Seismic microzonation of Chania,Crete (Greece) based on SASW measurements and non-linear site response analyses

Following a brief overview of past applications of, and more recent advances on seismic microzonation, the results of a seismic microzonation study for the city of Chania, Greece, are presented. The study was based on V_s vs. depth profiles obtained at 19 sites of the urban area by performing SASW measurements. The spatial distribution of V_s values was utilized in estimating V_s30 values, depth to bedrock and the fundamental ground period variation across the area of the city as well as for conducting 1-D finite element non-linear inelastic site response analyses. The input earthquake excitations employed in the response analyses were based on the results of an available seismic hazard study for the Chania Area. The results of analyses were utilized for establishing the spatial distribution of rock motion amplification, the expected ground motions and spectral values in the area of the city. Contour maps providing values of the expected ground motion in the urban area are given which may become a practical tool in assessing the seismic risk and expected damage in the Chania area. The maps can also be used in the design of new earthquake resistant structures or the seismic retrofitting of existing ones. Finally, the results were utilized to demonstrate the inadequacy of using V_s,30 values for classifying the soil conditions in the Chania area.     

A comparison of seismic risk maps for Italy

National seismic risk maps are an important risk mitigation tool as they can be used for the prioritization of regions within a country where retrofitting of the building stock or other risk mitigation measures should take place. The production of a seismic risk map involves the convolution of seismic hazard data, vulnerability predictions for the building stock and exposure data. The seismic risk maps produced in Italy over the past 10 years are compared in this paper with recent proposals for seismic risk maps based on state-of-the-art seismic hazard data and mechanics-based vulnerability assessment procedures. The aim of the paper is to open the discussion for the way in which future seismic risk maps could be produced, making use of the most up-to-date information in the fields of seismic hazard evaluation and vulnerability assessment.     

Seismic vulnerability assessment to slight damage based on experimental modal parameters

The aim of this paper is to adjust behaviour models for each class of structure for vulnerability assessment by using ambient vibration. A simple model based on frequencies, mode shapes and damping, taken from ambient vibrations, allows computation of the response of the structures and comparison of inter‐storey drifts with the limits found in the literature for the slight damage grade, considered here as the limit of elastic behaviour. Two complete methodologies for building fragility curves are proposed: (1) using a multi‐degree of freedom system including higher modes and full seismic ground‐motion and (2) using a single‐degree of freedom model considering the fundamental mode f₀ of the structure and ground‐motion displacement response spectra S_D(f₀). These two methods were applied to the city of Grenoble, where 60 buildings were studied. Fragility curves for slight damage were derived for the various masonry and reinforced concrete classes of buildings. A site‐specific earthquake scenario, taking into account local site conditions, was considered, corresponding to an M_L = 5.5 earthquake at a distance of 15 km. The results show the benefits of using experimental models to reduce variability of the slight damage fragility curve. Moreover, by introducing the experimental modal model of the buildings, it is possible to improve seismic risk assessment at an overall scale (the city) or a local scale (the building) for the first damage grade (slight damage). This level of damage, of great interest for moderate seismic‐prone regions, may contribute to the seismic loss assessment. Copyright © 2011 John Wiley & Sons, Ltd.     

Seismic damage estimation for low- and mid-rise buildings in japan

While much effort has been spent on analysis of individual structures, building class seismic damage estimators, of value in disaster planning, code-writing, city planning, national hazards policy formulation, etc., have been little investigated. Based largely on data from Sendai City, Japan in the 12 June, 1978 Miyagiken-oki earthquake (M_L = 7.4), estimators of seismic damage for low- and mid-rise buildings in urban Japan have been determined. For low-rise buildings, based on damage to over 60,000 buildings, damage ratios for onset of damage and collapse and for cost of damage are found to correlate best with response spectra at 0.75 s. Using published test data and average building properties, a seismic damage model explains the low-rise building behaviour and permits examination of the effect of structural changes on the estimated damage. For mid-rise buildings, damage states (0: none, 4: total) are determined as a function of maximum storey displacement, based on published natural period determinations (pre- and post-earthquake) for 189 mid-rise buildings in Sendai. The effects of structural changes on expected damage can also be estimated. With these two building class damage estimators, a large part of future seismic damage to urban Japan can be estimated, as well as the effects of various mitigation measures.     

Regional and local seismic hazard assessment

Recent earthquakes such as the M_JMA 7.2 Hyogo-ken Nambu earthquake and the M 7.4 Kocaeli earthquake demonstrate once again the need to include detailed soil investigation into hazard evaluation, that is the need of microzonation. Seismic hazard assessment evaluated at a regional scale generally does not consider soil effects but only in a limited way using an attenuation law that can be ‘soft soil’ or ‘rock’. However, the relevant role of seismic hazard in the assessment of seismic coefficients for the definition of the actions in seismic codes must be properly considered. That is to say, the level of protection of buildings is proportional to a definite level of hazard (generally considered to be the ground motion with 10% probability of exceedence in 50 years). When a microzonation is performed, this criterion cannot be ignored, therefore, a clear linkage must be established between hazard (regional scale) and microzonation. The crucial point is represented by the reference motion (or input motion) to be used for site effects analysis, that must be compatible with the regional seismic hazard. In this paper, three different approaches for reference motion evaluation are analysed: probabilistic; stochastic; and deterministic. Through the case history of Fabriano microzonation the three approaches are compared. It is shown that each approach presents advantages and disadvantages with respect to the others. For example, the probabilistic approach (the reference motion is directly derived from the expected response spectra for a given return period) is linked with hazard, but produces an overestimation in short periods range, while the deterministic approach correctly simulates the wave propagation, but it ends with a kind of conditional probability. Until now, clear criteria to choose the right approach do not appear to exist and the expert experience is of fundamental importance.     

Performance-based methodology for assessing seismic vulnerability and capacity of buildings

This paper presents a performance-based methodology for the assessment of seismic vulnerability and capacity of buildings. The vulnerability assessment methodology is based on the HAZUS methodology and the improved capacity- demand-diagram method. The spectral displacement (Sd) of performance points on a capacity curve is used to estimate the damage level of a building. The relationship between Sd and peak ground acceleration (PGA) is established, and then a new vulnerability function is expressed in terms of PGA. Furthermore, the expected value of the seismic capacity index (SCev) is provided to estimate the seismic capacity of buildings based on the probability distribution of damage levels and the corresponding seismic capacity index. The results indicate that the proposed vulnerability methodology is able to assess seismic damage of a large number of building stock directly and quickly following an earthquake. The SCev provides an effective index to measure the seismic capacity of buildings and illustrate the relationship between the seismic capacity of buildings and seismic action. The estimated result is compared with damage surveys of the cities of Dujiangyan and Jiangyou in the M8.0 Wenchuan earthquake, revealing that the methodology is acceptable for seismic risk assessment and decision making. The primary reasons for discrepancies between the estimated results and the damage surveys are discussed.     

Rock-physics-based carbonate pore type characterization and reservoir permeability heterogeneity evaluation, Upper San Andres reservoir, Permian Basin, west Texas

The use of the shear wave velocity data as a field index for evaluating the liquefaction potential of sands is receiving increased attention because both shear wave velocity and liquefaction resistance are similarly influenced by many of the same factors such as void ratio, state of stress, stress history and geologic age. In this paper, the potential of support vector machine (SVM) based classification approach has been used to assess the liquefaction potential from actual shear wave velocity data. In this approach, an approximate implementation of a structural risk minimization (SRM) induction principle is done, which aims at minimizing a bound on the generalization error of a model rather than minimizing only the mean square error over the data set. Here SVM has been used as a classification tool to predict liquefaction potential of a soil based on shear wave velocity. The dataset consists the information of soil characteristics such as effective vertical stress (σ′_v0), soil type, shear wave velocity (V_s) and earthquake parameters such as peak horizontal acceleration (a_max) and earthquake magnitude (M). Out of the available 186 datasets, 130 are considered for training and remaining 56 are used for testing the model. The study indicated that SVM can successfully model the complex relationship between seismic parameters, soil parameters and the liquefaction potential. In the model based on soil characteristics, the input parameters used are σ′_v0, soil type, V_s, a_max and M. In the other model based on shear wave velocity alone uses V_s, a_max and M as input parameters. In this paper, it has been demonstrated that Vs alone can be used to predict the liquefaction potential of a soil using a support vector machine model.     

IM‐based and EDP‐based decision models for the verification of the seismic collapse safety of buildings

Decision models for the verification of seismic collapse safety of buildings are introduced. The derivations are based on the concept of the acceptable (target) annual probability of collapse, whereas the decision making involves comparisons between seismic demand and capacity, which is familiar to engineering practitioners. Seismic demand, which corresponds to the design seismic action associated with a selected return period, can be expressed either in terms of an intensity measure (IM) or an engineering demand parameter (EDP). Seismic capacity, on the other hand, is defined by dividing the near‐collapse limit‐state IM or EDP by an appropriate risk‐targeted safety factor (γ _im or γ _edp ), which is the only safety factor used in the proposed decision model. Consequently, the seismic performance assessment of a building should be based on the best possible estimate. For a case study, it is shown that if the target collapse risk is set to 10^?4 (0.5% over a period of 50 years), and if the seismic demand corresponds to a return period of 475 years (10% over a period of 50 years), then it can be demonstrated that γ _im is approximately equal to 2.5 for very stiff buildings, whereas for buildings with long periods the value of γ _im can increase up to a value of approximately 5. The model using γ _edp is equal to that using γ _im only if it can be assumed that displacements, with consideration of nonlinear behavior, are equal to displacements from linear elastic analysis.     

Vulnerability of floating tunnel shafts for increasing earthquake loading

Fragility curves constitute the cornerstone in seismic risk evaluations and performance-based earthquake engineering. They describe the probability of a structure to experience a certain damage level for a given earthquake intensity measure, providing a relationship between seismic hazard and vulnerability. In this paper a numerical approach is applied to derive fragility curves for tunnel shafts built in clays, a component that is found in several critical infrastructure such as urban metro networks, airport facilities or water and waste water projects. The seismic response of a representative tunnel shaft is assessed using tridimensional finite difference non-linear analyses carried out with the program FLAC^3D, under increasing levels of seismic intensity. A hysteretic model is used to simulate the soil non-linear behavior during the seismic event. The effect of soil conditions and ground motion characteristics on the soil-structure system response is accounted for in the analyses. The damage is defined based on the exceedance of the concrete wall shaft capacity due to the developed seismic forces. The fragility curves are estimated in terms of peak ground acceleration at a rock or stiff soil outcrop, based on the evolution of damage with increasing earthquake intensity. The proposed fragility models allows the characterization of the seismic risk of a representative tunnel shaft typology and soil conditions considering the associated uncertainties, and partially fill the gap of data required in performing a risk analysis assessment of tunnels shafts.     

Three‐term inversion of prestack seismic data using a weighted l2, 1 mixed norm

We present a new inversion method to estimate, from prestack seismic data, blocky P‐ and S‐wave velocity and density images and the associated sparse reflectivity levels. The method uses the three‐term Aki and Richards approximation to linearise the seismic inversion problem. To this end, we adopt a weighted mixed l_{2, 1}‐norm that promotes structured forms of sparsity, thus leading to blocky solutions in time. In addition, our algorithm incorporates a covariance or scale matrix to simultaneously constrain P‐ and S‐wave velocities and density. This a priori information is obtained by nearby well‐log data. We also include a term containing a low‐frequency background model. The l_{2, 1} mixed norm leads to a convex objective function that can be minimised using proximal algorithms. In particular, we use the fast iterative shrinkage‐thresholding algorithm. A key advantage of this algorithm is that it only requires matrix–vector multiplications and no direct matrix inversion. The latter makes our algorithm numerically stable, easy to apply, and economical in terms of computational cost. Tests on synthetic and field data show that the proposed method, contrarily to conventional l₂‐ or l₁‐norm regularised solutions, is able to provide consistent blocky and/or sparse estimators of P‐ and S‐wave velocities and density from a noisy and limited number of observations.     

Application of diffracted wave analysis to time‐lapse seismic data for CO2 leakage detection

Time‐lapse seismic analysis is utilized in CO₂ geosequestration to verify the CO₂ containment within a reservoir. A major risk associated with geosequestration is a possible leakage of CO₂ from the storage formation into overlaying formations. To mitigate this risk, the deployment of carbon capture and storage projects requires fast and reliable detection of relatively small volumes of CO₂ outside the storage formation. To do this, it is necessary to predict typical seepage scenarios and improve subsurface seepage detection methods. In this work we present a technique for CO₂ monitoring based on the detection of diffracted waves in time‐lapse seismic data. In the case of CO₂ seepage, the migrating plume might form small secondary accumulations that would produce diffracted, rather than reflected waves. From time‐lapse data analysis, we are able to separate the diffracted waves from the predominant reflections in order to image the small CO₂ plumes. To explore possibilities to detect relatively small amounts of CO₂, we performed synthetic time‐lapse seismic modelling based on the Cooperative Research Centre for Greenhouse Gas Technologies (CO2CRC) Otway project data. The detection method is based on defining the CO₂ location by measuring the coherency of the signal along diffraction offset‐traveltime curves. The technique is applied to a time‐lapse stacked section using a stacking velocity to construct offset‐traveltime curves. Given the amount of noise found in the surface seismic data, the predicted minimum detectable amount of CO₂ is 1000–2000 tonnes. This method was also applied to real data obtained from a time‐lapse seismic physical model. The use of diffractions rather than reflections for monitoring small amounts of CO₂ can enhance the capability of subsurface monitoring in CO₂ geosequestration projects.     

The road map for seismic risk analysis in a Mediterranean city

The seismic risk analysis evaluation in the Mediterranean area is one of the main tasks for the preservation of Cultural Heritage and for the sustainable development of Mediterranean cities. The Mediterranean area is characterised by a medium–high level of seismic risk, so that earthquakes are the major cause for the destruction of monuments, residential and industrial buildings. A case history regarding the seismic risk analysis for the city of Catania (Italy) is presented, since the city has been heavy damaged in the past by strong earthquakes such as the 1169 earthquake (XI MCS), the 1542 earthquake (IX MCS), the 1693 earthquake (XI MCS) and the 1818 earthquake (VIII MCS) etc., which caused several thousands of deaths. Fault modelling, attenuation laws, synthetic accelerograms, recorded accelerograms and site effects are considered for the evaluation of the seismic action. Vulnerability of physical environment, related to the presence of cavities and to seismic-induced landslides and liquefaction has been analysed, with special reference to the new modelling of such phenomena and to the application of models to given areas. Soil–structure Interaction has been analysed for some geotechnical works, such as shallow foundation and retaining wall, by means of physical and numerical modelling. The paper deals with the vulnerability of physical environment (landslides, liquefaction, etc.), while the road map continues with the analysis of vulnerability of monuments and buildings, with the aim of the estimation of the seismic resistance required to defend against the seismic action given by the scenario earthquake. For the mitigation of seismic risk, structural improvements of R.C. buildings with different methodology and techniques have been analysed, as well as the guideline for the strengthening of buildings. The work shows that the seismic risk of the city is not a summation of the seismic risk of each building, because the vulnerability of the urban system plays an important role on the seismic risk evaluation of a given city. To this aim the vulnerability of the road infrastructures, lifelines, and urban framework have been also analysed in the project.     

Topographic effects on the Monte Po Hill in Catania (Italy)

The study regards an evaluation of site effects on and near the Monte Po hill, located in the north-eastern part of the city of Catania (Italy), an area at high seismic risk. At the beginning of 2007 a seismic station was located in a school building, situated at the slope toe, but no seismic events have been recorded as yet. Therefore, synthetic seismograms have been used to evaluate the ground response analysis at the surface. Because the average slope is moderate (less than 15°), 1-D computer codes have been used to model the equivalent-linear earthquake site response analyses of layered hill deposits, as generally performed by professionals. However, the slope response has also been analysed in greater detail, using a 2-D computer code and the soil characterisation has been evaluated accurately by means of borings, Down-Hole tests, SDMT tests and laboratory tests. Comparing 1-D with 2-D results the stratigraphic site amplification and the Topographic Aggravation Factor (TAF) have also been computed. The aim of the study is that it will form a basis for the design of works to remediate the damage caused by a landslide reactivated by the earthquake in Eastern Sicily on December 13, 1990 (M_L=5.4).     

Seismic microzoning from synthetic ground motion parameters: case study, Santiago de Cuba

Synthetic seismograms (P-SV and SH waves) have been calculated along six profiles in Santiago de Cuba basin, with a cutoff frequency of 5 Hz, by using a hybrid approach (modal summation for a regional 1D structure plus finite differences for a local 2D structure embedded in the first). They correspond to a scenario earthquake of M_S=7 that may occur in Oriente fault zone, directly south of the city. As initial data for a seismic microzoning, the characterisation of earthquake effects has been made considering several relative (2D/1D) quantities (PGDR, PGVR, PGAR, DGAR, I_AR—ratios of peak ground values of displacement, velocity and acceleration, and of design ground acceleration and Arias intensity-, etc.) and functions representative of the ground motion characteristics in soil (2D) with respect to bedrock (1D). The functions are the response spectra ratio RSR(f), already routinely used in this kind of work, and the elastic energy input ratio E_IR(f), defined, for the first time, in this paper. These data, sampled at 115 sites within all the profiles have been classified in two steps, using logical combinatory algorithms: connected components and compact sets. In the first step, from the original ground motion parameters or functions extracted from the synthetic seismograms, nine sets have been classified and the partial results show the spatial distribution of the soil behaviour as a function of the component of motion. In the second step, the results of the classification of the nine sets have been used as input for a further classification that shows a spatial distribution of sites with a quasi-homogeneous integral ground motion behaviour. By adding the available geological surface data, a microzoning scheme of Santiago de Cuba basin has been obtained.     

Seismic risk of circum-Pacific earthquakes: II. Extreme values using Gumbel's third distribution and the relationship with strain energy release

In a previous paper (Makropoulos andBurton, 1983) the seismic risk of the circum-Pacific belt was examined using a whole process technique reduced to three representative parameters related to the physical release of strain energy, these are:M ₁, the annual modal magnitude determined using the Gutenberg-Richter relationship;M ₂, the magnitude equivalent to the total strain energy release rate per annum, andM ₃, the upper bound magnitude equivalent to the maximum strain energy release in a region.The risk analysis is extended here using the part process statistical model of Gumbel's IIIrd asymptotic distribution of extreme values. The circum-Pacific is chosen being a complete earthquake data set, and the stability postulate on which asymptotic distributions of extremes are deduced to give similar results to those obtained from whole process or exact distributions of extremes is successfully checked. Additionally, when Gumbel III asymptotic distribution curve fitting is compared with Gumbel I using reduced chi-squared it is seen to be preferable in all cases and it also allows extensions to an upper-bounded range of magnitude occurrences. Examining the regional seismicity generates several seismic risk results, for example, the annual mode for all regions is greater thanm(1)=7.0, with the maximum being in the Japan, Kurile, Kamchatka region atm(1)=7.6. Overall, the most hazardous areas are situated in this northwestern region and also diagonally opposite in the southeastern circum-Pacific. Relationships are established between the Gumbel III parameters and quantitiesm ₁(1),X ₂ and , quantities notionally similar toM ₁,M ₂ andM ₃ although is shown to be systematically larger thanM; thereby giving a physical link through strain energy release to seismic risk statistics. Inall regions of the circum-Pacific similar results are obtained forM ₁,M ₂ andM ₃ and the notionally corresponding statistical quantitiesm ₁(1),X ₂ and , demonstrating that the relationships obtained are valid over a wide range of seismotectonic enviroments.     

Seismic hazard model for loss estimation and risk management in Taiwan

We developed a seismic hazard model for Taiwan that integrates all available tectonic, seismicity, and seismic hazard information in the region to provide risk managers and engineers with a model they can use to estimate earthquake losses and manage seismic risk in Taiwan. The seismic hazard model is composed of two major components: a seismotectonic model and a ground-shaking model. The seismotectonic model incorporates earthquakes that are expected to occur on the Ryukyu and Manila subduction zones, on the intermediate-depth Wadati-Benioff seismicity zones, on the active crustal faults, and within seismotectonic provinces. The active crustal faults include the Chelungpu fault zone, the source of the damaging M_W 7.6 Chi-Chi earthquake, and the Huangchi-Hsiaoyukeng fault zone that forms the western boundary of the Taipei Basin. The ground-shaking model uses both US, worldwide, and Taiwanese attenuation relations to provide robust estimates of peak ground acceleration and response spectral acceleration on a reference site condition for shallow crustal and subduction zone earthquakes. The ground shaking for other site conditions is obtained by applying a nonlinear soil-amplification factor defined in terms of the average shear-wave velocity in the top 30 m of the soil profile, consistent with the methodology used in the current US and proposed Taiwan building codes.     

Winkler model for dynamic response of composite caisson–piles foundations: Seismic response

A simplified method with a dynamic Winkler model to study the seismic response of composite caisson–piles foundations (CCPF¹) is developed. Firstly, with the dynamic Winkler model, the kinematic response of the CCPF subjected to vertically propagating seismic S-wave is analyzed by coupling the responses of caisson part and pile part. Secondly, a simplified model for the foundation–structure system is created with the structure simplified as a lumped mass connected to the foundation with an elastic column, and through the Fast Fourier Transformation (FFT) this model is enabled to solve transient seismic problems. Thirdly, the proposed method for the seismic response of CCPF-structure systems is verified by comparison against 3D dynamic finite element simulation, in which the Domain Reduction Method (DRM²) is utilized. Lastly, the mechanism and significance of adding piles in improving the earthquake resistance of the foundation and structure is analyzed through an example with different soil conditions. Discovered in this study is that adding piles under the caisson is an efficient way to increase seismic resistant capability of the soil–foundation–structure system, and the main mechanism of that is the elimination of the pseudo-resonance.     

Idealisation of soil–structure system to determine inelastic seismic response of mid-rise building frames

In this study, a novel and enhanced soil–structure model is developed adopting the direct analysis method using FLAC 2D software to simulate the complex dynamic soil–structure interaction and treat the behaviour of both soil and structure with equal rigour simultaneously. To have a better judgment on the inelastic structural response, three types of mid-rise moment resisting building frames, including 5, 10, and 15 storey buildings are selected in conjunction with three soil types with the shear wave velocities less than 600 m/s, representing soil classes C_e, D_e and E_e, according to Australian Standards. The above mentioned frames have been analysed under two different boundary conditions: (i) fixed-base (no soil–structure interaction) and (ii) flexible-base (considering soil–structure interaction). The results of the analyses in terms of structural displacements and drifts for the above mentioned boundary conditions have been compared and discussed. It is concluded that considering dynamic soil–structure interaction effects in seismic design of moment resisting building frames resting on soil classes D_e and E_e is essential.     

Seismic amplitude inversion for the transversely isotropic media with vertical axis of symmetry

Transverse isotropy with a vertical axis of symmetry is a common form of anisotropy in sedimentary basins, and it has a significant influence on the seismic amplitude variation with offset. Although exact solutions and approximations of the PP-wave reflection coefficient for the transversely isotropic media with vertical axis of symmetry have been explicitly studied, it is difficult to apply these equations to amplitude inversion, because more than three parameters need to be estimated, and such an inverse problem is highly ill-posed. In this paper, we propose a seismic amplitude inversion method for the transversely isotropic media with a vertical axis of symmetry based on a modified approximation of the reflection coefficient. This new approximation consists of only three model parameters: attribute A, the impedance (vertical phase velocity multiplied by bulk density); attribute B, shear modulus proportional to an anellipticity parameter (Thomsen's parameter ε−δ); and attribute C, the approximate horizontal P-wave phase velocity, which can be well estimated by using a Bayesian-framework-based inversion method. Using numerical tests we show that the derived approximation has similar accuracy to the existing linear approximation and much higher accuracy than isotropic approximations, especially at large angles of incidence and for strong anisotropy. The new inversion method is validated by using both synthetic data and field seismic data. We show that the inverted attributes are robust for shale-gas reservoir characterization: the shale formation can be discriminated from surrounding formations by using the crossplot of the attributes A and C, and then the gas-bearing shale can be identified through the combination of the attributes A and B. We then propose a rock-physics-based method and a stepwise-inversion-based method to estimate the P-wave anisotropy parameter (Thomsen's parameter ε). The latter is more suitable when subsurface media are strongly heterogeneous. The stepwise inversion produces a stable and accurate Thomsen's parameter ε, which is proved by using both synthetic and field data.     

The design of visco‐acoustic weighted L1‐error frequency–space wavefield extrapolators

Seismic imaging is an important step for imaging the subsurface structures of the Earth. One of the attractive domains for seismic imaging is explicit frequency–space (f – x) prestack depth migration. So far, this domain focused on migrating seismic data in acoustic media, but very little work assumed visco‐acoustic media. In reality, seismic exploration data amplitudes suffer from attenuation. To tackle the problem of attenuation, new operators are required, which compensates for it. We propose the weighted L ₁ ‐error minimisation technique to design visco‐acoustic f – x wavefield extrapolators. The L ₁ ‐error wavenumber responses provide superior extrapolator designs as compared with the previously designed equiripple L ₄ ‐norm and L_∞‐norm extrapolation wavenumber responses. To verify the new compensating designs, prestack depth migration is performed on the challenging Marmousi model dataset. A reference migrated section is obtained using non‐compensating f – x extrapolators on an acoustic dataset. Then, both compensating and non‐compensating extrapolators are applied to a visco‐acoustic dataset, and both migrated sections are then compared. The final images show that the proposed weighted L ₁ ‐error method enhances the resolution and results in practically stable images.