GNGTS 2021 - Atti del 39° Convegno Nazionale

GNGTS 2021 S essione 3.1 388 nstrained Inversion (Viezzoli et al. 2009; Vignoli et al. 2014; Ley-Cooper et al. 2015; his framework, the value of λ is chosen a posteriori in order to meet the following hi-squared value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) of time gates (Vignoli et al. 2021). to perform a similar task with respect to the minimization of objective functional in n functions at each unit of the network making its output ( m ) a continuous and he input ( d ); this, in turn, defines a continuous and differentiable error function for the ce between the network and the target outputs. Hence, the error function can be Dataset (TD) using a relatively simple gradient-based procedure. So, the problem of to map the recorded measurements into resistivity vector is reduced to the nctional: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) the elements of the (data, model) couples ( d t , m t ) of the TD. Of course, in this case, the g the optimal weights w of the connections between the network units. Thus, the NN K on with respect to w . Once K is built based on the TD, it can be applied to the elements t to infer the corresponding conductivity models m . The TD should be based on the prior ailable about the investigated area. This might sound self-referential, but it is actually the theory (and, clearly, also of any NN approach) (Bai et al. 2020). ure consists of a multilayer perceptron with (a) an input layer with 54 (the number of ee hidden layers of 100, 500, 200 units, respectively, and (c) an output layer mber of conductivity model parameters) units. ils concerning the crucial preparation of the TD, are discussed in-depth in Bai et al. d TD consisted of around 12,000 ( d t , m t ) couples. eterministic inversion (Eq. 1), in the inversion performed through K (Eq. 4), no lateral d each sounding is inverted separately. NN approach, we applied the NN on a known verification dataset. nductivity sections whose 1D models were used to generate the noise-free synthetic . 1a (together with its associated data) shows a portion of our verification dataset. of the conductivity sections inferred by the proposed NN. In turn, the reconstructed ave been used to calculate their associated electromagnetic response; the comparison etic data and the calculated ones is shown, model-by-model, with a red dot (red axis his data misfit estimation makes clear that the conductivity distribution recovered by atible with the observations within 4%. So, the NN result is capable to retrieve almost ginal model. Moreover, Fig. 1b demonstrates that the proposed approach is robust lateral coherence of the conductivity sections despite each model has been inverted 0). compare the results obtained with the NN — exactly the same used for the previous a more traditional 1D deterministic inversion (Eq. 1) when applied on an existing (3) with strained Inversion (Viezzoli et al. 2009; Vignoli et al. 2014; Ley-Cooper et al. 2015; s framework, the value of λ is chosen a posteriori in order to meet the following -squared value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) f time gates (Vignoli et al. 2021). o perform a similar task with respect to the minimization of objective functional in functions at each unit of the network making its output ( m ) a continu us and e input ( d ); this, i turn, defines a continuous and differentiable error function for the e between the network and the target outputs. Hence, the error function can be Dataset (TD) using a relatively simple gradient-based procedure. So, the problem of to map the recorded measurements into resistivity vector is reduced to the ctional: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) he el ments of the (data, model) couples ( d t , m t ) of the TD. Of co rse, in this case, the the optimal weights w of the connections between the network units. Thus, the NN K n with resp ct to w . Once K is built based on the TD, it can be applied to the elements o inf r the corr sponding conductivity models m . The TD should be ased on the prior lable about the investigated area. This might sound self-referential, but it is actually the eory (and, clearly, also of any NN approach) (Bai et al. 2020). e consists of a multilayer perceptron with (a) an input layer with 54 (the number of e hidden layers of 100, 500, 200 units, respectively, and (c) an output layer ber of conductivity model parameters) units. concerning the crucial preparation of the TD, are discussed in-depth in Bai et al. TD consisted of around 12,000 ( d t , m t ) couples. erministic i version (Eq. 1), in the inversion performed through K (Eq. 4), no lateral each sounding is inverted separately. N approach, we applied the NN on a known verification dataset. ductivity sections whose 1D models were used to generate the noise-free synthetic 1a (together with its associated data) shows a portion of our verification dataset. the co d ctivity se tions inferred by the proposed NN. In turn, the reconstructed been used to calculate their associated electromagnetic response; the comparison tic data and the calculated ones is shown, model-by-model, with a red dot (red axis is data misfit estimation makes clear that the conductivity distribution r covered by ible with the observations within 4%. So, the NN result is capable to retrieve almost inal model. Mor over, Fig. 1b demonstrates that the proposed approach is robust ateral coherence of the conductivity sections despite each model has been inverted ). ompare the results obtained with the NN — exactly the same used for the previous more traditional 1D deterministic inversion (Eq. 1) when applied on an existing being the number of time gates (Vignoli et al. 2021). NNs can be built in order to performa similar task with respect to theminimization of objective functional in Eq. 1. NNs use activation functions at each unit of the network making its output ( , in general, more accurate in terms of reconstruction of the values. Here, we discuss a novel approach, based on Neural ing resistivity models with a quality comparable with the nds on a laptop versus hours on a computational server. We pproach on synthetic and field datasets (Bai et al., 2020). by minimizing an bj c ive functional consisting of a data nctional to be minimized is often written: d ( d obs − F m ) ) ‖ L 2 2 + λ s ( m ) , (1) (b) m is the vector of the model parameters; (c) F is the to the corresponding data; F takes into account the physics sition system; (d) W d is the data weighting matrix usually s the st bilizer incorporating the prior knowledge about the ier λ controls the balance between the data and the prior using here to assess the performances of the alternative ional model parameterization. Hence, m and F rely on the ding, and each associated model, is handled independently forward modelling F is always one-dimensional, a lateral osed by means of the regularization term. In this respect, y common option of s ( m ) equal to the minimum gradient m )= ‖ ∇ m ‖ L 2 2 . (2) cally 1D, the stabilizer acts both along the vertical ( z ) and the aterally coherency (without being truly 2D/3D). This is the ) a continuous and differentiable function of the input ( inversion strateg es ven if th latte approaches are, in general, more accurate in terms of recon d pth of the targets and of etrieval of true resistivity values. Here, we discuss a novel approach, Network (NN) techniques, capable of reconstructing resistivity models with a quality comp inversion strategy, but in a fraction of the time: seconds on a laptop versus hours on a computati demonstrate the advantages of th proposed novel approach on synthetic and field datasets (Bai et Method lo ies. ATEM d ta are usually inverted by minimizing an objective functional cons misf t term plus a stabilizer. Hence, the objective functional to be imized is often written: P ( λ ) ( d , m ) = ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 + λ s ( m ) , where (a) d obs is the vect r of the measurements; (b) m is the vector of the model paramete forwar modelling operator connecting the model m to the corresponding data; F takes into acc of the process and the characteristics of the acquisition system; (d) W d is the data weighting based on the measur ment u certainty; (e) s ( m ) is the stabilizer incorporating the prior know re istivity model to b recovered; (f) the multiplier λ controls the balance between the dat information. In the deterministic scheme we are using here to assess the performances of approach based on NN, we consider a one-dim nsional model parameterization. Hence, m an local 1D assumpti n. So, e ch ind vidual data sounding, and each associated model, is handle from the adjacent ones. Mor precisely, while th forward modelling F is always one-dimen co train betwee the neighboring models is imposed by means of the regularization term. concerning the stabilizer c oice, we adopt th very common option of s ( m ) equal to the mi norm stabilizer s ( m )= ‖ ∇ m ‖ L 2 2 . H nce, even though the conductivi y distribution is locally 1D, the s abilizer acts both along the ver horizontal ( x ) direction, enforcing some level of laterally coherency (without being truly 2D/3 ); this, in turn, defines a continuous and differentiable error function for the evaluation of the difference between the network and the target outputs. Hence, the error function can be minimized over a Training Dataset (TD) using a relatively simple gradient-based pro edure. So, the problem of buildi g an effective NN to map the recorded measurements int resistivity vector is r duced to the minimization of an error functio al: ework, the value of λ is chosen a posteriori in order to meet the following red value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) gates (Vignoli et al. 2021). form a similar task with respect to th mi imization of objective func ional in tions at each unit of the network making its output ( m ) a continuous and t ( d ); this, in turn, define a continuous and differe tiable error function for the ween the network and the target outputs. Hence, the error function can be et (TD) using a relatively simple gradi nt-based procedure. So, the problem of ap the recorded measurements into resistivity vect r is reduced to the al: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) ments of the (data, model) couples ( d t , m t ) of the TD. Of course, in this case, the ptimal weights w of the connections between the network units. Thus, the NN K respect to w . Once K is built based on the TD, it can be applied to the elements r the corresponding conductivity models m . The TD should be based on the prior about the investig ted area. This might sound self-referential, but it is actually the (and, clearly, also of any NN approach) (Bai et al. 2020). sists of a multilayer perceptron with (a) an input layer with 54 (the number of den layers of 100, 500, 200 units, respectively, and (c) an output layer of conductivity model parameters) units. cerning the crucial preparation of the TD, are discussed in-depth in Bai et al. onsisted of around 12,000 ( d t , m t ) couples. istic inversion (Eq. 1), in the inversion performed through K (Eq. 4), no lateral sounding is inverted separately. proach, we applied the NN on a known verification dataset. ity sections whose 1D models were used to generate the noise-free synthetic gether with its associated data) shows a portion of our verification dataset. conductivity sections inferred by the proposed NN. In turn, the reconstructed n used to calculate their associated electromagnetic response; the comparison ta and the calculated ones is shown, model-by-model, with a red dot (red axis a misfit estimation makes clear that the conductivity distribution recovered by ith the observations within 4%. So, the NN result is capable to retrieve almost odel. Moreover, Fig. 1b demonstrates that the proposed approach is robust coherence of the conductivity sections despite each model has been inverted re the results obtained with the NN — exactly the same used for the previous traditional 1D deterministic inversion (Eq. 1) when applied on an existing (4) where Inversion (Viezzoli et al. 2009; Vign li et l. 2014; L y-Coop r et al. 2015; ork, th value of λ is chosen a posteriori in order to meet the following value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) tes (Vignoli et al. 2021). m a similar task with respect to the inimiza ion of objective fu ctional in ns at each unit f the network making its output ( a continuou and d ); this, in turn, d fines a continuous and differ nti ble error function for the n the network and the targ t outputs. Hence, the error func ion can be (TD) using a elatively simple gradie -b sed proc dure. So, the problem of p the recorded measurements into resistivity vector is reduced to the E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) nts of the (data, m del) coupl s ( d t , m t ) of the TD. Of course, in this case, the imal weights w of the connections between the ne work units. Thus, the NN K spect t w . Once K is built based on the TD, it can be applied to the elements he corresponding conductivity models m . The TD should be based on the prior ut the investigated area. This might sound self-referential, but it is actually the d, clearly, also of any NN approach) (Bai et al. 2020). ts of multilayer perceptron with (a) an input layer with 54 (the number of layers of 100, 500, 200 units, respectively, and (c) an output layer onductivity mode arameters) uni s. ni g the crucial preparation of the TD, are discussed in-depth in Bai et al. sisted of arou d 12,000 ( d t , m t ) couples. ic inversion (Eq. 1), in the inversion performed through K (Eq. 4), no lateral unding is inverted separately. ach, we applied the NN on a known verification dataset. sections whose 1D mo els were used to generate the noise-free synthetic ther with its associated data) shows a portion of our verification dataset. ductivity sections infer ed by th proposed NN. In turn, th reconstructed used to cal ulate their associated electr magnetic response; the compa ison nd the calcul ted ones i shown, model-by-model, with a red dot (red axis isfit estimation makes clear that the conductivity distribution recover d by the observations within 4%. So, the NN result is capable to retrieve almost del. Moreover, Fig. 1b demonstrates that the proposed approach is robust herence of the conductivity sections despite each model has been inverted the results obtained with the NN — exactly the same used for the previous aditional 1D deterministic inversion (Eq. 1) when applied on an existing and nversion (Viezzoli et al. 2009; Vignoli et al. 2014; Ley-Cooper et al. 2015; rk, the valu of λ is chos n posteriori n rder to meet he following alue 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) s (Vignoli et al. 2021). a similar task w h resp ct o the minimization f objective functio al in at each unit of the twork making its output ( m ) a continuous and ); this, in turn, efines a continuous and differ ntiable rror function f r th the network and the target outputs. Hence, he error function can be D) using a relatively simple gr dient-based procedure. So, the probl m of the recorded measurements into resi tiv ty vector is reduced to the E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) ts of the (dat , model) couples ( d t , m t ) of the TD. Of course, in this case, the al weigh s w of the connections between the network units. Thus, the N K ect to w . Once K is built based on the TD, it can be applied to the l ments corresponding conductiv ty models m . The TD should be based on the prior t the investigated are . This might sound self-refer ntial, but it is actually the , clearly, also f ny N approach) (Bai et al. 2020). of a multilayer perceptron with (a) an input layer with 54 (the number of layers of 100, 50 , 200 units, respectively, and (c) an output layer nductiv y model par met rs) units. g the crucial prepar tion of the TD, are discussed in-depth in Bai et al. sted of ar und 12,000 ( d t , m t ) couples. invers on (Eq. 1), in the inv rsion performed through K (Eq. 4), no lateral ding is inverted separ tely. ch, we applied the N on a known verif cati n dat set. ections w ose 1D models were used to generate the noise-free syn hetic er with its as o iated at ) shows a portion of our verif cation dat set. uctiv ty se tions inferred by the proposed NN. In turn, the reconstructed ed to calcul te their associated el ctromagnetic response; the comparison d the calculated ones is hown, model-by-model, with a red dot (red axis sfit estimation makes clear that the conductivity distribution recover d by he observations within 4%. So, the N result is capable to retrieve almost l. Moreover, Fig. 1b emonstrates that the proposed approach is robust er nce of the conductiv ty sections despite each model has been inverted e esults obtaine with the N — exactly the same used for the previous itional 1D det rmin stic inversion (Eq. 1) when ap lied on an existing consist of the elements of the (data, model) couples ( condition concerning the chi-squared value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , with N d being the number of time gates (Vignoli et al. 2021). NNs can be built in order to perform a similar task with respect to the minimization of object Eq. 1. NNs use activation functions at each unit of the network making its output ( m ) a differentiable function of the input ( d ); this, in turn, defines a con nuous and differentiable erro evalu tion of the difference between the ne work and the target outputs. Hence, the error minimized over a Training Dataset (TD) using a relatively simple gradient-based procedure. So building an effective NN to map the record d meas rements i to resistivity vector is minimization of an error functional: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , where D and M consist of the elements of the (data, model) coupl s d t , m t ) of the TD. Of course minimization aims at finding the optimal weights w of the connections between the network units. is found via the minimization with respect to w . Once K is built based on the TD, it can be applie d obs of the observ d datas to inf r th corresponding conduc ivity models m . The TD should be (geological) knowledg availa l about the investigated area. This might sound self-referential, bu key point of regularization the ry (and, clearly, als of any NN approach) (Bai et al. 2020). The c sider d NN structur consists of a multilayer perc ptron with (a) an input layer with 5 time g te ) units, (b) three hidden layers of 100, 500, 200 units, respectively, and (c) characterized by 30 (the number of conductivit model param ters) units. These, and the other details concerning the crucial preparati n of the TD, are discussed in-de (2020). In total, the utilized TD consisted of around 12,000 ( d t , m t ) couples. Differently from the 1D deterministic inversion (Eq. 1), in the inversion performed through K ( inform tion is included, and each sounding is inverted separately. Synthetic test. To test the NN approach, we applied the NN on a known verification dataset. Fi . 1a shows the true conductivity sections whose 1D models were used to generate the noi data to be invert d. So, Fig. 1a (together w th its associated data) shows a portion of our verificat Instead, Fig. 1b consists of th con uctivity sections inferred by the proposed NN. In turn, t conductivities (Fig. 1b) have been used to c lculate their associ t d electromagnetic response; between the orig al synthetic data and the calcula ed ones is shown, model-by-model, with a on he right in Fig. 1b). This ata misfit estimation makes clear hat the conductivity distributi the NN is g nerally compatibl with the observations wit in 4%. So, the NN result is capable t all the features of the original model. Moreover, F g. 1b monstrates that the proposed ap en ugh o reconstru t the l eral herence of the conductivity section despite each model h individually (Ba et al. 2020). Field example. Here, we compare the results obtained with the NN — exactly the same used synthetic test — against a more traditional 1D deterministic inversion (Eq. 1) when applie VTEM dataset. . Of course, in this case, the minimization aims at fi ding the optimal weights es ence f the Spatially Constrained Inve sion (V ezzoli et al. 2009; Vigno Vignoli et l. 2017). I this framework, the value of λ is chosen a post condition concerning the chi-squared value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , with N d being the number of time gates (Vignoli et al. 2021). NNs c n be b il in ord r t perform a similar task with respect to the m Eq. 1. NNs use activation functions at each uni of the network maki differ tiable functio of the input ( d ); this, in turn, defines a co tinuous a evaluation of the difference etween th network and the target utput minimized over a Training Dataset (TD) using a relatively simple gradien building an effective NN to map the recorded measurements into r inimization of an error functional: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , where D a d M consist of the elements of the (data, model) couples ( d t , m t minimizat on aims t find ng t e optimal weights w of the connections betw is found via the minimization with r spect to w . Once K is built based on th d obs of the observed d taset to infer the corresponding conductivity models (geological) knowledge available about the investigated area. This might sou key point of regulariza ion theory (and, cle rly, also of any NN approach) (B The consider d NN structure consists of a multilayer perceptron with (a) t m gates) units, (b) three hidden l y rs of 100, 500, 200 units, r charact ized by 30 (the number of conductivity model parameters) units. Th e, and the ot er d tails concerning the crucial preparation of the T (2020). In total, the utilized TD c nsist d of around 12,000 ( d t , m t ) couple Differently from the 1D deterministic inversion (Eq. 1), in the inversion pe information is included, and each sounding is inverted separately. Synthe ic t st. To t st the NN approach, we applied the NN on a known ve Fig. 1a sh ws the true onductivity sections whose 1D models were use data to be inverted. So, Fig. 1a (t gether with its associated data) shows a p Instead, Fig. 1b consists of the conductivity sections inferred by the pro conductivities (Fig. 1b) have been used to calculate their associated elect betwee the original synthetic data an the calculated ones is shown, mo on the right in Fig. 1 ). This data misfit estimation makes clear that the c th NN is g ne ally com atible with the observations within 4%. So, the all the features of the original model. Moreover, Fig. 1b demonstrates t enough to reconstruct the lateral coherence of the conductivity sections individually (Bai et al. 2020). Field example. Here, w comp r the results obtained with the NN — e synthetic test — against a more traditional 1D deterministic inversion VTEM dataset. of the connections between th netw rk units. Thus, the NN is found via the minimization with respect t essence of the Spatially Constrained Inversion (Viezzoli et al. Vignoli et al. 2017). In this framework, the value of λ is c condition concerning the chi- quar d value χ 2 = ( 1 N d ) ‖ W d ( d obs − F ( with N d being the number of time gates (Vignoli et al. 2021). NNs can be built in order to p rform a similar task with resp Eq. 1. NNs use activatio functions t each unit of the n differentiabl functi n o the input ( d ); this, in turn, d fines a evaluation of the difference between the network and th t minimized over a Training Dataset (TD) using a rela ively si building an effective NN to map the recorded measure minimization of an error functional: E ( w ) = ‖ K ( D , w ) where D and M consist of the elements of the (data, model) co minimizat on aims at finding the optimal weights w of the conn is found via the mi imization with respect to w . Once K is buil d obs of the observed dataset to infer the corresponding conductiv (geo ogical) knowl dge available about the investigated area. T key poi t of regularization theory (and, clearly, also of any NN The considered NN structure consists of a multilayer perceptr time gates) units, (b) three hidden lay rs of 100, 500, charact riz d by 30 (the number of conductivity model parame These, and the other details concerning the crucial preparati (2020). In total, the utilized TD consisted of around 12,000 ( d t Differently from the 1D determ n stic inversion (Eq. 1), in the information is included, and each sounding is inverted separate Synthetic test. To test the NN approach, we appli d th NN o Fig. 1a shows the true conductivity ections whose 1D mode data to be inverted. So, Fig. 1a (tog ther with its associated dat Instead, Fig. 1b consists of he conductivity sections inferre conductivities (Fig. 1b) have been used to calculate their ass between the origin l synthe ic data and the calculated ones is on the right in Fig. 1b). This data misfit estimation makes cle the NN is generally compatible with the observations within 4 all the fea ures of the origi al model. Moreover, Fig. 1b de enough to reconstruct the lateral coherence of the conductivi individually (Bai et al. 2020). Field ex mple. He , we compare the r sults obtained with t synthetic test — against a more traditional 1D deterministi VTEM dataset. . O ce ion (Viezzoli et al. 2009; Vignoli et al. 2014; Ley-Cooper t al. 2015; the value of λ is chosen a posteriori in order to meet the following ( 1 N d ) ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 ≈ 1 , (3 ) ignoli et al. 2021). milar task with respect to the minimiz tion of objective functional in each unit of the network m king its output ( m ) a continuous and s, in turn, defines a continuo s and differentiable error function for the network and the target outputs. H nce, the error function can be using a relatively simple gradient-based procedure. So, the problem of recorded measu ements into re istivi y vector is reduced to th E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , (4) the (da a, model) couples ( d t , m t ) of the TD. Of cours , in this case, the eights w of the connections between the network units. Thus, the NN K to w . Once K is built based on the TD, it can be applied t the elements responding conductivity models m . The TD shoul be based on th prior investigated area. This might sound s lf-refer ntial, but it is actually the arly, also of any NN approach) (Bai et al. 2020). multilayer per eptron with (a) an input lay r with 54 (the number of rs of 100, 500, 200 units, respectively, and (c) an output layer tivity model parameters) units. he crucial preparation of the TD, are disc ss d in-depth in Bai et al. of around 12,000 ( d t , m t ) coupl s. rsion (Eq. 1), in the inv rsion performed through K (Eq. 4), no lateral g is inverted separately. e applied the NN on a known verification dataset. ons whose 1D models were used to generate the noise-free synthetic i its associated data) shows a portion of our verification dataset. ity sections inferred by the proposed NN. In turn, the reconstructed o calculate their associated el ctromagnetic response; the comparison e calculated ones is shown, model-by-model, with a red dot (red axis estimation makes clear that the conductivity distribution recovered by bservations within 4%. So, the NN result is capable to ret ieve almost oreover, Fig. 1b demonstrates that the proposed approach is robu t ce of the conductivity sections despite each model has been inverted sults obt ined with the NN — exactly the same used for the previou nal 1D deterministic inversion (Eq. 1) when applie on an existing is built based on the TD it can be applied to the elements The possibility of getting reliable results very quickly after, or even during, the d ta collecti not merely for quality check, but also for adjusting the ocation of the prop sed flight line time-domain (ATEM) acquisition. This kind of readiness could have a large impact in term the Value of Information of the measurements to be acquired. Besides, the relevanc rec nstructing resistivity mod ls from ATEM data is demonstrated by the routine use of Imaging (CDI) methodologies in miner l exploration. In fact, CDIs are extremely efficient f perspective, and, at the same time, they prese ve a very high lateral resolution. Hence, they a inversion strategies ven if the latter approaches are, in general, more accurate in terms f depth of the targets and of retrieval of true resistivity values. Here, w discuss a novel appro Network (NN) te hniques, capable of reconstructing r s stivity mod ls with a quality c inversion s rategy, but in a fraction of the time: seconds on a laptop versus hours on a comp demonstrate the advantages of he proposed novel approach n synthetic and field datasets ( M thodologies. ATEM data ar usu lly inverted by minimizing an objective functi nal sfit term plus a stabilizer. Hen e, h objective fu ctional to be minimized is ften written P ( λ ) ( d , m ) = ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 + λ s ( m ) , where (a) d obs is the vector of the measure en s; (b) m is the vector of the model para forward modelling operator conn cting he m del m to t corresponding data; F takes into of the process nd the ch racteristics of the acquisition system; (d) W d is the data weig based on the measurement uncertainty; (e) s ( m ) is the stabilizer incorporating the prior r sistivity mod l to be recover d; (f) the mu tiplier λ controls the b lance between th information. In he determini tic scheme we are using here t as ess the performance approach based on NN, we consider a one-dimensional mod l parameterization. Hence, local 1D assumption. So, each individual data sounding, and each associated model, is ha from the adjacent ones. M re p ecisely, while the forward modelling F is always one-d c straint betw en the neighb ring models is imposed by means of the regularization t conc rning the s abilizer choice, we adopt the very common option of s ( m ) equal to th norm stabilizer s ( m )= ‖ ∇ m ‖ L 2 2 . Hence, even though the conductivity di tribution is l cally D, the s abilizer acts both long th h rizontal ( x ) direction, enforcing some l v l of lat rally coherency (without being truly of the observed dataset to infer the corresponding c nductivity models niv. of C gliari, It ly ility f getting rel able es lts very quickly after, or even during, the data collection would be crucial, for quality check, bu also for adjusting t locat on of the proposed fl ght lin s during an airborne in (ATEM) acquisition. This kind f readi ess could have a large impact in terms of maximization of of Information of the measurements to be acquired. Besides, the relevance of fast to ls for ing resistivity models from ATEM data is demonstrated by th routine use of Conductivity-Dep h DI) ethodologies in m n ral exploration. In fact, CDIs are extremely efficien fro computational , and, at the same time, they preserve a very high lateral resolution. Hence, they are often pr ferred to trategies even if the latter approaches are, in general, more accurate in terms of reconstruction of the e targe s and of retriev l of true resistivity values. H r , w discuss a ov l appro ch, based on Neural NN) techniques, capable of r constr cting r s stivity mod ls with a quality c mpa able with the trategy, but in a fraction of the time: seconds on a laptop versus hours on a computational server. We e the advantages of th proposed novel appr ach on syntheti and field datasets (Bai et al., 2020). gies. ATEM data are usually inv rted by minimizing a objective functional c n st ng of a data plus a stabilize . Hence, he objec ive f nctional to be minimized is ofte written: P ( λ ) ( d , m ) = ‖ W d ( d obs − F ( m ) ) ‖ L 2 2 + λ s ( m ) , (1) d obs i the vector of the measur ments; (b) m is the vector of the model parameters; (c) F is the delling operator connecting the model m to the corresponding dat ; F akes into ac unt the physics ess and the characteristics of the cqu sition system; (d) W d is the dat weighting matrix usually he m asurement uncertainty; (e) s ( m ) is the stabilizer incorporating the prior knowledge about the model to be recovered; (f) the multiplier λ controls the balance between the data and the prior . In the deterministi sch me we are using here to assess the performances of the alternative ased on NN, we onsider a one-dimensional model parameterization. Hence, m and F rely on the ssumption. So, each i dividual data sounding, and each ass ciated model, is handled independently djacent nes. More precisely, while the forward mod lling F is always one-dimensional, a later l between the neighbori g models is imposed by means of the regularization term. In this respect, the stabilizer ch ic , we adopt the very common op ion of s ( m ) equ l to the minimum gradient izer s ( m )= ‖ ∇ m ‖ L 2 2 . (2) n though the conductivity distributio is locally 1D, the st biliz r acts both al ng th vertical ( z ) and the ( x ) direction, enforcing some level of laterally coherency (with ut being truly 2D/3D). This i the . The TD s ould b ba ed on the prior (ge logi al) knowledge available about the i vestigated area. This might sound self-referential, but it is actually the key point of regularization theory (and, clearly, also f any NN approach) (Bai et al. 2020). The c nsidered NN structure co sists of a multilayer perceptron with (a) n input layer with 54 (the number of time gates) units, (b) three hidden layers of 100, 500, 200 units, respectively, and (c) an output lay r cha acteriz d by 30 (the numb f conductivity model parameters) units. These, and the other t il oncerning the crucial preparation of the TD, a e discussed in- depth in Bai et al. (2020). In total, the util zed TD consisted of around 12,000 ( essence of the Spatially Constrained Inversion (Viezzoli et al. 2009; Vignoli et al. 2014; Ley- Vignoli et al. 2017). In this framework, he value of λ is chos n a p steriori in order to condition conc rning the chi-squared value χ 2 = ( 1 N d ) ‖ W d ( obs − F ( m ) ) ‖ L 2 2 ≈ 1 , with N d being the number of tim gates (Vignoli et al. 2021). NN can b built in order t perform a simil r task with r spect to the minimization of obj Eq. 1. NNs use activation functi ns at each unit f th network making its output ( m differentiable function of t e input ( d ); this, in turn, defi es a continuous and differentiable evaluation of the difference between t e etwork an the target outputs. Hence, the err minimiz d over a Training Dataset (TD) using a relatively simple gradient-b sed procedure. building an effective NN to map the recorded measurements into resistivity vector minimization f an error fu ctional: E ( w ) = ‖ K ( D , w )− M ‖ L 2 2 , where D and M consist of the elements of the (dat , model) couples d t , m t ) of the TD. Of co minimizati n aims a finding the optimal weights w of the connections b tween the network un is o nd via the minimiza ion with respect to w . Onc K is built b sed on the TD, it can be ap d obs f th observed dataset t inf th corresponding c nductivity models m . The TD sh uld (geological) knowl dg available about the investigated area. This might sound self-referential, key point of regularization th ry (and, clearly, also of any NN approach) (Ba et al. 2020). The consi ere NN structure consists of a multilayer percep ron ith (a) an input lay with time gates) units, (b) thre hid en layers of 100, 500, 200 un ts, respectively, and ( characterized by 30 (t e numb r of c nductivity model p rameters) uni s. These, and the oth r details concerning the crucial pr parati n of he TD, are discussed in (2020). I total, the utiliz TD consisted of arou d 12,000 ( d t , m t ) couples. Differently f om the 1D deter inistic inv sion (Eq. 1), i the inversion performed through inform ion is included, a d each sounding is inverted separately. Synthetic test. To test the NN approac , we a plied the NN on a known verification dataset. Fig. 1a shows the true conductivity sections whose 1D models wer used to generate the data to be inverted. So, Fig. 1a (tog ther with its associated data) shows a portion of our verifi Instead, Fig. 1b co ists of the conductivity sections inferred by the proposed NN. In tur conductivities (Fig. 1b) have been used to calculat thei associated electromagnetic respon between the original sy thet c data and the calculated ones is shown, model-by-model, with on the right in Fig. 1b). This data misfit estimation makes clear that the conductivity distrib the NN is generally compatible with the observations within 4%. So, the NN result is capabl all the features of the original model. Moreover, Fig. 1b demonstrates that the proposed enough to reconstruct the lateral coherence of the conductivity sections despite each mode individually (Bai et al. 2020). couples. Diff rently fr m t 1 et r inistic inv rsion (Eq. 1), in the inversion per ormed h ough (Eq. 4), no lateral inf rmation is included, and e ch sounding is inverted separ tely. Synthetic tes To test the NN approach, w applied the NN on a k own verification dataset. Fig. 1a shows the true conductivity sections whose 1D m dels re used to ge erate the noise-free synthetic data to be inverted. So, Fig. 1a (together with its ssoci ted data) shows a po tion of our verification d t s t. Inst ad, Fig. 1b consists of the condu tivity sections inferred by the proposed NN. In turn, the r constructed conductivities (Fig. 1b) havebeenused to calcul te their associ tedelectromagnetic response; the comparison between the original synthetic data and the calculated ones is shown, model-by-model, wit a red dot (red axis on the right in Fig. 1b). Thi data misfit estimation makes clear that the c nductivity di r bution recove d by the NN is generally compatible with the observations within 4%. So, the NN result is capable to retrieve almost all the f atures of the original model. Moreover, Fig. 1b demonstrates that the pr posed approach is r ust en ugh to reco struct the lateral coherence of the conductivity sections despite each model has been inverted individually (Bai e al. 2020). Fiel exa ple Here, we compare the results obtained with the NN — exactly the same used for the previous synthetic test — against a more traditional 1D deterministic inversion (Eq. 1) when applied on an existing VTEM dataset.