Spatial Sensitivity Functions Research Articles

Fast-forward modeling of borehole nuclear magnetic resonance measurements acquired in deviated wells and spatially heterogeneous formations

High-angle and horizontal (HAHz) wells are routinely drilled to enhance reservoir exposure and increase fluid productivity. The interpretation of nuclear magnetic resonance (NMR) measurements entails major technical challenges in horizontal layers penetrated by HAHz wells or through dipping formation layers penetrated by a vertical well. Three-dimensional geometric effects, coupled with spatially and petrophysically heterogeneous rocks, may bias petrophysical estimates obtained from borehole NMR measurements using methods designed for vertical wells and horizontal layers. Reliable interpretation requires an accurate and rapid forward-modeling method that integrates NMR physics and tool/instrument specifications, borehole, and 3D formation geometric properties. The latter is possible with the recent development of a forward-modeling algorithm that accurately and efficiently simulates NMR measurements based on 3D spatial sensitivity functions (SSFs). We implement and quantify this forward-modeling approximation across dipping formations penetrated by deviated wells in the presence of mud-filtrate invasion. We validate and verify the fast approximation against a 3D multiphysics model using challenging synthetic cases, in highly apparent dip wells with the varying radial front of the mud-filtrate invasion. Our work compares the effect of radial length of investigation from the three distinct NMR acquisition shells to better understand borehole NMR measurements acquired in 3D complex geometries. On average, the fast approximation via SSFs reproduces NMR measurements in 3 s of central processing unit time with maximum root-mean-square errors below 1% and can therefore be used for real-time calculations and interpretations. Modeling results indicate that thinly bedded formations and their petrophysical properties can be resolved with relatively low measurement resolution in HAHz wells and dipping formations.

Rapid modeling of borehole measurements of nuclear magnetic resonance via spatial sensitivity functions

Borehole measurements of nuclear magnetic resonance (NMR) are routinely used to estimate in situ rock and fluid properties. Conventional NMR interpretation methods often neglect bed-boundary and layer-thickness effects in the calculation of fluid volumetric concentrations and NMR relaxation-diffusion correlations. Such effects introduce notable spatial averaging of intrinsic rock and fluid properties across thinly bedded formations or in the vicinity of boundaries between layers exhibiting large property contrasts. Forward modeling and inversion methods can mitigate the aforementioned effects and improve the accuracy of true layer properties in the presence of mud-filtrate invasion and borehole environmental effects across spatially complex formations. We have developed a fast and accurate algorithm to simulate borehole NMR measurements using the concept of spatial sensitivity functions (SSFs) that honor NMR physics and incorporate tool, borehole, and formation geometry. Tool sensitivity maps are derived from a 3D multiphysics forward model that couples NMR tool properties, magnetization evolution, and electromagnetic propagation. In addition, a multifluid relaxation model based on Brownstein-Tarr’s equation is introduced to estimate layer NMR porosity decays and relaxation-diffusion correlations from pore-size-dependent rock and fluid properties. The latter model is convolved with the SSFs to reproduce borehole NMR measurements. The results indicate that NMR spatial sensitivity is controlled by porosity, electrical conductivity, excitation pulse duration, and tool geometry. We benchmark and verify the SSF-derived forward approximation against 3D multiphysics simulations for a series of synthetic cases with variable bed thickness and petrophysical properties, and in the presence of mud-filtrate invasion in a vertical well. Results indicate that the approximation can be executed in a few seconds in a central processing unit, by a factor of 1000 times faster than rigorous multiphysics calculations, with maximum root-mean-square errors of 1%.

Calibrationless joint compressed sensing reconstruction for rapid parallel MRI

3D magnetic resonance imaging (3D MRI) or multi-slice MRI involves significant data acquisition time. Traditionally, scanning rate of conventional MRI is restricted due to inherent physiological and instrumental limitations. In multi-slice parallel MRI (multi-slice pMRI) adjacent slices are highly correlated, so one can interpolate missing k-space data of any slice from its adjacent slices. Moreover, images corresponding to different coils are also highly correlated as they represent the same field of view (FoV) with different spatial sensitivity functions. Exploiting above redundancies in multi-slice pMRI, we propose a joint compressed sensing (CS) reconstruction model, which contains two joint sparsity promoting regularization terms based on wavelet forest sparsity and joint total variations (JTV). Extensive experiments are carried out for performance evaluations and compared with existing methods using real multi-slice pMRI datasets. Results in terms of CS reconstruction quality show significant improvements compared to the state-of-the-art. Additionally, implementation of the proposed method is done using multi-core CPU and GP-GPU in a hybrid computing environment to study its clinical feasibility in terms of computational time.

Interpretation of borehole flexural measurements in high-angle wells using 3D spatial sensitivity functions

Undulating wells are routinely drilled to improve reservoir exposure across hydrocarbon-bearing zones. Although conventional acoustic-log interpretation methods are reliable in vertical wells, they often yield inaccurate results when applied to high-angle or horizontal wells due to azimuthal asymmetry, spatial averaging effects, and wave-mode interference. Three-dimensional finite-difference and finite-element algorithms are typically used to quantify the aforementioned effects on acoustic logs, but they are extremely demanding on the central processing unit’s (CPU) time and memory. We develop a fast algorithm to simulate flexural slownesses acquired in high-angle wells using 3D linear spatial sensitivity functions. Spatial sensitivity functions quantify the variation of phase slowness measured by the sonic tool due to spatial perturbations of elastic properties. First, we construct 3D frequency-dependent sensitivity functions of flexural modes using the product of 1D axial, radial, and azimuthal sensitivity functions obtained from first-order approximations. Then, we simulate frequency-domain flexural logs acquired with wireline tools and dipole sources in isotropic and vertical transversely isotropic (VTI) formations penetrated by high-angle wells. For the examined examples, simulated flexural logs exhibit root-mean-square errors less than [Formula: see text] when compared with those calculated with a 3D time-domain finite-difference (3D-TDFD) algorithm. We found that in isotropic formations with layers thinner than the length of the receiver array, flexural slownesses acquired separately with cross dipoles are different because of geometric asymmetry, whereas in VTI formations we observed differences between cross-dipole slownesses because of effective anisotropy. Furthermore, the flexural logs are affected by spatial averaging introduced by the acoustic wireline tool, especially in the vicinity of layer boundaries. Three-dimensional sensitivity functions reduce the computation time of the flexural logs from an average of 15 h of CPU time per depth (using 3D-TDFD methods) to less than 3 min. The fast simulation algorithm disregards wave reflections, wave-mode conversion, and wave-mode interference occurring at layer boundaries.

Inversion-based interpretation of borehole sonic measurements using semianalytical spatial sensitivity functions

One of the major challenges in the interpretation of sonic logs is the presence of bed-boundary (shoulder-bed) effects, which can decrease the accuracy of the estimated elastic properties due to spatial averaging caused by the sonic receiver array across thin beds. A reliable approach to account for shoulder-bed effects in the interpretation of sonic logs is the implementation of forward modeling and inversion techniques. We have developed an efficient inversion-based interpretation method for the estimation of in situ rock formation elastic properties from sonic logs acquired in vertical wells. The method uses semianalytical spatial sensitivity functions for fast forward modeling of sonic-slowness logs and estimates formation elastic properties by iteratively matching the available logs with numerical simulations. Due to the intrinsic vertical resolution limits of sonic logs, the inversion method requires predefined bed boundaries using other high-resolution well logs. We first developed a workflow to estimate shear slowness of thin beds from flexural mode dispersions measured at multiple discrete frequencies. Then, we extended the method to estimate layer-by-layer compressional and shear slownesses by combining the slowness logs of nondispersive P- and S-waves. The method was first successfully verified using data numerically simulated for synthetic rock formations consisting of multiple thin beds and exhibiting large elastic property contrasts. Maximum relative errors and maximum uncertainty in the estimates were less than 2% and 6%, respectively. Field data acquired in wells penetrating multiple horizontal thin beds were also used to appraise the reliability and resolution improvement of the method in the estimation of formation compressional and shear slownesses. Our results indicated that our inversion-based interpretation method improved the vertical resolution of sonic logs by 70%. The new inversion-based interpretation method opens up the possibility of integrating modern borehole acoustic measurements with nuclear and resistivity logs for improved poroelastic and petrophysical interpretations.

Spatial sensitivity functions for rapid numerical simulation of borehole sonic measurements in vertical wells

Borehole sonic measurements are routinely used to measure dynamic elastic properties of rock formations surrounding a wellbore. However, shoulder-bed effects, mud-filtrate invasion, and near-wellbore damage often influence the measurements and bias the interpretations. Inversion-based methods can reduce the influence of complex geometric conditions in the estimation of formation properties from sonic logs, but they require fast-forward modeling algorithms. We have developed new spatial sensitivity functions for rapid modeling of borehole sonic measurements, which were equivalent to the Green’s functions of borehole modal wave measurements. An adaptive finite-element (FE) method was used to calculate 1D axial (vertical) and 2D (axial-radial) sensitivity functions, which combined the dependence of sonic logs on the wave mode, frequency, formation properties, and logging-instrument geometry. Semianalytical formulations were also used to calculate the sensitivity functions, which involved modeling the wave propagation through an effective layered medium. The 1D axial sensitivity functions were also extended to quantify propagation properties of nondispersive modes in a borehole. Calculated semianalytical sensitivity coefficients were efficient for rapid simulation of modal frequency dispersion in the presence of a logging tool. Simulated sonic logging measurements across synthetic thinly bedded and invaded formations were compared with numerical simulations obtained with the finite-element method. Our results confirmed the efficiency, reliability, and accuracy of the approximate sonic simulation method. The maximum relative error of the rapid simulation method was consistently less than 4%, with only 2% of the central processing unit time and 2% of the memory usage compared with FE simulations.

Analytical and numerical investigations into hemisphere-shaped electrostatic sensors.

Electrostatic sensors have been widely used in many applications due to their advantages of low cost and robustness. Their spatial sensitivity and time-frequency characteristics are two important performance parameters. In this paper, an analytical model of the induced charge on a novel hemisphere-shaped electrostatic sensor was presented to investigate its accurate sensing characteristics. Firstly a Poisson model was built for electric fields produced by charged particles. Then the spatial sensitivity and time-frequency response functions were directly derived by the Green function. Finally, numerical calculations were done to validate the theoretical results. The results demonstrate that the hemisphere-shaped sensors have highly 3D-symmetrical spatial sensitivity expressed in terms of elementary function, and the spatial sensitivity is higher and less homogeneous near the hemispherical surface and vice versa. Additionally, the whole monitoring system, consisting of an electrostatic probe and a signal conditioner circuit, acts as a band-pass filter. The time-frequency characteristics depend strongly on the spatial position and velocity of the charged particle, the radius of the probe as well as the equivalent resistance and capacitance of the circuit.

Fast modeling of borehole neutron porosity measurements with a new spatial transport-diffusion approximation

The quantitative integration of nuclear measurements into the in situ petrophysical and geophysical evaluation of rock formations has been elusive because of the lack of efficient algorithms to simulate them. We discovered a new method for rapid numerical simulation of borehole neutron measurements using Monte Carlo (MC)-derived spatial flux sensitivity functions (FSFs) and diffusion flux-difference (DFD) approximations. The method calculates spatial sensitivity flux perturbations using flux-difference approximations of one-group neutron diffusion models. By invoking appropriate boundary conditions, the one-group, time-independent neutron diffusion solution is implemented for nonmultiplying systems in 2D and 3D cylindrical coordinates. The solution is differentiated with respect to the neutron cross section to obtain an expression for flux-difference due to cross-section perturbations. Constant transport-correction coefficients for cross-section parameters are calculated with a flux-fitting method to account for deviations of borehole neutron measurements from the physics of diffusion. Thereafter, spatial FSF responses are rapidly and accurately calculated using a first-order Rytov DFD approximation. Estimated flux-differences are next used to calculate lumped higher order perturbation terms. The DFD technique is tested and benchmarked against MC calculations in the presence of standoff, invasion, and well deviation for wireline and logging-while-drilling tools. Benchmark examples and application in highly deviated wells indicate that neutron porosity logs can be accurately and efficiently simulated with the new DFD method, even in complex geometrical and physical conditions, with errors lower than 1.2 porosity units.

Regularized Least Squares Estimating Sensitivity for Self-calibrating Parallel Imaging

Calibration of the spatial sensitivity functions of coil arrays is a crucial element in parallel magnetic resonance imaging (pMRI). The self-calibrating technique for sensitivity extraction has complemented the common calibration technique that uses a separate pre-scan. In order to improve the accuracy of sensitivity estimate from small number of self-calibrating data, which is extracted from a fully sampled central region of a variable-density k-space acquisition in self-calibrating parallel images, a novel scheme for estimating the sensitivity profiles is proposed in the paper. On consideration of truncation error and measurement errors in self-calibrating data, the issue of calculating sensitivity would be formulated as a regularized least squares estimation problem, which is solved by the preconditioned conjugate gradients algorithm. When applying the estimated coil sensitivity to reconstruct full field-of-view(FOV) image from the under-sampling simulated and in vivo data, the normalized signal-to-noise ratio (NSNR) of reconstruction image is evidently improved, and meanwhile the normalized mean squared error (NMSE) is remarkably reduced, especially when a rather large accelerate factor is used.

Self-calibrating Coil Sensitivity Profiles for Parallel Imaging Based on Anisotropic Diffusion

Calibration of the spatial sensitivity functions of coil arrays is a crucial element in parallel magnetic resonance imaging (PMRI).In order to overcome the sensitivity miscalibration errors introduced by pre-scan calibration method and decrease the total examination time , a new self-calibrating method for estimating the coil sensitivity profiles was proposed i n th e paper . F rom a fully - sampled central region of a variable-density k -space acquisition, t he sensitivity calibration images were directly extracted , and then were adaptively smoothed by a new anisotropic diffusion scheme to extract encoding effects of coil sensitivities. When the estimated coil sensitivity profiles were applied to reconstruct the full Field-Of-View image from the uniformly under-sampled MR data when under-sampl e rate was greater than 2 , the experimental results showed that the quality of reconstruct ion image was evidently improv ed. Therefore, t h is self-calibrating method for calculating sensitivity profiles is suitable for much faster imaging in parallel magnetic imaging. Meanwhile, the paper suggest that the suitable number of central k-space lines for generating the internal reference images should be chosen between 8 to 20.

Sensitivity of the Dual‐Probe Heat‐Pulse Method to Spatial Variations in Heat Capacity and Water Content

Dual‐probe heat‐pulse (DPHP) sensors are useful for measuring soil heat capacity (C) and water content (θ), yet little is known about their effective measurement volume. We have adapted previous work on well testing to investigate the spatial sensitivity of the DPHP method for measuring both C and θ. Spatial sensitivity functions were derived by using a perturbation expansion approach in which C and θ vary with position, but differ only slightly from their uniform values C0 and θ0, respectively. Because the dimensionless forms of these spatial sensitivity functions are identical, the spatial sensitivity of the DPHP method is the same for both C and θ measurements. The spatial sensitivity function is not radially symmetric about the heater probe. Instead, the locations of the temperature and heater probes are of equal importance in defining the spatial sensitivity. The spatial sensitivity is greatest in small areas immediately outside of the heater and temperature probes. Far from the sensor, contours of equal spatial sensitivity approach the shapes of a family of ellipses. For a sensor with a probe spacing of 6 mm, the boundary containing 99% of the total spatial sensitivity is closely approximated by an ellipse with area 168 mm2 and a major axis 15.6 mm in length. The spatial sensitivity for C is unaffected by the magnitude of the soil thermal properties near the sensor, as long as the thermal properties are uniform. Likewise, for θ measurements, the spatial sensitivity is independent of the value of θ near the sensor, as long as θ is uniform.

A spline-based approach for computing spatial impulse responses

Computer simulations are an essential tool for the design of phased-array ultrasonic imaging systems. FIELD II, which determines the two-way temporal response of a transducer at a point in space, is the current de facto standard for ultrasound simulation tools. However, the need often arises to obtain two-way spatial responses at a single point in time, a set of dimensions for which FIELD II is not well optimized. This paper describes an analytical approach for computing the two-way, far-field, spatial impulse response from rectangular transducer elements under arbitrary excitation. The described approach determines the response as the sum of polynomial functions, making computational implementation quite straightforward. The proposed algorithm, named DELFI, was implemented as a C routine under Matlab and results were compared to those obtained under similar conditions from the well-established FIELD II program. Under the specific conditions tested here, the proposed algorithm was approximately 142 times faster than FIELD II for computing spatial sensitivity functions with similar amounts of error. For temporal sensitivity functions with similar amounts of error, the proposed algorithm was about 1.7 times slower than FIELD II using rectangular elements and 19.2 times faster than FIELD II using triangular elements. DELFI is shown to be an attractive complement to FIELD II, especially when spatial responses are needed at a specific point in time.

Spatial Spread of Visual Attention while Tracking a Moving Object

We conducted three experiments to investigate the spatial spread of visual attention. In Experiment 1, we measured the contrast sensitivities at various locations (spatial sensitivity function) relative to the moving target that the observer attended to track in an attentive tracking display. A probe was presented at a distance from the target at a location randomly chosen from within a certain range. The range of probe presentation location varied to examine whether the observer changes the area of attention to cope with this range. The results show that the probe range influenced the shape of spatial sensitivity function. The change in shape of this function suggests that the observer covers a wider area with attention for large probe ranges than small probe ranges. In the following experiments, we investigated the effect of the distance between the tracking target and a probe at a fixed location relative to the target (Experiment 2), or between the target and the center of a probe range of fixed size (Experiment 3). Since the relative probe location in a session was fixed in the experiments, the observer would pay attention to the target and probe locations independently of the relative distance if he/she could focus attention at multiple locations. Spatial sensitivity functions obtained in Experiments 2 and 3 showed that this was not the case. In both experiments the sensitivity to the probe decreased with increase in the relative distance as in Experiment 1, where the probe was presented at a location randomly chosen within each range. This indicates that attention cannot be divided among multiple locations, at least under the present experimental conditions. We will discuss a possible interpretation of the present results with a limited attentional resource and its spatial distribution.

Self-calibrating parallel imaging with automatic coil sensitivity extraction.

Calibration of the spatial sensitivity functions of coil arrays is a crucial element in parallel magnetic resonance imaging (PMRI). The most common approach has been to measure coil sensitivities directly using one or more low-resolution images acquired before or after accelerated data acquisition. However, since it is difficult to ensure that the patient and coil array will be in exactly the same positions during both calibration scans and accelerated imaging, this approach can introduce sensitivity miscalibration errors into PMRI reconstructions. This work shows that it is possible to extract sensitivity calibration images directly from a fully sampled central region of a variable-density k-space acquisition. These images have all the features of traditional PMRI sensitivity calibrations and therefore may be used for any PMRI reconstruction technique without modification. Because these calibration data are acquired simultaneously with the data to be reconstructed, errors due to sensitivity miscalibration are eliminated. In vivo implementations of self-calibrating parallel imaging using a flexible coil array are demonstrated in abdominal imaging and in real-time cardiac imaging studies.

Spatio-temporal requirements for binocular correlation in stereopsis

We measured sensitivity to binocular correlation in dynamic random-dot stereograms that defined moving sinusoidal gratings-in-depth. At a range of spatial frequencies and drift rates we established sensitivity by adding Gaussian distributed disparity noise to the modulation of disparity that defined a cyclopean grating, and finding the noise amplitude that rendered the grating just detectable. This permitted correlation thresholds to be measured at a range of suprathreshold disparity amplitudes. Spatial requirements for binocular correlation depend little on temporal frequency, and vice versa. This suggests that binocular correlation mechanisms can be characterized by independent spatial and temporal sensitivity functions. The temporal frequency function has a low pass characteristic. Sensitivity declines above about 1 c/sec, reaching its limit at 4–8 c/sec. The spatial characteristic depends greatly on the amplitude of disparity modulation, changing from band pass at low amplitude to low pass at high amplitude. The maximum resolvable spatial frequency is 4–6 c/deg, but declines sharply for relatively high amplitudes. The interaction between amplitude and spatial frequency cannot be explained by fixed high or low limits on detectable disparity gradients.

Reduced spatial sensitization on nonuniform backgrounds.

Spatial sensitivity (Westheimer) functions, when measured on nonuniform backgrounds made up of light dots of 12 min arc, were found to differ in shape, depending on the polarity of the central area on which the test spot was placed. When thresholds were measured on the dark centre between light dots, ie on the adapting-field illumination, the resulting curve was similar to the control curve, measured on a uniform background equated for flux. In comparison, thresholds measured on a central light dot, serving as a pedestal, peaked at larger background diameters and showed much less sensitization compared to the control function.

Transducer modeling using axial profile fitting techniques

Axial profiles of three piezoelectric, nonfocused, immersion pulse-echo transducers were measured using a small spherical reflector and an automatic scanning device. The scanning distance range included the first and second maxima. Measured data were compared with rigid disk radiation theory. The frequency-dependent active radius was used as a parameter in the curve fitting. In addition, three different approaches were used in the curve fitting. These included (1) fitting the entire range of the scanning distances, (2) fitting the first maximum, and (3) fitting the second maximum. The results of fitting the first maximum agree extremely well with the entire curve fitting, while the results obtained from the second maximum contain more errors. The results of modeling the transducers will be discussed in terms of the active radii and spatial sensitivity functions. In general, good agreement between the theoretical and experimental results was obtained. [Work supported by Center for NDE, Ames, Iowa.]

Behavioral studies on the effect of abnormal early visual experience in monkeys: Spatial modulation sensitivity

Spatial modulation sensitivity functions have been investigated by behavioral methods in two monkeys reared with normal visual experience and 12 monkeys reared with abnormal, early visual experience. Experimental treatments were initiated when the animals were approximately one month of age. Two monkeys were each treated with one of the following procedures: (1) long-term monocular lid suture. (2) short-term monocular lid suture. (3) surgically induced esotropia. (4) surgically induced exotropia. (5) optical dissociation of binocular vision with ophthalmic prisms (6) chronic monocular cycloplegia. The results of the studies showed a severe loss of contrast sensitivity of the treated eyes compared to the control eyes for monkeys reared with monocular lid suture or surgically induced esotropia. Surgically induced exotropia resulted in a moderate reduction in sensitivity of the deviated eye while optical dissociation resulted in a mild reduction in sensitivity of one eye compared to the other. One of the two monkeys reared for seven months with chronic monocular cyclopegia had a relative reduction in contrast sensitivity of the treated eye, but the other monkey had equal sensitivities in the two eyes. However, binocular summation experiments showed that even though the relative difference between the monocular sensitivities was small or absent for the monkeys reared with optical dissociation or chronic monocular cycloplegia, none of them demonstrated binocular vision in these experiments.