertion that " grids [are] the format typically used." Line 172. If this is true in some contexts, then I question the appropriateness of such an approach outside traditional FFT based analysis of potential field data. For physical modelling based inversions (such as Tomofast-X and equivalent codes), I see no reason why would perform such an interpolation. Looking at Figure 2 the authors have generated 40,000 "measurements" out of 700 actual readings, i.e. 98% of the interpolated dataset are not based on reality. Furthermore, the interpolation is purely mathematical and thus does not obey the physics of potential fields. As a consequence the interpolated points will also violate potential field physics and do not contain any significant information. The authors rectify this problem by introducing their spatially varying error estimate, but I would argue that you should not interpolate to begin with. I would also like to note that the paper by Lelievre et al. 2009 given as a reference for the use of gridded data does not state that they use gridded data. In fact they state "direct observations should be considered more reliable than interpolations" and the other reference (Fullagar et al. 2000) is not part of the reference list. All modern potential field inversion codes I am aware of can deal with gravity data in irregular locations and use actual measurements, so I cannot see the value in interpolating and then penalizing.
5. As explained in 4. I doubt the application of the spatial uncertainty estimation on gridded data is very useful as we first create artificial data only to then penalize it with high uncertainty to exclude it from the inversion. A more useful and very interesting application of this method would be to use a relatively dense gravity survey (e.g. airborne) and see if spatial uncertainties can be derived from such a dataset without interpolation. The fundamental issue dealt with here is significant: Traditionally we would ascribe the same error to all measurements but for various reasons (corrections, measurement issues) this might not be the case. Thus any progress in this direction would be very helpful. I hope my comments help to achieve this.
6. A more minor comment: The paper is written very much from a mineral exploration perspective. For Solid Earth a more broad view would invite a broader readership.
Citation: https://doi.org/10.5194/egusphere-2024-3754-RC1 -
RC2: 'Comment on egusphere-2024-3754', Anonymous Referee #2, 03 Jun 2025
In this paper, the authors are tackling the important issue of estimating data uncertainties in the context of potential data inversion when sparse datasets are processed. In general, constant/same errors for all data are taken into account in the inversion processes, but in fact this can not be the case in all cases due to spatially lacks of data measurments. Therefore, the authors propose to estimate and add error covariance matrices to counterbalance the uncertainties in some spatial areas where data distributions are sparse. And this is one of my main concerns. Why do the authors interpolate data and then penalize them ? I do not see why the authors are doing this. Firstly, because there is an error in the measurments and secondly because additional errors are introduced by using interpolated data. It would be more interesting to consider only the real data where they actually are and adapt the density model grid (different SRTM topography grid resolutions for instance) to the actual location of the data points. Furthermore, the computational resources used to invert interpolated data can be much more important than when using spatially sparse data distributions. Besides, depending on the degree of sparsity of the data (dense or sparse data regions), the wavelengths that can be solved are different. So, what can be interesting is to perform a first FFT analysis to estimate and separate the different wavelengths that can be solved by inversion etc ...
- Besides, it seems that the authors are trying to provide a protocol of finding the best models for interpolated or not interpolated data on grids. However, the authors are not providing a real mathematical or physical criterion to discriminate those best models. It seems that only a visual discrimination procedure is depicted. For instance authors are claiming this at lines 429-433 in the first paragraph of section 3.4 "Inversion Visualisation" and particularly in the sentence "The thresholds for the relative density values were chosen mainly to aid visualisation of interesting geobodies rather than by some sophisticated statistical measure. Note while the range of values for each inversion ....". Furthermore part of everything presented after this paragraph is based on this procedure, which is not a mathematical objective criterion. All the discussion and results are mainly based on trial and error tests without substantial rigorous statistical and mathematical procedures.
- The results and discussions detailed in sections 3 and 4 (please replace "1. Results" by "3. results" in page 10, and "2. Discussion" by "4. Discussion") should be better presented. Sometimes it is very confusing to understand in which case [ i.e for interpolated data or not, and with or without uncertainties (spatial error constraints) taken into account in the inversions ] the solutions are the best. More particularly what are the objective (i. not subjective) mathematical criteria providing the highest probability of having a coherent and physical solution distribution or not ? An effort of synthesis of the results for the different cases should be done. For instance, a table summarizing/synthetizing them would be a suited way to clarify all the long discussions of the results in sections "3.4 Inversion Visualisation”, "3.5 Misfit” and “4. Discussion”. Besides, many results of the models obtained after inversion are discussed on the basis of visual arguments. This looks like somewhat sloppy. The authors should propose a suitable metric to estimate which model would have the highest probability to be acceptable physically. From a mathematical and physical point of view this will give more consistency to the results obtained.
- In the abstract, the introduction and also in the text body the INLA method is used. However, the method is never described and the parametrization of this method for the special geophysical case studied here is never provided. So, how the results could be reproduced by different users ? Furthermore, the authors are not clearly explaining the choice of this method nor justifying the Gaussian approximation applied to the data used here.
- What do authors mean exactly by "extrema of these alternatives" at line 224 page 6 ? Authors should state more clearly what this term (i.e "extrema") mean since it appears several times in the rest of the manuscript. This term should be clarified everywhere in the text.
- Lines 294 page 9 : What do "lower limit" and "upper limit" grids mean in practice ? Is this related to small or huge grids in size with large or small spatial discretization steps for data or density model grids (or for both data and density model grids) ? This should be better explained because the reader can not find out what those terms are covering exactly.
- At lines 538-542, authors are saying that boundary effects are immediately recognizable and can be ignored. So, if those artefacts are present, do they have an impact on the solutions in the most inner part of the computational domain or not ? What is the impact on the solutions and what is the criterion used to justify that these artefacts can be ignored ? This claim is very weird. This should be clarified.
- In the discussion section (by the way this should be section 4 and not section 2 !!!!), at lines 574-579, authors write :
" The next step is to establish the impact this has on the use of geophysics for mineral exploration. From a practical perspective, one may not care that the inversion has overfitted, especially since visual inversion results are quite similar (Figure 8) and the data cost for the overfitted results are very low. For some, this argument is valid, but it depends on where and what they are interested in."
So, if "from a practical perspective one may not care the inversion has overfitted, especially since visual inversion results are quite similar", why do the authors are performing all this present study ? It does not make any sense.
- Besides, the legends are too tiny and very hard to read in almost all the figures.
- In Figure 11-a, there are red and black (contour) lines. What do those colors represent ? In particular, what is the red line crossing the Figure. This is never mentioned. The units of the misfits in 11b and 11-c are not provided, please add them to the legends.
For all those reasons, and before any publication of this study, I suggest that major revisions of the manuscript have to be made and further efforts in terms of mathematical formalism and clarifications must be provided.
Best regards
The reviewer
Citation: https://doi.org/10.5194/egusphere-2024-3754-RC2
Data sets
A geophysical study using spatial error as inversion constraints Mark Lindsay, Vitaliy Orgarko, Jeremie Giraud, and Mosayeb Khademi https://data.csiro.au/collection/csiro:64073
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