**3. Results**

growing corn, soybeans, winter wheat, dairy herds, beef cattle, horse farms, apple orchards, and hardwood lumber. Surrounding the lake is a residential development, enclosed by the rural environmental agricultural matrix. The lake is used for recreational boating, summer

fishing, and winter ice fishing.

272 Land Use - Assessing the Past, Envisioning the Future

C2 = Carlson Index score E1 = evaporation in meters

L2 = maximum lake depth in meters

M2 = mean lake depth in meters P1 = prescription in meters

P2 = present phosphorus level in mg/l P3 = predicted phosphorus level in mg/l O1 = areal water load in meters/year

S1 = surface area of lake in square meters T1 = point source phosphorus pollution input

W6, W7, W8 = woodland in square meters P6, P7, P8 = grassland area in square meters U6, U7, U8 = urban savanna in square meters

I6, I7, I8 = industrial in square meters B6, B7, B8 = agriculture in square meters

6 = clayey soils, 7 = loamy soils, 8 = sandy soils.

**Table 2.** The list of variables employed to calculate non-toxic water quality.

V1 = lake volume in cubic meters

R1 = volume of watershed runoff to lake in cubic meters

F = flushing rate

L4 = nutrient supply

R = retention coefficient

C1 = terrestrial and other water body phosphorus supply in kg/ yr.

Ultra-oligotrophic <5 Oligo-mesotrophic 5–10 Meso-eutrophic 10–30 Eu-polytrophic 30–100 Polytrophic >100

**Table 1.** Trophic state lake classification and the associated Carlson Index Score.

**Figure 5** illustrates the land cover types within the study area; while **Figure 6** presents the soil types for the study area. A series of 11 equations (Eq. (1)–(11)) resulted in the final calculated Carlson Index Score, which was 92.52, placing the lake at the high end of a eu-polytrophic lake. The predicted results match the classification results for the lake. Lake volume in meter cube

$$\text{V1} = \left( (1.047) \ast \text{S1} / 3.14 \right) \ast \text{(L2)} \tag{1}$$

V1 = 1760827.261 m3

where S1 = 1,320,200 m2 (derived from CAD and public maps); L2 = 4 m.

Mean lake depth in meters

$$\mathbf{M2} = \mathbf{V1}/\mathbf{S1} \tag{2}$$

$$\text{M2 = 1.33 m (calculator)}.$$

where V1 = 1760827.261 m<sup>3</sup> ; S1 = 1320200.00 m2 .

Runoff calculations

$$\begin{aligned} \text{R1} &= \{ (\text{U}6^{\circ} \, 0.8) + (\text{U}7^{\circ} \, 0.6) + (\text{U}8^{\circ} \, 0.7) + (\text{B6}^{\circ} \, 0.5) + (\text{B7}^{\circ} \, 0.1) + (\text{B8}^{\circ} \, 0.3) + (\text{P} \, 0.4) \} + (\text{P} \, 0.4) \\ &+ (\text{P}7^{\circ} \, 0.2) + (\text{P}8^{\circ} \, 0.225) + (\text{W6}^{\circ} \, 0.2) + (\text{W7}^{\circ} \, 0.3) + (\text{W8}^{\circ} \, 0.125) \\ &+ (\text{I6}^{\circ} \, 0.9) + (\text{I7}^{\circ} \, 0.85) + (\text{I8}^{\circ} \, 0.8) + \text{O1} \}) \ast \text{P1} \end{aligned} \tag{3}$$

$$\text{R1 = } \text{38229613.000.}$$

where Yearly P1 = 33.22 inches = 0.844 m and **Table 3**.

Phosphorus contribution in calculations

$$\begin{aligned} \text{C1} &= \{ (\text{(U6} + \text{U7} + \text{U8)} \ast 4) + (\text{(B6} + \text{B7} + \text{B8)} \ast 0.35) + (\text{(P6} + \text{P7} + \text{P8)} \ast 0.25) \\ &+ (\text{(W6} + \text{W7} + \text{W8)} \ast 0.1) + (\text{(I6} + \text{I7} + \text{I8)} \ast 4) + (\text{O1}^{\ast} \ast 0.46) \} / 10^4 \end{aligned} \tag{4}$$

C1 = 10753.8.

Further calculations

$$\mathbf{O2} = \mathbf{R1} + (\mathbf{S1}^\ast (\mathbf{P1} - \mathbf{E1})) \tag{5}$$

Total present nutrient supply

where S1 = 1,320,200 m2

Permission).

Predicted phosphorus level

L4 = 8244.93 mg/m<sup>2</sup>

P3 = L4<sup>∗</sup> 0.6/1.33<sup>∗</sup> 14.54.

P3 = 458.57 mg/m3

where L4 = 8244.93; R = 0.73; M2 = 1.33 m; F = 3.65.

L4 = (((S1<sup>∗</sup> 0.46∗(10−4)) + C1) ∗ 10<sup>6</sup>) + (P2<sup>∗</sup> V1∗(10−3))/S1 (9)

(CAD and map); C1 = 10753.8; P2 = 40 mg/m<sup>3</sup>

P3 = L4∗(1 − R)/(M2<sup>∗</sup> F) (10)

Carlson index = 14.42<sup>∗</sup> Ln(P3) + 4.15 = 92.52 (11)

**Figure 5.** This map presents the land cover types in the study area, with Rainbow Lake in the lower right. Beneath the lake is the Maple River and the Maple River Game Management area. This convergence was the general location of outlet of Glacial Lake Saginaw with flowage from right to left (copyright 2017 Zhen Wu, all rights reserved, used by

.year.

Comparative Approaches in Managing Wetland Environments and Land Uses: Rainbow Lake…

.

; V1 = 1760827.261 m<sup>3</sup>

http://dx.doi.org/10.5772/intechopen.79323

.

275

O2 = 6144742.000 + (1,320,200∗(0.844–0.635)).

$$\bullet 2 = 6\,\text{420}\,\text{664.}$$

where R1 = 6144742.000; S1 = 1,320,200 m2 (CAD and map); P1 = 0.844 m/year; E1 = 25 inches = 0.635 m/year.

$$\mathbf{Q1 = O2/S1} \tag{6}$$

 Q1 = 6, 420, 664/1320200. Q1 = 4.86. R = 13/(13 + Q1) (7) R = 13/(13 + 4.86). R = 0.73. F = O2/V1 (8)

F = 6, 420, 664/1760827.261.

F = 3.65.

Comparative Approaches in Managing Wetland Environments and Land Uses: Rainbow Lake… http://dx.doi.org/10.5772/intechopen.79323 275

Total present nutrient supply

R1 = 38229613.000.

C1 = (((U6 + U7 + U8) ∗ 4) + ((B6 + B7 + B8) ∗ 0.35) + ((P6 + P7 + P8) ∗ 0.25) +

O2 = R1 + (S 1∗(P1 − E1)) (5)

Q1 = O2/S1 (6)

R = 13/(13 + Q1) (7)

F = O2/V1 (8)

((W6 + W7 + W8) ∗ 0.1) + ((I6 + I7 + I8) ∗ 4) + (O1<sup>∗</sup> 0.46))/104 (4)

(CAD and map); P1 = 0.844 m/year; E1 = 25

where Yearly P1 = 33.22 inches = 0.844 m and **Table 3**.

C1 = 10753.8.

O2 = 6,420,664.

Q1 = 6, 420, 664/1320200.

Q1 = 4.86.

R = 13/(13 + 4.86).

F = 6, 420, 664/1760827.261.

R = 0.73.

F = 3.65.

where R1 = 6144742.000; S1 = 1,320,200 m2

O2 = 6144742.000 + (1,320,200∗(0.844–0.635)).

Phosphorus contribution in calculations

274 Land Use - Assessing the Past, Envisioning the Future

Further calculations

inches = 0.635 m/year.

$$\mathbf{L4} = \{ \left( \mathbf{[S1^\ast 0.46^\ast (10^{-4})) + C1} \\ \mathbf{1^\ast 10^\circ} \right) + \left( \mathbf{P2^\ast V1^\ast (10^{-3})} \right) / \mathbf{S1} \tag{9}$$

$$\textbf{L4 = } 8244.93 \text{ mg/m}^2 \text{.year.}$$

where S1 = 1,320,200 m2 (CAD and map); C1 = 10753.8; P2 = 40 mg/m<sup>3</sup> ; V1 = 1760827.261 m<sup>3</sup> . Predicted phosphorus level

$$
\mathcal{P}3 = \mathcal{L}4^\*(1 - \mathcal{R})/(\mathcal{M}2^\*\mathcal{F})\tag{10}
$$

$$
\mathcal{P}3 = \mathcal{L}4^\* 0.6/1.33^\* 14.54.
$$

$$
\mathcal{P}3 = \mathcal{L}58.57 \text{ mg/m}^3.
$$

where L4 = 8244.93; R = 0.73; M2 = 1.33 m; F = 3.65.

$$\text{Carkson index} = 14.42^{\circ} \,\text{Ln(P3)} + 4.15 = 92.52 \tag{11}$$

**Figure 5.** This map presents the land cover types in the study area, with Rainbow Lake in the lower right. Beneath the lake is the Maple River and the Maple River Game Management area. This convergence was the general location of outlet of Glacial Lake Saginaw with flowage from right to left (copyright 2017 Zhen Wu, all rights reserved, used by Permission).

**Figure 6.** A map of the soil types in the study area (copyright 2017 Zhen Wu, all rights reserved, used by Permission).

**4. Discussion and conclusion**

**Classification Soil Area (km2**

Industrial 3.52 1.00 0.01

Agriculture 203.12 1.00 0.81

Water (O1) 2.80 1.00 0.01

**Table 3.** The area of land in square meters for various land-cover types by soil types.

Sand (I8) 1.48 0.42 0.004 Loam (I7) 2.04 0.58 0.006 Clay (I6) 0.00 0.00 0.000

Sand (B8) 40.62 0.20 0.162 Loam (B7) 162.50 0.80 0.648 Clay (B6) 0.00 0.00 0

Sand 1.37 0.49 0.005 Loam 1.43 0.51 0.005 Clay 0.00 0.00 0.000

"The student who conducted the study of Rainbow Lake was surprised how available information was to make the prediction and conduct the modeling," stated Dr. Burley. "He stated that such information is not as freely available in other parts of the world," commented Dr. Burley. "It is true that information can equate to power. But in the United States and Canada, such information is supported by the public and the public has the right to access such information. Farmers, citizens, and researchers are free to access the information to make calculations to refute or support the findings of others. It is expected if not demanded," said Dr. Burley. "In many respects it is comforting that there are checks and balances in the use and application of information. If there are disputes, they can be openly addressed in public forums," mentioned Dr. Burley. "Especially at the township, county and state level, public employees responsible for natural resource management work together with citizens and comparatively, there is a

**) Percentage in each land use Percentage in total research area**

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277

Comparative Approaches in Managing Wetland Environments and Land Uses: Rainbow Lake…

"The largest disputes that I have witnessed have been amongst hydrological experts who may debate methods of sampling or the accuracy of equations to predict hydrological variables. These academics often have varying opinions. But any equation is simply an approximation and estimation of physical phenomena. In the engineering field, if the approximation generally works, then it is accepted with no theoretical explanation or search for a better equation. This was true of the Manning formula of confined water flow in swales and pipes. This equation is over 140 years old and is unquestioned by many, but it is simply a mathematical approximation of water flow," observes Dr. Burley. "I am sure that today investigators could develop improved

fair amount of respect and trust amongst everyone.," reports Dr. Burley.

**4.1. Rainbow Lake, Michigan**


Comparative Approaches in Managing Wetland Environments and Land Uses: Rainbow Lake… http://dx.doi.org/10.5772/intechopen.79323 277


**Table 3.** The area of land in square meters for various land-cover types by soil types.
