Theoretically- and empirically-derived cooling rates for active pāhoehoe lava flows show that surface cooling is controlled by conductive heat loss through a crust that is thickening with the square root of time. The model is based on a linear relationship that links log(time) with surface cooling. This predictable cooling behavior can be used assess the age of recently emplaced sheet flows from their surface temperatures. Using a single thermal image, or image mosaic, this allows quantification of the variation in areal coverage rates and lava discharge rates over 48 hour periods prior to image capture. For pāhoehoe sheet flow at Kīlauea (Hawai`i) this gives coverage rates of 1–5 m2/min at discharge rates of 0.01–0.05 m3/s, increasing to ∼40 m2/min at 0.4–0.5 m3/s. Our thermal chronometry approach represents a quick and easy method of tracking flow advance over a three-day period using a single, thermal snap-shot.
 Using thermal images collected on Kīlauea Volcano, Hawai`i, we examine the cooling properties of currently- and recently-active pāhoehoe sheet flow over time-scales of seconds-to-days. This builds on Hon et al.  in that we examine the relationships between surface temperature, conductive cooling, and crust thickness, as well as areal coverage and discharge rates, as functions of time. These relationships allow conversion of thermal-image-generated temperature maps into flow age maps for all flow units emplaced within ∼48 hours of image acquisition. This makes it possible to recreate the recent emplacement history of an active pāhoehoe flow using a single infrared image.
2. Thermal Image Data and Lava Flow Surface Temperature Maps
 Our thermal image data were collected using a FLIR S40 camera during February 21–24, 2006. The FLIR camera is sensitive to thermal radiation in the 7.5 to 13 μm waveband and allows collection of a 240 × 320 pixel image covering one of three temperature ranges: low (−40 to 120°C), medium (0 to 500°C) or high (∼300 to 1500°C). The total field of view is 18 × 24° with an angular resolution of 1.3 mrad. Atmospheric correction of all output pixel temperatures is achieved by inputting atmospheric temperature and humidity, as well as line-of-sight distance to the target [Ball and Pinkerton, 2006]. For our ground-based measurements, distance to the target was measured using a tape measure. For vertically orientated helicopter-based images height above the ground was obtained using the helicopter altimeter. Likewise, temperature (20°C) and humidity (50%) were measured and input in-situ at the time of each image acquisition, along with a 7.5–13 μm emissivity (ɛ) for basalt of 0.92 [from Salisbury and D'Aria, 1992].
 On February 21 we targeted a localized zone of active pāhoehoe that advanced ∼30 m in 2 hours (Figure 1a). Advance of this flow was by extension of pāhoehoe lobes across a ∼10 m wide flow front, leaving behind them an inflating pāhoehoe sheet (through which continued flow fed the advancing flow front). Lava core temperature, obtained using a k-type thermocouple, was 1141°C. A kinematic GPS survey completed on the following day revealed that the flow advanced a further 30 m, passing between two tumuli before dying out. The thermal camera was mounted on a tripod atop a tumulus affording a synoptic view of this active sheet (Figure 1a). A series of images was collected to cover the active flow front, as well as the cooling and inflating sheet that marked the track of this flow. These images enabled the construction of a mosaic covering the entire flow with a spatial resolution varying from ∼4 cm/pixel at the proximal section of the flow (30 m line-of-sight distance to point B; Figure 1a) to ∼1 cm/pixel at the flow front (10 m line-of-sight distance to point A; Figure 1a).
 Thermal images were also collected during a helicopter over-flight on February 24. At this time there were two zones of active sub-aerial pāhoehoe emplacement. The first zone was located just south of the vent area (Pu`u `Ō`ō flow) and the second zone (Hook flow) was ∼7 km from the vent. Both pāhoehoe flows fed by a tube-system, which extended all the way to the ocean to feed a third zone of ocean entry flows. Images were collected by targeting the camera vertically downwards from elevations of between 750 m (Pu`u `Ō`ō flow) and 500 m (Hook flow) above the flow field. From these data two image mosaics were constructed for the two flows (Figures 1b and 1c). High resolution (26 cm/pixel) image mosaics of specific sections of interest were obtained from lower elevations (∼200 m) and used for this analysis (Figure 2).
 The mosaics represent surface temperature maps for each flow (e.g., Figure 2a). By thresholding each image we can map zones above a given surface temperature. Surface temperature variations document the recent emplacement history of the flow as revealed by its cooling history. The zones of highest temperature, for example, represent the currently active, or most recently active zones. As the temperature threshold is relaxed, increasing areas of lava with lower surface temperatures (thus older) become apparent. By successively reducing the temperature threshold we can identify progressively cooler and, thus, older flow zones, allowing the recent advance of the flow field to be obtained from a single thermal image. To convert this temperature information to time we next examine the relationship between time and surface temperature forced by conductive cooling.
3. Surface Temperature and Discharge Rates
 For pāhoehoe sheet flows at Kīlauea, Hon et al.  showed that once surface temperature drops below 1070°C a crust forms and begins to cool due to conduction. Measurement of surface temperature (Tsurf, in °C) through time (t, in hours) showed that surface cooling could be expressed empirically by [Hon et al., 1994]:
By re-arranging equation (1), this predictable behavior can be used to assess the age of a recently-emplaced sheet flow from its surface temperature [t = 146 exp (−0.0164 Tsurf)]. We use this to convert each pixel temperature in the image to a value of time since flow emplacement. The equation (1) relationship is derived from surface temperature data spanning the first ∼100 hours of emplacement [Hon et al., 1994, Figure 9]. The fit begins to diverge from field data between 24 and 48 hours; and between 48 and 100 hours calculated surface temperatures begin to approach likely ambient temperatures. We thus apply the relationship within a 48 hour limit. This allows us to delineate areas of lava emplaced over time periods (Figure 2b), extending up to 48 hours prior to image acquisition (Figure 3). Area values can be converted to volumes by multiplying by lava thickness (H, in meters) using the empirically-derived relationship between thickness and time of Hon et al. :
This will give the total bulk volume emplaced between time t and image acquisition. Differencing volumes between two time (or temperature) steps allows the bulk volume over a given time window to be calculated. When divided by the duration of the time window, time-averaged discharge rate during that period is obtained.
 Our ground-based check on this method involved the flow observed being emplaced on February 21. The flow advanced ∼30 m between 14:00 and 16:00 (Figure 1a). It attained a typical width of ∼10 m and thickness of ∼0.65 m, giving a volume and time-averaged discharge rate of ∼195 m3 and ∼0.027 m3/s over the 2 hour emplacement period. Using the flow front zone across which there was little variation in pixel size we obtain an area of lava with surface temperatures greater than 450°C of 24 m2. Following equation (1) this represents the area emplaced in the previous 5.5 minutes. The empirically-derived mean thickness of 0.39 m gives a volume of 9 m3 for a time averaged discharge rate of 0.03 m3/s.
4. Steady and Waning Flow
 Using this image-based thermal chronometry approach we are able to identify and quantify the areal coverage and volumetric trends associated with steady and waning flow.
 The Pu`u `Ō`ō flow shows steady conditions. The image mosaic reveals that the flow had advanced ∼450 m across a ∼100 m wide zone in the ∼8 hours prior to image acquisition. At the time of acquisition, pāhoehoe lobes were active at the southern extremity of the sheet, as well as along its western edge (Figures 1b and 2). Over the hour prior to image acquisition, areal coverage rates were stable at ∼40 m2/min, so that the cumulative area increases steadily from time zero (Figure 3a). These convert to discharge rates of 0.3–0.5 m3/s.
 Our analysis of the Hook flow revealed that it comprised three active pāhoehoe flows (Figure 1c) across which activity was waning. Each flow was traced back to sources which, following our field surveys of February 21–23, were represented by tumuli at the head of each pāhoehoe sheet. The southern-most flow displayed highest surface temperatures, and was therefore the most recent flow (Figure 1c). However, maximum temperatures were ∼210°C. Converting this to time reveals that activity died out ∼5 hours prior to image acquisition. This is consistent with our field observations. During the afternoon of February 21, breakouts of S-type pāhoehoe were observed at the distal ends of all three flows, with core temperatures of up to 1141°C. By the afternoon of February 23 activity was confined to the distal end of just one flow. Here activity was restricted to effusion of late-stage, spiny pāhoehoe with maximum core temperatures of 1131°C. The image acquired the following morning (Figure 1c) suggests that activity finally died out around 06:00 on February 23. Our plots of areal coverage (Figure 3b) show a steady decline in coverage rates from ∼13 m2/min around 48 hours prior to image acquisition (i.e., on February 22) through ∼2 m2/min around 24 hours prior to image acquisition (i.e., on February 23). This converts to a decline in discharge rate from ∼0.14 m3/s to ∼0.03 m3/s.
 The February 21 flow exhibits an apparent waxing flow (Figure 3c). However, this is likely an artifact of the oblique viewing geometry where the spatial scaling of the pixels changes across the image [James et al., 2006]. We use pixel sizes appropriate to the flow front; applying these to greater distances underestimates the areal contribution of the flow's cooler regions, thereby inducing an artificial waxing trend (Figure 3c).
5. Theoretical Treatment: The Stefan Problem
 In the analysis presented, empirical relations defined for Hawaiian pāhoehoe sheet flow by Hon et al.  have been used. However, the heat transfer to the surface of the flow is controlled by the formation and growth of the crust. In order to extend the analysis to other volcanoes we can use instead an analytical thermal model (Figure 4, inset) where radiation and convection occur in parallel after conduction supplies heat from the center of the flow to the surface. The equation of this model can be written as [e.g., Kreith and Bohn, 1993]:
where Q is heat flux, Th is the hot lava temperature (in K), Ta is ambient temperature, and Rk, Rc, and Rr are the thermal resistance terms for conduction, convection and radiation, respectively. These are defined as:
in which y is crust thickness, k is thermal conductivity, A is the area over which the measurement is made, hc is the convective heat transfer coefficient, and σ is the Stefan-Boltzmann constant. Crust thickness can be determined using the Stefan  cooling problem:
where λ is a dimensionless scaling value, κ is lava thermal diffusivity (in mm s−1) and t is time; dividing by 1000 places units of y in meters. Value λ is determined by balancing latent heat of fusion with the heat capacity and temperature of the medium and is commonly between 0 and 1. Combining Equations (3)(4) and (5) so that
now allows surface temperature to be expressed as a function of time (Figure 4).
 To determine a cooling curve, we used an image time series acquired on February 21 spanning 40 minutes from initial emplacement. The average temperature of a 4-pixel square (initially centered on the flow front) was recorded. We compared these data with the predicted cooling curve of Hon et al.  and our modeled values (Figure 4). The advantage of using our thermal model is that it remains valid from the maximum lava temperature to ambient temperatures and can be adjusted for the thermodynamic properties of different lavas; that is, it can be applied given calculation of an appropriate λ value and input of a suitable thermal diffusivity.
 The instantaneous heat flux from active lava can be calculated given surface temperature [e.g., Crisp and Baloga, 1990; Keszthelyi and Denlinger, 1996; Harris et al., 2005b]. Such heat budget-based treatments can be used to constrain instantaneous effusion rate [e.g., Harris et al., 2005a] in a treatment the links effusion rate with lava area through heat flux [Wright et al., 2001]. However, in the treatment applied here the time-dependent cooling properties of the lava are used to obtain lava spreading and time-averaged discharge rates. This approach is based on a conductive cooling model whereby conduction across the surface crust (Qcond) can be related to core temperature (Tcore), surface temperature (Tsurf) and thermal diffusivity (κ) in [Qcond = k (Tcore − Tsurf)/√πκt], so that Qcond varies with the square root of time [Turcotte and Schubert, 2002]. As a result isotherms in a cooling lava will propagate downwards as a function of √t [e.g., Peck, 1978; Hon et al., 1994] and surface cooling will follow a predictable trend, Tsurf declining with log(t) (e.g., Figure 4). This allows flow areas of given age to be tracked from their time-dependent cooling properties (i.e., a given surface temperature will be tied to a unique age). This, in turn, allows flow coverage and discharge rates (averaged over known time windows) to be measured, where the time-dependant cooling behavior of pāhoehoe lava allows the development of an advancing pāhoehoe sheet to be traced in time backwards for up to 48 hours from a single thermal image.
 Thanks to David Okita for helicopter support. This work was supported by NASA grant NNG04GO64G “New Tools for Advanced Hot Spot Tracking”.