the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Interpolation of magnetic anomalies over an oceanic ridge region using an equivalent source technique and crust age model constraint
Duan Li
Jinsong Du
Chao Chen
Qing Liang
Shida Sun
Abstract. Marine magnetic surveys over oceanic ridge regions are of great interest for investigations of structure and evolution of oceanic crust, and have played a key role in developing the theory of plate tectonics (Dyment, 1993; Maus et al, 2007; Vine and Matthews, 1963). In this study, we propose an interpolation approach based on the duallayer equivalent source model for the generation of a magnetic anomaly map based on sparse survey line data over oceanic ridge areas. In this approach, information from an ocean crust age model is utilized as constraint for the inversion procedure. The constraints can affect the magnetization distribution of equivalent sources following crust age. The results of synthetic tests show that the obtained magnetic anomalies have higher accuracy than those obtained by other interpolation methods. Meanwhile, considering the unclear on the true magnetization directions of sources and the background field in the synthetic model, well interpolation result can still be obtained. We applied the approach to magnetic data obtained from five survey lines east of the Southeast Indian Ridge. This prediction result is useful to improve the lithospheric magnetic field models WDMAMv2 and EMAG2v3, in the terms of spatial resolution and the consistency with observed data.
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Duan Li et al.
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RC1: 'General comments on se2021117', Anonymous Referee #1, 18 Nov 2021
General comments
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
 The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
 At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
 The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
 Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
 The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Citation: https://doi.org/10.5194/se2021117RC1 
AC2: 'Reply on RC1', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #1:
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
(1) The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
Reply: We agree with this comment. Insufficient data is a limitation for any method, and our work is to compare the accuracy of each method for data prediction in the same situation. The equivalent source (ES) method is to transform observed data into source, and then make data prediction through the source. Thus, the ES method is better in physical principle than the method based on morphological characteristics of data, and the calculation results also support the conclusion.
(2) At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
Reply: It is an expression of the research result of Li et al. (2020, GRL), indicating that the calculation accuracy can be effectively improved by improving the structure and distribution of the ES. Therefore, in our work, a similar technique is expected to achieve better interpolation result, which has also been proved in the synthetic model test.
(3) The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
Reply: The crustal age model is only used to provide constraints on the direction trend, so that the equivalent source could extends in a specific direction, which does not affect the data fitting. What we provided in this work is a constraint method or idea. In addition to crustal age, other directional constraint information can also be converted into the weighting factors to participate in the inversion. Moreover, we want to recover the magnetic anomaly field which is helpful to construct the global lithospheric magnetic field, such as the EMAG2v3 (Dyment et al., EPSL, 2015; Lesur et al., EPS, 2016) and WDMAMv2 (Meyer et al., G3, 2017). These models also used the crustal age model of Müller et al. (2008).
(4) Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
Reply: Since the isochron or boundary of lineation is discretized and corresponds to the equivalentsource cell one by one. Whether the lineation is aligned with the x or y direction, large values of w_{x} or w_{y} are taken for cell inside the lineation, small values of w_{x} or w_{y} are taken for cell at the boundary of lineation, or small values are taken for both w_{x} or w_{y}.
(5) The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Reply: We agree with this comment. As we answered in the last question, the constraint principle is the same regardless of whether the lineation changes are complex or not.
Citation: https://doi.org/10.5194/se2021117AC2

RC2: 'Specific comments on se2021117', Anonymous Referee #1, 18 Nov 2021
Specific comments
I have also some specific comments/recommendations:
 It seems that the method uses a topocentric Cartesian system with x pointing to North, y to East, and z pointing down, but I could not find this information in the manuscript.
 On page 6 is written that “Regularization and precondition techniques were utilized to stabilize the inversion process and balance the decay of the potential field”. I understand that a preconditioning technique, in this case, does not introduce a priori information about the parameter vector m (eq. 1), but only controls the convergence. So, could you please explain what is the a priori information introduced by matrix P (eq. 1) and how it contributes to stabilizing the inversion?
 I think that the elements forming matrices G and P (eq. 1) must be clearly defined in the manuscript. Note that, without specifying the elements of matrix G, the reader cannot know what type of equivalent sources (prisms, dipoles, etc) form the equivalent layer.
 I recommend using a tool model to illustrate how matrices Wx and Wy (eq. 3) are defined.
 According to page 7, matrices Wx and Wy (eq. 3) impose smoothness only on the physical property distribution of the shallow layer. Why they are not used to also impose smoothness on the deep layer?
 What are the criteria to define the depth/geometry of shallow and deep layers?
 On page 7, it is written that “A layer with larger ES cell sizes at larger depth was utilized to simulate the background magnetic field.”. I understand that "changing cell sizes" is possible only if the layer is formed by 3D sources. How to change the cell sizes of a layer formed, for example, by dipoles?
 Apparently, the weights wx and wy (elements of matrices Wx and Wy, eq. 3) do not have any normalization. In this case, it is expected that their numerical values depend on the particular characteristics of the study area and the interpretation model. As a consequence, it is not possible to use a fixed 10^4 in all situations. I recommend including some discussion about this.
 What is the “geophysical meaning” of the synthetic magnetic interface presented in Section 3? Could it be related to the Curie isotherm? In this case, I think it should be smooth. It seems that this simulated magnetic interface is a purely mathematical way of generating longwavelength data.
 The simulated main geomagnetic field presented in Section 3 is constant, with intensity, inclination, and declination of 35000 nT, 40°, and 3°, respectively. The crust model, however, covers an area of approximately 5° x 5°. Is it reasonable to consider that the main field is constant throughout this area?
 In my opinion, a detailed description of the parameters used to generate the results shown in Figure 2 with all methods must be included in the manuscript. Otherwise, it is not possible to obtain a proper comparison.
 The study area in Section 4 covers an approximately 5° x 5° area. Is it reasonable to consider that the main geomagnetic field is constant throughout this area? How the variability of the main field affects the results?
 I think that Figure 3 should be improved. I could not understand the relationship between the axes “Northing” and “Distance” in panel (b). Apparently, panel (e) shows the two layers, their equivalent sources, and the weights wx and wy (elements of matrices Wx and Wy in eq. 3) associated with them, but it is not clear for me.
Citation: https://doi.org/10.5194/se2021117RC2 
AC1: 'Reply on RC2', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #1:
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
(1) The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
Reply: We agree with this comment. Insufficient data is a limitation for any method, and our work is to compare the accuracy of each method for data prediction in the same situation. The equivalent source (ES) method is to transform observed data into source, and then make data prediction through the source. Thus, the ES method is better in physical principle than the method based on morphological characteristics of data, and the calculation results also support the conclusion.
(2) At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
Reply: It is an expression of the research result of Li et al. (2020, GRL), indicating that the calculation accuracy can be effectively improved by improving the structure and distribution of the ES. Therefore, in our work, a similar technique is expected to achieve better interpolation result, which has also been proved in the synthetic model test.
(3) The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
Reply: The crustal age model is only used to provide constraints on the direction trend, so that the equivalent source could extends in a specific direction, which does not affect the data fitting. What we provided in this work is a constraint method or idea. In addition to crustal age, other directional constraint information can also be converted into the weighting factors to participate in the inversion. Moreover, we want to recover the magnetic anomaly field which is helpful to construct the global lithospheric magnetic field, such as the EMAG2v3 (Dyment et al., EPSL, 2015; Lesur et al., EPS, 2016) and WDMAMv2 (Meyer et al., G3, 2017). These models also used the crustal age model of Müller et al. (2008).
(4) Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
Reply: Since the isochron or boundary of lineation is discretized and corresponds to the equivalentsource cell one by one. Whether the lineation is aligned with the x or y direction, large values of w_{x} or w_{y} are taken for cell inside the lineation, small values of w_{x} or w_{y} are taken for cell at the boundary of lineation, or small values are taken for both w_{x} or w_{y}.
(5) The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Reply: We agree with this comment. As we answered in the last question, the constraint principle is the same regardless of whether the lineation changes are complex or not.
I have also some specific comments/recommendations:
(6) It seems that the method uses a topocentric Cartesian system with x pointing to North, y to East, and z pointing down, but I could not find this information in the manuscript.
Reply: Thank you for the suggestion. We have defined the coordinate system in the manuscript.
(7) On page 6 is written that “Regularization and precondition techniques were utilized to stabilize the inversion process and balance the decay of the potential field”. I understand that a preconditioning technique, in this case, does not introduce a priori information about the parameter vector m (eq. 1), but only controls the convergence. So, could you please explain what is the a priori information introduced by matrix P (eq. 1) and how it contributes to stabilizing the inversion?
Reply: The introduction of P is in the supplementary materials for the manuscript, which was uploaded simultaneously. The diagonal element of P is z^{ß}, where z is the central depth of the ES cell and ß is the weighting index, which can be determined based on the attenuation characteristics of the potential field generated by the corresponding ES cell (e.g., Liu et al., 2015; Li et al., 2020).
(8) I think that the elements forming matrices G and P (eq. 1) must be clearly defined in the manuscript. Note that, without specifying the elements of matrix G, the reader cannot know what type of equivalent sources (prisms, dipoles, etc) form the equivalent layer.
Reply: Thank you for your suggestion. The element forming matrices G and P, and the type of equivalent sources are introduced in the supplementary materials for the manuscript, which was uploaded simultaneously.
(9) I recommend using a tool model to illustrate how matrices Wx and Wy (eq. 3) are defined.
Reply: Thank you for the suggestion. The definition of W_{x} and W_{y} are illustrated in the supplementary materials for the revised manuscript.
(10) According to page 7, matrices Wx and Wy (eq. 3) impose smoothness only on the physical property distribution of the shallow layer. Why they are not used to also impose smoothness on the deep layer?
Reply: The deep layer of equivalent source is utilized to simulate the background magnetic field, which is unpredictable in practice. Therefore, the deep layer is not constrained. We have added relevant explain in the revised manuscript.
(11) What are the criteria to define the depth/geometry of shallow and deep layers?
Reply: The shallow layer is placed in the depth range of 1~6 times the observed data spacing, which is the previous research experience. The depth of deep layer is determined by the logarithmic power spectrum of observed data. We have supplemented to the paragraph in the revised manuscript.
(12) On page 7, it is written that “A layer with larger ES cell sizes at larger depth was utilized to simulate the background magnetic field.”. I understand that "changing cell sizes" is possible only if the layer is formed by 3D sources. How to change the cell sizes of a layer formed, for example, by dipoles?
Reply: In this work, we adopted the prism as the cell to construct the duallayer equivalent source, which is illustrated in the supplementary materials for the revised manuscript. Therefore, the geometric size of the cell can be changed.
(13) Apparently, the weights w_{x} and w_{y} (elements of matrices Wx and Wy, eq. 3) do not have any normalization. In this case, it is expected that their numerical values depend on the particular characteristics of the study area and the interpretation model. As a consequence, it is not possible to use a fixed 10^4 in all situations. I recommend including some discussion about this.
Reply: The values of w_{x} and w_{y} are still based on experience at present, and we have not been able to work out a quantitative setting standard. In fact, these values only serve to increase the difference of magnitude between weighted and unweighted cells. In our experience, 10^{4} or 10^{4 }can generally work. If the effect of this setting is not obvious in the real data application, the values of w_{x} and w_{y }can be increased according to the change of calculation results.
(14) What is the “geophysical meaning” of the synthetic magnetic interface presented in Section 3? Could it be related to the Curie isotherm? In this case, I think it should be smooth. It seems that this simulated magnetic interface is a purely mathematical way of generating longwavelength data.
Reply: The background field is simulated by magnetic interface with random fluctuation, in order to simulate the unpredictable longwavelength information in practical. In some cases, longwavelength information is more than just Curie fluctuations. If the calculation accuracy of method can be guaranteed in this case, the processing effect may be better if longwavelength is simpler.
(15) The simulated main geomagnetic field presented in Section 3 is constant, with intensity, inclination, and declination of 35000 nT, 40°, and 3°, respectively. The crust model, however, covers an area of approximately 5° × 5°. Is it reasonable to consider that the main field is constant throughout this area?
Reply: In our work, the IGRF model was used to obtain the intensity at each data point, and the magnetic anomaly was obtained by subtracting the intensity. The magnetization of equivalent source is obtained by the inversion of magnetic anomaly. The average values of inclination and declination of the study area were used to the calculation. According to our previous works, using constant inclination and declination has a great influence on reducing to the pole, but the influence on interpolation and continuation can be ignored. In further work, we will use the variation inclination and declination in the calculation to improve the calculation process.
(16) In my opinion, a detailed description of the parameters used to generate the results shown in Figure 2 with all methods must be included in the manuscript. Otherwise, it is not possible to obtain a proper comparison.
Reply: When using other methods, the parameters selected are common and default. In addition, the directional constraint is also considered in the calculation of these methods, so we think it is comparable.
(17) The study area in Section 4 covers an approximately 5° x 5° area. Is it reasonable to consider that the main geomagnetic field is constant throughout this area? How the variability of the main field affects the results?
Reply: Thank you for the question. As the answered in question (15), the IGRF model was used to obtain the intensity at each data point, and the magnetic anomaly was obtained by subtracting the intensity. The magnetization of equivalent source is obtained by the inversion of magnetic anomaly.
(18) I think that Figure 3 should be improved. I could not understand the relationship between the axes “Northing” and “Distance” in panel (b). Apparently, panel (e) shows the two layers, their equivalent sources, and the weights w_{x} and w_{y} (elements of matrices Wx and Wy in eq. 3) associated with them, but it is not clear for me.
Reply: The main purpose of using “distance” on the right side of the figure is to draw each survey line separately. The “distance” is the distance between other measuring points and the southern vertex of each survey line. Panel (e) shows the value and distribution of the w_{x} and w_{y} of each equivalent source cell. We also tried other forms of expression, but we think the proposed design is easier to convey more information by compared these forms.
Citation: https://doi.org/10.5194/se2021117AC1

RC3: 'Comment on se2021117', Anonymous Referee #2, 18 Jan 2022
General comment on manuscript se2021117 by Li et al.
The manuscript deals with the application of a duallayer equivalent source technique to magnetic anomaly data in order to improve their interpolation in areas where data are insufficient. The authors choose to test their method in an oceanic ridge region along the Southeast Indian Ridge. This approach has been successfully applied in the past by the authors (Li et al., 2020) to improve the accuracy of prediction of the three components of the magnetic field. In this case the authors use the same method over oceanic ridge regions that, from the geological point of view, are characterized by complex geometry where acting different processes and where the magnetic properties of the source rocks are extremely variable and poorly known. We have very few information on the sources that produces the magnetic anomalies in the oceanic ridge region
The manuscript is well organized and the figures are clear although in some case too small to read and appreciate the values (Fig. 3e). I do not judge the method, that could be successfully applied to obtain useful magnetic anomaly map in area with a non uniform data coverage, but I suggest to the authors to better clarify and discuss some points.
 As also reported by the other reviewer, the decision to use as constraint the model crustal age that is originally retrieved by magnetic data is at least controversial. If is true that other interpolation methods suffer the scattered data, the need to use constraints or a priori information in regions so complex could introduce additional errors. In addition to the dividing line of crust age, what other parameters concerning the magnetization of the source body did you use as a priori information?
 About the synthetic model experiment, the magnetic properties in terms of susceptibility (or magnetic remanence) of the cells are not specified. Moreover, I do not understand how the background magnetic field was calculated. In this geological setting which is for the authors the source of this long wavelength? How the use of the deeper layer improve the synthetic model results? Is it necessary to add it to the model?
 About the real data example, what magnetic properties did you get from the inversion of observed data? I do not understand if the obtained values are similar for both shallow and deep layers. This information imply the knowledge of the nature of the deep source or the origin of the background magnetic field. I suggest to discuss what represent the background magnetic field in the oceanic ridge region.
 I suggest to clarify how the authors chose the sizes of the model cells in both the synthetic and the real example. How the sizes of the cells influence the results? In particular, it is not clear the reasons behind the choice of the deeper layer; cells with size of 80 km*80km*40 km seem too big. Moreover, this level has been localized, as said by the authors, at the depth of the Curie point but in the oceanic ridge region the Curie point is shallow and cells with 40 km of thickness would fall in the mantle where the temperature is certainly higher than the Curie point (see Li et al., 2017, A global reference model of Curiepoint depths based on EMAG2, Scientific Reports, 7).
 In the chapter 3 of supplementary materials have been reported the sizes of the cells used in the experiment. Being different from those reported in the text I do not understand if they refer to the synthetic model or to the real model. I suggest to clarify this point.
Citation: https://doi.org/10.5194/se2021117RC3 
AC3: 'Reply on RC3', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #2:
The manuscript is well organized and the figures are clear although in some case too small to read and appreciate the values (Fig. 3e). I do not judge the method, that could be successfully applied to obtain useful magnetic anomaly map in area with a non uniform data coverage, but I suggest to the authors to better clarify and discuss some points.
Reply: We appreciate greatly your comments and suggestion that are very helpful for improving the manuscript. Below we clarify them point by point.
(1) As also reported by the other reviewer, the decision to use as constraint the model crustal age that is originally retrieved by magnetic data is at least controversial. If is true that other interpolation methods suffer the scattered data, the need to use constraints or a priori information in regions so complex could introduce additional errors. In addition to the dividing line of crust age, what other parameters concerning the magnetization of the source body did you use as a priori information?
Reply: Thank you for the question. The purpose of using crust age model as the constraint is to extract the direction information for extending the equivalent source in a certain direction, in which the trend information is used. The constraint information does not affect the fitting of observed data. The proposed method supports the application of a variety of prior information as constrain. In addition to the constraint information in the manuscript, other prior geological or geophysical data can also be converted into weighted factors to constrain equivalent source. Of course, only the crust age model was used in the work. More constraint information and methods will be tested in future work.
(2) About the synthetic model experiment, the magnetic properties in terms of susceptibility (or magnetic remanence) of the cells are not specified. Moreover, I do not understand how the background magnetic field was calculated. In this geological setting which is for the authors the source of this long wavelength? How the use of the deeper layer improve the synthetic model results? Is it necessary to add it to the model?
Reply: The background field is generated by a magnetic interface with random fluctuations in order to simulate the unpredictable longwavelength information in practice. In some cases, the longwavelength information not only contains Curie surface, but also other unknown signals. If the proposed method can obtain better results under the conditions in this manuscript, the calculation accuracy can also be guaranteed when the longwavelength signal is simple.
(3) About the real data example, what magnetic properties did you get from the inversion of observed data? I do not understand if the obtained values are similar for both shallow and deep layers. This information imply the knowledge of the nature of the deep source or the origin of the background magnetic field. I suggest to discuss what represent the background magnetic field in the oceanic ridge region.
Reply: Thank you for the suggestion. The magnetization of equivalent source was obtained from the inversion of observed data. For the equivalent source, it cannot directly correspond to geological significance in some cases. Especially, when the deep equivalent source is placed deeper, it should be understood that the equivalent source a group of model coefficients rather than a group of physical sources. This case is very similar with the research by Kother et al. (GJI, 2015). They set the equivalent source depth as 100 km, which in fact also has no geological reasons but for obtaining the optimal results.
(4) I suggest to clarify how the authors chose the sizes of the model cells in both the synthetic and the real example. How the sizes of the cells influence the results? In particular, it is not clear the reasons behind the choice of the deeper layer; cells with size of 80 km*80km*40 km seem too big. Moreover, this level has been localized, as said by the authors, at the depth of the Curie point but in the oceanic ridge region the Curie point is shallow and cells with 40 km of thickness would fall in the mantle where the temperature is certainly higher than the Curie point (see Li et al., 2017, A global reference model of Curiepoint depths based on EMAG2, Scientific Reports, 7).
Reply: Thank you for the suggestion. We place the top surface of deep layer near the Curie point, and its bottom extends to the deeper depth, in order to better rebuild the amplitude of magnetic anomaly. Relevant expressions have been revised in the updated manuscript.
(5) In the chapter 3 of supplementary materials have been reported the sizes of the cells used in the experiment. Being different from those reported in the text I do not understand if they refer to the synthetic model or to the real model. I suggest to clarify this point.
Reply: In the supplementary materials, the equivalent source was constructed for the theoretical model. In the report, the equivalent source was constructed for the real data. Because the different range and data spacing between these two data sets, these two equivalent sources are different.
Citation: https://doi.org/10.5194/se2021117AC3
Status: closed

RC1: 'General comments on se2021117', Anonymous Referee #1, 18 Nov 2021
General comments
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
 The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
 At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
 The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
 Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
 The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Citation: https://doi.org/10.5194/se2021117RC1 
AC2: 'Reply on RC1', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #1:
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
(1) The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
Reply: We agree with this comment. Insufficient data is a limitation for any method, and our work is to compare the accuracy of each method for data prediction in the same situation. The equivalent source (ES) method is to transform observed data into source, and then make data prediction through the source. Thus, the ES method is better in physical principle than the method based on morphological characteristics of data, and the calculation results also support the conclusion.
(2) At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
Reply: It is an expression of the research result of Li et al. (2020, GRL), indicating that the calculation accuracy can be effectively improved by improving the structure and distribution of the ES. Therefore, in our work, a similar technique is expected to achieve better interpolation result, which has also been proved in the synthetic model test.
(3) The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
Reply: The crustal age model is only used to provide constraints on the direction trend, so that the equivalent source could extends in a specific direction, which does not affect the data fitting. What we provided in this work is a constraint method or idea. In addition to crustal age, other directional constraint information can also be converted into the weighting factors to participate in the inversion. Moreover, we want to recover the magnetic anomaly field which is helpful to construct the global lithospheric magnetic field, such as the EMAG2v3 (Dyment et al., EPSL, 2015; Lesur et al., EPS, 2016) and WDMAMv2 (Meyer et al., G3, 2017). These models also used the crustal age model of Müller et al. (2008).
(4) Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
Reply: Since the isochron or boundary of lineation is discretized and corresponds to the equivalentsource cell one by one. Whether the lineation is aligned with the x or y direction, large values of w_{x} or w_{y} are taken for cell inside the lineation, small values of w_{x} or w_{y} are taken for cell at the boundary of lineation, or small values are taken for both w_{x} or w_{y}.
(5) The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Reply: We agree with this comment. As we answered in the last question, the constraint principle is the same regardless of whether the lineation changes are complex or not.
Citation: https://doi.org/10.5194/se2021117AC2

RC2: 'Specific comments on se2021117', Anonymous Referee #1, 18 Nov 2021
Specific comments
I have also some specific comments/recommendations:
 It seems that the method uses a topocentric Cartesian system with x pointing to North, y to East, and z pointing down, but I could not find this information in the manuscript.
 On page 6 is written that “Regularization and precondition techniques were utilized to stabilize the inversion process and balance the decay of the potential field”. I understand that a preconditioning technique, in this case, does not introduce a priori information about the parameter vector m (eq. 1), but only controls the convergence. So, could you please explain what is the a priori information introduced by matrix P (eq. 1) and how it contributes to stabilizing the inversion?
 I think that the elements forming matrices G and P (eq. 1) must be clearly defined in the manuscript. Note that, without specifying the elements of matrix G, the reader cannot know what type of equivalent sources (prisms, dipoles, etc) form the equivalent layer.
 I recommend using a tool model to illustrate how matrices Wx and Wy (eq. 3) are defined.
 According to page 7, matrices Wx and Wy (eq. 3) impose smoothness only on the physical property distribution of the shallow layer. Why they are not used to also impose smoothness on the deep layer?
 What are the criteria to define the depth/geometry of shallow and deep layers?
 On page 7, it is written that “A layer with larger ES cell sizes at larger depth was utilized to simulate the background magnetic field.”. I understand that "changing cell sizes" is possible only if the layer is formed by 3D sources. How to change the cell sizes of a layer formed, for example, by dipoles?
 Apparently, the weights wx and wy (elements of matrices Wx and Wy, eq. 3) do not have any normalization. In this case, it is expected that their numerical values depend on the particular characteristics of the study area and the interpretation model. As a consequence, it is not possible to use a fixed 10^4 in all situations. I recommend including some discussion about this.
 What is the “geophysical meaning” of the synthetic magnetic interface presented in Section 3? Could it be related to the Curie isotherm? In this case, I think it should be smooth. It seems that this simulated magnetic interface is a purely mathematical way of generating longwavelength data.
 The simulated main geomagnetic field presented in Section 3 is constant, with intensity, inclination, and declination of 35000 nT, 40°, and 3°, respectively. The crust model, however, covers an area of approximately 5° x 5°. Is it reasonable to consider that the main field is constant throughout this area?
 In my opinion, a detailed description of the parameters used to generate the results shown in Figure 2 with all methods must be included in the manuscript. Otherwise, it is not possible to obtain a proper comparison.
 The study area in Section 4 covers an approximately 5° x 5° area. Is it reasonable to consider that the main geomagnetic field is constant throughout this area? How the variability of the main field affects the results?
 I think that Figure 3 should be improved. I could not understand the relationship between the axes “Northing” and “Distance” in panel (b). Apparently, panel (e) shows the two layers, their equivalent sources, and the weights wx and wy (elements of matrices Wx and Wy in eq. 3) associated with them, but it is not clear for me.
Citation: https://doi.org/10.5194/se2021117RC2 
AC1: 'Reply on RC2', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #1:
The manuscript proposes a spacedomain method that uses a pair of equivalent layers for interpolating sparse totalfield anomaly data on the oceanic crust by using an age model as a constraint. Although not clearly specified in the manuscript, the method is developed in a topocentric Cartesian Coordinate system with x, y, and z axes pointing, respectively, to North, East, and down. The method consists in solving a constrainedlinear inverse problem for simultaneously estimating the physical property distribution on the two layers that yields an acceptable totalfield anomaly data fit. The method imposes smoothness along isochrons of oceanic crust only on the physical property distribution of the shallow layer with the purpose of filling the gaps of totalfield anomaly data. For me, the manuscript needs to be significantly improved before being considered for publication. The main problems are listed below:
(1) The equivalentlayer technique is offered as a better alternative to kriging, minimum curvature, cubic spline interpolation, and inverse distance weighting methods for interpolating sparse totalfield anomaly data on oceanic crust because “these methods might not be optimal for the data prediction in areas with insufficient data” (page 4). The problem here is that the equivalentlayer technique is also negatively affected by insufficient data.
Reply: We agree with this comment. Insufficient data is a limitation for any method, and our work is to compare the accuracy of each method for data prediction in the same situation. The equivalent source (ES) method is to transform observed data into source, and then make data prediction through the source. Thus, the ES method is better in physical principle than the method based on morphological characteristics of data, and the calculation results also support the conclusion.
(2) At the end of page 4, beginning of page 5, it is written that the equivalentlayer technique may provide a more accurate magnetic field because it is possible to improve its structure and distribution. In my opinion, this justification should be considerably improved. It is not clear how the structure and distribution of the equivalent layer can be modified to produce a more accurate field at the interpolating points. I understand that, by increasing the number of sources composing the equivalent layer, it is possible to obtain an exact data fit at the observation points because the inverse problem becomes underdetermined.
Reply: It is an expression of the research result of Li et al. (2020, GRL), indicating that the calculation accuracy can be effectively improved by improving the structure and distribution of the ES. Therefore, in our work, a similar technique is expected to achieve better interpolation result, which has also been proved in the synthetic model test.
(3) The proposed method uses the crustal age model of Müller et al. (2008) as a priori information for constraining the linear inversion of totalfield anomaly data on oceanic crust. This model, in turn, was obtained on the basis of marine magnetic anomaly identifications. It seems that there is a circular reasoning problem here. Because the age model depends on the magnetic data, it does not necessarily introduce new information into the inverse problem and apparently cannot be used as a constraint.
Reply: The crustal age model is only used to provide constraints on the direction trend, so that the equivalent source could extends in a specific direction, which does not affect the data fitting. What we provided in this work is a constraint method or idea. In addition to crustal age, other directional constraint information can also be converted into the weighting factors to participate in the inversion. Moreover, we want to recover the magnetic anomaly field which is helpful to construct the global lithospheric magnetic field, such as the EMAG2v3 (Dyment et al., EPSL, 2015; Lesur et al., EPS, 2016) and WDMAMv2 (Meyer et al., G3, 2017). These models also used the crustal age model of Müller et al. (2008).
(4) Matrices Wx and Wy (eq. 3) impose smoothness along x and y directions. However, the isochrons are not necessarily aligned with x or y directions. So, it is important to clearly explain how the proposed method deals with isochrons that are not aligned with the x or y directions.
Reply: Since the isochron or boundary of lineation is discretized and corresponds to the equivalentsource cell one by one. Whether the lineation is aligned with the x or y direction, large values of w_{x} or w_{y} are taken for cell inside the lineation, small values of w_{x} or w_{y} are taken for cell at the boundary of lineation, or small values are taken for both w_{x} or w_{y}.
(5) The simulated crust (Figure 1) has isochrons that are perfectly aligned with the NorthSouth direction (xaxis). In this case, matrix Wx (eq. 3) can be used to impose a strong smoothness along the xdirection. However, this model represents a very ideal situation. The simulated survey lines (Figure 2) are perfectly orthogonal to the simulated isochrons. This is also a very ideal situation. For me, the test with synthetic data presented in Section 3 should be used as an initial validation test. The conclusions obtained from this test cannot fully support the interpretation of real data. In my opinion, additional tests with synthetic data produced by models reproducing or at least approximating the complexity of a real magnetic survey on oceanic crust should be included in the manuscript.
Reply: We agree with this comment. As we answered in the last question, the constraint principle is the same regardless of whether the lineation changes are complex or not.
I have also some specific comments/recommendations:
(6) It seems that the method uses a topocentric Cartesian system with x pointing to North, y to East, and z pointing down, but I could not find this information in the manuscript.
Reply: Thank you for the suggestion. We have defined the coordinate system in the manuscript.
(7) On page 6 is written that “Regularization and precondition techniques were utilized to stabilize the inversion process and balance the decay of the potential field”. I understand that a preconditioning technique, in this case, does not introduce a priori information about the parameter vector m (eq. 1), but only controls the convergence. So, could you please explain what is the a priori information introduced by matrix P (eq. 1) and how it contributes to stabilizing the inversion?
Reply: The introduction of P is in the supplementary materials for the manuscript, which was uploaded simultaneously. The diagonal element of P is z^{ß}, where z is the central depth of the ES cell and ß is the weighting index, which can be determined based on the attenuation characteristics of the potential field generated by the corresponding ES cell (e.g., Liu et al., 2015; Li et al., 2020).
(8) I think that the elements forming matrices G and P (eq. 1) must be clearly defined in the manuscript. Note that, without specifying the elements of matrix G, the reader cannot know what type of equivalent sources (prisms, dipoles, etc) form the equivalent layer.
Reply: Thank you for your suggestion. The element forming matrices G and P, and the type of equivalent sources are introduced in the supplementary materials for the manuscript, which was uploaded simultaneously.
(9) I recommend using a tool model to illustrate how matrices Wx and Wy (eq. 3) are defined.
Reply: Thank you for the suggestion. The definition of W_{x} and W_{y} are illustrated in the supplementary materials for the revised manuscript.
(10) According to page 7, matrices Wx and Wy (eq. 3) impose smoothness only on the physical property distribution of the shallow layer. Why they are not used to also impose smoothness on the deep layer?
Reply: The deep layer of equivalent source is utilized to simulate the background magnetic field, which is unpredictable in practice. Therefore, the deep layer is not constrained. We have added relevant explain in the revised manuscript.
(11) What are the criteria to define the depth/geometry of shallow and deep layers?
Reply: The shallow layer is placed in the depth range of 1~6 times the observed data spacing, which is the previous research experience. The depth of deep layer is determined by the logarithmic power spectrum of observed data. We have supplemented to the paragraph in the revised manuscript.
(12) On page 7, it is written that “A layer with larger ES cell sizes at larger depth was utilized to simulate the background magnetic field.”. I understand that "changing cell sizes" is possible only if the layer is formed by 3D sources. How to change the cell sizes of a layer formed, for example, by dipoles?
Reply: In this work, we adopted the prism as the cell to construct the duallayer equivalent source, which is illustrated in the supplementary materials for the revised manuscript. Therefore, the geometric size of the cell can be changed.
(13) Apparently, the weights w_{x} and w_{y} (elements of matrices Wx and Wy, eq. 3) do not have any normalization. In this case, it is expected that their numerical values depend on the particular characteristics of the study area and the interpretation model. As a consequence, it is not possible to use a fixed 10^4 in all situations. I recommend including some discussion about this.
Reply: The values of w_{x} and w_{y} are still based on experience at present, and we have not been able to work out a quantitative setting standard. In fact, these values only serve to increase the difference of magnitude between weighted and unweighted cells. In our experience, 10^{4} or 10^{4 }can generally work. If the effect of this setting is not obvious in the real data application, the values of w_{x} and w_{y }can be increased according to the change of calculation results.
(14) What is the “geophysical meaning” of the synthetic magnetic interface presented in Section 3? Could it be related to the Curie isotherm? In this case, I think it should be smooth. It seems that this simulated magnetic interface is a purely mathematical way of generating longwavelength data.
Reply: The background field is simulated by magnetic interface with random fluctuation, in order to simulate the unpredictable longwavelength information in practical. In some cases, longwavelength information is more than just Curie fluctuations. If the calculation accuracy of method can be guaranteed in this case, the processing effect may be better if longwavelength is simpler.
(15) The simulated main geomagnetic field presented in Section 3 is constant, with intensity, inclination, and declination of 35000 nT, 40°, and 3°, respectively. The crust model, however, covers an area of approximately 5° × 5°. Is it reasonable to consider that the main field is constant throughout this area?
Reply: In our work, the IGRF model was used to obtain the intensity at each data point, and the magnetic anomaly was obtained by subtracting the intensity. The magnetization of equivalent source is obtained by the inversion of magnetic anomaly. The average values of inclination and declination of the study area were used to the calculation. According to our previous works, using constant inclination and declination has a great influence on reducing to the pole, but the influence on interpolation and continuation can be ignored. In further work, we will use the variation inclination and declination in the calculation to improve the calculation process.
(16) In my opinion, a detailed description of the parameters used to generate the results shown in Figure 2 with all methods must be included in the manuscript. Otherwise, it is not possible to obtain a proper comparison.
Reply: When using other methods, the parameters selected are common and default. In addition, the directional constraint is also considered in the calculation of these methods, so we think it is comparable.
(17) The study area in Section 4 covers an approximately 5° x 5° area. Is it reasonable to consider that the main geomagnetic field is constant throughout this area? How the variability of the main field affects the results?
Reply: Thank you for the question. As the answered in question (15), the IGRF model was used to obtain the intensity at each data point, and the magnetic anomaly was obtained by subtracting the intensity. The magnetization of equivalent source is obtained by the inversion of magnetic anomaly.
(18) I think that Figure 3 should be improved. I could not understand the relationship between the axes “Northing” and “Distance” in panel (b). Apparently, panel (e) shows the two layers, their equivalent sources, and the weights w_{x} and w_{y} (elements of matrices Wx and Wy in eq. 3) associated with them, but it is not clear for me.
Reply: The main purpose of using “distance” on the right side of the figure is to draw each survey line separately. The “distance” is the distance between other measuring points and the southern vertex of each survey line. Panel (e) shows the value and distribution of the w_{x} and w_{y} of each equivalent source cell. We also tried other forms of expression, but we think the proposed design is easier to convey more information by compared these forms.
Citation: https://doi.org/10.5194/se2021117AC1

RC3: 'Comment on se2021117', Anonymous Referee #2, 18 Jan 2022
General comment on manuscript se2021117 by Li et al.
The manuscript deals with the application of a duallayer equivalent source technique to magnetic anomaly data in order to improve their interpolation in areas where data are insufficient. The authors choose to test their method in an oceanic ridge region along the Southeast Indian Ridge. This approach has been successfully applied in the past by the authors (Li et al., 2020) to improve the accuracy of prediction of the three components of the magnetic field. In this case the authors use the same method over oceanic ridge regions that, from the geological point of view, are characterized by complex geometry where acting different processes and where the magnetic properties of the source rocks are extremely variable and poorly known. We have very few information on the sources that produces the magnetic anomalies in the oceanic ridge region
The manuscript is well organized and the figures are clear although in some case too small to read and appreciate the values (Fig. 3e). I do not judge the method, that could be successfully applied to obtain useful magnetic anomaly map in area with a non uniform data coverage, but I suggest to the authors to better clarify and discuss some points.
 As also reported by the other reviewer, the decision to use as constraint the model crustal age that is originally retrieved by magnetic data is at least controversial. If is true that other interpolation methods suffer the scattered data, the need to use constraints or a priori information in regions so complex could introduce additional errors. In addition to the dividing line of crust age, what other parameters concerning the magnetization of the source body did you use as a priori information?
 About the synthetic model experiment, the magnetic properties in terms of susceptibility (or magnetic remanence) of the cells are not specified. Moreover, I do not understand how the background magnetic field was calculated. In this geological setting which is for the authors the source of this long wavelength? How the use of the deeper layer improve the synthetic model results? Is it necessary to add it to the model?
 About the real data example, what magnetic properties did you get from the inversion of observed data? I do not understand if the obtained values are similar for both shallow and deep layers. This information imply the knowledge of the nature of the deep source or the origin of the background magnetic field. I suggest to discuss what represent the background magnetic field in the oceanic ridge region.
 I suggest to clarify how the authors chose the sizes of the model cells in both the synthetic and the real example. How the sizes of the cells influence the results? In particular, it is not clear the reasons behind the choice of the deeper layer; cells with size of 80 km*80km*40 km seem too big. Moreover, this level has been localized, as said by the authors, at the depth of the Curie point but in the oceanic ridge region the Curie point is shallow and cells with 40 km of thickness would fall in the mantle where the temperature is certainly higher than the Curie point (see Li et al., 2017, A global reference model of Curiepoint depths based on EMAG2, Scientific Reports, 7).
 In the chapter 3 of supplementary materials have been reported the sizes of the cells used in the experiment. Being different from those reported in the text I do not understand if they refer to the synthetic model or to the real model. I suggest to clarify this point.
Citation: https://doi.org/10.5194/se2021117RC3 
AC3: 'Reply on RC3', Jinsong Du, 20 Feb 2022
We are also grateful to the reviewers for the assessments, comments, suggestion and recommendations. All of them are carefully considered while revising the manuscript. Below we provide a pointbypoint response to all pieces of suggestion and comments.
Reviewer #2:
The manuscript is well organized and the figures are clear although in some case too small to read and appreciate the values (Fig. 3e). I do not judge the method, that could be successfully applied to obtain useful magnetic anomaly map in area with a non uniform data coverage, but I suggest to the authors to better clarify and discuss some points.
Reply: We appreciate greatly your comments and suggestion that are very helpful for improving the manuscript. Below we clarify them point by point.
(1) As also reported by the other reviewer, the decision to use as constraint the model crustal age that is originally retrieved by magnetic data is at least controversial. If is true that other interpolation methods suffer the scattered data, the need to use constraints or a priori information in regions so complex could introduce additional errors. In addition to the dividing line of crust age, what other parameters concerning the magnetization of the source body did you use as a priori information?
Reply: Thank you for the question. The purpose of using crust age model as the constraint is to extract the direction information for extending the equivalent source in a certain direction, in which the trend information is used. The constraint information does not affect the fitting of observed data. The proposed method supports the application of a variety of prior information as constrain. In addition to the constraint information in the manuscript, other prior geological or geophysical data can also be converted into weighted factors to constrain equivalent source. Of course, only the crust age model was used in the work. More constraint information and methods will be tested in future work.
(2) About the synthetic model experiment, the magnetic properties in terms of susceptibility (or magnetic remanence) of the cells are not specified. Moreover, I do not understand how the background magnetic field was calculated. In this geological setting which is for the authors the source of this long wavelength? How the use of the deeper layer improve the synthetic model results? Is it necessary to add it to the model?
Reply: The background field is generated by a magnetic interface with random fluctuations in order to simulate the unpredictable longwavelength information in practice. In some cases, the longwavelength information not only contains Curie surface, but also other unknown signals. If the proposed method can obtain better results under the conditions in this manuscript, the calculation accuracy can also be guaranteed when the longwavelength signal is simple.
(3) About the real data example, what magnetic properties did you get from the inversion of observed data? I do not understand if the obtained values are similar for both shallow and deep layers. This information imply the knowledge of the nature of the deep source or the origin of the background magnetic field. I suggest to discuss what represent the background magnetic field in the oceanic ridge region.
Reply: Thank you for the suggestion. The magnetization of equivalent source was obtained from the inversion of observed data. For the equivalent source, it cannot directly correspond to geological significance in some cases. Especially, when the deep equivalent source is placed deeper, it should be understood that the equivalent source a group of model coefficients rather than a group of physical sources. This case is very similar with the research by Kother et al. (GJI, 2015). They set the equivalent source depth as 100 km, which in fact also has no geological reasons but for obtaining the optimal results.
(4) I suggest to clarify how the authors chose the sizes of the model cells in both the synthetic and the real example. How the sizes of the cells influence the results? In particular, it is not clear the reasons behind the choice of the deeper layer; cells with size of 80 km*80km*40 km seem too big. Moreover, this level has been localized, as said by the authors, at the depth of the Curie point but in the oceanic ridge region the Curie point is shallow and cells with 40 km of thickness would fall in the mantle where the temperature is certainly higher than the Curie point (see Li et al., 2017, A global reference model of Curiepoint depths based on EMAG2, Scientific Reports, 7).
Reply: Thank you for the suggestion. We place the top surface of deep layer near the Curie point, and its bottom extends to the deeper depth, in order to better rebuild the amplitude of magnetic anomaly. Relevant expressions have been revised in the updated manuscript.
(5) In the chapter 3 of supplementary materials have been reported the sizes of the cells used in the experiment. Being different from those reported in the text I do not understand if they refer to the synthetic model or to the real model. I suggest to clarify this point.
Reply: In the supplementary materials, the equivalent source was constructed for the theoretical model. In the report, the equivalent source was constructed for the real data. Because the different range and data spacing between these two data sets, these two equivalent sources are different.
Citation: https://doi.org/10.5194/se2021117AC3
Duan Li et al.
Duan Li et al.
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