The Shapes of Pandemic Curves

Getting monthly data for even 1957 and 1968 is tough sledding. We finally found in the CDC Vital Statistics Archives monthly data for these pandemics. Monthly data for surrounding years is available for 1968, but only 1957 and 58 for that period. Influenza is a separate line immediately above pneumonia. Our interest was the shapes of the curves, so rather than get into the modern morass of adding flu and pneumonic deaths and looking for “excess”, we report here only deaths ascribed solely to flu.

Corona deaths are the crude totals from Worldometers and there has only been one complete monthly bin so far. CDC will eventually parse these crude deaths into flu and pneumonic, likely reducing the trend. Germany recently reduced its corona deaths nearly 50%. It appears corona will be worse than ’57 and ’68, but nowhere near as bad as 1918.

We show here San Francisco data as a proxy for the entire US based on the following:

We got monthly data by tracing in Autocad according to the monthly grid and apportioning the areas under the curve. We originally were going to use Boston as more representative, but population data proved difficult and SF had a nice round 500k population in 1918.

The graph above is for all causes of deaths. We had previously determined that the shapes of the curves for all causes in ’57 and ’68 closely matched the influenza only curves. On this basis, we apportioned our SF data to match 670k US flu and pneumonia deaths and multiplied it by the ratio of flu only to pneumonia deaths in 1968 (.074) to get flu only deaths in 1918. We then subtracted 1000 to roughly align the start points.

Obviously this is not ideal, but it is at least a rational basis for comparing the shapes.

There is a notable similarity of the shapes, reaching initial peaks in about a month and a half. It is also notable that the fall from initial highs seems much faster than IHME projects.

We have shown above only the second and third peaks of the widely circulated triple whammy from England shown above. US data for the first wave is dismal. A handful of deaths at an army base and nearby towns. The CDC offers this curve as anecdotal without axis data.There is an interesting paper (Olsen et al 2005) arguing for a US beginning in New York as early as 1916.

Sobering image of a ravaged lung from a soldier who died from the 1918 flu from Smithsonian Magazine:

We now feel trying to force everything to actual deaths for influenza only was an error that distorts the true historical perspective, so we make this addendum April 12. Below we have converted everything to deaths per capita and added the IHME model in monthly bins.


From this perspective, it appears very unlikely that Corona will ever make the big time in the US. be about the same as 68-69. Current data suggests that US cases and deaths are at or very near their peaks, somewhat earlier than IHME prediction.

A separate axis is required for 1918. It is a completely different animal. US population was 1/3 current, and deaths were six times higher. Let’s not go there, ever.

Note* We made a spreadsheet error from prior dithering with virus only that incorrectly reduced the IHME model. The graphic and text were modified 4-16-20.

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Unified Geological Map of the Grand Canyon

We were puzzled about the Supergroup so we bought a pdf map from the Geological Society of America that covered the eastern part. We found that a forest of inconsequential faults, strikes and dips, and other notations was distracting us from the trees. We converted it to Autocad with the notion of moving these nuisances to a different layer. This tediously done, we still did not like the hatches so we set about creating our own with better contrast.

This proved to be astonishingly difficult as the unit boundaries proved to be multiple crudely overlapping splines and polylines that confounded our new hatches. We wound up erasing duplicates and retracing the boundaries. Lots of Supergroup exposures lie west of our GSA map, so we used a similar approach with scaled screen captures from the USGS “Mapview”. These we positioned in

Arcmap to create the previously posted image below.

Arcmap allows crude two point geolocation of Autocad linework that is good enough for views from the stratosphere as above, but when we dug into the schists we really wanted more. We had been unable to accurately position the GSA map in Autocad. Autocad warps the underlying Bing image to match the linework from a single geolocation point, and none of their projections matched. 

During the schist work we realized that the Autocad imagery was good enough that we could SEE the layers and began drawing directly on the imagery. 

Thus the Unified Map of the Grand Canyon was born.

It was a lot of work. It is a completely different kind of geological map based on the river runner’s encounter with the TOP of each layer. Top means when you can’t see it any more because it is buried by the layer above, not when there is air above it. If there is air above it, you are somewhere below the top and above the bottom; which here is the top of the layer below. Using only the top forces constant mindfulness of the stratigraphy. 

We use no regions or hatches. The imagery itself becomes the hatch, true to the nature of the layer. We ignore land slides and faults not critical for understanding. Recent lava is a distraction. It is plainly visible. We die into it when it is thick, but often find a line that telegraphs through.

The map is unified in that the USGS Mapview uses many different maps compiled over half a century with different groupings and different names. Our interpretation groups Esplanade with Supai, Surprise Canyon with Redwall, and Temple Butte with Muav. The latter two are discontinuous, and well, we didn’t need any more lines.

We will get further into the features of this map in the next post. 








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A River Runner’s Guide to Grand Canyon Geology VI: The Schists

Geologists have great fun with puns on the term “schist”. Technically, it refers to metamorphic rock; rock that has been altered by high temperature and pressure. There are several grades of schist, our former favorite was “blue schist” as a double pun. You can imagine our surprise to find a camp in the Grand Canyon called “Bull Schist”, dispensing with subtlety.

Bull Schist

We camped just below and across the river at a camp called “Schist”, but you can see the intrusion across the river at the “Bull Schist” camp. This striking basaltic schist is what piqued our interest in the schists, and prompted this post.

The first schist encountered by river runners is the Vishnu schist about mile 78. This is also the first Paleoproterozoic rock encountered. The overlying Bass formation, the lowest of the Mesoproterozoic and Neoproterozoic Supergroup, is about 1.25 billion years old. The Vishnu schist shows multiple episodes of folding and metamorphism, but it is shot through and through with granite that has been dated about 1.68 billion years old, so there is a very significant unconformity here, a time lapse of some 430 million years.

We originally intended to ignore the granites and focus on the schists, but it is telling that the three gorges in the Grand Canyon that expose Paleoproterozoic rocks are called the upper, middle, and lower granite gorges.

Above the top of the Vishnu is shown as a green line in the first river encounter, just below Hance Rapid. Granite intrusions are shown in pink. The Bass Formation extends to the top of the red cliff face above the Vishnu/granite.

The schists can be seen above in the “W” of the Grand Canyon. They are found on the downstream sides of two modern uplifts the river detoured around to create the “W”. Perhaps the downstream sides were uplifted a bit more.

Current thinking is that the Paleoproterozoic Grand Canyon area was something like  current Indonesia, shown below; a complex subduction trench, island arc, back arc basin arrangement.

The Vishnu Schist (green) would have been intermediate back arc basin sediments from the primary island arc volcanic mountains. The Rama Schist (purple), with more silica and aluminum, would have been basin sediments from more inboard volcanics; something like the current Mono volcanic field. The Brahma Schist (blue, and the dark Bull Schist above), loaded with Iron and Manganese, would have been essentially mid ocean ridge basalts from back arc basin spreading.

Granite forms deep in the wet conditions beneath subduction related volcanic mountain ranges and island arcs. It cools slowly, and varies considerably in composition and crystalline structure depending on the exact conditions of temperature and pressure. Many different plutons (blobs) of granite can be identified, notably the Elves Chasm pluton of (maroon) granodiorite above. It is the oldest rock exposed in Western United States.

We intend a separate post on these plutons, but above we have simplified.

The (pink) 1.685 to 1.68 Ga granite is what we found  intruding the Vishnu at the first river encounter above. This stuff intrudes just about everything down the granite gorges with veins too small to draw. It seems to increase in tendency to form larger intrusions as we proceed downriver. Some folks consider this time period to be the peak of metamorphism.

The (maroon) granodiorite spans a lot of time from the Elves Chasm 1.84 to 1.71 Ga. It is far more foliated (showing layers), and expresses as plutons more than micro intrusions. It probably represents a series of main island arc mountain ranges.

Farthest downriver we have the (salmon) 1.375 Ga granite. This is granular (not foliated) and is actually of Mesoproterozoic age, but it is over 100 million years older than the Mesoproterozoic Supergroup members upstream. This is the Quartermaster pluton and it tends to look more like Sierra Nevada granite.

The reality is the granites and schists of the Grand Canyon Paleoproterozoic gorges represent an inseparable dance of island arcs and back arc basins pressed against the North American Continent.

We have done enough hard work to earn a flight of fancy.

Well, there you have it. A fanciful attempt to reconstruct geography 200 million years before eukaryotic (with a nucleus) cells evolved. Our most serious suggestion is that the distribution of Vishnu suggests two uplifts that dictate the “W” of the Grand Canyon may be ancient basins.


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A River Runner’s Guide to Grand Canyon Geology V: Transects

River runners follow the water. The water is in our blood, and following it is what we do. We began this series by using the Colorado River as a transect, and following the drainages up to the South Rim as lateral transects showing the elevations of the layers.

We had promised at the end of the last post to next explore the schists, but we had an idea to follow the water from the last Supergroup exposure at Tapeats Creek up and over the Kaibab uplift. The original notion was to illuminate the Supergroup. We followed Tapeats Creek up to the North Rim and then up Quaking Aspen Creek to the divide or watershed high point. From there, the water did not naturally take us back to the southeast across the Supergroup as intended. It took us down North Canyon, continuing to the northeast.

We followed it.

The dark blue transect is quite natural, and the water may eventually cut through the Kaibab Uplift along this line. This would leave the first dogleg of the Grand Canyon “W” as a cutoff meander.

As we followed the water we first encountered a puzzlement in the extra layers of Toroweap and Kaibab Limestone near the top. This appears to be the result of complex faulting and folding that may or not extend to deeper layers. We may eventually dig into this at an appropriate scale. We found the elevations of the strata at significantly higher elevations as we followed the water down North Canyon. Where the grade leveled off we were surprised to find an inverted sequence, where as we went down we encountered the sequence one would expect going up.

The river follows a very different (and much longer) path. It has been known since the Geological Survey in 1923 that the river is pretty much in equilibrium, and can be approximated closely with a straight grade as we have done here. Distances of the members at river level are apportioned according to the actual river length and position.

We are unaware of any published transects approximating ours, and offer here a reasonable hypothesis accounting for the data. The Butte Fault, which separates the Neoproterozoic Supergroup members from the river, transitions to the East Kaibab Monocline, which our transect crosses nearly perpendicular.

The folding of the East Kaibab Monocline must have taken place after Kaibab time, so Supergroup members may also have been folded. We find it interesting that the Escalante Creek member, the highest exposure “around the corner” and where we leave it in Tapeats Creek, is also the first Mesoproterozoic member the Colorado River encounters, even though it is not the youngest Mesoproterozoic member.

This post will be updated as we follow the water in future transects. Hopefully they will shed some light. Likely they will require some modifications to our effort here.






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A River Runner’s Guide to Grand Canyon Geology IV: Whither the Supergroup?

We left off the last post having seen the complete Supergroup sequence of over two miles of Neoproterozoic and Mesoproterozoic sediments. The river never encounters the Neoproterozoic part, roughly half. These lie above the river on the North Rim side along the northeast shoulder of the Kaibab uplift that defines the first leg of the “W” of the Grand Canyon.

The river essentially makes an end run around the southern end of the Kaibab uplift. Where it turns the corner and heads towards the northwest, the river first encounters the Paleoproterozoic felsic intruded schists of the upper “Granite Gorge”. We don’t know how old these are. They have been baked at high temperature at a very considerable depth, possibly in several episodes. Each melt overprints the isotopic information.

We show above the river leaving the colored and hatched Mesoproterozoic Supergroup and entering the first Granite Gorge. The granite intrusions look like swimming amabae. Red Cardenas Basalt intrusions can be seen scattered about the Supergroup.This sequence from the green Escalante Creek member to Cardenas intruded Bass Formation foreshadows the rest of the Supergroup in the Grand Canyon.

Below we zoom in to the strange relationship between the Bass and the Cardenas intrusion. It is like there is some weak layer in the Bass the Cardenas easily intruded. We will see this same relationship at the very last Supergroup exposure in the Grand Canyon. A date of 1741 million years has been found the most recent melt of the Rama Schist near the left of the image.


The solid hatched Cardenas Basalt (Yc) can be seen sandwiched within the Bass. The Hatakai Shale lies above it; and finally the outlining top of Paleozoic Tapeats.

After turning the corner, the river never encounters the Supergroup again until Shinumo Creek; but up side canyons, particularly on the North Rim side, Supergroup rocks can be found. The shallower canyons have only the lowest Bass and Shinumo members, with maybe some Cardenas Intrusion. The deeper canyons expose the green Escalante Creek, but nothing higher in the Mesoproterozoic sequence is ever exposed in the Grand Canyon again.

Despite the names Bass Rapid and Bass Camp, the Bass Formation only just barely manages to reach the river at the Shinumo Creek Supergroup exposure below. The Bass lies over waning Vishnu Schist near the end of the Upper Granite Gorge, and only reaches the river with the help of a fault.

The Shinumo Creek exposure is tectonically complex. We have shown only the faults that control Supergroup extents. Here the Cardenas has intruded between the Bass and the Hatakai Shale. Over the Hatakai we get the namesake Shinumo and Escalante Creek (greenish blue here). A fault system has left more Shinumo above the Escalante Creek (Dox), including an outlier in the “Mordred Abyss” beyond the usual Tapeats boundary.


Below we zoom into the final Supergroup exposure in the Grand Canyon. Fittingly, it extends down to the river, where river runners may bid it proper farewell. You may recall a couple graphics back how the Cardenas intruded the bass in a ribbon, seemingly in some weak layer, next to the older Paleoproterozoic schists. Here the river decided to follow the ribbon of Cardenas.

The Middle Granite Gorge begins just below Specter Rapid, near the bottom of the graphic. We will explore the relationship between the Supergroup and the older schists and granites in the next post.

When we step back and contemplate the distribution of the Supergroup rocks in the Grand Canyon, we still find ourselves asking the same question that drove us to this exercise: What’s going on here?

Do the Supergroup rocks extend over the top of the Kaibab uplift where they will be exposed when the Paleozoic overburden eventually erodes away? We find no reason to believe they do not. The Mesoproterozoic members, sometimes called the Unkar Group, have no unconformities or intervals of time and erosion. They extend the entire length of the second leg of the Grand Canyon “W” at fairly consistent elevations up the southwest side of the Kaibab Uplift.

Do the Neoproterozoic members overlie the Mesoproterozoic rocks beneath the Paleozoic rocks in the Kaibab uplift? Maybe. Both Karlstrom (2012) and Huntoon (1999) draw sections that have the Neoproterozoic pinch out at the North Rim.

The above is from Karlstrom. It is a transect at about Nankoweap. It shows both the Mesoproterozoic (Y) and Neoproterozoic (Z) both pinching out somewhat up the Kaibab uplift towards the North Rim. It is unlikely that anyone has drilled to verify this and it strikes us as relying heavily on the notion of a flat surface before Paleozoic deposition.

We find the top of the Mesoproterozoic exposures up the creeks on the western side of the uplift routinely at 4200 feet, some 2000′ higher than shown for the Escalante and Shinumo members here.

Do the Supergroup members continue dipping down beneath the Paleozoic members on river left? The Karlstrom section suggests they do.

The next post will tackle another river runner’s bewilderment; the schists.





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A River Runner’s Guide to Grand Canyon Geology III: The Supergroup

A typical visitor to the Grand Canyon looks down from the rim through about a mile of sediments to the river. Most of these are Paleozoic sediments extending back about 515 million years. They appear to be level, although in a prior post we showed this levelness is an illusion of limited perspective.

This post is about over two miles of older sediments the typical visitor rarely sees. The Supergroup sediments are sloped at about 15 degrees due to uplift prior to the Paleozoic sediments deposited on top of them. River runners encounter these rocks in a very bewildering sequence we resolved to clarify on our last trip.

Above we show the eastern distribution of Supergroup rocks on an Arcmap image showing the grand “W” of the Grand Canyon. It can be seen that all of the Supergroup exposures are on the shoulders of the Kaibab Uplift, which reads dark green from the forest. The North and South Rim Visitor Centers are roughly centered on the eastern Supergroup.

Above we zoom into the first iteration on the river, and complete sequence of the Supergroup. This corresponds to the right hand first leg of the “W” before the river turns northwest in the first image. The Supergroup is composed of 17 members. Nine are Neoproterozoic (hatched horizontal) and eight are Mesoproterozoic (hatched diagonal). The colors follow the spectrum from red to violet

Sediments are difficult to date in general and Proterozoic sediments in particular because complex critters used for markers had not yet  evolved. A tiny volcanic ash deposit in the Walcott, the second Neoproterozoic member, dates to 742 million years ago. The Walcott is separated from the Sixty Mile formation above by an unknown amount of missing time and an unknown amount of erosion. The Sixty Mile Formation, currently a maximum of 200 feet thick, is preserved rarely, and is separated from Paleozoic rocks above by  unknown intervals of time and erosion as well.

The oldest and lowest Neoproterozoic member, the Nankoweap Formation, is also bounded by unknown intervals of time and erosion. Interestingly, the lowest Nankoweap contains wisps of sediments derived from the latest Mesoproterozoic Cardenas Basalt, showing that these intrusions got off to an early start.

The river never encounters the Neoproterozoic section (Sixty Mile through Nankoweap). These rocks lie up the Kaibab monocline and are separated at about the top of the Redwall from the river sequence by a large vertical offset along the Butte Fault. Like other places in the Grand Canyon, you can get multiple copies of the sequences where faults have made a mess of things. From the river, the Paleozoic sequence goes from Tapeats to top of Redwall. Above the top of Redwall and the Butte Fault lie members of Neoproterozoic Supergroup. Above the Supergroup it starts over again with Tapeats and climbs up the entire sequence to the North Rim.

The river first encounters the Supergroup in the Mesoproterozoic Escalante Creek Formation, which is not even the youngest. This is because a vertical offset along the Palisades Fault has allowed the younger strata to erode away. Below the fault the river marches through each of the inclined sedimentary strata beginning with the Ochoa Point Formation in turn.

The youngest Mesoproterozoic member, the Cardenas Basalt, is an intrusive rather than sedimentary layer. Below the Palisades Fault it intrudes all the sedimentary layers and only meets the river in dikes intruding the Bass and Shinumo members near the bottom of the second graphic above. The Cardenas is generically dated about 1.1 billion years ago. The oldest Mesoproterozoic member, the Bass Formation, contains some ash dated at 1.254 billion years ago.

In summary, the Supergroup extends from 1.25 billion years ago to some unknown time between 742 and 515 million years ago. Its current aggregate thickness is over two miles. It contains four unconformities, intervals of erosion, where an unknown amount of additional thickness was lost.

The lowest Paleozoic member, The Tapeats Sandstone, overlies the Supergroup and provides an outline. The bottom of the Tapeats is about 515 million years old. We show the top of the Tapeats as a tan line (close to its actual color). We extend it through the entire distribution of Supergroup rocks, even where it overlies even older schists and granites rather than the supergroup, because we feel it ties the narrative of Paleozoic deposition over a highly eroded Proterozoic surface together.

We have found many places where the Tapeats pinches out against the Supergroup, and also against older Paleoproterozoic rocks. We will argue that the Proterozoic surface was not entirely flat, and that these places where the Tapeats pinches out represent islands in the Tapeats Sea.

For context, we show above the distribution of middle and late Proterozoic rocks in the western United States. These formed at the same time as the Supergroup. The warmer colors are igneous granites and volcanics and the cooler colors are sediments. The “W” of the Grand Canyon is shown as a white path in the lower left. The large area of sediments at the top of the image extends well into Canada and is known as the “Belt Supergroup”. Correlation of these units with the Grand Canyon Supergroup is hindered by the same difficulties of dating sediments.

In the next post we will follow the river down through the remaining Supergroup exposures.





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A River Runner’s Guide to Grand Canyon Geology II: Muddy Creek and the Formation Problem

One might think that by now we would understand pretty well how the Grand Canyon formed. We don’t.

Central to the problems of how and when the current Grand Canyon formed is the Muddy Creek Formation, a bunch of freshwater lake sediments of late Miocene/early Pliocene age (5-6 my). These formed in a lake to the west of the Grand Wash Cliffs that form the western edge of the Grand Canyon and the Colorado Plateau.

The problem is that the Colorado River has cut through these sediments, limiting the age of the formation current river course (and the Grand Canyon) to sometime more recent.

A very good synopsis of the problem is presented by Joel Peterson (2008)

Click to access i1052-5173-18-3-4.pdf

A graphic from this paper above shows the distribution of the Muddy Creek Formation in relation to the “W” of the Grand Canyon. It also shows some ideas for the direction of Colorado River flow prior to the present. The Peterson paper rules out the “C” arrow because the Muddy Creek sediments match the Virgin River and not the Colorado River. An earlier “A” direction has been ruled out, leaving only “B”, the current course.

How do we resolve the current course with the Muddy Creek Formation? Where was the water going before 6 million years ago if not into the Muddy Creek Lake?

Humans are not the first Dam builders on the Colorado River. The infamous Lava Falls Rapid is formed by a lava dam. The latest work on Grand Canyon lava dams Crow et al (2015) identifies 17 lava dams within the last million years.

Click to access d8c55fbdfead6d612a6a0c2cf94ec07bde3e.pdf

The graphic above from Crow et al shows these extensive flows beginning at river mile 177.

These dams filled the river channel for  over a hundred miles in some cases. Volcanic remnants can be seen 1000 feet above the river. The effect of all this damming would be to slow the river and cause it to drop its sediment, keeping it out of Muddy Creek Lake even if significant Colorado River water leaked in to Muddy Creek lake through karst tunnels and around the dams. Karst tunnels dump impressive amounts of water from the canyon walls today at Thunder River, Tapeats Creek, and Vasey’s Paradise.

This scenario keeps the Colorado River  flowing in the only plausible direction, allows  the greater lateral erosion (and apparent geomorphic age) in the western Grand Canyon to take place above the lava lakes, fills Muddy Creek Lake with Colorado River water but not sediment, and allows the cutting of Muddy Creek sediments within the last few hundred thousand years after dam breaches. The only fly on this lovely picture is the lack of evidence for these sediments upstream.

We have a modern analogue for the fate of river sediments behind dams: the Lake Mead deposits exposed by the fall in lake level in the last few years. River runners have a great perspective on how fast this erosion takes place. The stuff is constantly falling in the river, and is often a dusty nuisance. We noticed significant erosion of these unstable sediments in a single year. One can easily imagine the whole lot washed away in a decade. Perhaps we should be looking for bathtub rings rather than sediments upstream.

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A River Runner’s Guide To Grand Canyon Geology

River runners have a different perspective on Grand Canyon geology. Remarkably, the put in at Lee’s Ferry is above the entire Grand Canyon sequence, and all the strata seem to emerge in turn from the river. This makes sense, but some strata disappear to the heights of the canyon walls, and others keep coming back to river level. The typical geological guide books offer vague mentions of undulations, but we wanted something more specific.

The Kaibab Limestone that visitors to both the North and South Rims stand on to admire the canyon emerges from the river at the end of the first mile. The six highest strata, from the Kaibab to the Redwall emerge, follow the river a while, and then disappear to the heights as the canyon deepens. They are never seen at river level again.

Here we have used geological maps with elevation contours to plot the elevations of the strata in transects from the river to the South Rim. The Hurricane Fault complex creates a second iteration of Vishnu to Redwall between the river and the South Rim.

The Muav Limestone emerges from the river, disappears to the heights, but comes back down to form the Muav Gorge before climbing back up for the rest of the canyon. The Bright Angel Shale returns to river level three times and the Tapeats Sandstone four. The Vishnu Complex of granite intruded bedrock makes four complete entrances and exits. The Supergroup is just too weird. It is a mile of sediments ranging from 1.25 billion to 725 million years old that overlies the Vishnu (1.75 to 1.68 billion years old) but suddenly disappears without explanation or further ado. The river encounters only a small part, briefly. The Supergroup will be the subject of a future post.

The 280 miles typically floated through the Grand Canyon basically makes a giant “W”. The first segment from Lee’s Ferry to Unkar, the second from Unkar to Dubendorf Rapid, the flattened middle segment from Dubendorf to Parashant Canyon, a segment from Parashant to Diamond Creek, and a final leg from Diamond Creek to the Grand Wash Cliffs.

In the first graphic we stretched the “W” into a straight line. It can be seen from the graphic that the river seems to cut through at least three ancient mountain ranges founded on elevated portions of Vishnu Complex. The first of these ranges corresponds very closely to the second segment of the “W” from Unkar to Dubendorf. This can be seen as the dark, forested region in the Google Earth image above where the highest rim elevations of the canyon are found. The roads to both the North and South Rims reach the canyon in this area, and it is commonly called the Kaibab uplift.

A second dark forested area can bee seen in Google Earth corresponding to the elevated area where the Kaibab through Hermit strata end, the Hurricane Fault Complex, and the beginning of the Lower Granite Gorge. It seems clear that ancient structures affect the course of the river.

It is peculiar that the ancient substructure telegraphs up through the younger strata. The current thinking is that the Vishnu (and Supergroup) were worn flat before the Cambrian Tapeats was laid down after 540 million years ago. Near Sockdolager Rapid, the Tapeats overlies both the Supergroup and adjacent Vishnu at the same altitude!

We are thinking that the recent uplift of the Colorado Plateau lifted the Vishnu cores of the ancient mountains more than other areas. Just another Grand Canyon mystery we will be exploring before returning to Radiative Altitudes.

Geological maps used for elevation Data:

Billingsley, G.H., and Priest, S.S., 2010, Geologic map of the House Rock Valley area, Coconino County, northern Arizona: U.S. Geological Survey, Scientific Investigations Map SIM-3108, scale 1:24,000.

  • Title: Geologic history and paleogeography of Paleozoic and early Mesozoic sedimentary rocks, eastern Grand Canyon, Arizona
  • Author(s): Blakey, R.C., and Middleton, L.T.
  • Publishing Organization: Geological Society of America
  • Series and Number: Special Paper 489, p. 81

Huntoon, P.W., Billingsley, G.H., Sears, J.W., Ilg, B.R., Karlstrom, K.E., Williams, M.L., and Hawkins, David, 1996, Geologic map of the eastern part of the Grand Canyon National Park, Arizona: Grand Canyon Association,  , scale 1:62,500

Billingsley, G.H., and Huntoon, P.W., 1983, Geologic map of Vulcan’s Throne and vicinity, western Grand Canyon, Arizona: Grand Canyon Association,  , scale 1:48,000

Billingsley, G.H., Clark, M.D., and Huntoon, P.W., 1981, Geologic map of the Hurricane fault zone and vicinity, western Grand Canyon, Arizona: Grand Canyon Association,  , scale 1:48,000

Huntoon, P.W., Billingsley, G.H., and Clark, M.D., 1982, Geologic map of the lower Granite Gorge and vicinity, western Grand Canyon, Arizona: Grand Canyon Association,  , scale 1:48,000



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The Temperature and Altitude of Radiation to Space III

We left of the last post having found that brightness temperature was hopelessly asymmetric to radiance in the main CO2 deviation from the Planck temperature. This discovery dashed all hope of using brightness temperature to determine the temperatures of the individual lines. We were able to replicate the form of the radiance deviation using the S-B equation, but the resulting temperatures fall well below atmospheric temperature in the deeper parts of the deviation.

Above the asymmetry of brightness temperature to tropical 70 km radiance.

Above excellent symmetry using S-B with emissivity of 2.25, but temperatures of ~80 K at the bottom of the deviation. According to the Planck curves the bottom of the deviation is about 220K.

Nothing about this effort has been easy, and the only check we can use on the accuracy of the Planck curves for determining the radiative temperatures of individual lines is to use the temperature of the MODTRAN tropical lapse rate. We report here the results of stepping up in altitude and comparing the strong lines as determined by eyechrometer against our densified Planck curves, with the lapse temperatures of the chosen altitudes.

Above we see the eyechrometer results from our 5 degree densified Planck curves. The two fairly flat lines at the top are the top corners of the CO2 deviation, 544 and 792. It can be easily seen from the second graphic below that these change little with altitude. They bound the CO2 deviation, are among the weaker strong lines, and abut “windows” on either side that radiate to space at surface temperature. The next two lines are 598 and 742. They both drop off rapidly with opposite curves through the troposphere, and flatline through the rest of the section. The next two are the stronger lines 618 and 720. They behave more like the by far strongest line (magenta 668). The last two are 648 and 688. They are the boundaries of the rotational bands that follow the fundamental bend at 668. They define the zone of zero transmission to the tropopause, and follow 668 (and the lapse) through the troposphere. At 20 km, where the prominent “spike” at 668 begins, they diverge.

An effort was made in the above and prior graphics to crudely use line thickness proportional to absorption intensity. This is the difference between the apparent Planck temperature and the lapse temperature.The strongest line (668) shows the least discrepancy and three strongest lines generally have the least discrepancy.

The discrepancy is least at one kilometer elevation. This is not surprising because at one kilometer the deviation from Planck is very small.

Above the CO2 deviations are compared at 1 kilometer, 5 Kilometers, 10 kilometers, and 70 kilometers. The strong absorption lines are easily traced up in altitude. As we progress upward in altitude, the apparent Planck temperature becomes increasingly different from the lapse temperature for most lines up to the tropopause. From the tropopause to the stratopause the differences all lines decrease and reverse sign. The sign of difference reverses again from the stratopause to maximum MODTRAN elevation at 70 km.

Shown again above for convenience, positive numbers mean apparent Planck temperatures above the lapse temperature, and negative numbers mean apparent radiance temperatures below the lapse temperature.

The radiances and lapse temperatures are both averaged over the tropics, so the systematic discrepancies should mean something. The striking feature is that the sign of the discrepancies follows the lapse rate.

When you see a trend of apparent Planck temperatures decreasing relative to the lapse temperature in the stratosphere as the lapse temperature increases with altitude, it must mean that a significant part of the radiance is coming from below. Reversing this logic explains the discrepancies in the cooling with altitude regimes above and below the stratosphere.

MODTRAN has output of total transmission and surface emission we used to try and sort this out. The MODTRAN explanation of total transmission is fairly straightforward:

Slit function direct transmittance for the line-of-sight (LOS) path including all sources of molecular and particulate extinction.

We take this to mean what you see is what you got from below at the virtual sensor at your chosen altitude. We believe this should exclude the radiance produced at your chosen altitude. This would seem to be everything needed, but we noticed peculiar structure in the output surface emission:

Surface emission directly transmitted to the sensor in units of W cm-2 sr-1 / cm-1. If the LOS terminates at the ground, this term is computed as the product of the Planck surface emission, the directional emissivity, and the path transmittance. If the LOS does not terminate at the ground but a positive temperature is specified for input TPTEMP, SURF_EMIS will contain the transmitted surface emission of a target object. If the LOS does not terminate at the ground and input TPTEMP is zero, then SURF_EMIS is zero.

This sounds a lot like total transmission, but it seems unclear whether the surface in direct line of sight must always be the ground, or whether a different altitude and temperature can be specified.

We tried the “Ground Temperature Offset” feature to set one kilometer of lapse as the “surface” TPTEMP, but this merely moved identical radiance 6 degrees (the lapse) down the apparent Planck curves.

Above we find that Surface Emission and Transmission are very similar except in the lower wave numbers of the CO2 deviation, with emission being somewhat stronger than transmission where they diverge. We have no explanation and find neither useful in determining the radiance from below that seems to drive the difference between apparent Planck and lapse temperatures. In the zero transmission and emission zones, which are the same, there is still radiance from below. This radiance must come from surfaces above the ground.

In the next post we will take a stab at gauging above ground radiance (and possibly emissivity) using downward radiance.



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The Temperature and Altitude of Radiation to Space II

We left off the last post having accommodated MODTRAN radiance to the Planck curves, and having promised to derive the temperatures of the individual lines from an inversion of the Planck radiance formula to give the Planck brightness temperature.

Unfortunately, we were unable to make this work. Several different formulae for brightness can be found that yield somewhat different results, but they all show conformance to the general shape of the CO2 radiance deviation.

We decided to use the following two because despite their different approaches, when scaled they become identical.

Below is what we get using the first equation above not scaled:

We really don’t know what to make of this. The systematic decrease in brightness from lower to higher wave numbers across the CO2 deviation in relation to radiance seemingly gives little hope that useful temperatures can be gained this way.

The good news is that a fresh look at the Stephan-Boltzmann approach to temperature using an emissivity slightly above .2 gives a far more satisfying result.

Staley and Jurica (1970) derived a full column emissivity for CO2 of .2. We are frankly astonished that by tweaking this slightly higher we were able to get such good agreement across the ~250 wave numbers of the CO2 deviation. Perhaps .225 is the correct column emissivity. At any rate, we will use this approach to get our temperatures henceforth.



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