Available online at www.rilem.net

Materials and Structures 38 (November 2005) 807-815

1359-5997 © 2005 RILEM. All rights reserved. doi:10.1617/14298

ABSTRACT Among present non-destructive methods for concrete evaluation, ultrasonic testing uses relatively short wavelengths and therefore has

particular potential for detailed assessment of concrete. Methods like SAFT (Synthetic Aperture Focusing Technique) and tomographic reconstruction are able to provide high-resolution images of concrete areas, which can be employed for tasks such as area imaging, duct localization, fault detection, and thickness measurement. This contribution is intended to give insight into some of the principles and possibilities of ultrasonic concrete imaging using SAFT and tomographic reconstruction. It thereby reviews progress that has been achieved at the author’s institute during the last years. For SAFT reconstruction, the processing steps are explained that are necessary to obtain an image that is easy to interpret, including the influence of transducers, their coupling, and image noise suppression. Quantitative evaluation of ultrasonic images enables the examination of tendon ducts for voids and the objective assessment of image quality. A field example demonstrates the possibilities of SAFT reconstruction. In a separate section, ultrasonic tomography is shown to have the capability to detect faults such as honeycombing in concrete pillars. Finally, the potential of ultrasonic imaging and remaining steps necessary to open broad practical application are described.

1359-5997 © 2005 RILEM. All rights reserved.

RÉSUMÉ Parmi les méthodes non destructives existantes pour le contrôle d’ouvrages en béton, les essais par ultrasons utilisent des longueurs

d’onde relativement courtes et ont par conséquent un potentiel particulier pour une évaluation détaillée du béton. Des méthodes comme la méthode par synthèse d’ouverture, dite SAFT (Synthetic Aperture Focusing Technique), et la reconstruction tomographique, sont capables de fournir des images de haute résolution de surfaces en béton. Ces images peuvent être utilisées pour des travaux tels que l’imagerie de surface, localisation de conduits, détection de défauts et mesure de l’épaisseur. Cette contribution est destinée à donner un aperçu de quelques principes et possibilités de l’imagerie ultrasonique pour le béton au moyen de la méthode SAFT et de la reconstruction tomographique. L’article fait donc le point sur les progrès réalisés au cours de ces dernières années à l’institut de l’auteur. Pour la reconstruction par la méthode SAFT, sont expliquées les différentes étapes nécessaires à l’obtention d’une image qui soit facile à interpréter, en incluant l’influence des transducteurs, leur couplage et la suppression du bruit de l’image. Une évaluation quantitative des images ultrasoniques permet l’examen des conduits de tendons pour les cavités et l’évaluation objective de la qualité de l’image. Un exemple concret montre les possibilités de reconstruction par la méthode SAFT. Dans une section séparée, la tomographie par ultrasons permet de pouvoir détecter les défauts tels que les nids d’abeilles (« honeycombing ») dans des piliers de béton. Enfin, le potentiel de l’imagerie par ultrasons et les étapes restantes nécessaires à l’élargissement de son application pratique sont décrits.

1. INTRODUCTION

Non-destructive testing employing ultrasonic pulses is an established technique for the detection of internal objects in metals and other materials [1]. Specifically ultrasonic pulse-echo testing, allowing for one-sided access of a component, is used for inspection tasks in industry on a regular basis. The application to concrete, however, faces problems induced by the inhomogeneous concrete structure. Aggregate and pores have acoustical properties much different from those of the cement matrix, which gives rise

to scattering and mode-conversion of propagating ultrasonic waves. Among the consequences are pulse attenuation and structural noise, which can mask reflections of objects to be detected. In addition, a wide divergence angle is caused by a low ratio of transducer diameter to wavelength, complicating localization of internal objects.

To apply pulse-echo testing to concrete inspection, it was necessary to develop low-frequency transducers with sufficiently short impulse responses. With the introduction of such transducers and instruments in the beginning of the 1990’s, recording of single (A-scan) and multiple (B-scan)

M. Schickert Institute of Materials Testing (MFPA Weimar) at the Bauhaus-University Weimar, Germany

Received: 18 November 2004; accepted: 6 April 2005

Progress in ultrasonic imaging of concrete

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measurements was made possible [2]. It soon turned out that the suppression of structural noise would be crucial. This was the starting point for adopting the SAFT reconstruction technique for this purpose [3-5]. SAFT images show object indications concentrated at their correct position, are therefore easy to interpret, and can suppress structural noise by exploiting spatially diversified measurements. The work described here is based on the SAFT implementation first published in [5]. To permit direct comparison of the proposed methods, most examples are illustrated using results obtained at a single test specimen. A more specific treatment of ultrasonic SAFT imaging of concrete can be found in [6].

In contrast to the pulse-echo technique, sending and receiving transducer are placed on opposite sides of a component for transmission measurements. If only the pulse is evaluated that first arrives, pulse attenuation and structural noise have much less influence. The smaller sensitivity to detect internal objects can be overcome by tomographic imaging incorporating many measurements [7]. This technique can be employed, e.g., for the detection of honeycombing in pillars.

2. SAFT IMAGING

2.1 Principle Ultrasonic SAFT (Synthetic Aperture Focusing

Technique) reconstruction is an imaging method utilizing the information content of several pulse-echo measurements [6]. The measurements are recorded on the surface of the concrete on a one- or two-dimensional grid, the so-called aperture. Since transducers or transducer arrays of the size of a whole aperture are presently not manageable, one or more transducers need to be moved to scan the grid. If a single ultrasonic transducer is used for pulse generation and reception, a measurement setup according to Fig. 1 results. Alternatively, one sending and one receiving transducer could be moved in parallel. Only transducers transmitting pressure (longitudinal) waves were used for the results discussed in this report. A special low-frequency ultrasonic instrument generates electrical pulses and amplifies and digitizes the received signals.

The set of measured signals is then processed using SAFT reconstruction. The underlying algorithm coherently superimposes the signals for each image element, thus synthesizing a transducer of the size of the aperture with variable focusing to each image element. Linear apertures lead to two-dimensional images, planar apertures to three-dimensional images. Indications in SAFT images are reconstructions of internal object boundaries. SAFT reconstructions provide detailed information about the imaged concrete section and can therefore be used for detection and localization tasks. The superposition process reduces structural noise, which can be a severe problem in single pulse-echo measurements. But since physical limitations and wave propagation specialties such as mode conversion may cause interfering indications, the resulting images need to be interpreted, utilizing additional information if necessary. The method is explained in greater detail in [6], where also ultrasonic wave propagation in concrete is qualitatively explained.

The two-hole test specimen shown in Fig. 2 will be used to illustrate the general behavior of SAFT reconstructions and some results of parameter influence. The test specimen has a maximum aggregate size of 8 mm, a width in y-direction of 300 mm, and contains two transverse drillings [5].

Ultrasonic pulse-echo measurements were taken alongside the top of the specimen at a one-dimensional grid. Since reflections from internal objects can not easily be seen from single measurements [6], the complete data set was processed by the SAFT algorithm to obtain a two-dimensional image. The result is displayed in Fig. 3. Two separate indications represent the top of the drillings. The back wall is also visible as a horizontal line, partially shadowed by the drillings. While this result can be regarded as a good SAFT image, it suffers from a number of drawbacks. First and most striking, the amplitudes of the indications decay with depth. Then, the image is corrupted by noise that also decays with depth. Finally, each drilling indication has a different brightness despite of their identical size, and both amplitudes are not logically related to that of the back wall indication, which should be a stronger reflector. Thus this SAFT image is only the starting point for some signal processing steps intended to increase image fidelity, make the image easier to interpret, and extract quantitative information about the imaged objects.

Fig. 1 - Schematic measurement setup for one-dimensional aperture scans.

Fig. 2 - Sectional view of the two-hole test specimen (dimensions are in mm).

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2.2 Depth correction To ease interpretation, reconstructed SAFT images should

have an even brightness distribution. Two approaches have been used to compensate for the decay of signal amplitudes with depth. One utilizes measurements taken at reference reflectors, the other is based on noise statistics.

Reference reflectors located at different depths can serve as a means to build a depth correction curve [8]. The amplitude of each reflector is measured, and the resulting depth-dependent curve is inverted to yield the depth correction. This scheme is known from ultrasonic fault detection in steel where it is used to size defects from amplitude readings. For two-dimensional SAFT images of concrete, suitable reference reflectors are objects elongated perpendicularly to the image plane. Experiments reported recently resulted in exponential correction curves [8]. Concerning the example in Fig. 3, equalization of the drilling amplitudes leads to the image in Fig. 4, above. Compared to the original SAFT reconstruction, physical relationships are reflected correctly in the amplitudes of the indications. Both drilling indications now have the same brightness, and the indication of the back wall is clearly visible at a greater amplitude. In general, depth correction based on reference reflectors yields curves valid for the type of concrete at hand. The amplitude relation between small, unknown objects and the back wall can be used to size the objects.

In the second approach to depth correction, statistical parameters describing structural noise are equalized [9, 10]. In a region of the original SAFT reconstruction only containing noise, each image row with z = const. is taken as a sample, which is then statistically analyzed. To this end, parametric modeling of the probability density function by Weibull and LogNormal functions has been used, but also simpler measures may be adequate [9]. The logarithmic course of the selected parameter over depth is approximated by a quadratic regression curve and then inverted to yield the depth correction. This procedure was applied to the original SAFT reconstruction in Fig. 3. In the result in Fig. 4, below [6], structural noise amplitudes are evenly distributed over the whole image, and all object indications are clearly visible. In this case, object amplitudes are a detection quality measure since they are directly connected to the signal-to-

noise ratio (SNR) over constant noise. In general, object size relations are not preserved, which becomes more important for greater depth ranges. Because depth correction based on noise statistics leads to images that are easy to interpret, it is used on all further images except where noted.

3. TRANSDUCERS AND COUPLING

3.1 Transducer frequency content Among all elements in the measurement setup, ultrasonic

transducers are most influential on amplitude and frequency content of the transmitted signal. The transducer used for an actual measurement needs to be chosen in accordance with the given concrete and the inspection task. Low frequency waves are less affected by attenuation and structural noise and may be employed for coarser, more porous concrete and larger depths than higher frequency waves. On the other hand, they have limited resolution and cannot be used to detect small objects.

Since pulse-echo testing relies on the excitation of short pulses, suitable transducers are required to have a large bandwidth. It is technically demanding to develop transducers possessing both a low frequency spectrum and a large bandwidth. While in 1995 the longitudinal transducer with the lowest frequency band in the reporting institute had a bandwidth from 330 to 680 kHz [5], now transducers with a 70 to 470 kHz frequency range are available. (All bandwidth specifications are –6 dB limits measured in pulse-echo mode at a glass calibration block.) With these transducers measurements at 50 kHz and lower are possible, so porous concrete and larger depths are accessible. Longitudinal transducers with nominal frequencies lower than 100 kHz are

Fig. 4 - SAFT reconstructions, depth corrected (above) using reference reflectors and (below) noise statistics.

Fig. 3 - SAFT reconstruction of the two-hole test specimen without further processing.

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available, but have considerably lower bandwidth and thus limited resolution. In any case, the electrical excitation of the transducers should be adjusted to fit its bandwidth. This can be done, e.g., by varying the width of a rectangular pulse.

An example of the influence of transducer frequency content on SAFT results is shown in Fig. 5. In its part above [6], the SAFT image of a two-hole test specimen is depicted that is similar to that in Fig. 2, but has a maximum aggregate size of 32 mm. Measured with a 330 to 680 kHz transducer, the SAFT reconstruction is indistinct, and holes and the back wall can only be suspected [5]. For the image displayed at Fig. 5, below, the same specimen was measured using a newer 70 to 470 kHz transducer. Here all internal objects are imaged. Indications and structural noise appear softer due to the lower resolution.

3.2 Transducer coupling On their way from the sending to the receiving transducer,

ultrasonic waves need to pass the boundary between transducer and concrete twice. Because of highly different acoustic impedances, any air gap would greatly reduce pulse amplitude and deteriorate pulse quality. This way transducer coupling directly influences image quality in many respects such as signal-to-noise ratio, resolution, and uniform amplitude distribution. Coupling transducers to concrete involves ensuring an as-good-as-possible sound transition by using a coupling agent or a different technique. Additionally, transducer coupling makes up the major part of the measurement time, directly determining whether ultrasonic SAFT imaging can be applied in an economic way.

The coupling problem is the most difficult issue during measurements, and so far it has not been fully solved. Traditional coupling agents like Vaseline and honey work satisfactorily on smooth concrete surfaces, but are slow to use, and at least Vaseline is difficult to remove. In practice, many concrete surfaces are not smooth, and these coupling agents are inadequate. Providing the transducer with a rubber membrane yields good results in transmission measurements, when the concrete is wetted with a thin film of water [11]. In pulse-echo applications, the distance between transducer and concrete surface should be minimized in order to maintain an acceptable signal quality. Dry coupling of transducers does away with most of the coupling problems. It was made practicable by minimizing the transducer contact area, and by using an abrasion-resistant ceramic material for the surface [12]. To date many successful measurements have been carried out employing so-called Dot Point Contact (DPC) transducers [13], including investigations involving SAFT reconstruction [14, 15]. Questions remain regarding the steadiness of coupling, which concerns amplitude and time delay variations introduced by the coupling. Also, bandwidth and sensitivity of DPC transducers are still not as good as actual conventional low-frequency transducers.

Another possibility is fluid coupling. Fig. 6 shows the SAFT reconstruction of the two-hole test piece of Fig. 2, measured with water coupling. The result is a clear image, in which holes and the back wall each stand out more than 22 dB against structural noise. Within each part of the back wall indication, amplitude variations are small. When comparing this result with Fig. 4, below, it should be noted that a different transducer with a 180 to 580 kHz frequency range was used. Liquid coupling seems to be an alternative if some handling problems can be solved.

4. IMAGE PROCESSING

4.1 Thresholding based on statistical noise modeling

As was pointed out before, indication amplitudes are a quality measure, if depth correction based on noise statistics was applied. To distinguish between signal and noise, a detection operation can be introduced that regards

Fig. 5 - SAFT reconstructions of a two-hole test specimen as in Fig. 2, but with 32 mm maximum aggregate size using (above) 330 to 680 kHz and (below) 70 to 470 kHz transducers.

Fig. 6 - SAFT reconstruction of measurements made using water coupling.

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amplitudes below an amplitude threshold as “noise” and discards them. In an analogy to Radar detection methods, this threshold can be calculated from the statistical noise distribution [16, 10].

Fig. 7 illustrates this procedure depicting SAFT amplitude histograms of noise (black) and signal plus noise (grey). A parametric model using Weibull or LogNormal probability density functions models the amplitude distribution of a noise region of the image [9]. Given this distribution and a chosen false alarm probability, the threshold can be computed. Finally, all image points with amplitudes below the threshold are discarded as “noise”.

Fig. 8 shows the SAFT image of Fig. 4, below, after the thresholding operation. A Weibull distribution and a false alarm probability of 1% were used. Application results so far suggest that noticeable noise reduction can be achieved without significantly changing the shape of indications. This method is also intended as a step toward quality control in image examination.

An extension to this method combines different types of information in a single image [17]. The threshold is again used to separate “objects” from “noise”. Amplitudes assigned to objects are depth corrected based on the method of reference reflectors, and are coded in a certain color set. Noise regions are coded in a different color set instead of being suppressed, and are depth corrected according to the results of noise statistics. In the resulting nonlinear representation, “objects” are distinguished from “structural

noise” by their principal color. The size of small objects can be estimated by their brightness. Image content regarded as noise is marked by its color, but can still be interpreted.

4.2 Quantitative image evaluation Quantitative image evaluation comprises methods to

gain additional information about indications in order to quantitatively compare connected object properties. Examples are parameters such as amplitude, signal-to-noise ratio (SNR), and resolution. Quantitative methods can be combined to represent the information content of images according to objective measures.

Indication amplitudes contain information about the reflection coefficient of imaged objects. This information can be used to distinguish faulty portions of objects from healthy, as is important for the detection of voids in tendon ducts. Fig. 9 displays the SAFT image of a tendon duct partially filled with injection mortar (test specimen by courtesy of Dr. Kroggel, University of Darmstadt, Germany). The metal duct with a diameter of 80 mm is covered by 128 mm concrete. For x = 0 to 330 mm, the duct is completely filled, while for x = 330 to 1.000 mm, the fraction of filling gradually decreases to zero. Except for an image artefact on the left side, the decline of mortar filling is, starting at a certain point, reflected by an increasing amplitude, which is displayed here as the mean over the indication in z-direction. Current work is directed towards developing a method for void detection in tendon ducts.

Regular use of ultrasonic diagnosis requires some sort of quality assurance. A number of parameters have been defined to objectively characterize the quality of SAFT reconstructions and other types of images [18]. Concerning SAFT reconstruction, the principal imaging properties are of special interest. Systematic measurements have been started with a test specimen made from concrete with 8 mm maximum aggregate size, which contains transverse objects of different shapes and sizes in various depths [17]. SAFT reconstructions of this specimen were evaluated to extract quantitative parameters. Some results are shown in Fig. 10. In the part above, it can be seen that the signal-to-noise ratio of different targets decreases exponentially with depth. Reinforcement of 8 mm diameter can be detected at up to 230 mm concrete cover with a signal-to-noise ratio of 12 dB. Fig. 10, below, contains lateral and axial resolution curves for the object with quadratic cross section. The lateral resolution remains about constant with depth as long as the object is covered by all aperture positions. The axial resolution is higher (corresponding to lower values), but decreases with depth due to the acoustic low pass effect of concrete. With the establishment of similar dependencies for a range of concrete types, a comprehensive grading of SAFT imaging quality becomes possible.

5. SAFT APPLICATION EXAMPLE

To date ultrasonic SAFT reconstruction has found a number of practical applications. The measurement equipment is already small enough to be carried to construction sites and can be operated by one to two

Fig. 7 - Principle of signal detection based on parametric noise modeling.

Fig. 8 - SAFT reconstruction of Fig. 4, below, thresholded using a Weibull probability density function and 1% false alarm probability.

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persons (Fig. 11). The total time for imaging a concrete element along an aperture of 80 cm is about 1.5 hours, including 45 minutes aperture preparation and scan time

and less than a minute for SAFT reconstruction. A rough surface, very poor quality and very coarse grain can sometimes prevent concrete from being measurable.

In order to examine a bridge box, measurements were made at concrete walls and the floor. One of the floor results is displayed in Fig. 12, above. To ease interpretation, signal detection based on parametric noise modeling was applied. Thresholding was used the same way as in Fig. 8, again employing a Weibull probability density function and 1% false alarm probability. The final image was interpreted adding information such as the expected tendon size (Fig. 12, below [6]). Back wall shadowing can be used for rough object sizing. Altogether, detected objects include tendon ducts, reinforcement, supports, and one unknown

Fig. 9 - SAFT reconstruction of a tendon duct in a test specimen. From x = 330 to 1000 mm, filling of the duct decreases causing an increasing indication amplitude. The small plot shows the mean calculated over indication depth.

Fig. 10 - Depth-dependence of (above) signal-to-noise ratio and (below) resolution of different 8 mm lateral objects in a test specimen with 8 mm maximum aggregate size.

Fig. 11 - Ultrasonic measurements inside a pre-stressed concrete bridge.

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object. Additional applications of one-dimensional, two-dimensional, and three-dimensional SAFT reconstructions can be found in [6].

6. TOMOGRAPHIC IMAGING OF PILLARS

The methods presented so far use one-sided pulse-echo measurements. This approach, although best suited to most imaging tasks at concrete, faces problems if heavy reinforcement is encountered. Reinforcement leads to strong reflections that can mask objects and also shadows regions behind it. If, in addition, unconsolidated concrete in the form of honeycombing is searched, which is a poorly reflecting accumulation of aggregate lacking the cement matrix, pulse-echo measurements may fail to unveil the faults.

This type of tasks may occur when examining pillars. Pillars often contain heavy reinforcement, and honeycombing is among the most serious manufacturing problems. Fortunately, a different approach can be used here. For this purpose, ultrasonic measurements are conducted in transmission, i.e., a sending and a receiving transducer are placed on either side of a pillar’s cross-section. In each received signal, arrival time and amplitude of the pulse that first arrived are examined. These values are then processed by a tomographic algorithm to yield an image of the cross-section of the pillar.

The basic arrangement of transmission tomography [19] is sketched in Fig. 13. For numerical simplification, only pillars with circular cross-section are considered. In the figure, a single sending transducer sends out a wave, which is received by a certain number of receiving transducers, 11 in this case. The arrangement used here is called fan beam projection according to the location of the transducers. The sender and each two neighboring receivers enclose the projection angle γ. For measurement, signals of all sender/receiver-combinations are recorded. Then the whole arrangement is rotated by an increment of the measurement angle, ∆ϕ, and the fan beam measurement is started again. This sequence is followed until one revolution has been completed.

To obtain imaging quantities, the extended path is exploited that waves have to travel around a fault. The resulting delay and attenuation of the first arriving pulse are assigned to the respected direct path between sender and receiver, and are reconstructed separately for pulse velocity and pulse attenuation [7]. This is in contrast to conventional tomography, where object density is imaged. The implementation of the tomographic reconstruction is based on the Filtered Backprojection algorithm [19]. Since the positions of the receiving transducers do not lie on a circle around the sending transducer, an outer medium has to be simulated numerically [7].

The feasibility of this method is demonstrated at the test specimen shown in Fig. 14. The test specimen simulates a pillar with 380 mm outer diameter containing a cylindrical cavity with circular cross-section of 76 mm diameter. Measurements were carried out using pairs of compressional and vertically polarized shear transducers at frequency bands around 250 and 50 kHz, respectively. Each data set consists of 40 projections of 11 transmission measurements each, totaling to 440 velocity and 440 attenuation values measured at the envelopes of the received signals. From these, tomographic reconstructions were computed.

The result obtained for compressional pulse velocity is shown in Fig. 15. A comparison with the marked outline of the test specimen shows that the cavity is correctly imaged. Whereas the contrast seems appropriate to the intended task, an increase in resolution could be provided by an enlarged number of measurements. The circular surface of the specimen to the simulated surrounding medium is also

Fig. 12 - SAFT reconstruction of a pre-stressed concrete slab (above) before and (below) after statistics-based thresholding and interpretation.

Fig. 13 - Tomographic fan-beam measurement arrangement.

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visible and can be used for sizing. Both compressional and shear pulse attenuation images are less clear and contain artefacts [7]. This is attributed to coupling variations, which directly effect image quality in the attenuation case. In contrast, velocity images depend on time delay measurements, which can generally be carried out with higher precision at concrete. Merely the direct surface contact of the shear transducers adds time fluctuations, which result in a similar, but slightly uneven shear pulse velocity image when compared to Fig. 15.

Prior to field application of this method, the effect of reinforcement needs to be investigated. Reinforcement bars impose delay and attenuation to a propagating pulse and can thus affect image quality. A large number of electronically switchable transducers would result in a reduced measurement time. Work directed toward this goal is under way.

7. CONCLUSION

The described SAFT and tomographic reconstruction algorithms as well as additional signal processing schemes have reached an adult state and are ready to be applied in the field. In contrast, parts of the measurement process, especially the tedious transducer coupling to concrete, still pose problems. Conventional coupling agents are difficult to apply or remove, and therefore do not seem to be adequate. On the other hand, alternative methods such as

water or dry [12, 11] coupling are under active research. First experiments have shown that even air-coupled transducers may be applied [20]. Such developments allow application of transducer arrays [14], mechanical scanners [21], or a combination of both. If any of these promising coupling techniques can be developed to regular use, measurement times are greatly reduced, and economic efficiency can be reached. Further prospects are an improved image quality and three-dimensional images computed on-site from two-dimensional aperture measurements. Then, fusion of data acquired with different testing techniques can be used to combine the benefits of the respective methods [22].

With established measurement techniques, field use of ultrasonic imaging of concrete can be approached on a regular base. Possible applications are general area imaging, fault detection, localization of tendon ducts, localization of back-side reinforcement in certain cases, and thickness measurement. Here SAFT reconstruction using appropriate measurement equipment is promising to be of easy use, and to yield reliable, high-resolution solutions. A number of tasks such as void localization in tendon ducts and measurement of crack depth are still under research. In addition, ultrasonic tomography of concrete is shown to have the capability to detect faults in pillars.

The regular use of these techniques requires adequate instrumentation. Also, reliability of the inspection results is of immediate concern. Comparative studies conducted in the FOR384 research initiative may be a starting point (see below). Then, codes of practice need to be developed, and training for measurement and result-interpretation personnel has to be supplied.

ACKNOWLEDGEMENTS

Part of this work was supported by the Deutsche Forschungsgemeinschaft (German Research Council). Cooperation within the framework of the research initiative FOR384 is particularly acknowledged (www.for384.uni-stuttgart.de). Some of the measurements were carried out by U. Tümmler and F. Wolfram. The transducers used for the tomography experiments were manufactured by F. Wolfram.

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