Appendix 2: Absolute and relative resolutions of the ashmeter
The absolute and relative resolution of the measurements realized with the ashmeter must be
calculated for each value to insure the quality of the results. Here is presented the detailed method
to calculate these resolutions.
1. Resolution for the tephra mass
1.1. Mass measurement with a 0.1 g resolution electronic scale
Absolute resolution for the mass measurement RA(M):
RA(M) = ΔM = 0.1 g
(1)
Where ΔM: balance resolution
Relative resolution for the mass measurement RR(M):
(2)
RR(M) = RA(M)/M
Where m: mass measurement
Example for 1 g: RR(M) = 10%
Example for 100 g: RR(M) = 0.1%
1.2. In situ tephra mass calculation:
(3)
M3 = M2 – M1
Where M3: mass of in situ tephra; M1: mass of inner gauge; M2: mass of inner gauge + tephra load
Absolute resolution for the in situ tephra mass calculation RA(M3):
RA(M3) = √(RA(M1)2 + RA(M2)2) = 0.14 g
(4)
Relative resolution for the tephra mass calculation RR(M3):
(5)
RR(M3) = RA(M3)/M3
Example for 1 g: RR(M3) = 14.14 %
Example for 100 g: RR(M3) = 0.14 %
The same calculation is used for M4 (mass of the recipient used in laboratory), M5 (mass of the
receptacle + tephra dry load), and M6 (mass of dry tephra).
2. Resolution for the accumulation area
2.1. Funnel diameter resolution for a 16.5 cm-diameter funnel measured with a 0.1 cm-uncertainty
Absolute resolution for the funnel diameter:
(7)
RA(d) = Δd = 0.1 cm
Where Δd: funnel diameter uncertainty
Relative resolution for the funnel diameter RR(d):
(8)
RR(d) = RA(d)/d = 0.61 %
Where d: funnel diameter
2.2. Accumulation area calculation A:
(9)
A = π(d/2)2 = 213.82 cm2
Relative resolution for the accumulation area RR(A):
(10)
RR(A) = √(2(RR(d))2) = 0.86 %
3. Resolution for the area density value
Here is presented the calculation for the resolution of the in situ area density. For the dry area
density (ρA(D)), substitute M3 with M6.
In situ area density calculation ρA(is):
(8)
ρA(is) = M3/A
Relative resolution for the mass per area RR(ρA(is)):
(9)
RR(ρA(is)) = √(RR(M3)2 + RR(A)2)
Example for 10 g/m2: RR(ρA(is)) = 66.14 %
Example for 1000 g/m2: RR(ρA(is)) = 1.08 %
4. Resolution for the thickness reading
With a gauge or a scale, the resolution of the reading is half the interval (I) between two
graduations.
Absolute resolution for the thickness reading RA(T):
(10)
RA(T) = I/2
Relative resolution for the thickness reading RRT:
(11)
RR(T) = RA(T)/T
Due to the shape of the inner gauge, its graduation is not constant. For this ashmeter model (6-liter
large bottle and 1.5-liter small bottle), the scale has been divided into five graduation intervals:
Range
0.3-0.95 mm
1-1.9 mm
2-2.875 mm
3-6.75 mm
7-20 mm
Interval
0.05 mm
0.1 mm
0.125 mm
0.25 mm
0.5 mm
RA(T)
0.025 mm
0.05 mm
0.063 mm
0.125 mm
0.25 mm
RR(T)
8.3-2.6%
5.0-2.6%
3.1-2.2%
4.2-1.9%
3.6-1.3%
166.7-52.6%
50.0-26.3%
25.0-17.4%
16.7-7.4%
7.1-2.5%
Tape ruler
resolution
5. Resolution for the bulk density
Here is presented the resolution calculation for the in situ bulk density. For the dry bulk density,
substitute ρA(is) with ρA(D).
Bulk density calculation ρB(is):
(12)
ρB(is) = ρA(is)/T
Relative resolution for the bulk density RRρB:
(13)
RR(ρB(is)) = √(RR(ρA(is)) + RR(T)2)
For a 1.325 g/cm3 ash deposit (average of 16 bulk densities from Tungurahua 2012 tephra deposit)
the relative resolution on the bulk density will be:
Range
0.3-0.95 mm
1-1.9 mm
2-2.875 mm
3-6.75 mm
7-20 mm
RR(ρB(is))
8.5-2.8%
5.1-2.8%
3.3-2.3%
4.3-2.0%
3.7-1.5%