Reduction

The Projection

Chapter 13: Reduction

The work of reduction began on a Monday. Silas sat at the drafting table in his study, the field notebook open to the first page of station data, a yellow legal pad beside it for the computations, the mechanical pencil in his hand. The window was open and the morning air was warm — sixty-two degrees, the warmest day of the summer so far — and the birch trees outside were fully leafed, the leaves a bright, thin green that filtered the light into patterns on the study floor.

Reduction was the process of converting raw field observations into final positions and elevations. The word had always seemed appropriate to Silas, because the process was exactly that — a reduction, a distillation, a stripping away of the raw measurements to reveal the clean positions beneath them, the way you might strip the bark from a log to reveal the wood, the essential material hidden inside the rough exterior. The raw measurement included the instrument error, the observer error, the atmospheric refraction, the centering error, the leveling error — all the small imperfections that accumulated in every observation and that had to be accounted for and removed before the measurement could be trusted.

He began with the baseline. The two endpoints, measured by GPS, their positions known to within two centimeters. He wrote the coordinates in UTM, Zone 5 North, NAD 83 datum: Baseline North, Easting 413,729.44, Northing 7,482,783.21, Elevation 647.98 meters. Baseline South, Easting 413,731.02, Northing 7,482,569.08, Elevation 647.41 meters. The baseline length: 214.70 meters. The baseline azimuth: 179 degrees 34 minutes 12 seconds. Nearly due south. The baseline nearly aligned with the creek, which flowed south toward the Koyukuk.

From the baseline, he worked outward. Station one. The horizontal angle from the baseline backsight to station one: 173 degrees 42 minutes 15 seconds. The distance from baseline north to station one, computed from the GPS positions: 187.3 meters. From these, the position of station one: Easting 413,712.18, Northing 7,482,596.42. He checked this against the GPS position from the notebook: Easting 413,712.31, Northing 7,482,596.58. The difference: thirteen centimeters in easting, sixteen centimeters in northing. Within tolerance. Acceptable.

He carried the computation to station two. Station three. Station four. Each station's position computed from the previous station's position and the measured angles and distances, the chain extending upstream, each link dependent on the links before it, the uncertainty growing with each computation, the traverse stretching toward the headwaters like a rope stretched taut, vibrating with the accumulated tension of the measurements.

The work was slow. Each station required twenty minutes of computation — writing the angles, applying the corrections, computing the sines and cosines, multiplying by the distances, adding the results to the previous station's coordinates. He could have done it on the computer. Jin had offered to run the computations through the GIS software, which would have taken minutes instead of days. Silas had declined. The hand computation was the check. The hand computation was the verification. If the hand and the software agreed, the positions were reliable. If they disagreed, something was wrong, and the nature of the disagreement would indicate where the error lay.

He worked through the morning. Stations one through eight, the lower reach of the tributary, the terrain gentle, the distances long, the angles open. The computations went smoothly. The hand positions agreed with the GPS positions to within twenty centimeters at every station. The traverse was sound.

At noon he stopped and made lunch — a sandwich, cheese and mustard on bread that was slightly stale because he had not been to the store since returning from the field three days ago. He ate at the kitchen table, looking out the window at the yard. The fireweed had come up while he was gone. The magenta spikes were six inches tall along the fence, the leading edge of the bloom that would move up the stalks through July and August, the progress of the bloom a phenological calendar that Alaskans used to mark the passage of summer, the fireweed blooming from bottom to top, and when the bloom reached the top the summer was over.

Margot had painted fireweed. Of course she had. She had painted everything that grew in the yard and the surrounding landscape, everything that bloomed and leafed and turned color and dropped and died and came back, the annual cycle of the boreal vegetation rendered in watercolors and oils, the paintings accumulating in the house the way the vegetation accumulated in the yard, layer on layer, year on year, the painted record paralleling the natural one.

He finished the sandwich. He went back to the study.

The afternoon was for stations nine through sixteen, the middle reach of the tributary, the terrain steepening, the distances shorter, the angles sharper. The computation at station fourteen — the station where the 1.3-meter discrepancy had appeared — took longer than the others because he worked it three times, checking each step, verifying the inputs, trying to identify the source of the field discrepancy. The hand computation gave a position that agreed with the GPS to within eighteen centimeters, which was consistent with the other stations and did not explain the 1.3-meter discrepancy he had measured in the field. The discrepancy had been real in the field but had not survived the final computation, which used the adjusted GPS positions and the averaged theodolite angles rather than the single observations. The averaging had smoothed it out. The 1.3 meters had been absorbed into the statistical noise, reduced to a residual, a footnote, a data point that the computation had considered and dismissed.

He wrote the final position for station fourteen in the notebook. He drew a small mark beside it — not a notation, not a symbol, just a mark, a graphite dot that acknowledged the 1.3 meters, the discrepancy that had meant something to him and meant nothing to the computation, the human moment absorbed into the mathematical process, the personal reduced along with the professional.

He worked until six, then stopped. Fourteen stations reduced. Twelve to go. Two more days of computation, maybe three, and then the plotting, and then the contouring, and then the drawing.

He heated a can of chili for dinner and ate it at the kitchen table and washed the pot and the bowl and the spoon and set them in the rack. He went to the living room. He sat on the sofa. He looked at the wall opposite, where three of Margot's paintings hung — the glacier painting, large, three feet by four feet, the ice and the moraine and the sky; a smaller painting of the Tanana in summer, the river blue-green between birch-covered banks; and a watercolor of Denali from the Parks Highway, the mountain white and enormous above the foothills, the sky the pale blue of early autumn.

He looked at the paintings. He had not looked at them in two years. He had walked past them, sat beneath them, eaten meals in front of them, but he had not looked at them, had not seen them, had not allowed his eyes to rest on them and his mind to apprehend what they contained. The not-looking had been a strategy, a survival technique, a way of living in a house full of Margot's work without being overwhelmed by Margot's absence. The paintings were evidence. The paintings were proof. The paintings were the physical record of a life that had ended, and looking at them was a form of measurement, and the measurements they returned were not distances and elevations but the precise dimensions of what he had lost.

He looked at the glacier painting.

The ice was blue. Not the blue of the sky, which was pale and atmospheric, but the deep, saturated blue of old glacial ice, the blue that came from the compression of snow over centuries, the air squeezed out, the ice crystals aligned, the structure of the ice selectively absorbing the red wavelengths of light and transmitting the blue, the physics of color made visible in the body of the glacier. Margot had captured this blue. She had mixed it from ultramarine and white and a trace of viridian, and the mixture was not the ice — no pigment was ice — but it was faithful to the ice in the way that mattered, which was the way it looked to a human eye standing on the moraine in the morning light of an August day in the Wrangells.

The moraine was brown and grey, the debris of the glacier's retreat, the rubble of a landscape ground down and deposited, and Margot had painted it with a loaded brush, the strokes thick and textured, the surface of the painting mimicking the surface of the moraine, rough and uneven and real.

The sky was the pale blue that shifted to lavender at the edges, the color of distance, and the mountains on either side of the glacier were grey-blue, the color of rock seen through atmosphere, the color diminished by the miles of air between the viewer and the mountain, the air not transparent but translucent, filtering the light, scattering the blue, the atmosphere performing its own kind of projection, its own distortion of the true color.

He saw all of this. He saw it because he was looking, because for the first time in two years he was allowing his eyes to rest on a painting and his mind to receive what the painting contained, and what the painting contained was not grief but beauty, not absence but presence, the presence of Margot's eye and hand and judgment, the presence of the human being who had stood on the moraine and seen the glacier and mixed the colors and laid them on the canvas and produced this — this record, this representation, this projection of a landscape that existed in the Wrangells and that also existed here, on this wall, in this room, in this house, preserved not in contour lines and elevations but in color and light and the texture of brushstrokes.

He looked at the painting for a long time. The evening light came through the window and fell on the painting and changed it — the colors warming, the blue deepening, the sky glowing, the painting responding to the ambient light the way the actual glacier responded to the actual light, the representation sensitive to the same variable that the original was sensitive to, the painting alive in its way, changing with the time of day, the way the landscape changed.

He stood up. He walked to the painting. He stood close to it, close enough to see the individual brushstrokes, the marks of the brush in the paint, the texture of the surface. He could see where Margot had laid the paint on thickly — the moraine, the rocks — and where she had laid it on thinly — the sky, the distant mountains. He could see where she had scraped the paint with a palette knife, creating ridges and furrows in the surface, the surface topography of the painting, a contour map of pigment, measurable with instruments if you wanted to measure it, the height of the paint above the canvas a datum, the brushstrokes contour lines, the palette knife marks hachures.

A map. The painting was a map. A different kind of map, made with different instruments, measuring different properties of the same terrain. But a map.

He reached out and touched the painting. He touched the moraine, the thick, textured paint, and felt the ridges of the brushstrokes under his fingertip. The paint was dry and hard and permanent, the oils oxidized and polymerized over the years since Margot had applied them, the pigment locked into the surface, the color fixed, the painting as stable as a benchmark, as durable as a brass cap set in concrete.

He took his hand away. He looked at the painting one more time. Then he turned off the light and went to bed.

In the morning he resumed the computations. Stations fifteen through twenty-two. The terrain was steep in this section, the angles sharp, the distances short, and the computations were more complex, requiring additional corrections for the vertical angles that affected the horizontal distances when the slope exceeded ten degrees. He worked carefully, checking each correction, each conversion, each multiplication, the pencil point moving across the legal pad with the slow, deliberate strokes of a man who had done this work for decades and who understood that speed was not the goal, that accuracy was the goal, that a computation done quickly and wrong was worse than a computation done slowly and right.

At station eighteen, the computation produced a position that disagreed with the GPS by forty-two centimeters. Larger than the average, but within tolerance. He noted it, considered it, decided it was acceptable. The theodolite sightline at station eighteen had been partially obstructed by willows, and the target had been a flagged stake rather than a prism, and the centering error was larger than usual. Forty-two centimeters was the price of imperfect conditions. He accepted it.

He finished the last station — station twenty-six, the headwaters, the saddle — at three in the afternoon. The final computation: Easting 413,482.17, Northing 7,484,921.83, Elevation 1,284.73 meters. He checked this against the GPS: Easting 413,482.28, Northing 7,484,921.71, Elevation 1,284.89 meters. The difference: eleven centimeters in easting, twelve centimeters in northing, sixteen centimeters in elevation. Consistent with the rest of the traverse. The closure, which he had measured in the field as eleven millimeters, confirmed.

He wrote REDUCTION COMPLETE at the bottom of the last page of computations and drew a line under it. He set the pencil down. He flexed his hand, which was stiff from two days of writing, the fingers cramped around the pencil's hexagonal barrel, the joints protesting the sustained grip the way his knees protested the sustained walking.

The positions were computed. The data was reduced. The raw measurements had been distilled into the clean coordinates that would become the map. The process had taken two days, eight hours each day, sixteen hours of computation that the software could have done in minutes.

But the sixteen hours had given him something the software could not. They had given him intimacy with the data. He had touched every number, checked every angle, verified every distance. He knew the traverse the way he knew a trail he had walked — by heart, by feel, by the accumulated sensation of each step and each turn. He knew where the positions were strong and where they were weak, where the data was clean and where it was noisy, where the terrain cooperated and where it resisted. He knew the data the way you knew a person you had spent time with — not completely, not exhaustively, but well enough to trust it, well enough to draw a map from it.

He put the legal pad on the desk beside the notebook. The notebook on one side, the legal pad on the other, the field observations and the reduced positions, the raw and the cooked. Between them, on the desk, the vellum tubes and the Rapidograph pens and the bottle of Higgins Eternal India ink, waiting.

Tomorrow he would begin plotting. He would tape the vellum to the drafting table and draw the grid and plot the stations and begin the work of interpolation, the work of drawing contour lines between the measured points, the work of turning the positions into a picture, the data into a map.

He looked at the vellum tubes. He looked at the pens. He looked at the ink.

He looked at the door at the end of the hall.

The door was white. The knob was brass. The door was closed. He had touched the knob three days ago, had put his hand on it and held it for thirty seconds and then taken his hand away. The knob had been cool. The studio was behind it.

He turned back to the desk. He capped the mechanical pencil. He stacked the legal pad on the notebook. He turned off the desk lamp.

Tomorrow. The plotting. The map.

He went to the kitchen and made instant coffee and drank it standing at the window. The fireweed had grown. The spikes were eight inches tall now, the first flowers beginning to open at the base of the stalk, the magenta petals emerging from the green calyx, the bloom beginning, the summer measured in fireweed, the way the terrain was measured in contour lines — from the bottom up, from the base to the summit, from the beginning to the end.