Table of contents

In the last two decades, the use of non-invasive methods for the study and conservation of cultural heritage, from artefacts and historical sites to recent architectural structures, has gained increasing interest. This is due to several reasons: (i) the improvement of performance and information resolution of sensors and devices; (ii) the increasing availability of user-friendly data/image analysis, and processing software and routines; (iii) the ever greater awareness of archaeologists and conservators of the benefits of these technologies, in terms of reduction of costs, time and the risk associated with direct and destructive investigations of archaeological sites (excavation) and monuments (i.e. masonry coring).

The choice of diagnostic strategy depends on the spatial and physical characteristics of the cultural objects or sites, the aim of the investigation (knowledge, conservation, restoration) and the issues to be addressed (monitoring, decay assessment, etc). This makes the set up and validation of ad hoc procedures based on data processing and post-processing methods necessary, generally developed to address issues in other fields of application.

This methodological perspective based on an integrated and multi-scale approach characterizes the papers of this special issue, which is focused on integrated non-invasive sensing techniques and geophysical methods for the study and conservation of architectural, archaeological and artistic heritage.

In particular, attention is given to the advanced application of the synthetic aperture radar (SAR) from the satellite-based platform for deformation monitoring thanks to the innovative differential SAR interferometry (DInSAR) technique; Zeni et al show the significant possibilities of the proposed methodology in achieving a global vision not only of cultural heritage but also of the embedding territory.

This collection also deals with the application of non-invasive diagnostics to archaeological prospecting, and cultural heritage monitoring and diagnostics. The paper by Conyers et al gives a very good review of the limitations and advantages of ground penetrating radar (GPR) in archaeological prospecting, where the focus is on aspects of data processing and image visualization. Kadioglu et al describe a case that exploits GPR to define buried archaeological remains at the Zeynel Bey Tomb in the ancient city of Hasankeyf, Turkey. In particular, attention is given to a new three-dimensional visualization of the scene which provides a clearer representation compared to the usual constant depth slices.

The paper by Bavusi et al is concerned with the use of GPR as a tool to monitor an important recent architectural masterpiece and critical piece of infrastructure, the Musmeci bridge in Potenza; in particular, the complexity of this civil engineering structure is tackled by adopting advanced data processing tools such as microwave tomography. A microwave tomographic approach is also exploited for GPR data processing in cultural heritage diagnostics by Brancaccio et al, where their survey is concerned with the masonry diagnostics at St Carmine's Church, one of the more ancient and important monuments of Naples.

This special issue also gives a good overview of state-of-the-art methodologies and appealing cases of data integration and correlation. Nordebo et al present a new and advanced approach to information fusion, used here to combine ERT data from different orientations. The approach is general, suitable for sensor integration, and opens up a new route to data fusion for diagnostics and monitoring.

Some very interesting cases of data integration in archaeological prospecting and cultural heritage monitoring are also presented. Loperte et al present an interesting case of an integrated geophysical approach based on magnetic, GPR and geoelectric techniques used to investigate the Greek and Roman settlement of Paestum in southern Italy; their aim is to undertake preventive archaeological research on an area of the ancient settlement affected by new construction.

Another interesting case of on-field integration of GPR and seismic techniques is presented by Leucci et al; their aim is to characterize the deterioration status of the pillars of the cathedral of Tricarico, in the Basilicata region of southern Italy, where the diagnostics have provided crucial information that is helping to drive the restoration work.

Another example of sensor integration is presented by Carlomagno et al, which is focused on the joint exploitation of infrared thermography and GPR for inspection of architectonic structures. Two important studies are presented: the first aims to evaluate the conservation of frescoes as well as the state of the underground masonry structure at Pompei (Naples); the second aims to evaluate the state of degradation of remnants of a theatre at a site in Nora (Cagliari).

The integration of different diagnostic techniques is shown to be important also for movable pieces such as the sculpture of the Pharaoh with the god Amun, one of the most important pieces in the 'Museo delle Antichitá Egizie' of Turin. In particular, Sambuelli et al demonstrate the joint use of GPR ultrasonic 2D tomographies to assess the integrity of the base of the statue. Capizzi et al propose the integration of a pacometer, high frequency GPR and ultrasonic techniques to investigate the internal extension of all the visible fractures (and to search for hidden ones) of a marble slab (second to third century AD) from the archaeological museum of Rome, which was damaged in transit for an exhibition.

Finally, this special issue also presents cases of surface and material characterization, which is a field of increasing interest in monumental heritage monitoring. Camaiti et al present a survey that uses a portable hyperspectral device to characterize the alteration affecting the marble facade of the Santa Maria Novella church (13th century) in Florence. Their analysis shows how hyperspectral instruments may be considered a powerful tool for characterizing historical surfaces in a non-destructive and non-invasive way. Fort et al describe an advanced petrophysical analysis of several types of natural stone used in Spain that can be used to determine their anisotropy, which is one of the key factors affecting the quality and resistance to decay caused by a variety of different agents, such as water.

Monitoring of deformation phenomena affecting urban areas and man-made structures is of key relevance for the preservation of the artistic, archaeological and architectural heritage. The differential SAR interferometry (DInSAR) technique has already been demonstrated to be an effective tool for non-invasive deformation analyses over large areas by producing spatially dense deformation maps with centimetre to millimetre accuracy. Moreover, by exploiting long sequences of SAR data acquired by different sensors, the advanced DInSAR technique referred to as the small baseline subset (SBAS) approach allows providing long-term deformation time series, which are strategic for guaranteeing the monitoring of urban area displacements. In this work, we investigate the effectiveness of the two-scale multi-sensor SBAS-DInSAR approach to detect and monitor displacements affecting historical and artistic monuments. The presented results, achieved by applying the full resolution SBAS technique to a huge set of ERS-1/2 and ENVISAT data, spanning the 1992–2010 time interval and relevant to the city of Rome (Italy), show the capability of this approach to detect and analyse the temporal evolution of possible deformation phenomena affecting historical buildings and archaeological sites. Accordingly, our analysis demonstrates the effectiveness of the full resolution multi-sensor SBAS approach to operate as a surface deformation tool for supporting the study and conservation strategies of the historical, cultural and artistic heritage.

Ground-penetrating radar is an extremely useful tool for the mapping and interpretation of buried cultural remains within 2–3 metres of the surface, especially when the stratigraphy is complex. Standard reflection profiles can be processed to correct for depth and distance, and also filtered and processed to make cultural features visible. When many profiles are collected in closely spaced transects in a grid, reflections can be re-sampled and displayed in amplitude slice-maps, and isosurface renderings to make buried features visible. Sometimes, however, the abundance and complexity of subsurface reflections is so complex that each individual profile must be interpreted manually, which necessitates an understanding of radar wave propagation, reflection, refraction and attenuation in the ground. In order to differentiate reflections from cultural features this understanding of radar energy must be merged with an understanding of the chemistry of the ground, soil and geological stratigraphy, and how those variables affect radar reflections. When taken as a package of visualization tools, GPR can be used as an effective tool for interpreting aspects of history and culture at buried sites in ways not possible using traditional archaeological methods.

This paper is the result of a joint work between geophysicists and archaeologists in which the authors have used geophysical techniques to investigate the Greek and Roman settlement of Paestum, southern Italy for preventive archaeological research (commonly termed 'rescue archaeology') on an area of the ancient settlement affected by new building work and infrastructure. Starting from a background analysis of the archaeological and geological features of the site, an integrated geophysical approach based on magnetic, GPR and geoelectrical surveys was carried out on the Santa Venera area, a site selected to build a car parking. High-density and high-resolution cross-correlated geophysical surveys were carried out in different parts of the area to better resolve the structures. Systematic excavations confirmed the clues suggested by geophysical prospecting about the presence of archaeological remains such as walls, canals and tombs. By the use of non-destructive geophysical techniques a two-fold aim was reached: to properly plan the building of the infrastructure and preserve the ancient artefacts according to the advanced European guidelines on the protection of archaeological heritage.

An extensive experimental and numerical investigation has been carried out to assess the status of the 'Ponte sul Basento' (1967–1976), in the town of Potenza (Basilicata region, southern Italy), better known as the Musmeci bridge. Architecturally, the bridge is a considerable reinforced 20th century concrete structure that was designed and built by the Italian architect Sergio Musmeci (1926–1981). Moreover, the bridge represents an important element of the infrastructural network, linking the city centre to the Potenza-Sicignano highway, crossing the Basento river and the railway close to the main train station of the city. Recently, due to ageing and continuous and significant traffic, the bridge started to be affected by several problems such as water infiltration. Within the presented study, a widespread ground penetrating radar (GPR) survey has been designed to investigate the geometrical characteristics of the bridge deck (Gerber saddles, internal stiffening walls, pillar supports) and detect the presence of defects or damage due to water infiltration and traffic fatigue. Concerning this, a 900 MHz 3D GPR survey has been performed along a zone of one of the lanes on the road surface. Moreover, a second 1500 MHz 3D survey has been carried out at the bottom of the bridge deck in order to gain detailed information about an important structural element of the bridge, the Gerber saddle. Both results have been processed following two approaches: the first a classical time-domain processing session based on commercial software and the use of migration; the second in microwave tomography, an advanced frequency domain automatic PC-based inversion algorithm. In this paper, we present a comparative interpretation of both kinds of processed results, and provide considerations about the investigated structures.

The problem of diagnosing the internal status of a masonry structure by means of a non-destructive electromagnetic technique, based on a microwave tomographic imaging algorithm, is dealt with. The study is first developed through numerical arguments under a two-dimensional scalar geometry. Then, the microwave imaging algorithm is checked against experimental data collected in situ at the Carmine's church belfry in Naples, Italy, by means of a ground penetrating radar system. The features of two different methods for clutter removal are also discussed.

Information fusion via multimodal inverse problems and different sensors is addressed using a Fisher information analysis approach. The Fisher information measure is inherently additive, and it facilitates an appropriate weighting of the measurement data that is statistically optimal and can hence be useful with reconstruction algorithms in geophysical sensing. Given that there exists proper knowledge about the sensor noise statistics, correlations and spectral contents, as well as a correct forward model, the Fisher information is a natural measure of information because it is closely linked to the statistical maximum likelihood principle. To illustrate the concept of data correlation based on statistical Fisher information analysis, two simple and generic examples are employed in electrical resistivity and electromagnetic tomography, which are motivated by geophysical applications, such as tunnel detection. The examples demonstrate that a properly weighted data fusion can be of crucial importance for an ill-posed multimodal inverse problem.

The aim of this paper is to show a new monitoring approximation for ground penetrating radar (GPR) data. The method was used to define buried archaeological remains inside and outside the Zeynel Bey tomb in Hasankeyf, an ancient city in south-eastern Turkey. The study examined whether the proposed GPR method could yield useful results at this highly restricted site, which has a maximum diameter inside the tomb of 4 m. A transparent three-dimensional (3D) half bird's eye view was constructed from a processed parallel-aligned two-dimensional GPR profile data set by using an opaque approximation instead of linear opacity. Interactive visualizations of transparent 3D sub-data volumes were conducted. The amplitude-colour scale was balanced by the amplitude range of the buried remains in a depth range, and appointed a different opaque value for this range, in order to distinguish the buried remains from one another. Therefore, the maximum amplitude values of the amplitude-colour scale were rearranged with the same colour range. This process clearly revealed buried remains in depth slices and transparent 3D data volumes. However, the transparent 3D half bird's eye views of the GPR data better revealed the remains than the depth slices of the same data. In addition, the results showed that the half bird's eye perspective was important in order to image the buried remains. Two rectangular walls were defined, one within and the other perpendicularly, in the basement structure of the Zeynel Bey tomb, and a cemetery was identified aligned in the east–west direction at the north side of the tomb. The transparent 3D half bird's eye view of the GPR data set  also determined the buried walls outside the tomb. The findings of the excavation works at the Zeynel Bey tomb successfully overlapped with the new visualization results.

In this paper, we present the results of a diagnostics survey, based on the exploitation of ground penetrating radar (GPR) and sonic prospecting, to characterize the deterioration status of the pillars of the cathedral of Tricarico, in the Basilicata region (Southern Italy). The prospecting falls within the more general framework of investigating the structural conditions of this monument, which is affected by heavy instability problems. This study case points out the great effectiveness of the two employed diagnostic methods, when used in an integrated way, for detecting cracks and inhomogeneities in the inner structure of masonry building elements. With regard to GPR prospecting, a comparison is made between the results obtained by a standard processing and those obtained by means of an inverse scattering algorithm. For one of the investigated pillars, the results obtained from non-invasive tests are compared with those of direct inspection. This is performed by coring the pillar and examining both the core and the hole (the latter by means of an endoscope). The seismic investigation allowed us to prove the mediocre or bad state of conservation of the pillars.

This work is focused on the integration of infrared thermography and ground penetrating radar for the inspection of architectonic structures. First, laboratory tests were carried out with both techniques by considering an ad hoc specimen made of concrete and with the insertion of anomalies of a different nature and at different depths. Such tests provided helpful information for ongoing inspections in situ, which were later performed in two important Italian archaeological sites, namely Pompeii (Naples) and Nora (Cagliari). In the first site, the exploration was devoted to the analysis of the wall paintings of Villa Imperiale with the aim of evaluating the state of conservation of frescoes as well of the underneath masonry structure. As main findings, the applied techniques allowed outlining some areas, which were damaged by ingression in-depth of moisture and/or by disaggregation of the constituent materials, and also for recognition of previous restoration. In the archaeological area of Nora, instead, the attention was driven towards the evaluation of the state of degradation of the theatre remnants. Our prospections show that the front side of the theatre, being more strongly affected by degradation, needs a massive restoration work. As a general result, we demonstrated that a joint interpretation of infrared thermography and ground penetrating radar data supplies detailed 3D information from near-surface to deep layers, which may assist in restoration planning.

By late 2008 one of the most important pieces of the 'Museo delle Antichità Egizie' of Turin, the sculpture of the Pharaoh with god Amun, was planned to be one of the masterpieces of a travelling exhibition in Japan. The 'Fondazione Museo delle Antichità Egizie di Torino', who manages the museum, was concerned with the integrity of the base of the statue which actually presents visible signs of restoration dating back to the early 19th century. It was required to estimate the persistence of the visible fractures, to search for unknown ones and to provide information about the overall mechanical strength of the base. To tackle the first question a GPR reflection survey along three sides of the base was performed and the results were assembled in a 3D rendering. As far as the second question is concerned, two parallel, horizontal ultrasonic 2D tomograms across the base were made. We acquired, for each section, 723 ultrasonic signals corresponding to different transmitter and receiver positions. The tomographic data were inverted using four different software packages based upon different algorithms. The obtained velocity images were then compared each other, with the GPR results and with the visible fractures in the base. A critical analysis of the comparisons is finally presented.

The archaeological museum of Rome asked our group about the physical consistency of a marble slab (second to third century AD) that recently fell during its travel as part of an exhibition. We decided to use different methodologies to investigate the slab: namely a pacometer (Protovale Elcometer) to individuate the internal coupling pins, and ground-penetrating radar (GPR) (2000 MHz) and ultrasonic (55 kHz) tomographic high-density surveys to investigate the internal extension of all the visible fractures and to search for the hidden ones. For the ultrasonic data, tests were carried out to optimize the inversion parameters, in particular the cell dimensions. Surely, the choice of cell size for the inversion process must take into account the size of the acquisition grid and the ray number acquired. We proposed to calculate a minimum Fresnel's radius using the sampling frequency instead of that of the probes. For every methodology used, the quality of the acquired data was relatively high. This was then processed and compared to provide information that was useful for some of the insurance problems of the museum. Later on, the data was processed in depth to see how to improve the data processing and interpretation. Finally, the results of this in-depth study were exposed in detail. Ultrasonic and GPR tomographies show a strong correlation, and in particular, the inhomogeneous areas are located in correspondence to the slab injuries.

A portable hyperspectral device (ASD-FieldSpec FR Pro) has been employed for the characterization of alterations affecting the marble facade of the Santa Maria Novella church (XIII cent.) in Florence (Italy). The ASD-FieldSpec FR Pro collects the reflectance spectra of a selected target area (about 1.5 cm²). The spectra of calcite, gypsum and other mineral phases commonly occurring on outdoor surfaces exposed to the urban atmosphere were collected and presented. The spectral features of alteration minerals (depth of reflectance minima) appear to be affected by grain size, phase abundance in addition to lightness (L*) of the target area. Notwithstanding these limitations, the spectra may be used for a qualitative screening of the alteration and, under reasonable assumptions, the reflectance band depth may be used also for quantitative estimation of phase abundance. The monitoring of the conservation state of outdoor surfaces is considered of fundamental importance to plan conservative interventions on historical buildings. Our results point out that portable hyperspectral instruments may be considered as powerful tools for characterizing historical surfaces in a nondestructive and noninvasive way.

Anisotropy is a petrophysical property of natural stone and other construction materials that determines their quality and resistance to decay due to a variety of agents, such as water. A study was conducted on nine types of stone widely used in Spain's built heritage, using six previously defined anisotropy indices. These indices can be used to determine the degree of anisotropy, which helps explain the differential decay observed in stone materials quarried in the same bed and used to build the same structure. The conclusion reached is that anisotropy should be determined in the natural stone used both to restore the architectural heritage and in new construction, since the appropriate choice of material quality ensures greater resistance to decay and, therefore, increased durability. Materials with the lowest possible anisotropy should be selected, as this property governs their hydraulic behaviour: the lower the anisotropy in a material, the better its behaviour in relation to water and the longer its durability.

This paper focuses on the interpretation of seismic reflection, gravimetric, topographic, deep seismic refraction and seismicity data to study the recently proposed Ota–Vila Franca de Xira–Lisbon–Sesimbra (OVLS) fault zone and the lower Tagus Cenozoic basin (LTCB). The studied structure is located in the lower Tagus valley (LTV), an area with over 2 million inhabitants that has experienced historical earthquakes which caused significant damage and economical losses (1344, 1531 and 1909 earthquakes) and whose tectonic sources are thought to be local but mostly remain unknown. This study, which is intended as a contribution to improve the seismic hazard of the area and the neotectonics of the region, shows that the above-proposed fault zone is probably a large crustal thrust fault that constitutes the western limit of the LTCB. Gravimetric, deep refraction and seismic reflection data suggest that the LTCB is a foreland basin, as suggested previously by some authors, and that the OVLS northern and central sectors act as the major thrusts. The southern sector fault has been dominated by strike-slip kinematics due to a different orientation to the stress field. Indeed, geological outcrop and seismic reflection data interpretation suggests that, based on fault geometry and type of deformation at depth, the structure is composed of three major segments. These data suggest that these segments have different kinematics in agreement with their orientation to the regional stress field. The OVLS apparently controls the distribution of the seismicity in the area. Geological and geophysical information previously gathered also points that the central segment is active into the Quaternary. The segment lengths vary between 20 and 45 km. Since faults usually rupture only by segments, maximum expectable earthquake magnitudes and other parameters have been calculated for the three sectors of the OVLS fault zone using empirical relationships between earthquake statistics and geological parameters available from the literature. Calculated slip rates are compatible with previous estimates for the area (0.33 mm yr^–1). A more accurate estimation of the OVLS throw in the Quaternary sediments is therefore of vital importance for a more accurate evaluation of the seismic hazard of the area.

The Hormuz Strait is an important area which is located between the Persian Gulf, with half the oil reserves of the world, and the Oman Sea south of Iran. Many geophysical studies have been done to analyse the structure of subsurface layers in this strait and explore many new gas reservoirs as these findings help to simulate images from reservoir stratigraphy and understand the paleo-depositional environments. Spectral decomposition, which is a quick and important method in seismic interpretation, provides a strong tool to study these types of subsurface layers. In this paper, a red–green–blue (RGB) multi-colour display technique has been performed as an effective interpretation tool. This technique stacks three different frequencies with three different colours for composite display. So, these frequency components show the distribution of channel thickness as we map the variation of thickness with a colour scale that is equal to the quantitative thickness according to its relationship with frequency. In addition, seismic attributes such as coherency, peak and instantaneous frequency, which are sensitive to the edges of stratigraphic events (such as channels), are shown with a composite plot too; although the RGB technique is better than other techniques, for thickness variation a combined application is the best method of study.

This study is an application of the 2D resistivity inversion of Schlumberger electrical sounding. The 2D resistivity inversion was performed using a code based on the finite-element technique and regularization method. Twenty-nine electrical soundings collected in the delta of Wadi El-Arish, Northern Sinai, were used as a case study. The main objective is to compare the 1D and 2D inversion results of the subsurface resistivity distribution in areas where the lateral resistivity variation cannot be neglected. The examination of the resulting 2D, 1D resistivity models and geological cross-section built from borehole information approximately at the same location showed a significant reliability of 2D models. Some subsurface geological and hydrogeological features could be interpreted such as the identification of clay lenses, potentiometric surface of the quaternary aquifer, potentiometric depression due to over-pumping, the high resistivity contrast due to lateral facies change, and basin-like structural features.

An inversion algorithm is developed to estimate the depth and the associated model parameters of the anomalous body from the gravity or self-potential (SP) whole measured data. The problem of the depth (z) estimation from the observed data has been transformed into a nonlinear equation of the form F(z) = 0. This equation is then solved for z by minimizing an objective functional in the least-squares sense. Using the estimated depth, the polarization angle and the dipole moment or the depth and the amplitude coefficient are computed from the measured SP or gravity data, respectively. The method is based on determining the root mean square (RMS) of the depths estimated from using all s-values for each shape factor. The minimum RMS is used as a criterion for estimating the correct shape and depth of the buried structure. When the correct shape factor is used, the RMS of the depths is always less than the RMS computed using wrong shape factors. The proposed approach is applicable to a class of geometrically simple anomalous bodies, such as the semi-infinite vertical cylinder, the dike, the horizontal cylinder and the sphere, and it is tested and verified on synthetic examples with and without noise. This technique is also successfully applied to four real datasets for mineral exploration, and it is found that the estimated depths and the associated model parameters are in good agreement with the actual values.

Rapid techniques for self-potential (SP) data interpretation are of prime importance in engineering and exploration geophysics. Parameters (e.g. depth, width) estimation of the ore bodies has also been of paramount concern in mineral prospecting. In many cases, it is useful to assume that the SP anomaly is due to an ore body of simple geometric shape and to use the data to determine its parameters. In light of this, we describe a rapid approach to determine the depth and horizontal width of a two-dimensional plate from the SP anomaly. The rationale behind the scheme proposed in this paper is that, unlike the two- (2D) and three-dimensional (3D) SP rigorous source current inversions, it does not demand a priori information about the subsurface resistivity distribution nor high computational resources. We apply the second-order moving average operator on the SP anomaly to remove the unwanted (regional) effect, represented by up to a third-order polynomial, using filters of successive window lengths. By defining a function F at a fixed window length (s) in terms of the filtered anomaly computed at two points symmetrically distributed about the origin point of the causative body, the depth (z) corresponding to each half-width (w) is estimated by solving a nonlinear equation in the form ξ(s, w, z) = 0. The estimated depths are then plotted against their corresponding half-widths on a graph representing a continuous curve for this window length. This procedure is then repeated for each available window length. The depth and half-width solution of the buried structure is read at the common intersection of these various curves. The improvement of this method over the published first-order moving average technique for SP data is demonstrated on a synthetic data set. It is then verified on noisy synthetic data, complicated structures and successfully applied to three field examples for mineral exploration and we have found that the estimated depth is in good agreement with the known value reported in the literature.

Seismic absorption is sensitive to gas saturation, and this can lead to a number of observable features, such as low dominant frequency, absence of high frequencies and dimming of reflections. We obtain the time–frequency distribution of seismic signals at different scales based on wavelet decomposition and try to analyse the response of high-frequency sections for the characterization of hydrocarbon reservoirs. We present a case study using multi-scale (dominant) frequency and multi-scale absorption, and demonstrate that the method is effective in recognizing a deep tight-sandstone gas reservoir. By this study, we can draw the conclusion that multi-scale frequency and absorption attributes may indicate the gassy areas in reservoirs, and we have obtained many good applications in the Sichuan basin of China.

The no-type curve with negative skin of a horizontal well has been found in the current research. Negative skin is very significant to transient well test and rate decline analysis. This paper first presents the negative skin problem where the type curves with negative skin of a horizontal well are oscillatory. In order to solve the problem, we propose a new model of transient well test and rate decline analysis for a horizontal well in a multiple-zone composite reservoir. A new dimensionless definition of r_D is introduced in the dimensionless mathematical modelling under different boundaries. The model is solved using the Laplace transform and separation of variables techniques. In Laplace space, the solutions for both constant rate production and constant wellbore pressure production are expressed in a unified formula. We provide graphs and thorough analysis of the new standard type curves for both well test and rate decline analysis; the characteristics of type curves are the reflections of horizontal well production in a multiple-zone reservoir. An important contribution of our paper is that our model removed the oscillation in type curves and thus solved the negative skin problem. We also show that the characteristics of type curves depend heavily on the properties of different zones, skin factor, well length, formation thickness, etc. Our research can be applied to a real case study.

We use the nonstationary equivalent of the Fourier shift theorem to derive a general one-dimensional integral transform for the application and removal of certain seismic data processing steps. This transform comes from the observation that many seismic data processing steps can be viewed as nonstationary shifts. The continuous form of the transform is exactly reversible, and the discrete form provides a general framework for unitary and pseudounitary imaging operators. Any processing step which can be viewed as a nonstationary shift in any domain is a special case of this transform. Nonstationary shifts generally produce coordinate distortions between input and output domains, and those that preserve amplitudes do not conserve the energy of the input signal. The nonstationary frequency distortions, time distortions and nonphysical energy changes inherent to such operations are predicted and quantified by this transform. Processing steps of this type are conventionally implemented using interpolation operators to map discrete data values between input and output coordinate frames. Although not explicitly derived to perform interpolation, the transform here assumes the Fourier basis to predict values of the input signal between sampling locations. We demonstrate how interpolants commonly used in seismic data processing and imaging approximate the proposed method. We find that our transform is equivalent to the conventional sinc interpolant with no truncation. Once the transform is developed, we demonstrate its numerical implementation by matrix–vector multiplication. As an example, we use our transform to apply and remove normal moveout.